diff --git a/.gitattributes b/.gitattributes index 4ab3921c8bd4fb4ff70dd1183290602bef59f97a..5be22e9ba9b2836ec46f7dda45ed6ccf78a911d2 100644 --- a/.gitattributes +++ b/.gitattributes @@ -123,3 +123,4 @@ pNE1T4oBgHgl3EQf2AWA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex CNFAT4oBgHgl3EQfsx5b/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text VNAyT4oBgHgl3EQf8vqZ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text 2tFKT4oBgHgl3EQf7y43/content/2301.11946v1.pdf filter=lfs diff=lfs merge=lfs -text +XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/0tFPT4oBgHgl3EQfTjTq/content/tmp_files/load_file.txt b/0tFPT4oBgHgl3EQfTjTq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a01baf9cd73b861070fd3ddc3c78f6a9a72a71d --- /dev/null +++ b/0tFPT4oBgHgl3EQfTjTq/content/tmp_files/load_file.txt @@ -0,0 +1,1085 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf,len=1084 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='13054v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='FL] 30 Jan 2023 Monadic Expressions and their Derivatives Samira Attou1, Ludovic Mignot2, Clément Miklarz2, and Florent Nicart2 1 Université Gustave Eiffel, 5 Boulevard Descartes — Champs s/ Marne 77454 Marne-la-Vallée Cedex 2 2 GR2IF, Université de Rouen Normandie, Avenue de l’Université, 76801 Saint-Étienne-du-Rouvray, France samira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='attou@univ-eiffel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='fr, {ludovic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='mignot,clement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='miklarz1, florent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='nicart}@univ-rouen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='fr Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We propose another interpretation of well-known derivatives computations from regular expres- sions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' sets or linear combinations) using the notion of monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As an example of this generalization advantage, we first introduce a new derivation technique based on the graded module monad and then show an application of this technique to generalize the parsing of expression with capture groups and back references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations (like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, using the Colcombet and Petrisan categorical definition of automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' These category theory concepts allow us to perform this construction in a unified way, whatever the underlying monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques of functional programming, and we provide a web interface to manipulate concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1 Introduction This paper is an extended version of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Regular expressions are a classical way to represent associations between words and value sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As an example, classical regular expressions denote sets of words and regular expressions with multiplicities denote formal series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' From a regular expression, solving the membership test (determining whether a word belongs to the denoted language) or the weighting test (determining the weight of a word in the denoted formal series) can be solved, following Kleene theorems [11,17] by computing a finite automaton, such as the position automaton [9,3,5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Another family of methods to solve these tests is the family of derivative computations, that does not require the construction of a whole automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The common point of these techniques is to transform the test for an arbitrary word into the test for the empty word, which can be easily solved in a purely syntactical way (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' by induction over the structure of expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Brzozowski [4] shows how to compute, from a regular expression E and a word w, a regular expression dw(E) denoting the set of words w′ such that ww′ belongs to the language denoted by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Solving the membership test hence becomes the membership test for the empty word in the expression dw(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Antimirov [1] modifies this method in order to produce sets of expressions instead of expressions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' defines the partial derivatives ∂w(E) as a set of expressions the sum of which denotes the same language as dw(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' If the number of derivatives is exponential w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' the length |E| of E in the worst case3, the partial derivatives produce at most a linear number of expressions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Lombardy and Sakarovitch [13] extends these methods to expressions with multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, Sulzmann and Lu [18] apply these derivation techniques to parse POSIX expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' It is well-known that these methods are based on a common operation, the quotient of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Furthermore, Antimirov’s method can be interpreted as the derivation of regular expression with multiplicities in the Boolean semiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' However, the Brzozowski computation does not produce the same expressions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' equality over the syntax trees) as the Antimirov one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Main contributions: In this paper, we present a unification of these computations by applying notions of category theory to the category of sets, and show how to compute categorical automata as defined in [7], by reinter- preting the work started in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We make use of classical monads to model well-known derivatives computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Furthermore, we deal with extended expressions in a general way: in this paper, expressions can support extended 3 as far as rules of associativity, commutativity and idempotence of the sum are considered, possibly infinite otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' operators like complement, intersection, but also any n-ary function (algebraic mean, extrema multiplications, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The main difference with [15] is that we formally state the languages and series that the expressions denote in an inherent way w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' the underlying monads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' More precisely, this paper presents: – an extension of expressions to any n-ary function over the value set, – a monadic generalization of expressions, – a solution for the membership/weight test for these expressions, – a computation of categorical derivative automata, – a new monad that fits with the extension to n-ary functions, – an illustration implemented in Haskell using advanced functional programming, – an extension to capture groups and back references expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Motivation: The unification of derivation techniques is a goal by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Moreover, the formal tools used to achieve this unification are also useful: Monads offer both theoretical and practical advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Indeed, from a theoretical point of view, these structures allow the abstraction of properties and focus on the principal mechanisms that allow solving the membership and weight problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Besides, the introduction of exotic monads can also facilitate the study of finiteness of derivated terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' From a practical point of view, monads are easy to implement (even in some other languages than Haskell) and allow us to produce compact and safe code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, we can easily combine different algebraic structures or add some technical functionalities (capture groups, logging, nondeterminism, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=') thanks to notions like monad transformers [10] that we consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In Section 2, we gather some preliminary material, like algebraic structures or category theory notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We also introduce some functions well-known to the Haskell community that can allow us to reduce the size of our equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We then structurally define the expressions we deal with, the associated series and the weight test for the empty word in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In order to extend this test to any arbitrary word, we first state in Section 4 some properties required by the monads we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Once this so-called support is determined, we show in Section 5 how to compute the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The computation of derivative automata is explained in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A new monad and its associated derivatives computation is given in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' An implementation is presented in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, we show how to (alternatively to [18]) compute derivatives of capture group expressions in Section 9 and show that as far as the same operators are concerned, the derivative formulae are the same whatever the underlying monad is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 2 Preliminaries We denote by S → S′ the set of functions from a set S to a set S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The notation λx → f(x) is an equivalent notation for a function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A monoid is a set S endowed with an associative operation and a unit element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A semiring is a structure (S, ×, +, 1, 0) such that (S, ×, 1) is a monoid, (S, +, 0) is a commutative monoid, × distributes over + and 0 is an annihilator for ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A starred semiring is a semiring with a unary function ⋆ such that k⋆ = 1 + k × k⋆ = 1 + k⋆ × k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A K-series over the free monoid (Σ∗, ·, ε) associated with an alphabet Σ, for a semiring K = (K, ×, +, 1, 0), is a function from Σ∗ to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The set of K-series can be endowed with the structure of semiring as follows: 1(w) = � 1 if w = ε, 0 otherwise, 0(w) = 0, (S1 + S2)(w) = S1(w) + S2(w), (S1 × S2)(w) = � u·v=w S1(u) × S2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Furthermore, if S1(ε) = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' S1 is said to be proper), the star of S1 is the series defined by (S1)⋆(ε) = 1, (S1)⋆(w) = � n≤|w|,w=u1···un,uj̸=ε S1(u1) × · · · × S1(un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, for any function f in Kn → K, we set: (f(S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Sn))(w) = f(S1(w), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Sn(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (1) A functor 4 F associates with each set S a set F(S) and with each function f in S → S′ a function F(f) from F(S) to F(S′) such that F(id) = id, F(f ◦ g) = F(f) ◦ F(g), 4 More precisely, a functor over a subcategory of the category of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' where id is the identity function and ◦ the classical function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A monad5 M is a functor endowed with two (families of) functions – pure, from a set S to M(S), – bind, sending any function f in S → M(S′) to M(S) → M(S′), such that the three following conditions are satisfied: bind(f)(pure(s)) = f(s), bind(pure) = id, bind(g)(bind(f)(m)) = bind(λx → bind(g)(f(x)))(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The Maybe monad associates: – any set S with the set Maybe(S) = {Just(s) | s ∈ S} ∪ {Nothing}, where Just and Nothing are two syntactic tokens allowing us to extend a set with one value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – any function f with the function Maybe(f) defined by Maybe(f)(Just(s)) = Just(f(s)), Maybe(f)(Nothing) = Nothing – is endowed with the functions pure and bind defined by: pure(s) = Just(s), bind(f)(Just(s)) = f(s), bind(f)(Nothing) = Nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The Set monad associates: – with any set S the set 2S, – with any function f the function Set(f) defined by Set(f)(R) = � r∈R{f(r)}, – is endowed with the functions pure and bind defined by: pure(s) = {s}, bind(f)(R) = � r∈R f(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The LinComb(K) monad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' for K = (K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ×,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' associates: – with any set S the set of K-linear combinations of elements of S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' where a linear combination is a finite (formal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' commutative) sum of couples (denoted by ⊞) in K × S where (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' s) ⊞ (k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' s) = (k + k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – with any function f the function LinComb(K)(f) defined by LinComb(K)(f)(R) = ⊞ (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r)∈R (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' f(r)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – is endowed with the functions pure and bind defined by: pure(s) = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' bind(f)(R) = ⊞ (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r)∈R k ⊗ f(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' where k ⊗ R = ⊞ (k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r)∈R (k × k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' To compact equations, we use the following operators for any monad M: f <$> s = M(f)(s), m >>= f = bind(f)(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' If <$> can be used to lift unary functions to the monadic level, >>= and pure can be used to lift any n-ary function f in S1 × · · · × Sn → S, defining a function liftn sending S1 × · · · × Sn → S to M(S1) × · · · × M(Sn) → M(S) as follows: liftn(f)(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , mn) =m1 >>= (λs1 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' mn >>= (λsn → pure(f(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=') Let us consider the set 1 = {⊤} with only one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The images of this set by some previously defined monads can be evaluated as value sets classically used to weight words in association with classical regular expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As an example, Maybe(1) and Set(1) are isomorphic to the Boolean set, and any set LinComb(K)(1) can be converted into the underlying set of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' This property allows us to extend in a coherent way classical expressions to monadic expressions, where the type of the weights is therefore given by the ambient monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 5 More precisely, a monad over a subcategory of the category of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 3 Monadic Expressions As seen in the previous section, elements in M(1) can be evaluated as classical value sets for some particular monads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Hence, we use these elements not only for the weights associated with words by expressions, but also for the elements that act over the denoted series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In the following, in addition to classical operators (+, · and ∗), we denote: – the action of an element over a series by ⊙, – the application of a function by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let M be a monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' An M-monadic expression E over an alphabet Σ is inductively defined as follows: E = a, E = ε, E = ∅, E = E1 + E2, E = E1 · E2, E = E∗ 1, E = α ⊙ E1, E = E1 ⊙ α, E = f (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En) , where a is a symbol in Σ, (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a function from (M(1))n to M(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We denote by Exp(Σ) the set of monadic expressions over an alphabet Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As an example of functions that can be used in our extension of classical operators, one can define the function ExtDist(x1, x2, x3) = max(x1, x2, x3) − min(x1, x2, x3) from N3 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Similarly to classical regular expressions, monadic expressions associate a weight with any word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Such a relation can be denoted via a formal series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' However, before defining this notion, in order to simplify our study, we choose to only consider proper expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us first show how to characterize them by the computation of a nullability value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let M be a monad such that the structure (M(1), +, ×, ⋆, 1, 0) is a starred semiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The nullability value of an M-monadic expression E over an alphabet Σ is the element Null(E) of M(1) inductively defined as follows: Null(ε) = 1, Null(∅) = 0, Null(a) = 0, Null(E1 + E2) = Null(E1) + Null(E2), Null(E1 · E2) = Null(E1) × Null(E2), Null(E∗ 1) = Null(E1)⋆, Null(α ⊙ E1) = α × Null(E1), Null(E1 ⊙ α) = Null(E1) × α, Null(f(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En)) = f(Null(E1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Null(En)), where a is a symbol in Σ, (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a function from (M(1))n to M(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' When the considered semiring is not a starred one, we restrict the nullability value computation to expressions where a starred subexpression admits a null nullability value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In order to compute it, let us consider the Maybe monad, allowing us to elegantly deal with such a partial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The partial nullability value of an M-monadic expression E over an alphabet Σ is the element PartNull(E) of Maybe(M(1)) defined as follows: PartNull(ε) = Just(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(∅) = Just(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(a) = Just(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(E1 + E2) = lift2(+)(PartNull(E1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(E2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(E1 · E2) = lift2(×)(PartNull(E1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(E2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(E∗ 1) = � Just(1) if PartNull(E1) = Just(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Nothing otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(α ⊙ E1) = (λE → α × E) <$> PartNull(E1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(E1 ⊙ α) = (λE → E × α) <$> PartNull(E1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' PartNull(f(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En)) = liftn(f)(PartNull(E1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , PartNull(En)), where a is a symbol in Σ, (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a function from (M(1))n to M(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' An expression E is proper if its partial nullability value is not Nothing, therefore if it is a value Just(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' in this case, v is its nullability value, denoted by Null(E) (by abuse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring, and E be a M-monadic proper expression over an alphabet Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The series S(E) associated with E is inductively defined as follows: S(ε)(w) = � 1 if w = ε, 0 otherwise, S(∅)(w) = 0, S(a)(w) = � 1 if w = a, 0 otherwise, S(E1 + E2) = S(E1) + S(E2), S(E1 · E2) = S(E1) × S(E2), S(E∗ 1) = (S(E1))⋆, S(α ⊙ E1)(w) = α × S(E1)(w), S(E1 ⊙ α)(w) = S(E1)(w) × α, S(f(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En)) = f(S(E1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , S(En)), where a is a symbol in Σ, (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a function from (M(1))n to M(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' From now on, we consider the set Exp(Σ) of M-monadic expressions over Σ to be endowed with the structure of a semiring, and two expressions denoting the same series to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The weight associated with a word w in Σ∗ by E is the value weightw(E) = S(E)(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The nullability of a proper expression is the weight it associates with ε, following Definition 3 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let M be a monad such that the structure (M(1), +, ×, 1, 0) is a semiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let E be an M-monadic proper expression over Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then: Null(E) = weightε(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The previous proposition implies that the weight of the empty word can be syntactically computed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' inductively computed from a monadic expression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Now, let us show how to extend this computation by defining the computation of derivatives for monadic expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 4 Monadic Supports for Expressions A K-left-semimodule, for a semiring K = (K, ×, +, 1, 0), is a commutative monoid (S, ±, 0) endowed with a function ⊲ from K × S to S such that: (k × k′) ⊲ s = k ⊲ (k′ ⊲ s), (k + k′) ⊲ s = k ⊲ s ± k′ ⊲ s, k ⊲ (s ± s′) = k ⊲ s ± k ⊲ s′, 1 ⊲ s = s, 0 ⊲ s = k ⊲ 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A K-right-semimodule can be defined symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' An operad [12,14] is a structure (O, (◦j)j∈N, id) where O is a graded set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' O = � n∈N On), id is an element of O1, ◦j is a function defined for any three integers (i, j, k)6 with 0 < j ≤ k in Ok × Oi → Ok+i−1 such that for any elements p1 ∈ Om, p2 ∈ On, p3 ∈ Op: ∀0 < j ≤ m, id ◦1 p1 = p1 ◦j id = p1, ∀0 < j ≤ m, 0 < j′ ≤ n, p1 ◦j (p2 ◦j′ p3) = (p1 ◦j p2) ◦j+j′−1 p3, ∀0 < j′ ≤ j ≤ m, (p1 ◦j p2) ◦j′ p3 = (p1 ◦j′ p3) ◦j+p−1 p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Combining these compositions ◦j, one can define a composition ◦ sending Ok × Oi1 × · · · × Oik to Oi1+···+ik: for any element (p, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , qk) in Ok × Ok, p ◦ (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , qk) = (· · · ((p ◦k qk) ◦k−1 qk−1 · · · ) · · · ) ◦1 q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Conversely, the composition ◦ can define the compositions ◦j using the identity element: for any two elements (p, q) in Ok × Oi, for any integer 0 < j ≤ k: p ◦j q = p ◦ (id, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , id � �� � j−1 times , q, id, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , id � �� � k−j times ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As an example, the set of n-ary functions over a set, with the identity function as unit, forms an operad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A module over an operad (O, ◦, id) is a set S endowed with a function ⋇ from On × Sn to S such that f ⋇ (f1 ⋇ (s1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , s1,i1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , fn ⋇ (sn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn,in)) = (f ◦ (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , fn)) ⋇ (s1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , s1,i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn,in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 6 every couple (i, k) unambiguously defines the domain and codomain of a function ◦j The extension of the computation of derivatives could be performed for any monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Indeed, any monad could be used to define well-typed auxiliary functions that mimic the classical computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' However, some properties should be satisfied in order to compute weights equivalently to Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Therefore, in the following we consider a restricted kind of monads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A monadic support is a structure (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇) satisfying: – M is a monad, – R = (M(1), +, ×, 1, 0) is a semiring, – M = (M(Exp(Σ)), ±, 0) is a monoid, – (M, ⋉) is a Exp(Σ)-right-semimodule, – (M, ⊲) is a R-left-semimodule, – (M, ⊳) is a R-right-semimodule, – (M(Exp(Σ)), ⋇) is a module for the operad of the functions over M(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' An expressive support is a monadic support (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇) endowed with a function toExp from M(Exp(Σ)) to Exp(Σ) satisfying the following conditions: weightw(toExp(m)) = m >>= weightw (2) toExp(m ⋉ F) = toExp(m) · F, (3) toExp(m ± m′) = toExp(m) + toExp(m′), (4) toExp(m ⊲ x) = toExp(m) ⊙ x, (5) toExp(x ⊳ m) = x ⊙ toExp(m), (6) toExp(f ⋇ (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , mn)) = f(toExp(m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , toExp(mn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (7) Let us now illustrate this notion with three expressive supports that will allow us to model well-known derivatives computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 5 (The Maybe support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' toExp(Nothing) = 0, toExp(Just(E)) = E, Nothing + m = m, m + Nothing = m, Just(⊤) + Just(⊤) = Just(⊤), Nothing × m = Nothing, m × Nothing = Nothing, Just(⊤) × Just(⊤) = Just(⊤), Nothing ± m = m, m ± Nothing = m, Just(E) ± Just(E′) = Just(E + E′), 1 = Just(⊤), 0 = Nothing, 0 = Nothing, m ⋉ F = (λE → E · F) <$> m, m ⊲ m′ = m >>= (λx → m′), m ⊳ m′ = m′ >>= (λx → m), f ⋇ (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , mn) = pure(f(toExp(m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , toExp(mn))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 6 (The Set support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' toExp({E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En}) = E1 + · · · + En, + = ∪, × = ∩, ± = ∪, 1 = {⊤}, 0 = ∅, 0 = ∅, m ⋉ F = (λE → E · F) <$> m, m ⊲ m′ = m >>= (λx → m′), m ⊳ m′ = m′ >>= (λx → m), f ⋇ (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , mn) = pure(f(toExp(m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , toExp(mn))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 7 (The LinComb(K) support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' toExp((k1, E1) ⊞ · · · ⊞ (kn, En)) = k1 ⊙ E1 + · · · + kn ⊙ En, + = ⊞, (k, ⊤) × (k′, ⊤) = (k × k′, ⊤), 1 = (1, ⊤), 0 = (0, ⊤), ± = ⊞, 0 = (0, ⊤), m ⋉ F = (λE → E · F) <$> m, m ⊲ m′ = m >>= (λx → m′), m ⊳ k = (λE → E ⊙ k) <$> m, f ⋇ (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , mn) = pure(f(toExp(m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , toExp(mn))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 5 Monadic Derivatives In the following, (M, +, ×, 1, 0, ±, 0, ⋉, ⊲, ⊳, ⋇, toExp) is an expressive support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The derivative of an M-monadic expression E over Σ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a symbol a in Σ is the element da(E) in M(Exp(Σ)) inductively defined as follows: da(ε) = 0, da(∅) = 0, da(b) = � pure(ε) if a = b, 0 otherwise, da(E1 + E2) = da(E1) ± da(E2), da(E∗ 1) = da(E1) ⋉ E∗ 1, da(E1 · E2) = da(E1) ⋉ E2 ± Null(E1) ⊲ da(E2), da(α ⊙ E1) = α ⊲ da(E1), da(E1 ⊙ α) = da(E1) ⊳ α, da(f(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En)) = f ⋇ (da(E1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , da(En)) where b is a symbol in Σ, (E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En) are n M-monadic expressions over Σ, α is an element of M(1) and f is a function from (M(1))n to M(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The link between derivatives and series can be stated as follows, which is an alternative description of the classical quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let E be an M-monadic expression over an alphabet Σ, a be a symbol in Σ and w be a word in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then: weightaw(E) = da(E) >>= weightw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us proceed by induction over the structure of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' All the classical cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' the function operator left aside) can be proved following the classical methods ([1,4,13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Therefore, let us consider this last case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' da(f(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En)) >>= weightw = weightw(toExp(da(f(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En)))) (Eq (2)) = weightw(toExp(f ⋇ (da(E1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , da(En))) (Def 5)) = weightw(f(toExp(da(E1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , toExp(da(En)))) (Eq (7)) = f(weightw(toExp(da(E1))), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , weightw(toExp(da(En)))) (Def 4, Eq (1)) = f(da(E1) >>= weightw, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , da(En) >>= weightw) (Eq (2)) = f(weightaw(E1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , weightaw(En)) (Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' hyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=') = weightaw(f(E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , En)) (Def 4, Eq (1)) Let us define how to extend the derivative computation from symbols to words, using the monadic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The derivative of an M-monadic expression E over Σ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a word w in Σ∗ is the element dw(E) in M(Exp(Σ)) inductively defined as follows: dε(E) = pure(E), da·v(E) = da(E) >>= dv, where a is a symbol in Σ and v a word in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, it can be easily shown, by induction over the length of the words, following Proposition 2, that the derivatives computation can be used to define a syntactical computation of the weight of a word associated with an expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let E be an M-monadic expression over an alphabet Σ and w be a word in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then: weightw(E) = dw(E) >>= Null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Notice that, restraining monadic expressions to regular ones, – the Maybe support leads to the classical derivatives [4], – the Set support leads to the partial derivatives [1], – the LinComb support leads to the derivatives with multiplicities [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us consider the function ExtDist defined in Example 4 and the LinComb(N)-monadic expression E = ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' da(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + a∗) daa(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 2 ⊙ a∗) daaa(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 3 ⊙ a∗) daab(E) = ExtDist(b∗, b∗, b∗a∗) weightaaa(E) = daaa(E) >>= Null = ExtDist(1 + 1, 1, 1 + 3) = 4 − 1 = 3 weightaab(E) = daab(E) >>= Null = ExtDist(1, 1, 1) = 0 In the next section, we show how to compute the derivative automaton associated with an expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 6 Automata Construction A category C is defined by: – a class ObjC of objects, – for any two objects A and B, a set HomC(A, B) of morphisms, – for any three objects A, B and C, an associative composition function ◦C in HomC(B, C) −→ HomC(A, B) −→ HomC(A, C), – for any object A, an identity morphism idA in HomC(A, A), such that for any morphisms f in HomC(A, B) and g in HomC(B, A), f ◦C idA = f and idA ◦C g = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Given a category C, a C-automaton is a tuple (Σ, I, Q, F, i, δ, f) where – Σ is a set of symbols (the alphabet), – I is the initial object, in Obj(C), – Q is the state object, in Obj(C), – F is the final object, in Obj(C), – i is the initial morphism, in HomC(I, Q), – δ is the transition function, in Σ −→ HomC(Q, Q), – f is the value morphism, in HomC(Q, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The function δ can be extended as a monoid morphism from the free monoid (Σ∗, ·, ε) to the morphism monoid (HomC(Q, Q), ◦C, idQ), leading to the following weight definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The weight associated by a C-automaton A = (Σ, I, Q, F, i, δ, f) with a word w in Σ∗ is the morphism weight(w) in HomC(I, F) defined by weight(w) = f ◦C δ(w) ◦C i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' If the ambient category is the category of sets, and if I = 1, the weight of a word is equivalently an element of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Consequently, a deterministic (complete) automaton is equivalently a Set-automaton with 1 as the initial object and B as the final object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Given a monad M, the Kleisli composition of two morphisms f ∈ HomC(A, B) and g ∈ HomC(B, C) is the morphism (f >=> g)(x) = f(x) >>= g in HomC(A, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' This composition defines a category, called the Kleisli category K(M) of M, where: – the objects are the sets, – the morphisms between two sets A and B are the functions between A and M(B), – the identity is the function pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Considering these categories: – a deterministic automaton is equivalently a K(Maybe)-automaton, – a nondeterministic automaton is equivalently a K(Set)-automaton, – a weighted automaton over a semiring K is equivalently a K(LinComb(K))-automaton, all with 1 as both the initial object and the final object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Furthermore, for a given expression E, if i = pure(E), δ(a)(E′) = da(E′) and f = Null, we can compute the well-known derivative automata using the three previously defined supports, and the accessible part of these automata are finite ones as far as classical expressions are concerned [4,1,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' More precisely, extended expressions can lead to infinite automata, as shown in the next example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Considering the computations of Example 8, it can be shown that dan(E) = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Hence, there is not a finite number of derivated terms, that are the states in the classical derivative automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' This infinite automaton is represented in Figure 1, where the final weights of the states are represented by double edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The sink states are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗) ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + a∗) ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + 2 ⊙ a∗) ExtDist(b∗, b∗, b∗a∗) ExtDist(0, 0, a∗) ExtDist(b∗ + b∗a∗, b∗a∗b∗ + b∗, b∗a∗) ExtDist(b∗ + b∗a∗, b∗a∗b∗ + 2 ⊙ b∗, b∗a∗) ExtDist(a∗, a∗b∗, a∗) ExtDist(0, b∗, 0) ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗) ExtDist(b∗ + b∗a∗, b∗a∗b∗ + n ⊙ b∗, b∗a∗) 1 1 2 n 1 1 2 1 n a b b a b a b a b a b a b a b a b a Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The (infinite) derivative weighted automaton associated with E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In the following section, let us show how to model a new monad in order to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 7 The Graded Module Monad Let us consider an operad O = (O, ◦, id) and the association sending: – any set S to � n∈N On × Sn, – any f in S → S′ to the function g in � n∈N On × Sn → � n∈N On × S′n: g(o, (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn)) = (o, (f(s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , f(sn))) It can be checked that this is a functor, denoted by GradMod(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Moreover, it forms a monad considering the two following functions: pure(s) = (id, s), (o, (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn)) >>= f = (o ◦ (o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , on), (s1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , s1,i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn,in)) where f(sj) = (oj, sj,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sj,ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' However, notice that GradMod(O)(1) cannot be easily evaluated as a value space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Thus, let us compose it with another monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As an example, let us consider a semiring K = (K, ×, +, 1, 0) and the operad O of the n-ary functions over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Hence, let us define the functor7 GradComb(O, K) that sends S to GradMod(O)(LinComb(K)(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 7 it is folk knowledge that the composition of two functors is a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' To show that this combination is a monad, let us first define a function α sending GradComb(O, K)(S) to GradMod(O)(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' It can be easily done by converting a linear combination into an operadic combination, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' an element in GradMod(O)(S), with the following function toOp: toOp((k1, s1) ⊞ · · · ⊞ (kn, sn)) = (λ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , xn) → k1 × x1 + · · · + kn × xn, (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn)), α(o, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) = (o ◦ (o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , on), (s1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , s1,i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sn,in)) where toOp(Lj) = (oj, (sj,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , sj,ij)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Consequently, we can define the monadic functions as follows: pure(s) = (id, (1, s)), (o, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) >>= f = α(o, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) >>= f where the second occurrence of >>= is the monadic function associated with the monad GradMod(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us finally define an expressive support for this monad: toExp(o, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) = o(toExp(L1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , toExp(Ln)), (o, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) + (o′, (L′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , L′ n′)) = (o + o′, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln, L′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , L′ n′)) (o, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) × (o′, (L′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , L′ n′)) = (o × o′, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln, L′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , L′ n′)) ± = +, 1 = (id, (1, ⊤)), 0 = (id, (0, ⊤)), 0 = (id, (0, ⊤)), m ⋉ F = pure(toExp(m) · F), (o, (M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Mk)) ⊲ (o′, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) = (o(M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Mk) × o′, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)), (o, (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) ⊳ (o′, (M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Mk)) = (o × o′(M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Mk), (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln)) f ⋇ ((o1, (L1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , L1,i1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , (on, (Ln,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln,in))) = (f ◦ (o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , on), (L1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , L1,i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , Ln,in)) where (o + o′)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , xn+n′) = o(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , xn) + o′(xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , xn+n′) (o × o′)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , xn+n′) = o(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , xn) × o′(xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' , xn+n′) Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us consider that two elements in GradComb(O, K)(Exp(Σ)) are equal if they have the same image by toExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us consider the expression E = ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗) of Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' da(E) = ExtDist ⋇ ((+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (a∗b∗a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗))) = (ExtDist ◦ (+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' +),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗)) daa(E) = (ExtDist ◦ (+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' + ◦ (+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' id)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗)) = (ExtDist ◦ (+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' + ◦ (id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 2×)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗)) daaa(E) = (ExtDist ◦ (+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' + ◦ (id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 3×)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗b∗a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a∗)) daab(E) = (ExtDist ◦ (+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' +),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' b∗a∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ∅)) = (ExtDist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' b∗a∗)) weightaaa(E) = daaa(E) >>= Null = ExtDist ◦ (+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' id,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' +)(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 3) = ExtDist(1 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1 + 3) = 4 − 1 = 3 weightaab(E) = daab(E) >>= Null = ExtDist(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1) = 0 Using this monad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' the number of derivated terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' that is the number of states in the associated derivative automaton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Indeed, the computations are absorbed in the transition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' This automaton is represented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Notice that the dashed rectangle represent the functions that are composed during the traversal associated with a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The final weights are represented by double edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The sink states are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The state b∗ is duplicated to simplify the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗) ExtDist + + ExtDist + b∗a∗b∗ + a∗b∗a∗ + a∗b∗ a∗ b∗ b∗a∗ b∗ 1 1 1 1 1 1 1 1 a b a b b a b a a b b b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The Associated Derivative Automaton of ExtDist(a∗b∗ + b∗a∗, b∗a∗b∗, a∗b∗a∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' However, notice that not every monadic expression produces a finite set of derivated terms, as shown in the next example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us consider the expression E of Example 8 and the expression F = E · c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' It can be shown that dan(F) = toExp(dan(E)) · c∗ = ExtDist(a∗b∗ + a∗, a∗b∗, a∗b∗a∗ + n ⊙ a∗) · c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The study of the necessary and sufficient conditions of monads that lead to a finite set of derivated terms is one of the next steps of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 8 Haskell Implementation The notions described in the previous sections have been implemented in Haskell, as follows: – The notion of monad over a sub-category of sets is a typeclass using the Constraint kind to specify a sub- category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – n-ary functions and their operadic structures are implemented using fixed length vectors, the size of which is determined at compilation using type level programming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – The notion of graded module is implemented through an existential type to deal with unknown arities: Its monadic structure is based on an extension of heterogeneous lists, the graded vectors, typed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' the list of the arities of the elements it contains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – The parser and some type level functions are based on dependently typed programming with singletons [8], allowing, for example, determining the type of the monads or the arity of the functions involved at run-time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – An application is available here [16] illustrating the computations: the backend uses servant to define an API;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' the frontend is defined using Reflex, a functional reactive programming engine and cross compiled in JavaScript with GHCJS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As an example, the monadic expression of the previous examples can be entered in the web application as the input ExtDist(a*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='b*+b*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='a*,b*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='a*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='b*,a*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='b*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='a*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 9 Capture Groups Capture groups are a standard feature of POSIX regular expressions where parenthesis are used to memorize some part of the input string being matched in order to reuse either for substitution or matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We give here an equivalent definition along with derivation formulae and a monadic definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The semantic of this definition conforms to those of POSIX expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Precisely, when a capture group has been involved more than one time due to a stared subexpression, the value of the corresponding variable corresponds to the last capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='1 Syntax of Expressions with Capture Groups A capture-group expression E over a symbol alphabet Σ and a variable alphabet Γ (or Σ, Γ-expression for short) is inductively defined as E = a, E = ε, E = ∅, E = F + G, E = F · G, E = F ∗, E = (F)x, E = x, where F and G are two Σ, Γ-expressions, a is a symbol in Σ, u is in Σ∗ and x is a variable in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In the POSIX syntax, capture groups are implicitly mapped with variables respectively with the order of the opening parenthesis of a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Here, each capture group is associated explicitly to a variable by indexing the closing parenthesis with the name of this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='2 Contextual Expressions and their Contextual Languages In order to define the contextual language and the derivation of capture-group expressions, we need to extend the syntax of the expressions in order to attach to any capture group the current part of the input string captured during an execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' A contextual capture-group expression E over a symbol alphabet Σ and a variable alphabet Γ (or Σ, Γ-expression for short) is inductively defined as E = a, E = ε, E = ∅, E = F + G, E = F · G, E = F ∗, E = (F)u x, E = x, where F and G are two Σ, Γ-expressions, a is a symbol in Σ, u is in Σ∗ and x is a variable in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Notice that a Σ, Γ-expression is equivalent to a contextual capture-group expression where u = ε for every occurrence of capture group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In the following, we consider that a context is a function from Γ to Maybe(Σ∗), modelling the possibility that a variable was initialized (or not) during the parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The set of contexts is denoted by Ctxt(Γ, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Using these notions of contexts, let us now explain the semantics of contextual capture-group expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' While parsing, a context is built to memorize the different affectations of words to variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Therefore, a (contextual) language associated with an expression is a set of couples built from a language and the context that was used to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The classic atomic cases (a symbol, the empty word or the empty set) are easy to define, preserving the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Another one is the case of a variable x: the context is applied here to compute the associated word (if it exists) and is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The recursive cases are interpreted as such: – The contextual language of a sum of two expressions is the union of their contextual languages, computed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – The contextual language of a catenation of two expressions F and G is computed in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' First, the contextual language of F is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Secondly, for each couple (L, ctxt) of this contextual language, the function ctxt is considered as the new context to compute the contextual language of G, leading to new couples (L′, ctxt′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, for each of these combinations, a couple (L·L′, ctxt′) is added to form the resulting contextual language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – The contextual language of a starred expression is, classically, the infinite union of the powered contextual languages, computed by iterated catenations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – The contextual language of a captured expression (F)u x is computed in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' First, the contextual language of F is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then, for each couple (L, ctxt) of it, a word w is chosen in L and the context ctxt must be updated coherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' More formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' the contextual language of a Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Γ-expression E associated with a context ctxt in Ctxt(Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Σ) is the subset Lctxt(E) of 2Σ∗ × Ctxt(Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Σ) inductively defined as follows: Lctxt(a) = {({a},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Lctxt(ε) = {({ε},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Lctxt(∅) = ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Lctxt(x) = � ∅ if ctxt(x) = Nothing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' {({w},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)} otherwise if ctxt(x) = Just(w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Lctxt(F + G) = Lctxt(F) ∪ Lctxt(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Lctxt(F · G) = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1(G) {(L1 · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Lctxt(F ∗) = � n∈N (Lctxt(F)) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Lctxt((F)u x) = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' w∈L1 {({w},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' [ctxt1]x←uw)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' where F and G are two Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Γ-expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a is a symbol in Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' x is a variable in Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' u is in Σ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Ln is defined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' for any set L of couples (language,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' context) by Ln = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � (L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt)∈L {({ε},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)} if n = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Ln−1 {(L1 · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' and [ctxt]x←w is the context defined by [ctxt]x←w(y) = � Just(w) if x = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt(y) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The contextual language of an expression E is the set of couples obtained from an uninitialised context, where nothing is associated with any variable, that is the set Lλ_→Nothing(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, the language denoted by an expression E is the set of words obtained by forgetting the contexts, that is the set � (L,_)∈Lλ_→Nothing(E) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us consider the three following expressions over the symbol alphabet {a, b, c} and the variable alphabet {x}: E = E1 · E2, E1 = ((a∗)xbx)∗, E2 = cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The language denoted by E2 is empty, since it is computed from the empty context, where nothing is associated with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' However, parsing E1 allows us to compute contexts that define word values to affect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us thus show how is defined the contextual language of E1: – the contextual language of (a∗)x is the set� n∈N {({an}, λx → Just(an))} where each word an is recorded in a context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – the contextual language of (a∗)xbx is the set � n∈N {({anban}, λx → Just(an))} where each word an is recorded in a context applied to evaluate the variable x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – the contextual language of E1 is the union of the two following sets S1 and S2: S1 = {({ε}, λx → Nothing)} S2 = {({anban | n ∈ N}∗ · {ambam}, λx → Just(am)) | m ∈ N} where each iteration of the outermost star produces a new record for the variable x in the context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' however, notice that only the last one is recorded at the end of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, the language of E is obtained by considering the contexts obtained from the parsing of E1 to evaluate the occurrence of x in E2, leading to the set� m∈N ({anban | n ∈ N}∗ · {ambamcam}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Obviously, some classical equations still hold with these computations: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let E, F and G be three Σ, Γ-expressions and ctxt be a context in Ctxt(Γ, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The two following equations hold: Lctxt(E · (F + G)) = Lctxt(E · F + E · G) Lctxt(F ∗) = Lctxt(ε + F · F ∗) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us proceed by equality sequences: Lctxt(E · (F + G)) = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (F +G) {(L1 · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (F )∪Lctxt1(G) {(L1 · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (F ) {(L1 · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} ∪ � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1(G) {(L1 · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = Lctxt(E · F) ∪ Lctxt(E · G) = Lctxt(E · F + E · G) Lctxt(F ∗) = � n∈N (Lctxt(F)) n = (Lctxt(F)) 0 ∪ � n∈N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='n≥1 (Lctxt(F)) n = (Lctxt(F)) 0 ∪ � n∈N Lctxt(F) · (Lctxt(F)) n = (Lctxt(F)) 0 ∪ Lctxt(F) · � n∈N (Lctxt(F)) n = Lctxt(ε + F · F ∗) In order to solve the membership test for the contextual capture-group expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' let us extend the classical derivation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' But first, let us show how to extend the nullability predicate, needed at the end of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='3 Nullability Computation The nullability predicate allows us to determine whether the empty word belongs to the language denoted by an expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As far as capture groups are concerned, a context has to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Therefore, the nullability predicate can be represented as a set of contexts the application of which produces a language that contains the empty word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' As we have seen, the nullability depends on the current context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Given an expression and a context ctxt, the nullability predicate is a set in 2Ctxt(Γ,Σ), computed as follows: Nullctxt(ε) = {ctxt} Nullctxt(∅) = ∅ Nullctxt(a) = ∅ Nullctxt(x) = � {ctxt} if ctxt(x) = Just(ε) ∅ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Nullctxt(E + F) = Nullctxt(E) ∪ Nullctxt(F) Nullctxt(E · F) = � ctxt′∈Nullctxt(F ), ctxt′′∈Nullctxt′ (G) {ctxt′′} Nullctxt(E∗) = {ctxt} Nullctxt((E)u x) = � ctxt′∈Nullctxt(F ) {[ctxt′]x←u} where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us consider the three expressions of Example 12: E = E1 · E2, E1 = ((a∗)xbx)∗, E2 = cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' For any context ctxt, Nullctxt(E1) = {ctxt}, Nullctxt(E2) = ∅, Nullctxt(E) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The nullability predicate allows us to determine whether there exists a couple in the contextual language of an expression such that its first component contains the empty word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let E be a Σ, Γ-expression and ctxt be a context in Ctxt(Γ, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then the two following conditions are equivalent: – Nullctxt(E) ̸= ∅, – ∃(L, _) ∈ Lctxt(E) | ε ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' By induction over the structure of E: – If E = a ∈ Σ or E = ∅, the property holds since Nullctxt(E) is empty and since there is no couple (L, ctxt′) in Lctxt(E) with ε in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = ε, the following two conditions hold, Nullctxt(E) = {ctxt}, Lctxt(E) = {({ε}, ctxt)}, satisfying the stated condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = F + G, the following two conditions hold: Nullctxt(F + G) = Nullctxt(F) ∪ Nullctxt(G), Lctxt(F + G) = Lctxt(F) ∪ Lctxt(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Since, by induction hypothesis, the following two conditions hold Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L, Nullctxt(G) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(G) | ε ∈ L, the proposition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = F · G, the two following conditions hold: Nullctxt(F · G) = � ctxt′∈Nullctxt(F ), ctxt′′∈Nullctxt′(G), {ctxt′′}, Lctxt(F · G) = � (L,ctxt′)∈Lctxt(F ), (L′,ctxt′′)∈Lctxt′(G), {(L · L′, ctxt′′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Since, by induction hypothesis, the two following conditions hold, Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L, Nullctxt′(G) ̸= ∅ ⇔ ∃(L, ctxt′′) ∈ Lctxt′(G) | ε ∈ L, the proposition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = F ∗, since the two following conditions hold Nullctxt(F ∗) = {ctxt}, Lctxt(F) 0 = {({ε}, ctxt)} ∈ Lctxt(F ∗), the stated condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = (F)u x, both following conditions hold: Nullctxt((F)u x) = � ctxt′∈Nullctxt(F ) {[ctxt′]x←u}, Lctxt((F)u x) = � (L,ctxt′)∈Lctxt(F ), w∈L {({w}, [ctxt′]x←uw)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then, following induction hypothesis, Nullctxt(F) ̸= ∅ ⇔ ∃(L, ctxt′) ∈ Lctxt(F) | ε ∈ L, the stated condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = x, both following conditions hold: Nullctxt(x) = � {ctxt} if ctxt(x) = Just(ε) ∅ otherwise, Lctxt(x) = � ∅ if ctxt(x) = Nothing, {({w}, ctxt)} otherwise if ctxt(x) = Just(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Therefore, the proposition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='4 Derivation formulae Similarly to the nullability predicate, the derivation computation builds the context while parsing the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' the derivative of an expression with respect to a context is a set of couples (expression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' context),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' induc- tively computed as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' for any Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Γ-expression and for any context ctxt in Ctxt(Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Σ): dctxt a (ε) = ∅ dctxt a (∅) = ∅ dctxt a (b) = � ∅ if a ̸= b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' {(ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)} otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' dctxt a (x) = � dctxt a (w) if ctxt(x) = Just(w) ∅ otherwise dctxt a (F + G) = dctxt a (F) ∪ dctxt a (G) dctxt a (F · G) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ) {(F ′ · G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)} ∪ � ctxt′∈Nullctxt(F ) dctxt′ a (G) dctxt a (F ∗) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ) {(F ′ · F ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)} dctxt a ((F)u x) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ) {((F ′)u·a x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)} where F and G are two Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Γ-expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a is a symbol in Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' x is a variable in Γ and u is in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us consider the three expressions of Example 12: E = E1 · E2, E1 = ((a∗)xbx)∗, E2 = cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then, for any context ctxt, dctxt a (E) = {((a∗)a xbx((a∗)xbx)∗cx, ctxt)}, dctxt b (E) = {(x((a∗)xbx)∗cx, λx → ε)}, dctxt c (E) = {(x, ctxt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The derivation of an expression allows us to syntactically express the computation of the quotient of the language components in contextual languages, where the quotient w−1(L) is the set {w′ | ww′ ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and a be a symbol in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then: � (E′,ctxt′)∈dctxt a (E) Lctxt′(E′) = � (L′,ctxt′)∈Lctxt(E) {(a−1(L′), ctxt′)} Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' By induction over the structure of E, assimilating ∅ and {(∅, ctxt)} for any context ctxt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = ε or E = ∅, the property vacuously holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = b ∈ Σ, � (E′,ctxt′)∈dctxt a (b) Lctxt′(E′) = � ∅ if b ̸= a, {({ε}, ctxt)} otherwise, = {(a−1({b}), ctxt)} = � (L′,ctxt′)∈Lctxt(b) {(a−1(L′), ctxt′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = F + G,� (E′,ctxt′)∈dctxt a (F +G) Lctxt′(E′) = � (E′,ctxt′)∈dctxt a (F )∪dctxt a (G) Lctxt′(E′) = � (E′,ctxt′)∈dctxt a (F ) Lctxt′(E′) ∪ � (E′,ctxt′)∈dctxt a (G) Lctxt′(E′) = � (L′,ctxt′)∈Lctxt(F ) {(a−1(L′), ctxt′)} ∪ � (L′,ctxt′)∈Lctxt(G) {(a−1(L′), ctxt′)} = � (L′,ctxt′)∈Lctxt(F )∪Lctxt(G) {(a−1(L′), ctxt′)} = � (L′,ctxt′)∈Lctxt(F +G) {(a−1(L′), ctxt′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = F · G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' � (E′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈dctxt a (F ·G) Lctxt′(E′) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ) Lctxt′(F ′ · G) ∪ � ctxt′∈Nullctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (G′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′′)∈dctxt′ a (G) Lctxt′′(G′) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1(G) {(L1 · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} ∪ � ctxt′∈Nullctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (G′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′′)∈dctxt′ a (G) Lctxt′′(G′) = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (G) {(a−1(L1) · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} ∪ � ctxt1∈Nullctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (G) {(a−1(L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (G) {(a−1(L1) · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} ∪ � ∃(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F )|ε∈L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1(G) {(a−1(L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (G) {(a−1(L1) · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} ∪ � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ε∈L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (G) {(a−1(L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1 (G) {(a−1(L1 · L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈Lctxt(F ·G) {(a−1(L′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – If E = F ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' � (E′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈dctxt a (F ∗) Lctxt′(E′) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ) Lctxt′(F ′ · F ∗) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1(F ∗) {(L1 · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1(F ∗) {(a−1(L1) · L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' (L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt2)∈Lctxt1(F ∗) {(a−1(L1 · L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt2)} = � (L′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈Lctxt(F ·F ∗) {(a−1(L′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)} = � (L′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈Lctxt(ε+F ·F ∗) {(a−1(L′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)} = � (L′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈Lctxt(F ∗) {(a−1(L′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)} – If E = (F)u x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' � (E′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈dctxt a ((F )u x) Lctxt′(E′) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ) Lctxt′((F ′)u·a x ) = � (ctxt′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='F ′)∈dctxt a (F ) (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt′(F ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' w∈L1 {({w},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' [ctxt1]x←uaw)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' w∈a−1(L1) {({w},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' [ctxt1]x←uaw)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' aw∈L1 {({w},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' [ctxt1]x←uaw)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' aw∈L1 {(a−1({aw}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' [ctxt1]x←uaw)} = � (L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt1)∈Lctxt(F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' w∈L1 {(a−1({w}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' [ctxt1]x←uw)} = � (L′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈Lctxt((F )u x) {(a−1(L′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)} – If E = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' � (E′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈dctxt a (x) Lctxt′(E′) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 � (E′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈dctxt a (w) Lctxt′(E′) if ctxt(x) = Just(w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ∅ otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 � (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt)∈dctxt a (aw) Lctxt(w) if ctxt(x) = Just(aw),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ∅ otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' = � {({w},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)} if ctxt(x) = Just(aw),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ∅ otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' = � {(a−1({aw}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)} if ctxt(x) = Just(aw),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ∅ otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' = � {(a−1({w}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)} if ctxt(x) = Just(w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ∅ otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' = � (L′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt′)∈Lctxt(x) {(a−1(L′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt′)} The derivation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a word is, as usual, an iterated application of the derivation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a symbol, recursively defined as follows, for any Σ, Γ-expression E, for any context ctxt in Ctxt(Γ, Σ), for any symbol a in Σ and for any word v in Σ∗: dctxt ε (E) = {(E, ctxt)}, dctxt a·v (E) = � (E′,ctxt′)∈dctxt a (E) dctxt′ v (E′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Example 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us consider the three expressions of Example 14: E = E1 · E2, E1 = ((a∗)xbx)∗, E2 = cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then, for any context ctxt, dctxt ab (E) = dctxt b ((a∗)a xbx((a∗)xbx)∗cx) = {(x((a∗)xbx)∗cx, λx → a)} dctxt aba (E) = dλx→a a (x((a∗)xbx)∗cx) = {(((a∗)xbx)∗cx, λx → a)} dctxt abac(E) = dλx→a c (((a∗)xbx)∗cx) = {(x, λx → a)} dctxt abaca(E) = dλx→a a (x) = {(ε, λx → a)} Such an operation allows us to syntactically compute the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and w be a word in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then: � (E′,ctxt′)∈dctxt w (E) Lctxt′(E′) = � (L′,ctxt′)∈Lctxt(E) {(w−1(L′), ctxt′)} Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' By a direct induction over the structure of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, the membership test of a word w can be performed as usual by first computing the derivation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' w, and then by determining the existence of a nullable derivative, as a direct corollary of Proposition 3 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let E be a Σ, Γ-expression, ctxt be a context in Ctxt(Γ, Σ) and w be a word in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then the two following conditions are equivalent: – ∃(L, _) ∈ Lctxt(E) | w ∈ L, – ∃(E′, ctxt′) ∈ dctxt w (E) | Nullctxt′(E′) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We have shown how to compute the derivatives and solve the membership test in a classical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us show how to embed the context computation in a convenient monad, in order to generalize the definitions to other structure than sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='5 The StateT Monad Transformer Monads do not compose well in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' However, ones can consider particular combinations of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Among those, well-known patterns are the monad transformers like the StateT Monad Transformer [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' This combination allows us to mimick the use of global variables in a functional way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In our setting, it allows us to embed the context computation in an elegant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let S be a set and M be a monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We denote by StateT(S, M) following the mapping: StateT(S, M)(A) = S → M(A × S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' In other terms, StateT(S, M)(A) is the set of functions from S to the monadic structure M(A×S) based on couples in the cartesian product (A × S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' The mapping StateT(S, M) can be equipped by a structure of functor, defined for any function f from a set A to a set B by StateT(S, M)(f)(state)(s) = M(λ(a, s) → (f(a), s))(state(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' It can also be equipped with the structure of monad, defined for any function f from a set A to the set StateT(S, M)(B): pure(a) = λs → pure(a, s) bind(f)(state)(s) = state(s) >>= λ(a, s′) → f(a)(s′) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='6 Monadic Definitions The previous definitions associated with capture-group expressions can be equivalently restated using the StateT monad transformer specialised with the Set monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Let us first consider the following claims where M = StateT(Ctxt(Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Set),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' allowing us to bring closer M and the previous notion of monadic support: – R = (M(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ×,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 0) is a semiring by setting: f1 + f2 = λs → f1(s) ∪ f2(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' f1 × f2 = f1 >>= λ_ → f2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 1 = λs → {(⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' s)} = pure(⊤),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 0 = λs → ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – M = (M(Exp(Σ)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 0) is a monoid by setting: ± = +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 0 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ⋉) is a Exp(Σ)-right-semimodule by setting: f ⋉ F = λs → � (E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='ctxt)∈f(s) {(E · F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ctxt)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' – (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' ⊲) is a R-left-semimodule by setting: f1 ⊲ f2 = f1 >>= λ_ → f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Then, the nullable predicate formulae can be equivalently restated as an element in StateT(Ctxt(Γ, Σ), Set)(1), which is equal by definition to Ctxt(Γ, Σ) → Set(1 × Ctxt(Γ, Σ)), isomorphic to Ctxt(Γ, Σ) → Set(Ctxt(Γ, Σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' It can inductively be computed as follows: Null(ε) = 1 Null(∅) = 0 Null(a) = 0 Null(E + F) = Null(E) + Null(F) Null(E · F) = Null(E) × Null(F) Null(E∗) = 1 Null(x)(ctxt) = � pure((⊤, ctxt)) if ctxt(x) = Just(ε), ∅ otherwise, Null((E)u x)(ctxt) = Set(λ(⊤, ctxt′) → (⊤, [ctxt′]x←u))(Null(F)(ctxt)), where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Notice that these formulae are the same that the ones in Definition 2 as far as classical operators are concerned, and that these formulae can be easily generalized to other convenient monads than Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Moreover, the derivative of an expression is an element in StateT(Ctxt(Γ, Σ), Set)(Exp(Σ, Γ)): da(ε) = 0 da(∅) = 0 da(b) = � 0 if a ̸= b, pure(ε) otherwise, da(E + F) = da(E) ± da(F) da(E · F) = da(E) ⋉ F + Null(E) ⊲ da(F) da(E∗) = da(E) ⋉ E∗ da((E)u x) = StateT(Ctxt(Γ, Σ), Set)(λF → (F)ua x )(da(E)) da(x)(ctxt) = � pure((w, ctxt)) if ctxt(x) = Just(aw), ∅ otherwise, where E and F are two Σ, Γ-expressions, a is a symbol in Σ, x is a variable in Γ and u is in Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Once again, notice that these formulae are the same that the ones in Definition 5 as far as classical operators are concerned, and that these formulae can be easily generalized to other convenient monads than Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, the derivation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' a word is monadically defined as in previous sections: dε(E) = pure(E), dav(E) = da(E) >>= dv, and the membership test of a word w can be equivalently rewritten as follows: (dw(E) >>= Null)(λ_ → Nothing) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' 10 Conclusion and Perspectives In this paper, we achieved the first step of our plan to unify the derivative computation over word expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Monads are indeed useful tools to abstract the underlying computation structures and thus may allow us to consider some other functionalities, such as capture groups via the well-known StateT monad transformer [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' We aim to study the conditions satisfying by monads that lead to finite set of derivated terms, and to extend this method to tree expressions using enriched categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Finally, we plan to extend monadic derivation to other underlying monads for capture groups, linear combinations for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFPT4oBgHgl3EQfTjTq/content/2301.13054v1.pdf'} +page_content=' Antimirov, V.' metadata={'source': 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b/4NE1T4oBgHgl3EQfSgOT/content/tmp_files/2301.03067v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..00cc8198271c94658f36e7d97dd7c2bcbe77f581 --- /dev/null +++ b/4NE1T4oBgHgl3EQfSgOT/content/tmp_files/2301.03067v1.pdf.txt @@ -0,0 +1,897 @@ +arXiv:2301.03067v1 [astro-ph.HE] 8 Jan 2023 +Neutron stars in the context of f(T, T ) gravity +Cl´esio E. Mota1,∗ Luis C. N. Santos2,† Franciele M. da Silva3,‡ +Cesar V. Flores4,5,§ Iarley P. Lobo6,7,¶ and Valdir B. Bezerra2∗∗ +1Departamento de F´ısica, CFM - Universidade Federal de Santa +Catarina; C.P. 476, CEP 88.040-900, Florian´opolis, SC, Brasil. +2Departamento de F´ısica, CCEN-Universidade Federal da Para´ıba; +C.P. 5008, CEP 58.051-970, Jo˜ao Pessoa, PB, Brazil +3N´ucleo Cosmo–ufes & Departamento de F´ısica, Universidade Federal do Esp´ırito Santo, +Av. +Fernando Ferrari, 540, CEP 29.075-910, Vit´oria, ES, Brazil +4Centro de Ciˆencias Exatas, Naturais e Tecnol´ogicas, +CCENT - Universidade Estadual da Regi˜ao Tocantina do Maranh˜ao; C.P. 1300, +CEP 65901-480, Imperatriz, MA, Brasil. +5Departamento de F´ısica, CCET - Universidade Federal do Maranh˜ao, +Campus Universit´ario do Bacanga; CEP 65080-805, S˜ao Lu´ıs, MA, Brasil. +6Department of Chemistry and Physics, Federal University of Para´ıba, +Rodovia BR 079 - Km 12, 58397-000 Areia-PB, Brazil. and +7Physics Department, Federal University of Lavras, +Caixa Postal 3037, 37200-000 Lavras-MG, Brazil. +In this work, we investigate the existence of neutron stars (NS) in the framework of f(T, T ) +gravity, where T is the torsion tensor and T is the trace of the energy-momentum tensor. The +hydrostatic equilibrium equations are obtained, however, with p and ρ quantities passed on by +effective quantities ¯p and ¯ρ, whose mass-radius diagrams are obtained using modern equations of +state (EoS) of nuclear matter derived from relativistic mean field models and compared with the +ones computed by the Tolman-Oppenheimer-Volkoff (TOV) equations. Substantial changes in the +mass-radius profiles of NS are obtained even for small changes in the free parameter of this modified +theory. The results indicate that the use of f(T, T ) gravity in the study of NS provides good results +for the masses and radii of some important astrophysical objects, as for example, the low-mass X-ray +binary (LMXB) NGC 6397 and the pulsar of millisecond PSR J0740+6620. In addition, radii results +inferred from the Lead Radius EXperiment (PREX-2) can also be described for certain parameter +values. +Keywords : general relativity, modified gravity, neutron stars. +I. +INTRODUCTION +In recent years, there have been a growing number +of ideas exploring modifications and alternative formu- +lations of General Relativity (GR) emerging of different +contexts. In fact, GR is a theory well tested, providing +an interesting description of the space-time nature as a +dynamical stage where physical phenomena takes place. +In parallel to the advances in GR, the quantization of +the gravitational field remains an open problem. With +respect to this issue, it was pointed out that the action +for gravity should be constructed with higher-order cur- +vature terms in the context of renormalization at one +loop level [1]. In the literature there are some formula- +tions of gravity where the usual Einstein-Hilbert action +is supplemented by higher-order curvature terms, as for +example in the context of the f(R) theory in which case +∗ clesio200915@hotmail.com +† luis.santos@ufsc.br +‡ franmdasilva@gmail.com +§ cesarovfsky@gmail.com +¶ iarley˙lobo@fisica.ufpb.com +∗∗ valdirbarbosa.bezerra@gmail.com +the Ricci scalar R in the action is replaced by a general +function f(R) [2]. +On the other hand, there are questions concerning the +content of energy and matter in the universe that, at +the moment, are not satisfactorily explained in the scope +of standard theories. +The observed rotation curves of +galaxies [3] and the “missing mass” of galaxy clusters +[4] suggest the dark matter hypothesis, while the ac- +celerated expansion of the universe observed today can +be interpreted as an effect of the so-called dark energy +[5, 6]. Unexpectedly these observations reveals that the +ordinary baryonic matter corresponds to only 4% of con- +tent of energy of the universe while the dark matter and +dark energy correspond to 20% and 76%, respectively. In +this sense, there are studies considering the possibility of +modified theories of gravity which may help to alleviate +the need for dark components of energy of the universe +beyond the scope of GR. +The late-time acceleration of the universe can be in- +terpreted under two points of view. In the first one, it +is introduced a dark energy sector in the energy content +of the universe through a type of field. In the second +one, the gravitational field itself is modified. +In addi- +tion, there may be combinations of both approaches de- +pending on the couplings between gravitational and non- + +2 +gravitational sectors of theory [7–10]. Thus, it is expected +that different formulations of gravity imply that standard +results in astrophysics suffer modifications. Compact ob- +jects as neutron stars (NS), have been studied consid- +ering effects of such modifications [11–20]. +NS in the +context of f(R) gravity were studied in [21–23] and in +f(R, T ) gravity in the papers [24–28]. In common, all +of these works have considered effects on NS due to the +modification of the gravitational field that include extra +terms in the action. In the scheme of nonconservative +gravity, the modification of the gravitational field can be +done through a reinterpretation of the conservation law, +as was considered in the papers [29, 30] (for a review on +non-conservative theories of gravity, see [31]). Usually, +the non-conservation of the stress-energy tensor is pro- +portional to the matter density and pressure themselves. +For this reason, an environment such as a compact ob- +ject like a NS turns out to be an appealing laboratory for +testing such theories. +In the context of modified theories of gravity, the so- +called f(T,T ) gravity is a class of such theories, free of +ghosts and instabilities which, when applied to cosmo- +logical problems, leads to interesting results [32]. In this +formulation, the action depends on the torsion scalar T +and on the trace of the energy-momentum tensor T . As +in the case of f(T) gravity where the action is an ar- +bitrary function of the torsion, in f(T, T ) gravity, the +action is a arbitrary function of both the trace of the +energy-momentum tensor and the torsion scalar. +In this paper, we study an important context, not yet +explored in the literature, that are the implications of +the f(T,T ) gravity on NS. In particular, we obtain the +mass-radius relation of NS in the context of this modified +gravity and compare our results with recent astrophysical +observations and experiments. +This work is organized as follows: In Section II we ex- +pose a summary of the f(T,T ) gravity. In Section III we +derive the equations describing static, spherically sym- +metric stars in this modified theory of gravity. In Section +IV we present our results and in Section V we close with +our final remarks. +II. +GRAVITATIONAL FIELD EQUATIONS OF +f(T, T ) GRAVITY +Given a line element describing a space-time we want +to study +ds2 = gµνdxµdxν = ηABeA +µeB +νdxµdxν +(1) +where gµν and {eA +µ} are respectively the metric tensor +and the components of the tetrad associated to space- +time geometry, and ηAB = diag(1, −1, −1, −1) is the +Minkowski metric. The signature (+ − − −) and ge- +ometrized units, that is, G = c = 1, will be taken into +account. +In GR we assume that gravity is associated +with the curvature of the space-time and thus we use the +Levi-Civita’s connection +◦ +Γρ +µν = 1 +2gρσ (∂νgσµ + ∂µgσν − ∂σgµν) +(2) +to compute quantities associated with the curvature such +as the Ricci scalar, R, that is present in the GR’s action. +On the other hand, in teleparallel theory one assumes +that gravity is associated to the torsion of the space-time +and thus the Weizenbock’s connection +Γλ +µν = e λ +A ∂µeA +ν = −eA +µ∂νe λ +A +(3) +is used to construct quantities associated with the tor- +sion, as the torsion scalar T that appears in the telepar- +allel gravity action. In the modified teleparallel theories +it is assumed that the action depends on a arbitrary func- +tion of T. In our case, we are going to consider a modified +action given by [32] +S = +� +d4x e +�T + f(T, T ) +16π ++ Lm +� +, +(4) +where e is the determinant of the tetrads e = det(eA +µ) = +√−g and T += gµνTµν is the trace of the energy- +momentum tensor Tµν, which can be obtained from the +Lagrangian for the matter distribution Lm in the follow- +ing way +Tµν = gµνLm − 2∂Lm +∂gµν . +(5) +Let us assume that the function f(T, T ) is given by +f (T, T ) = ω Tn T − 2Λ , +(6) +where ω, n and Λ are arbitrary constants, specifically ω +can be interpreted as a coupling constant of geometry +with matter fields, n is a pure number (assumed to be +unity here) and Λ can be recognized as the cosmological +constant as discussed in [32, 33]. +We are interested in matter that can be described by +a perfect fluid, so that Tµν is given by: +Tµν = −pgµν + (p + ρ)uµuν, +(7) +where p is the pressure and ρ is the energy density of +the fluid. By varying the action from Equation (4) with +respect to the tetrad we find the following field equation +Gµν = 8πT eff +µν , +(8) +where the effective energy-momentum tensor T eff +µν +is +T eff +µν += gµν + + +� +− ω(ρ − 3p) + 2Λ +� +16π ++ ωp +8π + ++Tµν +� +1+ ω +8π +� +. +(9) +Calculating the covariant derivative of the energy- +momentum tensor given by Equation (7), we obtain the +following result +∇µTνµ = +1 +� +4π + (1/2)ω +� +�ω +4 (∂νT ) − ω +2 ∂νp +� +. +(10) + +3 +In a cosmological context, equation 10 can be associated +to creation or destruction of matter throughout the uni- +verse evolution. As discussed in [26], the interpretation of +creation or destruction of matter particles in the NS level +encounters difficulties in a static framework as occurs +in the study of the hydrostatic equilibrium expression, +i.e, the Tolman-Oppenheimer-Volkof equation. Also, it +usually implies in the presence of a fifth force and non- +geodesic trajectory for free particles. Naturally, results +that depend on such imput would also be modified corre- +spondingly. However, this is not the case analyzed in the +present paper. In the next section we use Equations (8) +to (10) to obtain and analyse the mass-radius relation of +NS in the context of modified teleparallel gravity. +III. +STELLAR STRUCTURE EQUATIONS +In this section, we discuss some of the main procedures +that leads to the deduction of the hydrostatic equilibrium +equation in the context of f(T, T ) gravity. +To study compact stars, such as NS, magnetars and +other astrophysical structures, we assume these objects +as being homogeneous, static (no rotation), isotropic and +spherically symmetric [34]. Therefore, we must use the +appropriate metric in a convenient coordinate system +that describes the object being studied. The most gen- +eral metric describing the space-time under consideration +is given by the line element +ds2 = eν(r)dt2 − eλ(r)dr2 − r2(dθ2 + sin θ2dφ2), +(11) +where ν and λ are radial functions that we want to de- +termine based on the field equations (8). +Thus, using +Equation (11) and substituting appropriately into Equa- +tion (8),we obtain the following results +e−λ�λ′ +r − 1 +r2 +� ++ 1 +r2 = 8π + + + + + +� +− ω(ρ − 3p) + 2Λ +� +16π ++ ωp +8π + + + ρ +� +1 + ω +8π +� + + = 8π¯ρ, +(12) +e−λ�ν′ +r + 1 +r2 +� +− 1 +r2 = −8π + + + + + +� +− ω(ρ − 3p) + 2Λ +� +16π ++ ωp +8π + + − p +� +1 + ω +8π +� + + = 8π¯p, +(13) +e−λ +4r +� +2 +� +λ′ − ν′� +− +� +2ν′′ + ν′2 − ν′λ′� +r +� += −8π + + + + + +� +− ω(ρ − 3p) + 2Λ +� +16π ++ ωp +8π + + − p +� +1 + ω +8π +� + + = 8π¯p, +(14) +where, the prime denotes a derivative with respect to +the radial coordinate r. The quantities ¯ρ and ¯p are the +effective pressure and energy density, defined as +¯ρ = ρ + ωρ +16π + 5ω p +16π + Λ +8π, +(15) +¯p = p + ωρ +16π − 3ω p +16π − Λ +8π . +(16) +In addition to the field equations, we also need to consider +the conservation equation (10) in f(T, T ) gravity so that +we have a complete set of equations to be solved. +In +the case we are studying, Equation (10) has the form as +follows +−p′− ν′ +2 (ρ+p) = +1 +� +4π + (1/2)ω +� +�ωρ′ +4 +− 5ω p′ +4 +� +. (17) +Redefining the function λ(r) as +e−λ(r) = 1 − 2M(r) +r +, +(18) +and rearranging Equations (12) and and (17), we get the +equations required to describe static spherically symmet- +ric stellar structures in f(T, T ) gravity theory, which are +given by +dM +dr = 4πr2 ¯ρ, +(19) +and +d¯p +dr = −M ¯ρ +r2 +� +1 + ¯p +¯ρ +� � +1 + 4πr3¯p +M +� � +1 − 2M +r +�−1 +. +(20) +In the next section we show some results obtained by +solving Equations (19) and (20) for realistic EoS of NS. + +4 +IV. +RESULTS +In this section, we present the results obtained from the +solution of the field equations in the context of f(T, T ) +modified theory of gravity applied to NS. +As an input to the stellar hydrostatic equilibrium equa- +tions, we use two realistic EoS obtained from a relativis- +tic mean field (RMF) approach. Firstly, we consider the +IU-FSU [35] parametrization because it is able to explain +reasonably well both nuclear [36] and stellar matter prop- +erties [37]. We then compare the IU-FSU results with the +ones obtained with a stiffer EoS calculated with a model +of coupling of mesons and quarks, the quark–meson cou- +pling (QMC) model [38]. (For the EoS with the QMC +model, we refer the reader to refs. [38–42].) It is well +known that a stiffer EoS leads to a bigger NS maximum +mass in contrast to a softer one. In fact, using the EoS +QMC as an input to the stellar equilibrium equations +yields a maximum mass greater than 2.0 M⊙, and, there- +fore, we want to verify that we get the same qualitative +behavior for macroscopic properties (such as mass and +radius) with parameterizations that are substantially dif- +ferent. For the NS crust, we use the full BPS [43] EoS. +After defining the EoS, some boundary conditions are +required to solve the equations (19) and (20) along the +radial coordinate r, from the center towards the surface +of the star. At the star’s center r = 0 we take: +M(0) = 0 ; +¯ρ(0) = ¯ρc ; +¯p(0) = ¯pc. +(21) +The radius of the star (r = R) is determined as the +point where the pressure vanishes, i.e, ¯p(R) = 0. +At +this point, the interior solution connects softly with the +Schwarzschild vacuum solution, indicating that the po- +tential metrics of the interior and the exterior metric are +related as eν(R) = +1 +eλ(R) = 1 − 2M/R, being M the total +mass of the star. +Let us discuss and compare our results with recent +astrophysical observations and nuclear physics experi- +ments. At first, the NS in LMXB NGC 6397, depicted as +a green shaded area in all figures, provides a constraint +at 68% confidence level over the possible values of the +masses and corresponding radii of the NS [44, 45]. Simi- +larly, the millisecond pulsars are among the most useful +astrophysical objects in the Universe for testing funda- +mental physics, because they impose some of the most +stringent constraints on high-density nuclear physics in +the stellar interior [46]. +Recent measurements coming +from the Neutron Star Interior Composition Explorer +(NICER) mission reported pulsar observations for canon- +ical (1.4 M⊙) and massive (2.0 M⊙) NS. The mass mea- +surement and radius estimates provided for these objects, +are 11.80 km ≤ R1.4 ≤ 13.1 km for the 1.4M⊙ NS PSR +J0030+0451 (horizontal line segment in red colour shown +in all Figures) and 11.60 km ≤ R ≤ 13.1 km for a NS with +mass between 2.01M⊙ ≤ M ≤ 2.15M⊙ PSR J0740+6620 +(the rectangular region in orange colour shown in all Fig- +ures). +However, the authors of Ref. +[47] used the re- +cent measurement of neutron skin on 208Pb by PREX-2 + 0.5 + 1 + 1.5 + 2 + 2.5 + 4 + 6 + 8 + 10 + 12 + 14 + 16 +IU-FSU +M /MO• +R(km) +ϖ = 0.0 +ϖ = 0.01 +ϖ = 0.02 +ϖ = 0.08 +ϖ = 0.1 +ϖ = 0.2 +ϖ = - 0.01 +ϖ = - 0.02 +ϖ = - 0.2 + 1.9 + 1.95 + 10.5 + 11 + 11.5 + 12 + 12.5 +IU-FSU +FIG. 1. Mass-radius relation for families of NS’s described +by the IU-FSU EoS. We analyse the effect of varying the pa- +rameter ω of the f(T, T ) theory. The red and green line seg- +ment represent the radius range of the 1.4M⊙ NS for PSR +J0030 + 0451 and PREX-2, respectively. The orange rectan- +gular region corresponds to the range of radius estimates for +2.08 ± 0.07M⊙ NS PSR J0740+6620. Similarly, the blue and +pink horizontal lines stand, respectively, for the mass mea- +surements of NS PSR J1614 + 2230 and NS PSR J0348 + +0432. +The purple solid line curve is solution for the usual +TOV equation from GR. +to constrain the radius of NS, which leads to a predic- +tion of the radius of the canonical 1.4 M⊙ of 13.25 km +≲ R1.4 ≲ 14.26 km (horizontal line segment in green +colour shown in all Figures). +Likewise, we also com- +pare our results with two massive stars that had been +discovered in 2010 and 2013, namely, PSR J1614+2230 +[48] with mass 1.97 ± 0.04 M⊙ (horizontal line in blue +colour shown in all Figures) and PSR J0348+0432 [49] +with mass 2.01 ± 0.04 M⊙ (horizontal line in pink colour +shown in all Figures). Our results are discussed in the +next paragraphs. +We modelled the function f(T, T ) according to equa- +tion (6). This function model has already been used in +recent works as, for example, in [32, 33]. We explore the +values of the parameter ω which range from −0.2 to 0.2. +On the other hand, we check that the Λ parameter has no +significant effect on the mass-radius profiles of NS, since +it appears as a constant in the f(T, T ) function that we +have chosen. Therefore, we use Λ = 0. Note that we +recover the GR solution from f(T, T ) theory by assum- +ing that ω = Λ = 0. These plots are represented by the +continuous purple lines in the Figures. +In Figure 1 we show the effects of f(T, T ) theory on +NS properties obtained with the IU-FSU EoS. We can +see that the value of ω has a very small influence on the +maximum mass of the stars. The radius of the canon- +ical NS (M = 1.4M⊙) is considerably affected. Note a +bigger (smaller) radius for the most positive (negative) +values of ω. We can observe that the results of PREX-2 + +5 + 0.5 + 1 + 1.5 + 2 + 2.5 + 4 + 6 + 8 + 10 + 12 + 14 + 16 +QMC +M /MO• +R(km) +ϖ = 0.0 +ϖ = 0.01 +ϖ = 0.02 +ϖ = 0.08 +ϖ = 0.1 +ϖ = 0.2 +ϖ = - 0.01 +ϖ = - 0.02 +ϖ = - 0.2 + 1.92 + 2 + 2.08 + 2.16 + 10.5 + 12 + 13.5 +QMC +FIG. 2. Mass-radius relation for families of NS’s described by +the QMC EoS. We analyse the effect of varying the parameter +ω of the f(T, T ) theory. The red and green line segment repre- +sent the radius range of the 1.4M⊙ NS for PSR J0030 + 0451 +and PREX-2, respectively. The orange rectangular region cor- +responds to the range of radius estimates for 2.08 ± 0.07M⊙ +NS PSR J0740+6620. Similarly, the blue and pink horizontal +lines stand, respectively, for the mass measurements of NS +PSR J1614 + 2230 and NS PSR J0348 + 0432. The purple +solid line curve is the solution for the usual TOV equation +from GR. +cannot be described with IU-FSU EoS in the GR, but in +f(T, T ) theory the solutions with ω = 0.08 and ω = 0.1 +produce mass and radius that agree with this constraint. +However, the solutions obtained with IU-FSU EoS can- +not describe the mass and radius of PSR J0740+6620, +PSR J1614+2230 and NS PSR J0348+0432 neither on +GR nor on f(T, T ) theory. +In Figure 2 we show the mass-radius relation obtained +for QMC EoS in f(T, T ) gravity. Again, the effect of the +parameter ω is to increase the radius when its values in- +crease positively and to decrease the radius when its val- +ues increase negatively. At the same time, the maximum +mass changes very little with the variation of ω. We can +also see that the solutions obtained with the QMC EoS +in f(T, T ) can accommodate almost all the constraints +we are taking into consideration, and with a smaller ra- +dius than in GR, if we take ω = −0.01 or ω = −0.02. +The exception is NS PSR J0030+0451 which only can be +described with QMC EoS in f(T, T ) gravity if we take +ω = −0.2. We can note that for both EoS analysed we +could not find a configuration that satisfies all the con- +straints at the same time. +We can see that for both EoS’s the value of ω has a +very small influence on the maximum mass of the stars, +on the other hand, the value of the radius of the star with +maximum mass increases when we increase the value of +ω and decreases when ω decreases. Also for both EoS’s, +the case ω = −0.2 produces mass-radius curves that are +typical of quark stars. +V. +FINAL REMARKS +We have investigated the effects of f(T, T ) gravity on +NS assuming these compact objects as being homoge- +neous, static and isotropic. In this way, we have consid- +ered a spherically symmetric space-time and solved the +field equations and the hydrostatic equilibrium equation +in the context of this modified theory of gravity. This +type of system can be transformed into a system with +effective pressure and energy density which permitted +that the hydrostatic equilibrium equation was obtained +through known techniques. For the choice of the f(T, T ) +function used here, we obtained that this theory can pre- +dict NS with almost the same mass and smaller radius +than in GR, for a given EoS, that is an interesting result +in view of the recent observations. Considering the low- +mass X-ray binary (LMXB) NGC 6397 and the pulsar of +millisecond PSR J0740+6620, the results obtained using +the modified hydrostatic equilibrium equations present +good agreement with the observed masses and radii. +We particularize f(T, T ) gravity according to equation +(6). +The good results obtained in comparison to GR +suggest future extensions of this work, as for example, by +taking into consideration different choices of the f(T, T ) +function, which should be done in a near future. It can be +interesting to test, for example, high powers in T besides +and new couplings between T and T . In addition, we +can use different EoS as input to the stellar hydrostatic +equilibrium equations along the aforementioned choices +of f(T, T ) function. +ACKNOWLEDGEMENTS +L.C.N.S. would like to thank Conselho Nacional de +Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) for +partial financial support through the research Project +No. +164762/2020-5 and F.M.S. would like to thank +CNPq for financial support through the research Project +No. +165604/2020-4. +I. P. L. was partially supported +by the National Council for Scientific and Technologi- +cal Development - CNPq grant 306414/2020-1 and by +the grant 3197/2021, Para´ıba State Research Foundation +(FAPESQ). I. P. L. would like to acknowledge the contri- +bution of the COST Action CA18108. V.B.B. is partially +supported by CNPq through the Research Project No. +307211/2020-7. +[1] R. Utiyama and B. S. 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Driebe, et al., “A massive pulsar in a +compact relativistic binary,” Science, vol. 340, no. 6131, +p. 1233232, 2013. + diff --git a/4NE1T4oBgHgl3EQfSgOT/content/tmp_files/load_file.txt b/4NE1T4oBgHgl3EQfSgOT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..997219ace58cd431569e770ba83f7bad15960cb0 --- /dev/null +++ b/4NE1T4oBgHgl3EQfSgOT/content/tmp_files/load_file.txt @@ -0,0 +1,734 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf,len=733 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='03067v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='HE] 8 Jan 2023 Neutron stars in the context of f(T, T ) gravity Cl´esio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Mota1,∗ Luis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Santos2,† Franciele M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' da Silva3,‡ Cesar V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Flores4,5,§ Iarley P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Lobo6,7,¶ and Valdir B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Bezerra2∗∗ 1Departamento de F´ısica, CFM - Universidade Federal de Santa Catarina;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 476, CEP 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='040-900, Florian´opolis, SC, Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 2Departamento de F´ısica, CCEN-Universidade Federal da Para´ıba;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 5008, CEP 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='051-970, Jo˜ao Pessoa, PB, Brazil 3N´ucleo Cosmo–ufes & Departamento de F´ısica, Universidade Federal do Esp´ırito Santo, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Fernando Ferrari, 540, CEP 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='075-910, Vit´oria, ES, Brazil 4Centro de Ciˆencias Exatas, Naturais e Tecnol´ogicas, CCENT - Universidade Estadual da Regi˜ao Tocantina do Maranh˜ao;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 1300, CEP 65901-480, Imperatriz, MA, Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 5Departamento de F´ısica, CCET - Universidade Federal do Maranh˜ao, Campus Universit´ario do Bacanga;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' CEP 65080-805, S˜ao Lu´ıs, MA, Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 6Department of Chemistry and Physics, Federal University of Para´ıba, Rodovia BR 079 - Km 12, 58397-000 Areia-PB, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' and 7Physics Department, Federal University of Lavras, Caixa Postal 3037, 37200-000 Lavras-MG, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In this work, we investigate the existence of neutron stars (NS) in the framework of f(T, T ) gravity, where T is the torsion tensor and T is the trace of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The hydrostatic equilibrium equations are obtained, however, with p and ρ quantities passed on by effective quantities ¯p and ¯ρ, whose mass-radius diagrams are obtained using modern equations of state (EoS) of nuclear matter derived from relativistic mean field models and compared with the ones computed by the Tolman-Oppenheimer-Volkoff (TOV) equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Substantial changes in the mass-radius profiles of NS are obtained even for small changes in the free parameter of this modified theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The results indicate that the use of f(T, T ) gravity in the study of NS provides good results for the masses and radii of some important astrophysical objects, as for example, the low-mass X-ray binary (LMXB) NGC 6397 and the pulsar of millisecond PSR J0740+6620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In addition, radii results inferred from the Lead Radius EXperiment (PREX-2) can also be described for certain parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Keywords : general relativity, modified gravity, neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' INTRODUCTION In recent years, there have been a growing number of ideas exploring modifications and alternative formu- lations of General Relativity (GR) emerging of different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In fact, GR is a theory well tested, providing an interesting description of the space-time nature as a dynamical stage where physical phenomena takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In parallel to the advances in GR, the quantization of the gravitational field remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' With respect to this issue, it was pointed out that the action for gravity should be constructed with higher-order cur- vature terms in the context of renormalization at one loop level [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In the literature there are some formula- tions of gravity where the usual Einstein-Hilbert action is supplemented by higher-order curvature terms, as for example in the context of the f(R) theory in which case ∗ clesio200915@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='com † luis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='santos@ufsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='br ‡ franmdasilva@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='com § cesarovfsky@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='com ¶ iarley˙lobo@fisica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='ufpb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='com ∗∗ valdirbarbosa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='bezerra@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='com the Ricci scalar R in the action is replaced by a general function f(R) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' On the other hand, there are questions concerning the content of energy and matter in the universe that, at the moment, are not satisfactorily explained in the scope of standard theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The observed rotation curves of galaxies [3] and the “missing mass” of galaxy clusters [4] suggest the dark matter hypothesis, while the ac- celerated expansion of the universe observed today can be interpreted as an effect of the so-called dark energy [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Unexpectedly these observations reveals that the ordinary baryonic matter corresponds to only 4% of con- tent of energy of the universe while the dark matter and dark energy correspond to 20% and 76%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In this sense, there are studies considering the possibility of modified theories of gravity which may help to alleviate the need for dark components of energy of the universe beyond the scope of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The late-time acceleration of the universe can be in- terpreted under two points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In the first one, it is introduced a dark energy sector in the energy content of the universe through a type of field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In the second one, the gravitational field itself is modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In addi- tion, there may be combinations of both approaches de- pending on the couplings between gravitational and non- 2 gravitational sectors of theory [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Thus, it is expected that different formulations of gravity imply that standard results in astrophysics suffer modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Compact ob- jects as neutron stars (NS), have been studied consid- ering effects of such modifications [11–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' NS in the context of f(R) gravity were studied in [21–23] and in f(R, T ) gravity in the papers [24–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In common, all of these works have considered effects on NS due to the modification of the gravitational field that include extra terms in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In the scheme of nonconservative gravity, the modification of the gravitational field can be done through a reinterpretation of the conservation law, as was considered in the papers [29, 30] (for a review on non-conservative theories of gravity, see [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Usually, the non-conservation of the stress-energy tensor is pro- portional to the matter density and pressure themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' For this reason, an environment such as a compact ob- ject like a NS turns out to be an appealing laboratory for testing such theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In the context of modified theories of gravity, the so- called f(T,T ) gravity is a class of such theories, free of ghosts and instabilities which, when applied to cosmo- logical problems, leads to interesting results [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In this formulation, the action depends on the torsion scalar T and on the trace of the energy-momentum tensor T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' As in the case of f(T) gravity where the action is an ar- bitrary function of the torsion, in f(T, T ) gravity, the action is a arbitrary function of both the trace of the energy-momentum tensor and the torsion scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In this paper, we study an important context, not yet explored in the literature, that are the implications of the f(T,T ) gravity on NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In particular, we obtain the mass-radius relation of NS in the context of this modified gravity and compare our results with recent astrophysical observations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' This work is organized as follows: In Section II we ex- pose a summary of the f(T,T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In Section III we derive the equations describing static, spherically sym- metric stars in this modified theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In Section IV we present our results and in Section V we close with our final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' GRAVITATIONAL FIELD EQUATIONS OF f(T, T ) GRAVITY Given a line element describing a space-time we want to study ds2 = gµνdxµdxν = ηABeA µeB νdxµdxν (1) where gµν and {eA µ} are respectively the metric tensor and the components of the tetrad associated to space- time geometry, and ηAB = diag(1, −1, −1, −1) is the Minkowski metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The signature (+ − − −) and ge- ometrized units, that is, G = c = 1, will be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In GR we assume that gravity is associated with the curvature of the space-time and thus we use the Levi-Civita’s connection Γρ µν = 1 2gρσ (∂νgσµ + ∂µgσν − ∂σgµν) (2) to compute quantities associated with the curvature such as the Ricci scalar, R, that is present in the GR’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' On the other hand, in teleparallel theory one assumes that gravity is associated to the torsion of the space-time and thus the Weizenbock’s connection Γλ µν = e λ A ∂µeA ν = −eA µ∂νe λ A (3) is used to construct quantities associated with the tor- sion, as the torsion scalar T that appears in the telepar- allel gravity action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In the modified teleparallel theories it is assumed that the action depends on a arbitrary func- tion of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In our case, we are going to consider a modified action given by [32] S = � d4x e �T + f(T, T ) 16π + Lm � , (4) where e is the determinant of the tetrads e = det(eA µ) = √−g and T = gµνTµν is the trace of the energy- momentum tensor Tµν, which can be obtained from the Lagrangian for the matter distribution Lm in the follow- ing way Tµν = gµνLm − 2∂Lm ∂gµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (5) Let us assume that the function f(T, T ) is given by f (T, T ) = ω Tn T − 2Λ , (6) where ω, n and Λ are arbitrary constants, specifically ω can be interpreted as a coupling constant of geometry with matter fields, n is a pure number (assumed to be unity here) and Λ can be recognized as the cosmological constant as discussed in [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We are interested in matter that can be described by a perfect fluid, so that Tµν is given by: Tµν = −pgµν + (p + ρ)uµuν, (7) where p is the pressure and ρ is the energy density of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' By varying the action from Equation (4) with respect to the tetrad we find the following field equation Gµν = 8πT eff µν , (8) where the effective energy-momentum tensor T eff µν is T eff µν = gµν \uf8ee \uf8f0 � − ω(ρ − 3p) + 2Λ � 16π + ωp 8π \uf8f9 \uf8fb+Tµν � 1+ ω 8π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (9) Calculating the covariant derivative of the energy- momentum tensor given by Equation (7), we obtain the following result ∇µTνµ = 1 � 4π + (1/2)ω � �ω 4 (∂νT ) − ω 2 ∂νp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (10) 3 In a cosmological context, equation 10 can be associated to creation or destruction of matter throughout the uni- verse evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' As discussed in [26], the interpretation of creation or destruction of matter particles in the NS level encounters difficulties in a static framework as occurs in the study of the hydrostatic equilibrium expression, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='e, the Tolman-Oppenheimer-Volkof equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Also, it usually implies in the presence of a fifth force and non- geodesic trajectory for free particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Naturally, results that depend on such imput would also be modified corre- spondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' However, this is not the case analyzed in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In the next section we use Equations (8) to (10) to obtain and analyse the mass-radius relation of NS in the context of modified teleparallel gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' STELLAR STRUCTURE EQUATIONS In this section, we discuss some of the main procedures that leads to the deduction of the hydrostatic equilibrium equation in the context of f(T, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' To study compact stars, such as NS, magnetars and other astrophysical structures, we assume these objects as being homogeneous, static (no rotation), isotropic and spherically symmetric [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Therefore, we must use the appropriate metric in a convenient coordinate system that describes the object being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The most gen- eral metric describing the space-time under consideration is given by the line element ds2 = eν(r)dt2 − eλ(r)dr2 − r2(dθ2 + sin θ2dφ2), (11) where ν and λ are radial functions that we want to de- termine based on the field equations (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' using Equation (11) and substituting appropriately into Equa- tion (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='we obtain the following results e−λ�λ′ r − 1 r2 � + 1 r2 = 8π \uf8f1 \uf8f2 \uf8f3 \uf8ee \uf8f0 � − ω(ρ − 3p) + 2Λ � 16π + ωp 8π \uf8f9 \uf8fb + ρ � 1 + ω 8π �\uf8fc \uf8fd \uf8fe = 8π¯ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (12) e−λ�ν′ r + 1 r2 � − 1 r2 = −8π \uf8f1 \uf8f2 \uf8f3 \uf8ee \uf8f0 � − ω(ρ − 3p) + 2Λ � 16π + ωp 8π \uf8f9 \uf8fb − p � 1 + ω 8π �\uf8fc \uf8fd \uf8fe = 8π¯p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (13) e−λ 4r � 2 � λ′ − ν′� − � 2ν′′ + ν′2 − ν′λ′� r � = −8π \uf8f1 \uf8f2 \uf8f3 \uf8ee \uf8f0 � − ω(ρ − 3p) + 2Λ � 16π + ωp 8π \uf8f9 \uf8fb − p � 1 + ω 8π �\uf8fc \uf8fd \uf8fe = 8π¯p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (14) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' the prime denotes a derivative with respect to the radial coordinate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The quantities ¯ρ and ¯p are the effective pressure and energy density, defined as ¯ρ = ρ + ωρ 16π + 5ω p 16π + Λ 8π, (15) ¯p = p + ωρ 16π − 3ω p 16π − Λ 8π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (16) In addition to the field equations, we also need to consider the conservation equation (10) in f(T, T ) gravity so that we have a complete set of equations to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In the case we are studying, Equation (10) has the form as follows −p′− ν′ 2 (ρ+p) = 1 � 4π + (1/2)ω � �ωρ′ 4 − 5ω p′ 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (17) Redefining the function λ(r) as e−λ(r) = 1 − 2M(r) r , (18) and rearranging Equations (12) and and (17), we get the equations required to describe static spherically symmet- ric stellar structures in f(T, T ) gravity theory, which are given by dM dr = 4πr2 ¯ρ, (19) and d¯p dr = −M ¯ρ r2 � 1 + ¯p ¯ρ � � 1 + 4πr3¯p M � � 1 − 2M r �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (20) In the next section we show some results obtained by solving Equations (19) and (20) for realistic EoS of NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 4 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' RESULTS In this section, we present the results obtained from the solution of the field equations in the context of f(T, T ) modified theory of gravity applied to NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' As an input to the stellar hydrostatic equilibrium equa- tions, we use two realistic EoS obtained from a relativis- tic mean field (RMF) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Firstly, we consider the IU-FSU [35] parametrization because it is able to explain reasonably well both nuclear [36] and stellar matter prop- erties [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We then compare the IU-FSU results with the ones obtained with a stiffer EoS calculated with a model of coupling of mesons and quarks, the quark–meson cou- pling (QMC) model [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (For the EoS with the QMC model, we refer the reader to refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' [38–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=') It is well known that a stiffer EoS leads to a bigger NS maximum mass in contrast to a softer one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In fact, using the EoS QMC as an input to the stellar equilibrium equations yields a maximum mass greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='0 M⊙, and, there- fore, we want to verify that we get the same qualitative behavior for macroscopic properties (such as mass and radius) with parameterizations that are substantially dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' For the NS crust, we use the full BPS [43] EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' After defining the EoS, some boundary conditions are required to solve the equations (19) and (20) along the radial coordinate r, from the center towards the surface of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' At the star’s center r = 0 we take: M(0) = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' ¯ρ(0) = ¯ρc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' ¯p(0) = ¯pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' (21) The radius of the star (r = R) is determined as the point where the pressure vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='e, ¯p(R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' At this point, the interior solution connects softly with the Schwarzschild vacuum solution, indicating that the po- tential metrics of the interior and the exterior metric are related as eν(R) = 1 eλ(R) = 1 − 2M/R, being M the total mass of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Let us discuss and compare our results with recent astrophysical observations and nuclear physics experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' At first, the NS in LMXB NGC 6397, depicted as a green shaded area in all figures, provides a constraint at 68% confidence level over the possible values of the masses and corresponding radii of the NS [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Simi- larly, the millisecond pulsars are among the most useful astrophysical objects in the Universe for testing funda- mental physics, because they impose some of the most stringent constraints on high-density nuclear physics in the stellar interior [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Recent measurements coming from the Neutron Star Interior Composition Explorer (NICER) mission reported pulsar observations for canon- ical (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='4 M⊙) and massive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='0 M⊙) NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The mass mea- surement and radius estimates provided for these objects, are 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='80 km ≤ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='4 ≤ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='1 km for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='4M⊙ NS PSR J0030+0451 (horizontal line segment in red colour shown in all Figures) and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='60 km ≤ R ≤ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='1 km for a NS with mass between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='01M⊙ ≤ M ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='15M⊙ PSR J0740+6620 (the rectangular region in orange colour shown in all Fig- ures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' However, the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' [47] used the re- cent measurement of neutron skin on 208Pb by PREX-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 4 6 8 10 12 14 16 IU-FSU M /MO• R(km) ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='0 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='01 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='02 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='08 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='1 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='2 ϖ = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='01 ϖ = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='02 ϖ = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='95 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 IU-FSU FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Mass-radius relation for families of NS’s described by the IU-FSU EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We analyse the effect of varying the pa- rameter ω of the f(T, T ) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The red and green line seg- ment represent the radius range of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='4M⊙ NS for PSR J0030 + 0451 and PREX-2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The orange rectan- gular region corresponds to the range of radius estimates for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='07M⊙ NS PSR J0740+6620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Similarly, the blue and pink horizontal lines stand, respectively, for the mass mea- surements of NS PSR J1614 + 2230 and NS PSR J0348 + 0432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The purple solid line curve is solution for the usual TOV equation from GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' to constrain the radius of NS, which leads to a predic- tion of the radius of the canonical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='4 M⊙ of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='25 km ≲ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='4 ≲ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='26 km (horizontal line segment in green colour shown in all Figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Likewise, we also com- pare our results with two massive stars that had been discovered in 2010 and 2013, namely, PSR J1614+2230 [48] with mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='04 M⊙ (horizontal line in blue colour shown in all Figures) and PSR J0348+0432 [49] with mass 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='04 M⊙ (horizontal line in pink colour shown in all Figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Our results are discussed in the next paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We modelled the function f(T, T ) according to equa- tion (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' This function model has already been used in recent works as, for example, in [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We explore the values of the parameter ω which range from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' On the other hand, we check that the Λ parameter has no significant effect on the mass-radius profiles of NS, since it appears as a constant in the f(T, T ) function that we have chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Therefore, we use Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Note that we recover the GR solution from f(T, T ) theory by assum- ing that ω = Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' These plots are represented by the continuous purple lines in the Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In Figure 1 we show the effects of f(T, T ) theory on NS properties obtained with the IU-FSU EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We can see that the value of ω has a very small influence on the maximum mass of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The radius of the canon- ical NS (M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='4M⊙) is considerably affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Note a bigger (smaller) radius for the most positive (negative) values of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We can observe that the results of PREX-2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 4 6 8 10 12 14 16 QMC M /MO• R(km) ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='0 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='01 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='02 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='08 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='1 ϖ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='2 ϖ = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='01 ϖ = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='02 ϖ = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='92 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 12 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='5 QMC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Mass-radius relation for families of NS’s described by the QMC EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We analyse the effect of varying the parameter ω of the f(T, T ) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The red and green line segment repre- sent the radius range of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='4M⊙ NS for PSR J0030 + 0451 and PREX-2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The orange rectangular region cor- responds to the range of radius estimates for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='07M⊙ NS PSR J0740+6620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Similarly, the blue and pink horizontal lines stand, respectively, for the mass measurements of NS PSR J1614 + 2230 and NS PSR J0348 + 0432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The purple solid line curve is the solution for the usual TOV equation from GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' cannot be described with IU-FSU EoS in the GR, but in f(T, T ) theory the solutions with ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='08 and ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='1 produce mass and radius that agree with this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' However, the solutions obtained with IU-FSU EoS can- not describe the mass and radius of PSR J0740+6620, PSR J1614+2230 and NS PSR J0348+0432 neither on GR nor on f(T, T ) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In Figure 2 we show the mass-radius relation obtained for QMC EoS in f(T, T ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Again, the effect of the parameter ω is to increase the radius when its values in- crease positively and to decrease the radius when its val- ues increase negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' At the same time, the maximum mass changes very little with the variation of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We can also see that the solutions obtained with the QMC EoS in f(T, T ) can accommodate almost all the constraints we are taking into consideration, and with a smaller ra- dius than in GR, if we take ω = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='01 or ω = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The exception is NS PSR J0030+0451 which only can be described with QMC EoS in f(T, T ) gravity if we take ω = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We can note that for both EoS analysed we could not find a configuration that satisfies all the con- straints at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We can see that for both EoS’s the value of ω has a very small influence on the maximum mass of the stars, on the other hand, the value of the radius of the star with maximum mass increases when we increase the value of ω and decreases when ω decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Also for both EoS’s, the case ω = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='2 produces mass-radius curves that are typical of quark stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' FINAL REMARKS We have investigated the effects of f(T, T ) gravity on NS assuming these compact objects as being homoge- neous, static and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In this way, we have consid- ered a spherically symmetric space-time and solved the field equations and the hydrostatic equilibrium equation in the context of this modified theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' This type of system can be transformed into a system with effective pressure and energy density which permitted that the hydrostatic equilibrium equation was obtained through known techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' For the choice of the f(T, T ) function used here, we obtained that this theory can pre- dict NS with almost the same mass and smaller radius than in GR, for a given EoS, that is an interesting result in view of the recent observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' Considering the low- mass X-ray binary (LMXB) NGC 6397 and the pulsar of millisecond PSR J0740+6620, the results obtained using the modified hydrostatic equilibrium equations present good agreement with the observed masses and radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' We particularize f(T, T ) gravity according to equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' The good results obtained in comparison to GR suggest future extensions of this work, as for example, by taking into consideration different choices of the f(T, T ) function, which should be done in a near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' It can be interesting to test, for example, high powers in T besides and new couplings between T and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' In addition, we can use different EoS as input to the stellar hydrostatic equilibrium equations along the aforementioned choices of f(T, T ) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' ACKNOWLEDGEMENTS L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' would like to thank Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) for partial financial support through the research Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 164762/2020-5 and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' would like to thank CNPq for financial support through the research Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' 165604/2020-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' was partially supported by the National Council for Scientific and Technologi- cal Development - CNPq grant 306414/2020-1 and by the grant 3197/2021, Para´ıba State Research Foundation (FAPESQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQfSgOT/content/2301.03067v1.pdf'} +page_content=' would like to acknowledge the contri- bution of the COST Action CA18108.' metadata={'source': 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Infrared Light Sensor Material +Ahalapitiya H. Jayatissaa) and Madhav Gautam +Mechanical, Industrial, and Manufacturing Engineering (MIME) Department +The University of Toledo, OH 43606, USA +a)Correspondence: ahalapitiya.jayatissa@utoledo.edu + + +Abstract: The infrared (IR) photoresponse of graphene synthesized by atmospheric chemical vapor +deposition (CVD) system using a mixture of hydrogen and methane gases was studied. The IR sensor +devices were fabricated using graphene films transferred on to a SiO2 substrate by a lift off process. The +quality of graphene was investigated with the Raman spectroscopy and optical microscopy. The +photoresponse was recorded under the illumination of IR light of wavelength 850 nm and intensity of +around 2.16 µW/mm2. The effects of temperature and hydrogenation on photoconductivity were also +studied. It was found that the transient response and recovery times decreased with the increase of the +temperature. Hydrogenation effect also caused the significant decrease in the photoresponse of the device. +Although the net change in the photoresponse for IR light was lower at low illumination intensity levels, +the transient responses were observed around 100 times faster than the recently reported CNT-based IR +sensors. + +Key words: CVD graphene, single layer, Infra-Red light, photoconductivity, 2D sensor materials + + +1. Introduction + +Optoelectronic devices working in near infra-red (NIR) (800 - 2000 nm) are always demanding for +different applications [1-4]. There has been significant works reported on the fabrication of optoelectronic +devices using NIR materials [5-12]. In recent years, single walled carbon nanotubes (SWCNTs) have +been investigated extensively as a semiconducting material for IR sensors because of its strong absorption +behavior in NIR region [7-12]. One of the key challenges in developing NIR detectors is the finding of +ultra fast optical response in the sensor material [5-8]. Recently, strong absorption behavior in NIR region +has been reported for thermally reduced graphene oxides [1,2]. This provides a pathway to use graphene +as an optoelectronic material for IR detection. Although the optical properties of graphene in visible +region have been reported by many researchers [13-15], we have not found any research work related to +the photoresponse of graphene in IR region of the spectrum. In this paper, photoresponse of graphene film +on macro-scale has been reported in different conditions. +Graphene is a monolayered carbon film with a film thickness of around 0.32Å [13 - 15], where carbon +atoms are arranged in a two-dimensional hexagonal lattice structure. It can be thought of as a single layer +peeled off from the graphite stack. It has evolved as an interesting material due to its unique physical and +electrical properties [16]. This material is different from most of the conventional semiconductors because +of its zero bandgap semi-conducting behavior [17]. For example, graphene-based transistor devices may +operate very faster than traditional silicon devices due to high intrinsic carrier mobility (~ 2x105 cm2v-1s-1) +[1, 2, 18]. Being the material of high mechanical stress and low density (2.2 gm/cm3), it may lead to the +application in nano-robotics [19, 20]. +We have investigated the photoconductivity of graphene layers synthesized in atmospheric chemical +vapor deposition (CVD) of CH4 on a copper substrate. The devices were fabricated by transferred CVD +graphene onto a SiO2/Si substrate. The investigations were carried out to understand the temperature +dependence and hydrogenation effect on photoconductivity of graphene in NIR region. Although the net +change in the photoresponse for IR light was lower at low illumination intensity levels (2.16 µW/mm2), + +the transient responses were observed around 100 times faster than photoconductivity of CNT for NIR +lights. + +2. Experimental Procedures + +The growth of graphene films was carried out on a copper (Cu) substrate (25 µm thick) in an alumina +tube furnace system under the flow of methane (CH4) and hydrogen (H2) gases. Copper substrate +(99.999% pure, Alfa Aesar) was heated in a tube furnace under the 150 standard cubic centimeters per +minute (sccm) flow of mixture of hydrogen and Argon (10% H2, 90% Ar) and annealed at 1100 0C for +one hour. After annealing, graphene deposition was carried out by passing a mixture of methane and +argon (5% CH4, 95% Ar) followed by the immediate cooling. Graphene deposited on copper by CVD +method was transferred to SiO2/Si substrate by wet etching of Cu [15, 21-23]. The thickness of the +thermally-grown SiO2 was 118 nm as confirmed by UV spectrometry [24]. The Raman spectra of these +films were recorded with the excitation wavelength of 530 nm. + In order to fabricate the IR sensors, a thin layer of gold (about 100 nm) was coated onto the +transferred graphene film by a vacuum evaporation method. The gold electrodes were patterned by +lithography followed by etching of gold with aqueous KI/I2 solution. The spacing and the length of these +electrodes were 6 mm and 4 mm, respectively. Fig. 1 shows the schematic diagram of the fabricated IR +sensor and photoresponse measurement circuit. The device was biased with a constant voltage (1.0 V) +during collection of the data. To understand the reflection of light from graphene, reflectance from bi- +layer substrate (SiO2/Si) and tri-layer substrate (graphene/SiO2/Si) were measured with a double beam +UV/Visible spectrometer (Shimadzu). The reflectance spectra were investigated in the spectral range 300- +1100 nm. + +Au +A +V0 +Graphene +SiO2 +IR +Light +Si + +Fig.1: Schematic of photoresponse measurement system (V0= 1.0 V). + +3. Results and Discussions + +3.1. Surface Characterization +The Raman spectroscopy has been used to characterize the quality of graphene. The Raman spectrum +of Graphene gives for main bands corresponding to the vibration mode of graphene. Fig. 2 shows the as- +measured Raman spectra of graphene films produced on SiO2 surface. The spectrum was normalized with +respect to the intensity level of 2D band. The peak at around 1580 cm-1 and 2660 cm-1, respectively, +indicate the G band and the 2D band, which are characteristics Raman peaks of graphene. It has been +reported that the defect free monolayer graphene can be identified with characteristic features of Raman +band intensities [25]. The intensity of 2D band is ~2 times larger than the intensity of G band suggesting +that the presence of less defective graphene on SiO2 surface. This fact is also supported by the weak +intensity of D-band (1350 cm-1). + + + +Fig. 2: Raman spectra of graphene transferred to silicon wafer (SiO2 + Si) scaled with respect to +the maximum peak. + + +3.2. Photoconductivity + +3.2.1. Dynamic response +Fig. 3 shows the dynamic response of photoconductivity of graphene film for the NIR light at room +temperature. Fig. 3(a) shows the response and recovery of the device when the IR light was turned on and +off, respectively, whereas Fig. 3(b) indicates the same characteristic for one cycle only. The intensity of +the IR light source used was 2.16 µW/mm2 at the device surface. Although the intensity level was very +low, a clear photoresponse of device was measured. The photogeneration of carriers can be primarily +attributed to the creation of bands at the defect of graphene sheets. When graphene is deposited on a +copper plate, defects are developed at the grain boundary of polycrystalline copper films. We believe that +these defects are responsible for the creation of localized photoactive regions, which contribute to the +photogeneration of carriers [26,27]. The photoresponse could be characterized with a time step function. +In both the photocurrent increase and drop cases, the experimental data were fitted well into the +exponential form as [10], + + + + + + + + + − ++ += + +t +A +I +I +o +o +exp +. + + + + + + + (1) + +Here, I is the current, t is the response time and Io,  and A0 are constants. Fig. 4(a) and 4(b) show the fit +of the response in the form explained above. The data analysis indicated that the time constants were 10 +ms and 31 ms for rise and fall of the photocurrent, respectively. + + +1.2 +2D +nsityRatio (ll) +1 +8'0 +0.6 +G +0.4 +0.2 +D +G +人 +0 +1000 +1500 +2000 +2500 +3000 +Raman Shift (anl) +Fig. 3: The photoresponse of the device due to IR light for (a) different cycles and (b) for one cycle. + + + + Fig. 4: The photoresponse of the device due to IR light for (a) response and (b) recovery. + + + + + + + + + + + + + + + +Fig. 5: The photoresponse of the device due to IR light at (a) 50 0C and (b) 100 0C. + + +3.2.2. The effect of temperature on photoconductivity +Fig. 6 shows the effect of temperature on the photoconductivity of graphene. The photoconductivity +was tested at 50 0C and 100 0C, respectively. During the experiment, the device was heated to the desired + +(a)1,252 +1.2515 +b) +1.251 +1.2505 +1.25 +1.2495 +1.249 +0 +50 +100 +150 +200 +250 +300 +Time (ms):(a)(b)1.2888 +(a) +1.2882 +(vu) +1.2864 +20 +40 +60 +80 +100 +120 +140 +Time (ms)(b)temperature for 30 minutes to ensure the thermal equilibrium. Transient responses of the device were +10.26 ms and 6.57 ms and the transient recovery times were 12.55 ms and 5.91 ms at 50 0C and 100 0C, +respectively. A significant difference in transient response of the device was not found when the device +temperature was increased from room temperature to 50 0C and transient response time decreased by 40% +when the temperature was changed from 50 0C to 100 0C. Similarly, the transient recovery time decreased +by 60% when the temperature was changed from room temperature to 50 0C and it decreased by 50% +when the temperature was changed from 50 0C to 100 0C. +On the other hand, the amplitude of the photocurrent didn’t show any significant difference when the +temperature was changed from room temperature to 50 0C whereas it decreased by 50% when the +temperature was changed from 50 0C to 100 0C. A slight change in photocurrent at high temperature +measurement (100 0C) from low temperature (50 0C) can be attributed to the career generation is +influenced by thermal effect associated with defects. Furthermore, the increase in current due to the +thermal effect of IR light is less pronounced at elevated temperatures because the change in the +temperature by IR heating is negligible. Therefore, the total photocurrent generation can be attributed to +the photo generation of carriers in the graphene. + + + + + + + + + + + + +Fig. 6: The photoresponse of the device in IR light due to hydrogenation at 100sccm of hydrogen +flow for (a) difference cycle and (b) one cycle. + +On the other hand, the amplitude of the photocurrent didn’t show any significant difference when the +temperature was changed from room temperature to 50 0C whereas it decreased by 50% when the +temperature was changed from 50 0C to 100 0C. Smaller change in low temperature gradient can be +attributed to the fact that small bandgap in graphene. Furthermore, the increase in current due to the +thermal effect of IR light is less pronounced at elevated temperatures because the change in the +temperature by IR heating is negligible. Therefore, the total photocurrent generation can be attributed to +the photo generation of carriers in the graphene. + +3.2.3. The effect of hydrogenation on photoconductivity +The effect of hydrogenation on photoresponse of the device was tested at 100 0C for different +concentrations of hydrogen flow rates. The device was heated at 100 0C for 30 min to ensure the thermal +equilibrium followed by the constant hydrogen flow for more than one hour until reach of the saturation +of surface of graphene by hydrogen by adsorption. The saturation was confirmed by monitoring resistance +changes against time using two-point probe method. + + + + + + +0.15026 +LtzosT'o +(a) +0.150234 +0.150221 +0.150208 +0.150195 +400 +600 +800 +10000.15026 +(b) +(mA) +0.150221 +0.150208 +0.150195 +460 +500 +Time (ms)Fig. 7 shows the photoresponse of the device at different flow rates of hydrogen. Transient responses +of the device were 6.05 ms and 7.27 ms in 50 sccm and 100 sccm flow rate of hydrogen gas, respectively, +and the corresponding values during recovery process were 7.1 ms and 7.81 ms, respectively. The +transient response of the device was found to differ by 17% in going from 50 to 100 sccm of hydrogen +flow rates. Table 1 lists the transient response and recovery times at different temperatures to compare the +effect of hydrogenation. + + + +Fig. 7: The photoresponse of the device in IR light due to hydrogenation at (a) 50 sccm +and (b) 100 sccm flow rate of hydrogen gas at 100 0C. + +Table 1: Transient response and recovery times at different temperatures. +Temperature +(0C) +Transient response (1) +(ms) +Transient recovery (2) +(ms) + In vacuum +In hydrogen +(100 sccm) + In vacuum In hydrogen +(100 sccm) +Room Tem. + 10.04 +13.90 +31.26 +44.29 + 100 + 6.57 + 7.24 + 5.91 + 7.81 + +The photoresponse of the device in hydrogen was also calculated and compared with that of the +device in vacuum at different temperatures. Response of the device was calculated using the formula +given by [25], + +% +100 +* +2 +2 +1 + + + + + + +− += +I +I +I +S +. + + + + + + + + +(2) + +Where, I1 and I2 are the currents with and without IR light, respectively. Generally, response is calculated +in percentage. +Fig. 8 shows the comparison of the responses due to hydrogenation effect at 100 0C. The response +was found to decrease by around 57% when the device was hydrogenated at 50 sccm flow rate of +hydrogen gas while it decreased by around 68% when the flow rate was increased to 100 sccm. The effect +of hydrogenation was even seen substantial at room temperature compared with hydrogenation at 100 0C. +The flow of hydrogen was continued during cooling. The decrease in the response of the device due to +hydrogenation effect was observed as expected. The semiconducting Behaviour of graphene is attributed + +1.4933 +1.4932 +(a) +1.4931 +L.493 +mo +1.4929 +1.4928 +1.492 +1L.4926 +0 +10 +28 +30 +40 +Time (ws)1.5025 +1.5024 +(b) +1.5021 +1.502 +1.5019 +.0 +10 +20. +30 +40 +50 +Tine (ms)to the formation of bands at the defect sites [26]. When hydronation is occurred, the conductivity can be +reduced to a certain extent due to the passivation of defect sites with hydrogen. + + + +Fig. 8: The photoresponse of the device in IR light at 100 0C in (a) hydrogenation at 100 sccm of +hydrogen flow and (b) in ambient condition. + +4. Conclusion +In this paper, a graphene-based IR sensor was investigated in different conditions in terms of the +photoresponse in the presence of light. The device was fabricated between electrode materials and the +presence of a monolayer of graphene was confirmed by Raman Spectroscopy. The effect of temperature +on photoconductivity was recorded at different temperature conditions. The photoconductivity of +graphene films was interpreted as due to the creation of localized bands in defect sites at the gran +boundaries of CVD graphene. The device exhibited a temperature-dependent effect on the photoresponse +behavior. The transient response and recovery times were seen reduced in the high-temperature region, +indicating that the thermal effect due to heating was more pronounced than the heating effect caused by +the IR light. It also revealed the fact that the net photocurrent change due to IR light decreases as the +charge carriers responsible for conduction are already excited to the conduction band due to thermal +heating before IR light was used. The hydrogenation effect on photoconductivity was also studied. The +hydrogenation caused a significant decrease in the photoresponse of the device at high temperature as +expected because the hydrogen ions were believed to be adsorbed at the grain boundaries and passivate +the defects that are responsible for photoconductivity. As the device was illuminated with a low intensity +(~ 2.16 µW/mm2) of IR light, the net change in the photocurrent was not significant. However, the +transient responses were observed around 100 times faster than the recently reported CNT-based IR +sensor, which may lead to the application of graphene towards ultra-fast optical response devices. + +Acknowledgements: This research was supported by a grant (Grant #: ECCS 0925783) from National +Science Foundation (NSF) of USA. + +References +[1] +S A McDonald et al. Nat. Mater. 4 (2005) 138. +[2] B Pradhan, K Setyowati, H Liu, D H Waldeck and J Chen Nano Letters 8 (2008) 1142. +[3] +M Acik, G Lee, C Mattevi, M Chhowalla, K Cho and Y J Chabal Nature Mater. 9 (2010) 840. +[4] +M E Itkis, F Borondics, A Yu and R C Haddon Science 312 (2003) 413. +[5] +K Yoshino et al. Adv. Mater. 11 (1999) 1382. +[6] +C J Barber, C Winder, N S Sariciftci, J C Hummelen, A Dhanabalan and P A Hal Adv. Funct. Mater. 12 +(2002) 709. +[7] +Y Yao, Y Liang, V Shrotriya, S Xiao, L Yu and Y Yang Adv. Mater 19 (2007) 3979. +[8] +K Wai, C Lai, N Xi, K Carmen, F Fung, H Chen and T Tarn Appl. Phys. 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China 4 (2010) 45. +[20] A H Neto, F Guinea, N M R Peres, K S Novoselov and A K Geim Rev. Modern Phys. 81 (2009) 45. +[21] A Reina et al. J. Phys. Chem. C 112 (2008) 17741. +[22] D Wei, Y Liu, Y Wang, H Zhang, L Huang and G Yu Nano Letters 9 (2009) 1752. +[23] L Xuesong et al. Nano letters 9 (2009) 4359. +[24] A. Ferrari et al. Phys. Rev. Let. 97 (2006) 187401. +[25] M Gautam, AH Jayatissa, Materials Science and Engineering: C 31 (2011) 1405 +[26] L Liu, M Qing, Y Wang and S Chen J. Mater. Science & Technol. 31 (2015) 599 . +[27] J Sun, N Lin, Z Li, H Ren, C Tang and X Zhao Royal Soc. of Chemistry Adv. 6 (2016) 1090. + + + diff --git a/4tFAT4oBgHgl3EQfFByL/content/tmp_files/load_file.txt b/4tFAT4oBgHgl3EQfFByL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b8aff3fc3e4738e7f6e11987313daef513d0cac --- /dev/null +++ b/4tFAT4oBgHgl3EQfFByL/content/tmp_files/load_file.txt @@ -0,0 +1,290 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf,len=289 +page_content='Graphene as Infrared Light Sensor Material Ahalapitiya H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Jayatissaa) and Madhav Gautam Mechanical, Industrial, and Manufacturing Engineering (MIME) Department The University of Toledo, OH 43606, USA a)Correspondence: ahalapitiya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='jayatissa@utoledo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='edu Abstract: The infrared (IR) photoresponse of graphene synthesized by atmospheric chemical vapor deposition (CVD) system using a mixture of hydrogen and methane gases was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The IR sensor devices were fabricated using graphene films transferred on to a SiO2 substrate by a lift off process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The quality of graphene was investigated with the Raman spectroscopy and optical microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The photoresponse was recorded under the illumination of IR light of wavelength 850 nm and intensity of around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='16 µW/mm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The effects of temperature and hydrogenation on photoconductivity were also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' It was found that the transient response and recovery times decreased with the increase of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Hydrogenation effect also caused the significant decrease in the photoresponse of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Although the net change in the photoresponse for IR light was lower at low illumination intensity levels, the transient responses were observed around 100 times faster than the recently reported CNT-based IR sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Key words: CVD graphene, single layer, Infra-Red light, photoconductivity, 2D sensor materials 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Introduction Optoelectronic devices working in near infra-red (NIR) (800 - 2000 nm) are always demanding for different applications [1-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' There has been significant works reported on the fabrication of optoelectronic devices using NIR materials [5-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' In recent years, single walled carbon nanotubes (SWCNTs) have been investigated extensively as a semiconducting material for IR sensors because of its strong absorption behavior in NIR region [7-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' One of the key challenges in developing NIR detectors is the finding of ultra fast optical response in the sensor material [5-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Recently, strong absorption behavior in NIR region has been reported for thermally reduced graphene oxides [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' This provides a pathway to use graphene as an optoelectronic material for IR detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Although the optical properties of graphene in visible region have been reported by many researchers [13-15], we have not found any research work related to the photoresponse of graphene in IR region of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' In this paper, photoresponse of graphene film on macro-scale has been reported in different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Graphene is a monolayered carbon film with a film thickness of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='32Å [13 - 15], where carbon atoms are arranged in a two-dimensional hexagonal lattice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' It can be thought of as a single layer peeled off from the graphite stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' It has evolved as an interesting material due to its unique physical and electrical properties [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' This material is different from most of the conventional semiconductors because of its zero bandgap semi-conducting behavior [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' For example, graphene-based transistor devices may operate very faster than traditional silicon devices due to high intrinsic carrier mobility (~ 2x105 cm2v-1s-1) [1, 2, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Being the material of high mechanical stress and low density (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2 gm/cm3), it may lead to the application in nano-robotics [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' We have investigated the photoconductivity of graphene layers synthesized in atmospheric chemical vapor deposition (CVD) of CH4 on a copper substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The devices were fabricated by transferred CVD graphene onto a SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The investigations were carried out to understand the temperature dependence and hydrogenation effect on photoconductivity of graphene in NIR region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Although the net change in the photoresponse for IR light was lower at low illumination intensity levels (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='16 µW/mm2), the transient responses were observed around 100 times faster than photoconductivity of CNT for NIR lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Experimental Procedures The growth of graphene films was carried out on a copper (Cu) substrate (25 µm thick) in an alumina tube furnace system under the flow of methane (CH4) and hydrogen (H2) gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Copper substrate (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='999% pure, Alfa Aesar) was heated in a tube furnace under the 150 standard cubic centimeters per minute (sccm) flow of mixture of hydrogen and Argon (10% H2, 90% Ar) and annealed at 1100 0C for one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' After annealing, graphene deposition was carried out by passing a mixture of methane and argon (5% CH4, 95% Ar) followed by the immediate cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Graphene deposited on copper by CVD method was transferred to SiO2/Si substrate by wet etching of Cu [15, 21-23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The thickness of the thermally-grown SiO2 was 118 nm as confirmed by UV spectrometry [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The Raman spectra of these films were recorded with the excitation wavelength of 530 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' In order to fabricate the IR sensors, a thin layer of gold (about 100 nm) was coated onto the transferred graphene film by a vacuum evaporation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The gold electrodes were patterned by lithography followed by etching of gold with aqueous KI/I2 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The spacing and the length of these electrodes were 6 mm and 4 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 1 shows the schematic diagram of the fabricated IR sensor and photoresponse measurement circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The device was biased with a constant voltage (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='0 V) during collection of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' To understand the reflection of light from graphene, reflectance from bi- layer substrate (SiO2/Si) and tri-layer substrate (graphene/SiO2/Si) were measured with a double beam UV/Visible spectrometer (Shimadzu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The reflectance spectra were investigated in the spectral range 300- 1100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Au A V0 Graphene SiO2 IR Light Si Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='1: Schematic of photoresponse measurement system (V0= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='0 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Results and Discussions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Surface Characterization The Raman spectroscopy has been used to characterize the quality of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The Raman spectrum of Graphene gives for main bands corresponding to the vibration mode of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 2 shows the as- measured Raman spectra of graphene films produced on SiO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The spectrum was normalized with respect to the intensity level of 2D band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The peak at around 1580 cm-1 and 2660 cm-1, respectively, indicate the G band and the 2D band, which are characteristics Raman peaks of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' It has been reported that the defect free monolayer graphene can be identified with characteristic features of Raman band intensities [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The intensity of 2D band is ~2 times larger than the intensity of G band suggesting that the presence of less defective graphene on SiO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' This fact is also supported by the weak intensity of D-band (1350 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 2: Raman spectra of graphene transferred to silicon wafer (SiO2 + Si) scaled with respect to the maximum peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Photoconductivity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Dynamic response Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 3 shows the dynamic response of photoconductivity of graphene film for the NIR light at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 3(a) shows the response and recovery of the device when the IR light was turned on and off, respectively, whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 3(b) indicates the same characteristic for one cycle only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The intensity of the IR light source used was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='16 µW/mm2 at the device surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Although the intensity level was very low, a clear photoresponse of device was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The photogeneration of carriers can be primarily attributed to the creation of bands at the defect of graphene sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' When graphene is deposited on a copper plate, defects are developed at the grain boundary of polycrystalline copper films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' We believe that these defects are responsible for the creation of localized photoactive regions, which contribute to the photogeneration of carriers [26,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The photoresponse could be characterized with a time step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' In both the photocurrent increase and drop cases, the experimental data were fitted well into the exponential form as [10], \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 − + = \uf074 t A I I o o exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' (1) Here, I is the current, t is the response time and Io, \uf074 and A0 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 4(a) and 4(b) show the fit of the response in the form explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The data analysis indicated that the time constants were 10 ms and 31 ms for rise and fall of the photocurrent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content="2 2D nsityRatio (ll) 1 8'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='6 G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2 D G 人 0 1000 1500 2000 2500 3000 Raman Shift (anl) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 3: The photoresponse of the device due to IR light for (a) different cycles and (b) for one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 4: The photoresponse of the device due to IR light for (a) response and (b) recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 5: The photoresponse of the device due to IR light at (a) 50 0C and (b) 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The effect of temperature on photoconductivity Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 6 shows the effect of temperature on the photoconductivity of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The photoconductivity was tested at 50 0C and 100 0C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' During the experiment, the device was heated to the desired (a)1,252 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2515 b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='251 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2505 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2495 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='249 0 50 100 150 200 250 300 Time (ms):(a)(b)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2888 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2882 (vu) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2864 20 40 60 80 100 120 140 Time (ms)(b)temperature for 30 minutes to ensure the thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Transient responses of the device were 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='26 ms and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='57 ms and the transient recovery times were 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='55 ms and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='91 ms at 50 0C and 100 0C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' A significant difference in transient response of the device was not found when the device temperature was increased from room temperature to 50 0C and transient response time decreased by 40% when the temperature was changed from 50 0C to 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Similarly, the transient recovery time decreased by 60% when the temperature was changed from room temperature to 50 0C and it decreased by 50% when the temperature was changed from 50 0C to 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' On the other hand, the amplitude of the photocurrent didn’t show any significant difference when the temperature was changed from room temperature to 50 0C whereas it decreased by 50% when the temperature was changed from 50 0C to 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' A slight change in photocurrent at high temperature measurement (100 0C) from low temperature (50 0C) can be attributed to the career generation is influenced by thermal effect associated with defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Furthermore, the increase in current due to the thermal effect of IR light is less pronounced at elevated temperatures because the change in the temperature by IR heating is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Therefore, the total photocurrent generation can be attributed to the photo generation of carriers in the graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 6: The photoresponse of the device in IR light due to hydrogenation at 100sccm of hydrogen flow for (a) difference cycle and (b) one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' On the other hand, the amplitude of the photocurrent didn’t show any significant difference when the temperature was changed from room temperature to 50 0C whereas it decreased by 50% when the temperature was changed from 50 0C to 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Smaller change in low temperature gradient can be attributed to the fact that small bandgap in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Furthermore, the increase in current due to the thermal effect of IR light is less pronounced at elevated temperatures because the change in the temperature by IR heating is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Therefore, the total photocurrent generation can be attributed to the photo generation of carriers in the graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The effect of hydrogenation on photoconductivity The effect of hydrogenation on photoresponse of the device was tested at 100 0C for different concentrations of hydrogen flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The device was heated at 100 0C for 30 min to ensure the thermal equilibrium followed by the constant hydrogen flow for more than one hour until reach of the saturation of surface of graphene by hydrogen by adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The saturation was confirmed by monitoring resistance changes against time using two-point probe method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content="15026 LtzosT'o (a) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='150234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='150221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='150208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='150195 400 600 800 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='15026 (b) (mA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='150221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='150208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='150195 460 500 Time (ms)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 7 shows the photoresponse of the device at different flow rates of hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Transient responses of the device were 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='05 ms and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='27 ms in 50 sccm and 100 sccm flow rate of hydrogen gas, respectively, and the corresponding values during recovery process were 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='1 ms and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='81 ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The transient response of the device was found to differ by 17% in going from 50 to 100 sccm of hydrogen flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Table 1 lists the transient response and recovery times at different temperatures to compare the effect of hydrogenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 7: The photoresponse of the device in IR light due to hydrogenation at (a) 50 sccm and (b) 100 sccm flow rate of hydrogen gas at 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Table 1: Transient response and recovery times at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Temperature (0C) Transient response (\uf0741) (ms) Transient recovery (\uf0742) (ms) In vacuum In hydrogen (100 sccm) In vacuum In hydrogen (100 sccm) Room Tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='04 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='90 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='26 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='29 100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='91 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='81 The photoresponse of the device in hydrogen was also calculated and compared with that of the device in vacuum at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Response of the device was calculated using the formula given by [25], % 100 2 2 1 \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 − = I I I S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' (2) Where, I1 and I2 are the currents with and without IR light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Generally, response is calculated in percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 8 shows the comparison of the responses due to hydrogenation effect at 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The response was found to decrease by around 57% when the device was hydrogenated at 50 sccm flow rate of hydrogen gas while it decreased by around 68% when the flow rate was increased to 100 sccm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The effect of hydrogenation was even seen substantial at room temperature compared with hydrogenation at 100 0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The flow of hydrogen was continued during cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The decrease in the response of the device due to hydrogenation effect was observed as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The semiconducting Behaviour of graphene is attributed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='4933 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='4932 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='4931 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='493 mo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='4929 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='4928 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='492 1L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='4926 0 10 28 30 40 Time (ws)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='5025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='5024 (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='5021 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='502 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='5019 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='0 10 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 30 40 50 Tine (ms)to the formation of bands at the defect sites [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' When hydronation is occurred, the conductivity can be reduced to a certain extent due to the passivation of defect sites with hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 8: The photoresponse of the device in IR light at 100 0C in (a) hydrogenation at 100 sccm of hydrogen flow and (b) in ambient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' Conclusion In this paper, a graphene-based IR sensor was investigated in different conditions in terms of the photoresponse in the presence of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The device was fabricated between electrode materials and the presence of a monolayer of graphene was confirmed by Raman Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The effect of temperature on photoconductivity was recorded at different temperature conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The photoconductivity of graphene films was interpreted as due to the creation of localized bands in defect sites at the gran boundaries of CVD graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The device exhibited a temperature-dependent effect on the photoresponse behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The transient response and recovery times were seen reduced in the high-temperature region, indicating that the thermal effect due to heating was more pronounced than the heating effect caused by the IR light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' It also revealed the fact that the net photocurrent change due to IR light decreases as the charge carriers responsible for conduction are already excited to the conduction band due to thermal heating before IR light was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The hydrogenation effect on photoconductivity was also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' The hydrogenation caused a significant decrease in the photoresponse of the device at high temperature as expected because the hydrogen ions were believed to be adsorbed at the grain boundaries and passivate the defects that are responsible for photoconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' As the device was illuminated with a low intensity (~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content='16 µW/mm2) of IR light, the net change in the photocurrent was not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfFByL/content/2301.08425v1.pdf'} +page_content=' However, the transient responses were observed around 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We estab- +lish bounds, in some cases tight, on the mosaic numbers for the surface-links with ch-index up to 10. As +an application, we use mosaic diagrams to enhance the kei counting invariant for unoriented surface-links +as well as classical knots and links. +Keywords: Mosaic knots, Surface-links, Marked graph diagrams, kei homset enhancements +2020 MSC: 57K12 +1 +Introduction +Surface-links are compact surfaces smoothly embedded in R4 or S4, i.e. surfaces which are knotted and +linked in 4-space. Surface-links include many more distinct topological types of unknotted objects – spheres, +tori, projective planes, Klein bottles, etc. – compared with classical knots, and additionally include both +orientable and non-orientable cases. +Introduced in [12], marked graph diagrams are knot diagrams with marked vertices representing saddle +points of a surface-link. A marked graph diagram satisfying certain mild conditions determines a surface- +link up to ambient isotopy in R4, and marked graph diagrams together with the Yoshikawa moves provide +a convenient diagrammatic calculus for combinatorial computation with surface-links. Moreover, marked +graph diagrams and their Yoshikawa equivalence classes provide a diagrammatic way to represent cobordisms +between classical knots and links. +A mosaic diagram for a classical knot K is a rectangular (usually square) arrangement of square tiles +containing crossings, arcs or nothing such that the arcs join to form a diagram of K. Mosaics were used in +[11] to define quantum knots, elements of Hilbert spaces generated by mosaic diagrams. +In this paper we take the first steps toward extending these constructions to the case of surface-links by +considering mosaic presentations for surface-links using marked graph diagrams. We establish a set of tiles +and Yoshikawa moves for marked graph mosaics and provide mosaic diagrams for each of the surface-links in +the Yoshikawa table of surface-links with up to ch-index 10, establishing an upper bound on mosaic number +for these surface-links. As an application we use mosaic presentations to define a new enhancement of the kei +counting invariant for classical knots and links as well as for surface-links. As with mosaic number, we can +compute an upper bound with respect to a certain ordering on the new enhancement from a given diagram +of a surface-link or classical knot or link. +The paper is organized as follows. In Section 2 we review some preliminaries about knot mosaics and +marked graph diagrams. In Section 3 we introduce marked graph mosaics and obtain some results including +upper bounds, some tight, on the the mosaic numbers of both orientable and non-orientable surface-links +with ch-index less than or equal to 10. In Section 5 we define kei-colored mosaics and use them to enhance +the kei counting invariant for classical knots and links as well as surface-links. We conclude in Section 6 with +some questions for future research. +∗Email: smchoi@knu.ac.kr. Partially supported by Basic Science Research Program through the National Research Founda- +tion of Korea(NRF) funded by the Ministry of Education(2021R1I1A1A01049100) and the National Research Foundation of +Korea (NRF) grant funded by the Korean government (MSIT) (No. 2022R1A5A1033624). +†Email: Sam.Nelson@cmc.edu. Partially supported by Simons Foundation Collaboration Grant 702597. +1 +arXiv:2301.00287v1 [math.GT] 31 Dec 2022 + +2 +Preliminaries +We review knot mosaics and recall surface-links, marked graph diagrams and their relationships. +2.1 +Surface-links and marked graph diagrams +A surface-link is the image of a closed surface smoothly (piecewise linear and locally flatly) embedded in R4 +(or S4). If it is called a surface-knot, then the underlying surface is connected. A surface-link is orientable +if the underlying surface is orientable; otherwise, it is nonorientable or unorientable. An unoriented surface- +link is either an unorientable surface-link or an orientable surface link without a specified orientation. Two +surface-links F and F ′ are equivalent if there exists an orientation-preserving homeomorphism h : R4 → R4 +such that h(F) = F ′. There are many useful schemes for describing for surface-links since it is difficult to +directly deal with surface-links in 4-space for research. For example, broken surface diagrams, marked graph +diagrams, motion pictures etc. See [2, 5, 6, 15] for more information. +We use an effective tool for handling surface-links known as a marked graph diagram. A marked graph is +a spatial graph embedded in R3 possibly with 4-valent vertices decorated by a line segment like +. We call +such a line segment a marker and a vertex with a marker a marked vertex. +An orientation of edges incident with a marked vertex is one of two types of the orientation, such as +or +. A marked graph is said to be orientable if it admits an orientation. Otherwise, it is non-orientable. Two +(oriented) marked graphs are said to be equivalent if they are ambient isotopic in R3 keeping the rectangular +neighborhoods and markers (with orientation). In the same way as a link diagram, one can define a marked +graph diagram which is a diagram in R2 with classical crossings and marked vertices. +For each marked vertex +of a marked graph diagram D, the local diagram obtained by splicing in +a direction consistent with its marker (say + direction), looks like +. By applying this in the opposite +direction (called − direction), the resulting local diagram looks like +. Therefore one can obtain two classical +link diagrams, denoted by L+(D) and L−(D), from D by splicing every marked vertices in + direction and +− direction, respectively. We call L+(D) and L−(D) the positive and negative resolutions of D, respectively. +A marked graph diagram D is said to be admissible if both resolutions L−(D) and L+(D) are trivial. A +marked graph is said to be admissible if it has an admissible marked graph diagram. For example, it is easy +to check that a marked graph diagram D of the spun trefoil as follows is admissible. +D +L_(D) +L+(D) +Let D be a admissible marked graph diagram. Then a surface-link F(D) can be constructed and it is +uniquely determined from D up to equivalence. Conversely, every surface-link F can be expressed by an +admissible marked graph diagram D, that is, F(D) is equivalent to F. See [7, 12, 15] for more details. +For example, the correspondence between the marked graph diagram and the standard projective plane are +illustrated in the following figure. +R3×{0} +R3×{1} +R3×{-1} +R3×[1,∞) +R3×[-1,∞) +R4 +2 + +The equivalence moves Γ1, · · · , Γ8 for marked graph diagrams is called Yoshikawa moves [15]. +Γ1 +Γ2 +Γ3 +Γ4 +Γ5 +Γ8 +Γ'4 +Γ6 +Γ7 +Γ'6 +Proposition 1 ([8, 14, 15]). Two marked graph diagrams D and D′ present equivalent oriented surface- +links if and only if D can be obtained from D′ by a finite sequence of ambient isotopies in R2 and Yoshikawa +moves. +Definition 1. Let K be a marked graph diagram. The ch-index of K, denoted ch(K), is the total number +of crossings and marked vertices in K. +2.2 +Mosaic Knots +A mosaic (unoriented) tile is one of rectangles with arcs and possibly with one crossing, depicted as follows. +The set of mosaic tiles T0, T1, · · · , T10 is denoted by T(u) and there are exactly 5 tiles, up to rotation. The +endpoints of an arc on a mosaic tile are called connection points of the tile and are also located the center +of an edge. There are tiles with 0, 2 and 4 connection points in T(u). +4 connection points +0 connection points +2 connection points +An (m, n)-mosaic is an m × n matrix whose entries are mosaic tiles in T(u). If m = n, then it is simply +called an n-mosaic. The sets of (m, n)-mosaics and n-mosaics are denoted by M(m,n) and M(n), respectively. +Two tiles in a mosaic are said to be contiguous if they lie immediately next to each other in the same either +row or column. A tile in a mosaic is said to be suitably connected if all its connection points touch the +3 + +connection points of contiguous tiles. all its connection points meet the connection points of contiguous tiles. +Note that for 4-mosaic illustrated above, its (2, 2)-entry tile is suitably connected, but its (3, 3)-entry tile is +not suitably connected. +Definition 2. A knot (m, n)-mosaic is an (m, n)-mosaic in which all tiles are suitably connected. The set of +all knot (m, n)-mosaic is the subset of M(m,n), denoted by K(m,n). If m = n, then it is called a knot n-mosaic +and its set is denoted by K(n). +Example 1. The trefoil 31 has a knot 5-mosaic and 4-mosaic, as follows. +For the equivalence for mosaic knots, there are planar isotopy moves and Reidemeister moves by using +mosaic tiles. The non-deterministic tiles are necessary to define the moves, as follows : +Each non-deterministic tile means two types of tiles. +or +or +Non-deterministic tiles labeled by the same letter A or B are synchronized. + A + A + A + B + B + B + B + A +The equivalence of mosaic knots consists of 11 moves for planar isotopy, 2 moves for Reidemeister moves +I, 4 moves for Reidemeister moves II and 6 moves for Reidemeister moves III. +0. Planar isotopy moves : 11 types +P1 +P4 +P2 +P3 +P7 +P5 +P6 +P10 +P11 +P8 +P9 +4 + +1. Reidemeister moves I : 2 types +2. Reidemeister moves II : 4 types +3. Reidemeister moves III : 6 types + A + B + B + A + A + B + B + A + A + B + B + A + A + B + B + A + A + B + B + A + A + B + B + A +All mosaic moves are permutations on the set M(n) of n-mosaics. Indeed, they are also in the group of +all permutations of the set K(n) of knot n-mosaics. +Definition 3. The ambient isotopy group A(n) is the subgroup of the group of all permutations of the set +K(n) generated by all planar isotopy moves and all Reidemeister moves. +Two n-mosaics M and M ′ are said to be of the same knot n-type, denoted by M +n∼ M ′, if there exists an +element of A(n) such that it transforms M into M ′. Two n-mosaics M and M ′ are said to be of the same +knot type if there exists a non-negative integer k such that +ikM +n+k +∼ ikM ′, +where i : M(j) → M(j+1) is the mosaic injection by adding a row and a column consisting of only empty tiles. +In [11], Lomonaco and Kauffman conjectured that tame knot theory is equivalent to knot mosaic theory +and in [9], Kuriya and Shehab proved the conjecture. +Proposition 2. Let K and K′ be two knot mosaics of two tame knots k and k′, respectively. Then K and +K′ are of the same knot mosaic type if and only if k and k′ are of the same knot type. +Definition 4. The mosaic number of a knot (or a link) K, denoted by m(K), is the smallest integer n for +which K can be represented by a n-mosaic. +It is obvious that the mosaic number is an invariant for knots and links. For example, the mosaic number +of 31 is 4 and it is easy to show this. In the papers [13, 10], they calculated the mosaic number of knots up +to 8 crossings. +5 + +3 +Marked Graph Mosaics +Let T(u) +M denote the set of 2 Symbols, called mosaic (unoriented) tiles with markers, as follows : +Note that the two tiles are the same up to rotation and have 4 connection points. For constructing an +n-mosaic for marked graph diagrams, consider all tiles of T(u) ∪ T(u) +M as elementary tiles. +Other definitions can be defined in a manner such as mosaic knots, for instance, connection points, +contiguous, suitably connected. An (m, n)-mosaic is an m × n matrix M = (Mij) of tiles, with rows and +columns indexed 0, 1, · · · , m − 1 where each (i, j)-entry Mij is an element of T(u) ∪ T(u) +M . The set of (m, n)- +mosaics is denoted by M(m,n) +M +. It m = n, then an (n, n)-mosaic is a n-mosaic and its set is denoted by +M(n) +M . +Definition 5. A marked graph (m, n)-mosaic is a (m, n)-mosaic in which all tiles are suitably connected. +The set of all marked graph (m, n)-mosaic is the subset of M(m,n) +M +, denoted by K(m,n) +M +. If m = n, then it is +called a marked graph n-mosaic and its set is denoted by K(n) +M . +Example 2. The marked graph diagrams 01, 21 +1 and 60,1 +1 +have the marked graph mosaics as follows. +21 +1 +01 +61 +0,1 +For the equivalence for marked graph mosaics, there are planar isotopy moves and Yoshikawa moves by +using mosaic tiles in T(u) ∪ T(u) +M . The mosaic moves for planar isotopy are the same P1, · · · , P11 with knot +mosaic moves and 4 additional moves P ′ +8, P ′ +9, P ′ +10, P ′ +11 depicted as follows. +P10' +P11' +P8' +P9' +6 + +Yoshikawa moves Γ1, Γ2, Γ3 are the same with Reidemeister moves I, II, III. The mosaic moves for Yoshikawa +moves Γ4, · · · , Γ8 are as follows. + A + B + B + A + A + B + B + A + A + B + B + A + A + B + B + A +All marked graph mosaic moves are permutations on the set M(n) +M of n-mosaics. Indeed, they are also in +the group of all permutations of the set K(n) +M of marked graph n-mosaics. +Definition 6. The ambient isotopy group A(n) +M is the subgroup of the group of all permutations of the set +K(n) +M generated by all planar isotopy moves and all Yoshikawa moves. +Two marked graph n-mosaics M and M ′ are said to be of the same marked graph n-type, denoted by +M +n∼ M ′, if there exists an element of A(n) +M such that it transforms M into M ′. Two marked graph n-mosaics +M and M ′ are said to be of the same marked graph type if there exists a non-negative integer k such that +ikM +n+k +∼ ikM ′, +where i : M(j) → M(j+1) is the mosaic injection by adding a row and a column consisting of only empty +tiles. Therefore, we can obtain the following result. +Theorem 3. Let M and M ′ be two marked graph mosaics of two marked graphs K and K′, respectively. +Then M and M ′ are of the same marked graph mosaic type if and only if K and K′ are equivalent. +For oriented surface-links, consider original oriented mosaic tiles in T(o) (see in [11]) and add 4 oriented +mosaic tiles with markers as follows. Then we can deal with oriented marked graph mosaics similar to oriented +knot mosaics. +7 + +The definition of suitably connected when an orientation is given also considers only cases where the orien- +tation is well matched. Therefore, the oriented marked graph mosaics can also follow the same flow. +4 +Mosaic numbers +The marked graph diagram 81 can reduce the size of its marked graph mosaic using mosaic moves. +Definition 7. The mosaic number of a marked graph diagram K, denoted by m(K), is the smallest integer +n for which K can be represented by a marked graph n-mosaic. +It is obvious that the smallest number of the mosaic size of a marked graph diagram is an invariant for +surface-links. +Theorem 4. The mosaic number of a marked graph diagram is an invariant for surface-links. +It is obvious that the mosaic number of the standard sphere 01 is 2 and the mosaic numbers of both 21 +1 +and 2−1 +1 +are 4. +For finding the mosaic numbers, one can use twofold rule, introduced in [13]. For a given (m, n)-mosaic +D, since there are exactly two ways to connect adjacent connection points in the boundary of D, one can +obtain exactly two marked graph (m + 2, n + 2)-mosaics �D1 and �D2, where D is suitably connected except +the connection point of its boundary. The entry tiles of D are called inner tiles of �D1 or �D2. It is obvious +that a crossing and a marked vertex must be located in the position of inner tiles for the suitably connected +condition. +or +It is clear that if one of four inner corners has a crossing or a marked vertex and if one of two mosaics +by the twofold rule makes a kink, then the crossing or the marked vertex can be removed by Γ1 or Γ6, Γ′ +6, +respectively. +Γ'6 +Γ1 +Γ6 +Theorem 5. Let K be a marked graph K. If ch(K) ≥ 7, then m(K) ≥ 6 where ch(K) denotes the ch-index +of K. +8 + +Proof. Let K be a marked graph whose ch-index is greater than or equal to 7. If ch(K) ≥ 10, then m(K) ≥ 6 +because the number of inner tiles of a 5-mosaic diagram is 9. Similarly, it is easy to check that m(K) ≥ 5 if +ch(K) ≥ 7. +In the case that ch(K) = 8, we will show that m(K) ̸= 5. Suppose that m(K) = 5, that is, there is a +marked graph 5-mosaic diagram D of K such that the ch-index of D is 8. Since the number of inner tiles of +D is 9, there are 9 types for inner tiles. All cases have at least 1 row in the boundary of inner tiles, whose +all mosaic tiles are crossings or marked vertices, as follows up to rotation. + ? + ? + ? + ? + ? + ? + ? ? + ? + ? + ? ? + ? + ? + ? + ? + ? ? +or +By applying the twofold rule, the resulting mosaics have always at least one kink. Therefore, one can +remove the corresponding crossing or marked vertex. It contradicts that the ch-index is 8. Hence, m(K) ≥ 6. +Similar that ch(K) = 7, suppose that m(K) = 5. Let D be a marked graph 5-mosaic diagram of K with +ch-index 7. Then there are 36 cases of its inner tiles and they have at least 1 row as depicted above except 2 +cases. By applying the same argument of the case of ch(K) = 8, 34 cases are contradictory. In the remaining +2 cases, both have exactly two corners with no crossings and no marked vertices. Then for each cases, there +are 4 subcases as follows. + ? + ? +By the twofold rule, for each subcase, there two marked graph mosaics; one of them has always at least one +kink. Since we can reduce the ch-index of D, it contradicts that the ch-index is 7 and then m(K) ≥ 6. +or +or +or +or +9 + +The remaining diagrams of 4 subcase are the same shown as follows. +It has exactly one component. It contradicts that the number of components of 70,−2 +1 +has two components. +Hence, m(K) ≥ 6. +The following diagrams are marked graph mosaics of surface-links with ch-index ≤ 10. The size of some +mosaic diagrams are 6 as follows. By Theorem 5, we know that their mosaic numbers are exactly 6. +101 +1 +101 +0,0,1 +101 +1,1 +101 +0,1 +102 +0,1 +91 +91 +0,1 +101 +103 +91 +1,-2 +102 +81 +21 +1 +01 +61 +0,1 +81 +1,1 +21 +-1 +81 +-1,-1 +71 +0,-2 +101 +-2,-2 +101 +-1,-1 +101 +0,-2 +102 +0,-2 +We conclude this section with a table of mosaic numbers for surface-links of small ch-index. +10 + +K +m(K) +01 +2 +21 +1, 2−1 +1 +4 +60,1 +1 +5, 6 +70,−2 +1 +, 81,1 +1 , 8−1,−1 +1 +, 100,1 +2 +6 +81, 91, 90,1 +1 , 91,−2 +1 +, 101, 102, 100,1 +1 , 101,1 +1 , 100,−2 +2 +, 10−1,−1 +1 +6, 7 +103, 101 +1, 100,0,1 +1 +, 100,−2 +1 +, 10−2,−2 +1 +6, 7, 8 +5 +Kei-Colored Mosaic Diagrams +Recall that a kei is a set X with a binary operation ∗ satisfying the axioms +(i) For all x ∈ X, x ∗ x = x, +(ii) For all x, y ∈ X, we have (x ∗ y) ∗ y = x, and +(iii) For all x, y, z ∈ X we have (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z). +A map f : X → X′ between kei is a kei homomorphism if it satisfies +f(x ∗ y) = f(x) ∗ f(y) +for all x, y ∈ X. Kei are also called involutory quandles; see [3] for more. +Example 3. Every group is a kei under the operation x ∗ y = yx−1y, called the core kei of the group. +Every surface-link L (including classical knots and links, which can be regarded as trivial cobordisms) has +a fundamental kei K(L) whose presentation can be read from a diagram of the surface-link. More precisely, +the fundamental kei of a surface-link has generators corresponding to sheets, i.e., connected components of +a marked graph diagram representing L where we divide at classical undercrossings, together with relations +at the crossings as shown (suggestively as mosaic tiles) +The elements of the fundamental kei are then equivalence classes of kei words in these generators modulo +the equivalence relation generated by the crossing relations and the kei axioms. The isomorphism class of +the fundamental kei is a well-known invariant of unoriented surface-links. +Given a finite kei X, an assignment of elements of X to the sheets of an oriented marked graph diagram +(i.e., segments ending at undercrossing points or marked vertices) is a kei coloring (also called an X-coloring) +of the diagram if it satisfies the crossing condition pictured above at every crossing. +An X-coloring of a diagram D of a surface-link L defines and is defined by a unique element of the set +of kei homomorphisms Hom(K(L), X). This homset is an invariant of surface-links for every finite kei X, +from which useful computable invariants can be extracted. The simplest example is the cardinality of the +set, known as the kei counting invariant, denoted ΦZ +X(L) = |Hom(K(L), X)|. +Generally speaking, any invariant of kei-colored diagrams (or equivalently, homset elements) yields an +invariant known as an enhancement of the kei counting invariant. Examples include the celebrated cocyle +invariants studied in [1] and the more recent kei module invariants introduced in [4]. +We will use mosaic diagrams to enhance the kei counting invariant in the following way. Let L be a +surface-link with mosaic diagram D and let X be a finite kei. Assigning elements of X (called “kei colors”) +11 + +y +h* +C +yto each of the arcs on the tiles in D such that the colors match at connection points and satisfy the kei +coloring conditions at the crossings and marked vertices, we obtain an X-colored mosaic diagram. If we let +f ∈ Hom(K(L), X) be the homset element represented by this coloring, we may denote the colored diagram +by Df. +Definition 8. Let L be a surface-link represented by a marked graph diagram D and let X be a finite kei. +For each kei coloring f ∈ Hom(K(L), X) let us define the kei deficiency of Df as the difference between the +cardinality of the image subkei of f and the number of kei colors appearing in Df. Let φf be the minimal +kei deficiency over the set of minimal mosaic number diagrams Df representing f. Then the multiset +ΦMos,M +X +(L) = {φf | f ∈ Hom(K(L), X)} +is the mosaic deficiency enhancement multiset of the kei homset invariant. For ease of comparison we may +also convert this to polynomial form by summing over the multiset terms of the form uφf to define the +mosaic deficiency enhancement polynomial +ΦMos +X +(L) = +� +f∈Hom(K(L),X) +uφf . +Since there may be many distinct equivalent diagrams of L with minimal mosaic number, to get an +invariant we take for each coloring the minimal kei deficiency over the (finite) set of all diagrams of L with +minimal mosaic number. Then by construction, the multiset of φf-values forms an invariant of surface-links. +From a given minimal-mosaic number diagram of L we can obtain an upper bound on each of the φf-values; +to compute the invariant in general requires finding the complete set of minimal-mosaic number diagrams of +L, which can be computationally difficult. +Let us order the set of polynomials with nonnegative integer coefficients lexicographically by exponent. +That is, to compare two polynomials we first compare their constant terms and in case of a tie, we use +the linear term as a tiebreaker; if the constant and linear terms are equal, we use the quadratic term as +a tiebreaker etc. Then finding a new diagram which reduces the deficiency moves a coloring representative +from a higher exponent into a lower exponent, yielding a smaller lexicographical position; hence it follows +that any particular diagram yields an upper bound on the invariant. +To prove tightness of this bound, one can check exhaustively (which we have not done in the Example +below) that all other mosaic diagrams with the same or lesser mosaic number of the link or surface-link in +question have the same deficiencies for their colorings representing the nontrivial homset elements. +Remark 1. We observe that we can similarly define deficiency enhancements using crossing number or +ch-index in place of mosaic number. Generally speaking, on any diagram with nonzero deficiency we can +perform Reidemeister II moves to reveal “missing” colors in the image subkei. Since these moves increase +ch-index without changing the mosaic number, we expect that these should be different invariants. +Example 4. Consider the surface-knot 101 and the kei Core(Z5). Our python computations show that 101 +has 25 colorings by the kei Core(Z5). These include five monochromatic colorings which have deficiency zero +12 + +and 20 nontrivial colorings, each of which is surjective with deficiency 1 on this diagram, e.g. +. +Then from this diagram we obtain an upper bound 5 + 20u on the kei deficiency polynomial. +We end this section by defining another easy-to-define but difficult-to-compute invariant us surface-links +using mosaics and kei. +Definition 9. Let L be a surface-link and X a finite kei. For each f ∈ Hom(K(L), X) and each positive +integer n ≥ 2, let ρn +f be the minimal kei deficiency value over all n-mosaic diagrams of L. Then the sequence +{ρn +f }∞ +n=2 is the kei deficiency spectrum for f, and as before we have an invariant multiset of such spectra. +Remark 2. We note that since classical knots can be understood as surface-links with an empty set of +marked vertices (i.e., trivial cobordisms between two copies of the knot), the invariants defined in this +section are also invariants of classical knots and links. +6 +Questions +There remains much to be done on the topic of mosaic surface-links. Finding efficient ways to prove tightness +of bounds is of interest, as is extending the quantum knot constructions in [11]. +Say a surface-link L is X-deficiency heterogeneous if it has at least two homset elements which require +different minimal-mosaic number diagrams to realize their minimal X-deficiencies. Is there any such surface- +link? For a given kei X, which is the smallest ch-index of any link which is X-deficiency heterogeneous? For +a fixed surface-link L, for which finite kei X, if any, is L X-deficiency heterogeneous? +A question raised by Seiichi Kamada at a talk on this topic while this paper was in preparation is whether +the ordering of surface-links by ch-number agrees with that induced by mosaic number – e.g., does there +exist a surface-link whose minimal ch-diagram has greater mosaic number than its minimal mosaic diagram. +As mentioned in Remark 1, since there are moves which change the ch-index without changing the mosaic +number, it is not clear what is the relationship between these two notations of complexity of surface-links. +References +[1] J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, and M. Saito. State-sum invariants of knotted +curves and surfaces from quandle cohomology. Electron. Res. Announc. Amer. Math. Soc., 5:146–156 +(electronic), 1999. +13 + +4 +5 +2 +101 +4 +5[2] S. Carter, S. Kamada, and M. Saito. Surfaces in 4-space. Encyclopaedia of Mathematical Sciences. +Springer-Verlag, 2004. +[3] M. Elhamdadi and S. Nelson. Quandles—an introduction to the algebra of knots, volume 74 of Student +Mathematical Library. American Mathematical Society, Providence, RI, 2015. +[4] Y. Joung and S. Nelson. Biquandle module invariants of oriented surface-links. Proc. Amer. Math. Soc., +148(7):3135–3148, 2020. +[5] S. Kamada. Braid and knot theory in dimension four. Mathematical Surveys and Monographs. American +Mathematical Society, 2002. +[6] S. Kamada. +Surface-knots in 4-space. +Springer Monographs in Mathematics. Springer, 2017. +An +introduction. +[7] A. Kawauchi, T. Shibuya, and S. Suzuki. Descriptions on surfaces in four-space. i. normal forms. Math. +Sem. Notes Kobe Univ., 10:75–125, 1982. +[8] C. Kearton and V. Kurlin. +All 2-dimensional links in 4-space live inside a universal 3-dimensional +polyhedron. Algebr. Geom. Topol., 8:1223–1247, 2008. +[9] T. Kuriya and O. Shehab. +The lomonaco-kauffman conjecture. +J. Knot Theory Ramifications, +23:1450003, 20 pp., 2014. +[10] H. J. Lee, L. Ludwig, J. Paat, and A. Peiffer. Knot mosaic tabulation. Involve, 11:13–26, 2018. +[11] S. J. Lomonaco and L. H. Kauffman. Quantum knots and mosaics. In Quantum information science +and its contributions to mathematics, pages 177–208. American Mathematical Society, 2010. +[12] S. J. Lomonaco, Jr. The homotopy groups of knots. I. How to compute the algebraic 2-type. Pacific J. +Math., 95(2):349–390, 1981. +[13] S. Oh, K. Hong, H. Lee, and H. J. Lee. Quantum knots and the number of knot mosaics. Quantum Inf. +Process., 14:801–811, 2015. +[14] F. J. Swenton. +On a calculus for 2-knots and surfaces in 4-space. +J. Knot Theory Ramifications, +10:1133–1141, 2001. +[15] K. Yoshikawa. An enumeration of surfaces in four-space. Osaka J. Math., 31:497–522, 1994. +Nonlinear Dynamics and Mathematical Application Center +Kyungpook National University +Daegu, 41566, Republic of Korea +Department of Mathematical Sciences +Claremont McKenna College +850 Columbia Ave. +Claremont, CA 91711 USA +14 + diff --git a/6dAyT4oBgHgl3EQfcvcQ/content/tmp_files/load_file.txt b/6dAyT4oBgHgl3EQfcvcQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..97b20b25d974022bc777a3f59e06e3b73e367c75 --- /dev/null +++ b/6dAyT4oBgHgl3EQfcvcQ/content/tmp_files/load_file.txt @@ -0,0 +1,440 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf,len=439 +page_content='Marked Graph Mosaics Seonmi Choi∗ Sam Nelson† Abstract We consider the notion of mosaic diagrams for surface-links using marked graph diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We estab- lish bounds, in some cases tight, on the mosaic numbers for the surface-links with ch-index up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' As an application, we use mosaic diagrams to enhance the kei counting invariant for unoriented surface-links as well as classical knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Keywords: Mosaic knots, Surface-links, Marked graph diagrams, kei homset enhancements 2020 MSC: 57K12 1 Introduction Surface-links are compact surfaces smoothly embedded in R4 or S4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' surfaces which are knotted and linked in 4-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Surface-links include many more distinct topological types of unknotted objects – spheres, tori, projective planes, Klein bottles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' – compared with classical knots, and additionally include both orientable and non-orientable cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Introduced in [12], marked graph diagrams are knot diagrams with marked vertices representing saddle points of a surface-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A marked graph diagram satisfying certain mild conditions determines a surface- link up to ambient isotopy in R4, and marked graph diagrams together with the Yoshikawa moves provide a convenient diagrammatic calculus for combinatorial computation with surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Moreover, marked graph diagrams and their Yoshikawa equivalence classes provide a diagrammatic way to represent cobordisms between classical knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A mosaic diagram for a classical knot K is a rectangular (usually square) arrangement of square tiles containing crossings, arcs or nothing such that the arcs join to form a diagram of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Mosaics were used in [11] to define quantum knots, elements of Hilbert spaces generated by mosaic diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In this paper we take the first steps toward extending these constructions to the case of surface-links by considering mosaic presentations for surface-links using marked graph diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We establish a set of tiles and Yoshikawa moves for marked graph mosaics and provide mosaic diagrams for each of the surface-links in the Yoshikawa table of surface-links with up to ch-index 10, establishing an upper bound on mosaic number for these surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' As an application we use mosaic presentations to define a new enhancement of the kei counting invariant for classical knots and links as well as for surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' As with mosaic number, we can compute an upper bound with respect to a certain ordering on the new enhancement from a given diagram of a surface-link or classical knot or link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In Section 2 we review some preliminaries about knot mosaics and marked graph diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In Section 3 we introduce marked graph mosaics and obtain some results including upper bounds, some tight, on the the mosaic numbers of both orientable and non-orientable surface-links with ch-index less than or equal to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In Section 5 we define kei-colored mosaics and use them to enhance the kei counting invariant for classical knots and links as well as surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We conclude in Section 6 with some questions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ∗Email: smchoi@knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Partially supported by Basic Science Research Program through the National Research Founda- tion of Korea(NRF) funded by the Ministry of Education(2021R1I1A1A01049100) and the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 2022R1A5A1033624).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' †Email: Sam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='Nelson@cmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Partially supported by Simons Foundation Collaboration Grant 702597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='00287v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='GT] 31 Dec 2022 2 Preliminaries We review knot mosaics and recall surface-links, marked graph diagrams and their relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='1 Surface-links and marked graph diagrams A surface-link is the image of a closed surface smoothly (piecewise linear and locally flatly) embedded in R4 (or S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' If it is called a surface-knot, then the underlying surface is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A surface-link is orientable if the underlying surface is orientable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' otherwise, it is nonorientable or unorientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' An unoriented surface- link is either an unorientable surface-link or an orientable surface link without a specified orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Two surface-links F and F ′ are equivalent if there exists an orientation-preserving homeomorphism h : R4 → R4 such that h(F) = F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' There are many useful schemes for describing for surface-links since it is difficult to directly deal with surface-links in 4-space for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For example, broken surface diagrams, marked graph diagrams, motion pictures etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' See [2, 5, 6, 15] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We use an effective tool for handling surface-links known as a marked graph diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A marked graph is a spatial graph embedded in R3 possibly with 4-valent vertices decorated by a line segment like .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We call such a line segment a marker and a vertex with a marker a marked vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' An orientation of edges incident with a marked vertex is one of two types of the orientation, such as or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A marked graph is said to be orientable if it admits an orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Otherwise, it is non-orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Two (oriented) marked graphs are said to be equivalent if they are ambient isotopic in R3 keeping the rectangular neighborhoods and markers (with orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In the same way as a link diagram, one can define a marked graph diagram which is a diagram in R2 with classical crossings and marked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For each marked vertex of a marked graph diagram D, the local diagram obtained by splicing in a direction consistent with its marker (say + direction), looks like .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' By applying this in the opposite direction (called − direction), the resulting local diagram looks like .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Therefore one can obtain two classical link diagrams, denoted by L+(D) and L−(D), from D by splicing every marked vertices in + direction and − direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We call L+(D) and L−(D) the positive and negative resolutions of D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A marked graph diagram D is said to be admissible if both resolutions L−(D) and L+(D) are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A marked graph is said to be admissible if it has an admissible marked graph diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For example, it is easy to check that a marked graph diagram D of the spun trefoil as follows is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' D L_(D) L+(D) Let D be a admissible marked graph diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then a surface-link F(D) can be constructed and it is uniquely determined from D up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Conversely, every surface-link F can be expressed by an admissible marked graph diagram D, that is, F(D) is equivalent to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' See [7, 12, 15] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For example, the correspondence between the marked graph diagram and the standard projective plane are illustrated in the following figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' R3×{0} R3×{1} R3×{-1} R3×[1,∞) R3×[-1,∞) R4 2 The equivalence moves Γ1, · · · , Γ8 for marked graph diagrams is called Yoshikawa moves [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=" Γ1 Γ2 Γ3 Γ4 Γ5 Γ8 Γ'4 Γ6 Γ7 Γ'6 Proposition 1 ([8, 14, 15])." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Two marked graph diagrams D and D′ present equivalent oriented surface- links if and only if D can be obtained from D′ by a finite sequence of ambient isotopies in R2 and Yoshikawa moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let K be a marked graph diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The ch-index of K, denoted ch(K), is the total number of crossings and marked vertices in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='2 Mosaic Knots A mosaic (unoriented) tile is one of rectangles with arcs and possibly with one crossing, depicted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The set of mosaic tiles T0, T1, · · · , T10 is denoted by T(u) and there are exactly 5 tiles, up to rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The endpoints of an arc on a mosaic tile are called connection points of the tile and are also located the center of an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' There are tiles with 0, 2 and 4 connection points in T(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 4 connection points 0 connection points 2 connection points An (m, n)-mosaic is an m × n matrix whose entries are mosaic tiles in T(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' If m = n, then it is simply called an n-mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The sets of (m, n)-mosaics and n-mosaics are denoted by M(m,n) and M(n), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Two tiles in a mosaic are said to be contiguous if they lie immediately next to each other in the same either row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A tile in a mosaic is said to be suitably connected if all its connection points touch the 3 connection points of contiguous tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' all its connection points meet the connection points of contiguous tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Note that for 4-mosaic illustrated above, its (2, 2)-entry tile is suitably connected, but its (3, 3)-entry tile is not suitably connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A knot (m, n)-mosaic is an (m, n)-mosaic in which all tiles are suitably connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The set of all knot (m, n)-mosaic is the subset of M(m,n), denoted by K(m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' If m = n, then it is called a knot n-mosaic and its set is denoted by K(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The trefoil 31 has a knot 5-mosaic and 4-mosaic, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For the equivalence for mosaic knots, there are planar isotopy moves and Reidemeister moves by using mosaic tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The non-deterministic tiles are necessary to define the moves, as follows : Each non-deterministic tile means two types of tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' or or Non-deterministic tiles labeled by the same letter A or B are synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A A A B B B B A The equivalence of mosaic knots consists of 11 moves for planar isotopy, 2 moves for Reidemeister moves I, 4 moves for Reidemeister moves II and 6 moves for Reidemeister moves III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Planar isotopy moves : 11 types P1 P4 P2 P3 P7 P5 P6 P10 P11 P8 P9 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Reidemeister moves I : 2 types 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Reidemeister moves II : 4 types 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Reidemeister moves III : 6 types A B B A A B B A A B B A A B B A A B B A A B B A All mosaic moves are permutations on the set M(n) of n-mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Indeed, they are also in the group of all permutations of the set K(n) of knot n-mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The ambient isotopy group A(n) is the subgroup of the group of all permutations of the set K(n) generated by all planar isotopy moves and all Reidemeister moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Two n-mosaics M and M ′ are said to be of the same knot n-type, denoted by M n∼ M ′, if there exists an element of A(n) such that it transforms M into M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Two n-mosaics M and M ′ are said to be of the same knot type if there exists a non-negative integer k such that ikM n+k ∼ ikM ′, where i : M(j) → M(j+1) is the mosaic injection by adding a row and a column consisting of only empty tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In [11], Lomonaco and Kauffman conjectured that tame knot theory is equivalent to knot mosaic theory and in [9], Kuriya and Shehab proved the conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let K and K′ be two knot mosaics of two tame knots k and k′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then K and K′ are of the same knot mosaic type if and only if k and k′ are of the same knot type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The mosaic number of a knot (or a link) K, denoted by m(K), is the smallest integer n for which K can be represented by a n-mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' It is obvious that the mosaic number is an invariant for knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For example, the mosaic number of 31 is 4 and it is easy to show this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In the papers [13, 10], they calculated the mosaic number of knots up to 8 crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 5 3 Marked Graph Mosaics Let T(u) M denote the set of 2 Symbols, called mosaic (unoriented) tiles with markers, as follows : Note that the two tiles are the same up to rotation and have 4 connection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For constructing an n-mosaic for marked graph diagrams, consider all tiles of T(u) ∪ T(u) M as elementary tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Other definitions can be defined in a manner such as mosaic knots, for instance, connection points, contiguous, suitably connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' An (m, n)-mosaic is an m × n matrix M = (Mij) of tiles, with rows and columns indexed 0, 1, · · · , m − 1 where each (i, j)-entry Mij is an element of T(u) ∪ T(u) M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The set of (m, n)- mosaics is denoted by M(m,n) M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' It m = n, then an (n, n)-mosaic is a n-mosaic and its set is denoted by M(n) M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A marked graph (m, n)-mosaic is a (m, n)-mosaic in which all tiles are suitably connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The set of all marked graph (m, n)-mosaic is the subset of M(m,n) M , denoted by K(m,n) M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' If m = n, then it is called a marked graph n-mosaic and its set is denoted by K(n) M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The marked graph diagrams 01, 21 1 and 60,1 1 have the marked graph mosaics as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 21 1 01 61 0,1 For the equivalence for marked graph mosaics, there are planar isotopy moves and Yoshikawa moves by using mosaic tiles in T(u) ∪ T(u) M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The mosaic moves for planar isotopy are the same P1, · · · , P11 with knot mosaic moves and 4 additional moves P ′ 8, P ′ 9, P ′ 10, P ′ 11 depicted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=" P10' P11' P8' P9' 6 Yoshikawa moves Γ1, Γ2, Γ3 are the same with Reidemeister moves I, II, III." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The mosaic moves for Yoshikawa moves Γ4, · · · , Γ8 are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A B B A A B B A A B B A A B B A All marked graph mosaic moves are permutations on the set M(n) M of n-mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Indeed, they are also in the group of all permutations of the set K(n) M of marked graph n-mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The ambient isotopy group A(n) M is the subgroup of the group of all permutations of the set K(n) M generated by all planar isotopy moves and all Yoshikawa moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Two marked graph n-mosaics M and M ′ are said to be of the same marked graph n-type, denoted by M n∼ M ′, if there exists an element of A(n) M such that it transforms M into M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Two marked graph n-mosaics M and M ′ are said to be of the same marked graph type if there exists a non-negative integer k such that ikM n+k ∼ ikM ′, where i : M(j) → M(j+1) is the mosaic injection by adding a row and a column consisting of only empty tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Therefore, we can obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let M and M ′ be two marked graph mosaics of two marked graphs K and K′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then M and M ′ are of the same marked graph mosaic type if and only if K and K′ are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For oriented surface-links, consider original oriented mosaic tiles in T(o) (see in [11]) and add 4 oriented mosaic tiles with markers as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then we can deal with oriented marked graph mosaics similar to oriented knot mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 7 The definition of suitably connected when an orientation is given also considers only cases where the orien- tation is well matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Therefore, the oriented marked graph mosaics can also follow the same flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 4 Mosaic numbers The marked graph diagram 81 can reduce the size of its marked graph mosaic using mosaic moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The mosaic number of a marked graph diagram K, denoted by m(K), is the smallest integer n for which K can be represented by a marked graph n-mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' It is obvious that the smallest number of the mosaic size of a marked graph diagram is an invariant for surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The mosaic number of a marked graph diagram is an invariant for surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' It is obvious that the mosaic number of the standard sphere 01 is 2 and the mosaic numbers of both 21 1 and 2−1 1 are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For finding the mosaic numbers, one can use twofold rule, introduced in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For a given (m, n)-mosaic D, since there are exactly two ways to connect adjacent connection points in the boundary of D, one can obtain exactly two marked graph (m + 2, n + 2)-mosaics �D1 and �D2, where D is suitably connected except the connection point of its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The entry tiles of D are called inner tiles of �D1 or �D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' It is obvious that a crossing and a marked vertex must be located in the position of inner tiles for the suitably connected condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' or It is clear that if one of four inner corners has a crossing or a marked vertex and if one of two mosaics by the twofold rule makes a kink, then the crossing or the marked vertex can be removed by Γ1 or Γ6, Γ′ 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=" Γ'6 Γ1 Γ6 Theorem 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let K be a marked graph K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' If ch(K) ≥ 7, then m(K) ≥ 6 where ch(K) denotes the ch-index of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let K be a marked graph whose ch-index is greater than or equal to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' If ch(K) ≥ 10, then m(K) ≥ 6 because the number of inner tiles of a 5-mosaic diagram is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Similarly, it is easy to check that m(K) ≥ 5 if ch(K) ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In the case that ch(K) = 8, we will show that m(K) ̸= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Suppose that m(K) = 5, that is, there is a marked graph 5-mosaic diagram D of K such that the ch-index of D is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Since the number of inner tiles of D is 9, there are 9 types for inner tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' All cases have at least 1 row in the boundary of inner tiles, whose all mosaic tiles are crossings or marked vertices, as follows up to rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' or By applying the twofold rule, the resulting mosaics have always at least one kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Therefore, one can remove the corresponding crossing or marked vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' It contradicts that the ch-index is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Hence, m(K) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Similar that ch(K) = 7, suppose that m(K) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let D be a marked graph 5-mosaic diagram of K with ch-index 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then there are 36 cases of its inner tiles and they have at least 1 row as depicted above except 2 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' By applying the same argument of the case of ch(K) = 8, 34 cases are contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' In the remaining 2 cases, both have exactly two corners with no crossings and no marked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then for each cases, there are 4 subcases as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' By the twofold rule, for each subcase, there two marked graph mosaics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' one of them has always at least one kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Since we can reduce the ch-index of D, it contradicts that the ch-index is 7 and then m(K) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' or or or or 9 The remaining diagrams of 4 subcase are the same shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' It has exactly one component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' It contradicts that the number of components of 70,−2 1 has two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Hence, m(K) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The following diagrams are marked graph mosaics of surface-links with ch-index ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The size of some mosaic diagrams are 6 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' By Theorem 5, we know that their mosaic numbers are exactly 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 101 1 101 0,0,1 101 1,1 101 0,1 102 0,1 91 91 0,1 101 103 91 1,-2 102 81 21 1 01 61 0,1 81 1,1 21 1 81 1,-1 71 0,-2 101 2,-2 101 1,-1 101 0,-2 102 0,-2 We conclude this section with a table of mosaic numbers for surface-links of small ch-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 10 K m(K) 01 2 21 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 2−1 1 4 60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='1 1 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 6 70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='−2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 8−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='−1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='1 2 6 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='−2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 102,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='−2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 10−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='−1 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 7 103,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 101 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='−2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 10−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='−2 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 8 5 Kei-Colored Mosaic Diagrams Recall that a kei is a set X with a binary operation ∗ satisfying the axioms (i) For all x ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' x ∗ x = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' (ii) For all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' y ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' we have (x ∗ y) ∗ y = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' and (iii) For all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' z ∈ X we have (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A map f : X → X′ between kei is a kei homomorphism if it satisfies f(x ∗ y) = f(x) ∗ f(y) for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Kei are also called involutory quandles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' see [3] for more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Every group is a kei under the operation x ∗ y = yx−1y, called the core kei of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Every surface-link L (including classical knots and links, which can be regarded as trivial cobordisms) has a fundamental kei K(L) whose presentation can be read from a diagram of the surface-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' More precisely, the fundamental kei of a surface-link has generators corresponding to sheets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=', connected components of a marked graph diagram representing L where we divide at classical undercrossings, together with relations at the crossings as shown (suggestively as mosaic tiles) The elements of the fundamental kei are then equivalence classes of kei words in these generators modulo the equivalence relation generated by the crossing relations and the kei axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The isomorphism class of the fundamental kei is a well-known invariant of unoriented surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Given a finite kei X, an assignment of elements of X to the sheets of an oriented marked graph diagram (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=', segments ending at undercrossing points or marked vertices) is a kei coloring (also called an X-coloring) of the diagram if it satisfies the crossing condition pictured above at every crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' An X-coloring of a diagram D of a surface-link L defines and is defined by a unique element of the set of kei homomorphisms Hom(K(L), X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' This homset is an invariant of surface-links for every finite kei X, from which useful computable invariants can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' The simplest example is the cardinality of the set, known as the kei counting invariant, denoted ΦZ X(L) = |Hom(K(L), X)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Generally speaking, any invariant of kei-colored diagrams (or equivalently, homset elements) yields an invariant known as an enhancement of the kei counting invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Examples include the celebrated cocyle invariants studied in [1] and the more recent kei module invariants introduced in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We will use mosaic diagrams to enhance the kei counting invariant in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let L be a surface-link with mosaic diagram D and let X be a finite kei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Assigning elements of X (called “kei colors”) 11 y h* C yto each of the arcs on the tiles in D such that the colors match at connection points and satisfy the kei coloring conditions at the crossings and marked vertices, we obtain an X-colored mosaic diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' If we let f ∈ Hom(K(L), X) be the homset element represented by this coloring, we may denote the colored diagram by Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let L be a surface-link represented by a marked graph diagram D and let X be a finite kei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For each kei coloring f ∈ Hom(K(L), X) let us define the kei deficiency of Df as the difference between the cardinality of the image subkei of f and the number of kei colors appearing in Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let φf be the minimal kei deficiency over the set of minimal mosaic number diagrams Df representing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then the multiset ΦMos,M X (L) = {φf | f ∈ Hom(K(L), X)} is the mosaic deficiency enhancement multiset of the kei homset invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For ease of comparison we may also convert this to polynomial form by summing over the multiset terms of the form uφf to define the mosaic deficiency enhancement polynomial ΦMos X (L) = � f∈Hom(K(L),X) uφf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Since there may be many distinct equivalent diagrams of L with minimal mosaic number, to get an invariant we take for each coloring the minimal kei deficiency over the (finite) set of all diagrams of L with minimal mosaic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then by construction, the multiset of φf-values forms an invariant of surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' From a given minimal-mosaic number diagram of L we can obtain an upper bound on each of the φf-values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' to compute the invariant in general requires finding the complete set of minimal-mosaic number diagrams of L, which can be computationally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let us order the set of polynomials with nonnegative integer coefficients lexicographically by exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' That is, to compare two polynomials we first compare their constant terms and in case of a tie, we use the linear term as a tiebreaker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' if the constant and linear terms are equal, we use the quadratic term as a tiebreaker etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then finding a new diagram which reduces the deficiency moves a coloring representative from a higher exponent into a lower exponent, yielding a smaller lexicographical position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' hence it follows that any particular diagram yields an upper bound on the invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' To prove tightness of this bound, one can check exhaustively (which we have not done in the Example below) that all other mosaic diagrams with the same or lesser mosaic number of the link or surface-link in question have the same deficiencies for their colorings representing the nontrivial homset elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We observe that we can similarly define deficiency enhancements using crossing number or ch-index in place of mosaic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Generally speaking, on any diagram with nonzero deficiency we can perform Reidemeister II moves to reveal “missing” colors in the image subkei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Since these moves increase ch-index without changing the mosaic number, we expect that these should be different invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Consider the surface-knot 101 and the kei Core(Z5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Our python computations show that 101 has 25 colorings by the kei Core(Z5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' These include five monochromatic colorings which have deficiency zero 12 and 20 nontrivial colorings, each of which is surjective with deficiency 1 on this diagram, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then from this diagram we obtain an upper bound 5 + 20u on the kei deficiency polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We end this section by defining another easy-to-define but difficult-to-compute invariant us surface-links using mosaics and kei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Let L be a surface-link and X a finite kei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For each f ∈ Hom(K(L), X) and each positive integer n ≥ 2, let ρn f be the minimal kei deficiency value over all n-mosaic diagrams of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Then the sequence {ρn f }∞ n=2 is the kei deficiency spectrum for f, and as before we have an invariant multiset of such spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' We note that since classical knots can be understood as surface-links with an empty set of marked vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=', trivial cobordisms between two copies of the knot), the invariants defined in this section are also invariants of classical knots and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' 6 Questions There remains much to be done on the topic of mosaic surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Finding efficient ways to prove tightness of bounds is of interest, as is extending the quantum knot constructions in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Say a surface-link L is X-deficiency heterogeneous if it has at least two homset elements which require different minimal-mosaic number diagrams to realize their minimal X-deficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Is there any such surface- link?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For a given kei X, which is the smallest ch-index of any link which is X-deficiency heterogeneous?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' For a fixed surface-link L, for which finite kei X, if any, is L X-deficiency heterogeneous?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' A question raised by Seiichi Kamada at a talk on this topic while this paper was in preparation is whether the ordering of surface-links by ch-number agrees with that induced by mosaic number – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=', does there exist a surface-link whose minimal ch-diagram has greater mosaic number than its minimal mosaic diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' As mentioned in Remark 1, since there are moves which change the ch-index without changing the mosaic number, it is not clear what is the relationship between these two notations of complexity of surface-links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Carter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfcvcQ/content/2301.00287v1.pdf'} +page_content=' Jelsovsky, S.' metadata={'source': 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b/6tAyT4oBgHgl3EQfpvgE/content/tmp_files/2301.00529v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4db4fefcf53edf977e257b071159ac60d63fa6c --- /dev/null +++ b/6tAyT4oBgHgl3EQfpvgE/content/tmp_files/2301.00529v1.pdf.txt @@ -0,0 +1,6561 @@ +arXiv:2301.00529v1 [math.RT] 2 Jan 2023 +HARMONIC ANALYSIS ON THE FOURFOLD COVER OF THE SPACE OF +ORDERED TRIANGLES I: THE INVARIANT DIFFERENTIALS +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +Abstract. Denote by SLn(R) the group of n × n real matrices with determinant one, A the +subgroup consisting of the diagonal matrices with positive entries, and SLn(R)/A the manifold +of left cosets gA, g ∈ SLn(R). In this paper, we will be concerned with the harmonic analysis +on the homogeneous space SLn(R)/A when n = 3. In particular, we provide explicit generators +and their relations for the algebra of the invariant differential operators. Then we prove that +some of the non-central generators are essentially self-adjoint. +Contents +1. +Introduction +1 +2. +Structure of the Algebra of the Invariant Differentials +4 +2.1. +Generators of the invariant differentials on SLn(R)/A +4 +2.2. +Relations among the generators +7 +2.3. +The center of the algebra of the invariant differentials +11 +3. +Essential Self-Adjointness +19 +3.1. +Essential self-adjointness of the central elements +19 +3.2. +Reduction to the density of C∞ +c (X) in Dom(∆) +20 +3.3. +Coordinate charts induced by the Euler angles +21 +3.4. +Presence of left derivatives in D12 +22 +3.5. +Proof of the density and Theorem 1.3 +28 +Appendix A. +Computations in the Euler-Iwasawa Coordinates +36 +A.1. +Explicit formulas for the generators of the left derivatives on SL3(R)/A +36 +A.2. +Explicit formulas for the generators of the left-invariant differentials on SL3(R) +42 +References +46 +1. Introduction +Denote by SLn(R) the group of n × n real matrices with determinant one, A the subgroup +consisting of the diagonal matrices with positive entries, and SLn(R)/A the manifold of left +cosets gA, g ∈ SLn(R). In this paper, we will be concerned with the harmonic analysis on +the homogeneous space SLn(R)/A when n = 3. In particular, we will restrict attention to +the commutation relations and essential self-adjointness of the invariant differential operators +1 + +2 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +on SL3(R)/A; spectral decomposition of the invariant differential operators and its interaction +with Plancherel theorems are left for future research. +The space SL3(R)/A distinguishes itself among homogeneous spaces in various ways. Firstly, +it has a natural interpretation as the fourfold cover of the space of nondegenerate Schubert +triangles in the plane ([Sc]), of which the compactification is well studied as of a homogeneous +space of complexity 1 ([Sem], [Ti]). One may wish to investigate the Plancherel type formulas +(see [GGV], [GG] for the relation with the Radon transforms), via the Bernstein maps and +the Maass-Selberg relations (see [SV], [DKKS] for the spherical case). Secondly, SL3(R)/A is +one of the simplest examples of non-spherical homogeneous spaces. These spaces exhibit a very +different nature compared with spherical ones, and very little of their harmonic theory is known. +For instance, the finite multiplicity theorem for induction does not hold anymore ([KO]); the +algebra of invariant differential operators is a noncommutative algebra instead of a polynomial +ring ([Kn]); the representation theory of the ring of bi-invariant functions is mysterious as +well. For our case, at least one knows, a priori by the systematic work of Benoist-Kobayashi +([BK1], [BK2], [BK3], [BK4], [BIK]), that the natural unitary representation of SLn(R) in +L2 (SLn(R)/A) is tempered. +From the technical perspective, the investigation of the spectral theory of pseudo-Riemannian +manifolds is challenging, for the traditional elliptic theory is not applicable to the Laplace- +Beltrami operators with mixed signs. To remedy the situation, the analytic methods are always +fused with the peculiar geometry of the underlying manifold related to the representation theory. +The considerable efforts have been made mainly on the symmetric spaces, such as the group +manifolds ([HC1], [HC2]), the real hyperboloids ([RLN], [LNR1], [LNR2], [Sh], [St], [Ros], +[Fa], [Sek], [Mo]), special symmetric spaces ([Ma], [Sa], [DP], [KD], [BH], [Ha]), and general +symmetric spaces ([FJ], [OM], [OS], [Ba], [O], [De], [BS1], [BS2]), and, more recently, certain +locally symmetric spaces and spherical varieties ([KK1], [KK2]). +In the present situation, +SL3(R)/A has an inherent approachable geometry in spite of the rather involved analysis due +to its non-sphericity. For instance, the underlying manifold is diffeomorphic to SO3(R) × R3, +so that an extensive calculation is possible, just as the hyperboloid case, where the usage of +the spherical coordinates of the underlying product space Sp−1 × Sq−1 × R is crucial. +Before describing our results in more detail, we first set certain notations. To treat in a unified +fashion, let G denote SL3(R), g the Lie algebra of G, and U(g) the universal enveloping algebra +of the complexification of g. Denote by C∞(G) the space of complex-valued smooth functions +on G. Then, the infinitesimal action R on C∞(G) induced by the right regular representation +of G, maps U(g) into the algebra of algebraic differentials on G. More precisely, R acts on +u = X1X2 · · · Xk ∈ U(g) by +(Ruf) (g) := (R (X1 · · · Xk) f) (g) := +∂ +∂t1 +���� +t1=0 +· · · ∂ +∂tk +���� +tk=0 +f(g exp (t1X1) · · · exp (tkXk)). +(1) +Here X1, X2, · · · , Xk ∈ g, and f ∈ C∞(G); the exponential map is given by exp(X) := γ(1), +where γ : R → G is the one-parameter subgroup of G whose tangent vector at the identity is +equal to X. It is easy to verify that Ru, u ∈ U(g), is a left G-invariant differential operators +on G. Denote by D(G) the algebra of the left G-invariant differential operators on G. + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +3 +For a closed subgroup H ⊂ G, denote by D(G/H) the algebra of G-invariant differential +operators on the homogeneous space G/H. Denote by π : G → G/H the natural projection, +and h the Lie algebra of H. Define +DH(G) := {D ∈ D(G) | D(f ◦ Rh) ◦ R−1 +h += Df, ∀h ∈ H and f ∈ C∞(G)}, +(2) +where Rh : g �→ gh is the right translation of G for h ∈ H. Assuming G and H are reductive, +we have the standard isomorphism (Theorem 4.6 in Chapter 2 of [He]) +DH(G)/ +� +DH(G) ∩ D(G)h +� ∼= D(G/H). +(3) +It is induced by the map µ : DH(G) → D(G/H), such that for each D ∈ DH(G), µ(D) is the +element of D(G/H) such that +(µ(D)f) ◦ π = D(f ◦ π) for all smooth functions f on G/H. +(4) +Denote by Eij the 3 × 3 matrix unit with a 1 in the ith row and jth column. For distinct +i, j, k ∈ {1, 2, 3}, define differential operators on SL3(R)/A +Dij = µ +� +1 +2 +� +σ∈S2 +R +� +Eσ(i)σ(j)Eσ(j)σ(i) +� +� +, +(5) +and +Dijk = µ +� +1 +6 +� +σ∈S3 +R +� +Eσ(i)σ(j)Eσ(j)σ(k)Eσ(k)σ(i) +� +� +. +(6) +Note that Dij = Dji and Dijk = Djki = Dkij. +In the first part of the paper, we prove +Theorem 1.1. D (SL3(R)/A) is the noncommutative associative algebra generated over C by +{D12, D13, D23, D123, D213} with relations + + + + + + + + + + + +[D123, D213] = 0, +[Dij, Dik] = Dijk − Dikj, +i, j, k ∈ {1, 2, 3} are distinct, +[Dijk, Dij] = DjkDij − DijDik, +i, j, k ∈ {1, 2, 3} are distinct, +2 (D123D213 + D213D123 − D12D23D31 − D13D32D21) = (D23 − D13 − D12)2 . +(7) +The center of D (SL3(R)/A) is a polynomial ring in D123 + D213 and D12 + D23 + D13. +In the theory of harmonic analysis on homogeneous spaces, one of the central problems is +whether a symmetric invariant differential operator has a unique self-adjoint extension, as it +would enable a simultaneous study of spectral decomposition of both the regular representation +and the invariant differential operators (e.g. see [BS1], [BS2] for the case of reductive symmetric +spaces). There is considerable literature on essential self-adjointness for natural operators on a +complete Riemannian or Hermitian manifold. See for instance [Ga1], [Ga2], [Ga3], [Roe], [Co], +[Ri] for the Hodge-Laplace-Beltrami operator, [AV] for the ¯∂-Laplacian, [P], [W] for the Dirac +operator, and [Ch] for certain first order differential operators. For differential operators on +Rn, much more is known ([RS]), and we mention here that certain ellipticity ([IK]), or semi- +boundedness together with temperedness ([Dev]), guarantees the essential self-adjointness. + +4 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +For general homogeneous spaces, a classical result shows that the symmetric elements in the +image of the center of the universal enveloping algebra are essentially self-adjoint ([Seg], [NS], +or [Th]). As it can be shown that the center of D (SL3(R)/A) equals the image of the center of +U(sl3(R)), it follows +Proposition 1.2. Every symmetric differential operator in the center of D (SL3(R)/A) is es- +sentially self-adjoint. +For non-central elements, the most general result is due to Van den Ban [Ba], who established +the essential self-adjointness of the symmetric invariant differential operators for semi-simple +symmetric pairs, even if it is not a generalized Gelfand pair, semiboundedness is absent, or the +underlying manifold is non-Riemannian. Beyond that, to the best of our knowledge, there is no +general theory ensuring the essential self-adjointness in the pseudo-Riemannian setting, even +for the Laplacian operators (see [KK1], [KK2]). +The major part of the paper is to devoted to +Theorem 1.3. The differential operators D12, D13, D23 on SL3(R)/A are essentially self-adjoint. +We now briefly describe the basic ideas for the proofs. We exploit the universal enveloping +algebra to extract the algebraic structure of D(SL3(R)/A), and the normal form theory in [FH] +to determine the center of D(SL3(R)/A). To study the essential self-adjointness of symmetric +operators, we modify the scheme of [Ba]. +The elegant proof in [Ba] is to decompose the +differential operator into a bounded sum of left derivatives so that the wild growth of the +coefficients can be treated as bounded ones. +Unfortunately, in this non-spherical case, the +left derivatives are too degenerate to span the whole space of invariant differentials in a mild +way. We make the observation that by choosing the cutoff functions and the mollifiers in a +compatible way instead of separating the G˚arding type space as the operator core, one may +gain extra decays to balance the extraordinary growth of the coefficients. In fact, the chosen +cutoff functions are annihilated by the wildest terms and contribute desired decays thereafter. +The organization of the paper is as follows. +§2 is devoted to the algebraic structure of +D(SL3(R)/A). We study the presence of left derivatives in the coordinate form of D12 in §3.4. +In §3.5, we establish the density of C∞ +c (SL3(R)/A) in Dom(D12), and, as a consequence, prove +Theorem 1.3. The explicit formulas for the generators of the left derivatives and of the left +invariant differentials are given in Appendices A.1 and A.2, respectively. +Acknowledgement. +The authors appreciate greatly Professors N. Li, W. Li, Y. Xu and +J. Yu for many helpful discussion. This research is supported by National Key R& D Pro- +gram of China (No. 2022YFA1006700). The first author is partially supported by NSFC grant +(No. 12201012). +2. Structure of the Algebra of the Invariant Differentials +2.1. Generators of the invariant differentials on SLn(R)/A. Let S(g) be the symmetric +algebra over g. Then for a basis {X1, · · · , Xn} of g, S(g) can be identified with the algebra of +polynomials +� +(k1,··· ,kn)∈Nn +ak1···knXk1 +1 · · · Xkn +n , ak1···kn ∈ C. +(8) + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +5 +We have the following symmetrizer map λ : S(g) → D(G). +Theorem 2.1 (Theorem 4.3 in Chapter 2 of [He]). There is a unique linear bijection λ from +S(g) to D(G) such that λ(Xm) = R(Xm) for X ∈ g and m ∈ Z+. More precisely, +(λ(P)f)(g) := P +� ∂ +∂t1 +, · · · , ∂ +∂tn +� +f (g exp(t1X1 + · · · + tnXn)) +���� +t1=···=tn=0 +, +(9) +for P ∈ S(g) and f ∈ C∞(G). In particular (see Page 282 of [He]), +λ(Y1 · · · Yk) = 1 +k! +� +σ∈Sk +R +� +Yσ(1) · · · Yσ(k) +� +, +(10) +where Y1, · · · , Yk ∈ g, and Sk is the symmetric group of degree n. +Denote by Eij the n × n matrix unit with a 1 in the ith row and jth column. Define +Xij := Eij, 1 ≤ i ̸= j ≤ n, +Xll := Ell − Enn, 1 ≤ l ≤ n − 1, +(11) +which constitute a basis of sln(R). +Proposition 2.2. The algebra D(SLn(R)/A) is generated by +� +(µ ◦ λ) +� +Ei1i2Ei2i3 · · · Eik−1ikEiki1 +� ���� +2 ≤ k ≤ n, 1 ≤ i1, i2, · · · , ik ≤ n +i1, i2, · · · , ik are distinct +� +. +(12) +Proof of Proposition 2.2. +By (3), the invariant differential operators on SLn(R)/A are +induced from the left SLn(R) and right A invariant differential operators on SLn(R). Then by +(3), it suffices to prove that DA(SLn(R)) is generated by the elements in (12) and D(SLn(R))a, +where a is the Lie algebra of A. +Take D ∈ DA(SLn(R)). There is a polynomial PD such that λ(PD) = D by Theorem 2.1. +Arrange PD in the following lexicographic order, +PD = +∞ +� +k=1 +� +1≤i1≤i2≤···≤it≤···≤ik≤n; +1≤jt≤n and (it,jt)̸=(n,n) for 1≤t≤k; +if 1≤u 0 such that for all n = 1, 2, · · ·, +sup +x∈X +����(1 + |z12|)(1 + |z13| + |z23|) · ∂χn +∂z12 +(x) +���� + sup +x∈X +���� +� +|z12|2 + 1 +� +· +∂2χn +∂z12∂z12 +(x) +���� ≤ M. +(136) +Proof of Lemma 3.15. It is clear that (a), (b), (c), (d) follow from the definition in (132). +(e) follows from the computation that +z23 +∂χn +∂z13 +− z13 +∂χn +∂z23 += ξ +�ln (|z12|2 + 1) +n +� +· ξ′ +�|z13|2 + |z23|2 +n +� +· +�z23z13 +n +− z13z23 +n +� +≡ 0. +(137) +Computation yields +∂χn +∂z12 +(kx, z12, z13, z23) = ξ +�|z13|2 + |z23|2 +n +� +· ξ′ +�ln (|z12|2 + 1) +n +� +· +2z12 +|z12|2 + 1 · 1 +n. +(138) +Then, +sup +x∈X +����(1 + |z12|)(1 + |z13| + |z23|) · ∂χn +∂z12 +(x) +���� += sup +x∈X +�����ξ +�|z13|2 + |z23|2 +n +� +ξ′ +�ln (|z12|2 + 1) +n +����� · 2|z12|(1 + |z12|) +|z12|2 + 1 +· 1 + |z13| + |z23| +n +� +≤ 4 +sup +r∈[0,+∞) +|ξ′(r)| · sup +x∈X +�����ξ +�|z13|2 + |z23|2 +n +����� · 1 + |z13| + |z23| +n +� +. +(139) +Notice that when 1 + |z13| + |z23| ≥ 4n, +|z13|2 + |z23|2 +n +≥ (|z13| + |z23|)2 +2n +≥ (4n − 1)2 +2n +> 4, +(140) + +30 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +and hence ξ +� +|z13|2+|z23|2 +n +� += 0 by the definition. As a conclusion, we derive +sup +x∈X +����(1 + |z12|)(1 + |z13| + |z23|) · ∂χn +∂z12 +(x) +���� ≤ 16 +sup +r∈[0,+∞) +|ξ′(r)| , +(141) +which is a universal constant independent of n. Similarly, from the computation +∂2χn +∂z12∂z12 +(kx, z12, z13, z23) = ξ +�|z13|2 + |z23|2 +n +� +· +· +� +ξ′′ +�ln (|z12|2 + 1) +n +� +· +4z2 +12 +(|z12|2 + 1)2 · 1 +n2 + ξ′ +�ln (|z12|2 + 1) +n +� +· 2(1 − z2 +12) +(|z12|2 + 1)2 · 1 +n +� +, +(142) +we can conclude +sup +x∈X +���� +� +|z12|2 + 1 +� +· ∂2χn +∂z2 +12 +(x) +���� ≤ 4 +sup +r∈[0,+∞) +(|ξ′(r)| + |ξ′′(r)|) , +(143) +which is a universal constant independent of n. We thus establish (f). +As a conclusion, we complete the proof of Lemma 3.15. +Let {ψm}∞ +m=1 be a smooth approximate identity at the identity element Id of SL3(R) satis- +fying the following properties. +(1) ψm are smooth non-negative functions on SL3(R). +(2) ψm, are compactly supported in the open set Um ⊂ SL3(R), where ∩∞ +m=1Um = {Id}. +(3) +� +SL3(R) ψm(g) dg = 1, where dg is a fixed Haar measure on SL3(R). +For f ∈ L2(X), define convolutions ψm ∗ f by +(ψm ∗ f)(x) := +� +SL3(R) +ψm(g) · f +� +g−1x +� +dg, +m = 1, 2, · · · . +(144) +Lemma 3.16. The following holds for each f ∈ L2(X). +(a) ψm ∗ f are smooth functions on X for m = 1, 2, · · ·. +(b) ψm ∗ f ∈ L2(X) for m = 1, 2, · · ·, and ψm ∗ f → f in L2(X) as m → ∞. +(c) For each u ∈ U (sl3(R)), Lu(ψm ∗ f) ∈ L2(X). +Assume that ∆ is an SL3(R)-invariant differential operator on X, and that f, ∆f ∈ L2(X), +then +(d) ∆(ψm ∗ f) ∈ L2(X) for m = 1, 2, · · ·, and ∆(ψm ∗ f) → ∆f in L2(X) as m → ∞. +Proof of Lemma 3.16. +The arguments are standard and are given here for the sake of +completeness. For each u ∈ U(sl3(R)), it holds by definition that Lu(ψm ∗ f) = Lu(ψm) ∗ f. +This implies (a) since {Lu}u∈U(sl3(R)) provide all partial derivatives with respect to any local + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +31 +coordinates at each point of the manifold X. By the Cauchy-Schwarz inequality, +� +X +|(ψm ∗ f)(x)|2dµ = +� +X +���� +� +SL3(R) +ψm(g) · f +� +g−1x +� +dg +���� +2 +dµ +≤ +� +X + + + +� +SL3(R) +ψm (�g) d�g + + + + + + +� +SL3(R) +ψm(g) · +��f +� +g−1x +���2 dg + + + dµ = +� +X +|f (x)|2 dµ. +(145) +For each ǫ > 0, we can find continuous function fǫ with compact support such that +� +X +|fǫ (x) − f(x)|2 dµ < ǫ. +(146) +By the Cauchy-Schwarz inequality again, +� +X +|(ψm ∗ fǫ)(x) − fǫ(x)|2dµ = +� +X +���� +� +SL3(R) +ψm(g) +� +fǫ +� +g−1x +� +− fǫ(x) +� +dg +���� +2 +dµ +≤ +� +X +�� +SL3(R) +ψm (�g) d�g +� �� +SL3(R) +ψm(g) · +��fǫ +� +g−1x +� +− fǫ(x) +��2 dg +� +dµ += +� +SL3(R) +ψm(g) +�� +X +��fǫ +� +g−1x +� +− fǫ(x) +��2 dµ +� +dg ≤ sup +g∈Um +� +X +��fǫ +� +g−1x +� +− fǫ(x) +��2 dµ. +(147) +Since fǫ is uniform continuous with compact support, we conclude +lim +m→∞ +� +X +|(ψm ∗ fǫ)(x) − fǫ(x)|2dµ = 0. +(148) +Similarly to (145), we can derive by (146) +� +X +|(ψm ∗ f)(x) − f(x)|2dµ ≤ +� +X +|(ψm ∗ fǫ)(x) − fǫ(x)|2dµ + +� +X +|f (x) − fǫ(x)|2 dµ ++ +� +X +|(ψm ∗ (f − fǫ))(x)|2dµ ≤ +� +X +|(ψm ∗ fǫ)(x) − fǫ(x)|2dµ + 2ǫ. +(149) +Then, by(148) and (149), we have +lim +m→∞ +� +X +|(ψm ∗ f)(x) − f(x)|2dµ = 0. +(150) +(b) is now proved. Thanks to the fact that Lu(ψm ∗ f) = (Luψm) ∗ f, the same argument as in +(145) gives (c). +Next, assume f, ∆f ∈ L2(X). We first show that ∆(ψm ∗ f) = ψm ∗ (∆f) in the sense of +distribution. Recall the notion that {Xij}1≤i,j≤3, (i,j)̸=(3,3)is a basis of sl3(R) (see (11)) and the +lift �w of w ∈ C∞ (SL3(R)/A) is defined by �w := w ◦ π (see (3)). For clarity, in the following, +we denote the points of X by xA and the points of SL3(R) by x, g. + +32 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +For each invariant differential ∆, there exists a polynomial P∆ by Theorem 2.1 such that +(∆w)(xA) = P∆ +� +· · · , ∂ +∂tij +, · · · +� +�w + + + +x exp + + + + +� +1≤i,j≤3 +(i,j)̸=(3,3) +tijXij + + + + + + + + +�������� +t11=t12=···=t32=0 +=: P∆ +� ∂ +∂tij +� +�w +� +x exp +�� +tijXij +������ +tij=0 +, +w ∈ C∞ (SL3(R)/A) . +(151) +Let ∆∗ be the formal adjoint of ∆ (see Remark 2.7). Then ∆∗ is still an invariant differential +operator on X ([Ba]). For each h ∈ C∞ +c (X), +(∆(ψm ∗ f), h) = (ψm ∗ f, ∆∗h) += +� +X +�� +SL3(R) +ψm(g) · f +� +g−1xA +� +dg +� +P∆∗ +� ∂ +∂tij +� +�h +� +x exp +�� +tijXij +������ +tij=0 +dµ += +� +SL3(R) +ψm(g) +�� +X +f (xA) P∆∗ +� ∂ +∂tij +� +�h +� +gx exp +�� +tijXij +������ +tij=0 +dµ +� +dg += +� +X +f (xA) P∆∗ +� ∂ +∂tij +� �� +SL3(R) +ψ∗m(g)�h +� +g−1x exp +�� +tijXij +�� +dg +����� +tij=0 +. +(152) +where ψ∗ +m(g) := ψm(g−1). Noticing that ψ∗ +m ∗ �h = � +ψ∗ +m ∗ h, we have +� +X +f (xA) P∆ +� ∂ +∂tij +� � +� +ψ∗ +m ∗ h +� � +x exp +�� +tijXij +����� +tij=0 = (f, ∆∗(ψ∗ +m ∗ h)). +(153) +Substituting into (152), we have +(∆(ψm ∗ f), h) = (∆f, ψ∗ +m ∗ h) = +� +X +(∆f)(xA) · +�� +SL3(R) +ψm +� +g−1� +· h (g−1xA) dg +� +dµ += +� +X +�� +SL3(R) +ψm (�g) · (∆f) +� +�g−1�xA +� +d�g +� +· h (�xA) dµ = (ψm ∗ (∆f), h). +(154) +Similarly to (145), we have +|(∆(ψm ∗ f), h)| = |(ψm ∗ (∆f), h)| ≤ +� +X +|(∆f)(x)|2 dµ · +� +X +|h(x)|2 dµ. +(155) +Now ψm ∗ ∆f lies in L2(X) thanks to (b), we also have ∆(ψm ∗ f) = ψm ∗ ∆f in L2(X). Using +(b) again, ∆(ψm ∗f) = ψm ∗∆f → ∆f in L2(X) as m → ∞. We complete the proof of Lemma +3.16. +Now, we proceed to prove +Theorem 3.17. D12 is an SL3(R)-invariant, formally self-adjoint differential operator on X. +The graph of D12 on the domain C∞ +c (X) is dense in the graph of D12 on the domain Dom(D12), +with respect to the graph L2 × L2 norm. + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +33 +Proof of Theorem 3.17. Firstly, we show that D12 is formally self-adjoint. It suffices to +prove in each coordinate chart that D12 coincides with D∗ +12. This is can be done by a direct +computation, thanks to the invariant measure given by (100) and the explicit formula for D12 +given in Lemma 3.7 (or see Remark 2.7 for a coordinate-free approach). +For each f ∈ Dom(D12), consider functions χn · (ψm ∗ f), m, n = 1, 2, · · ·, where χn and +(ψm ∗ f) are defined by (132) and (144), respectively. It is clear that χn · (ψm ∗ f) ∈ C∞ +c (X) by +(a) in Lemma 3.15 and (a) in Lemma 3.16. Further, by (b) and (d) in Lemma 3.16, to prove +the density of C∞ +c (X) in Dom(D12), it suffices to prove that for fixed m = 1, 2, · · ·, +lim +n→∞ ∥χn · (ψm ∗ f) − ψm ∗ f∥L2(X) = 0, +(156) +and +lim +n→∞ ∥D12 (χn · (ψm ∗ f)) − D12(ψm ∗ f)∥L2(X) = 0. +(157) +(156) follows easily from (b) in Lemma 3.15, and Lebesgue’s dominated convergence theorem. +In the following, we shall establish (157) for m = 1, 2, · · ·. +By Lemma 3.14, computation yields that +D12 (χn · (ψm ∗ f)) = χn · D12 (ψm ∗ f) + + + +� +wj∈so3(R) +qjLwj +� ∂χn +∂z12 +� + · (ψm ∗ f) ++ + + +� +wj∈so3(R) +qjLwjχn + + · ∂ (ψm ∗ f) +∂z12 ++ ∂χn +∂z12 +· + + +� +wj∈so3(R) +qjLwj (ψm ∗ f) + + ++ +� +z2 +12 + 1 +� +∂2χn +∂z12∂z12 +· (ψm ∗ f) + 2 +� +z2 +12 + 1 +� ∂χn +∂z12 +· ∂ (ψm ∗ f) +∂z12 ++ +� +z23 +∂ +∂z13 +− z13 +∂ +∂z23 +� � ∂χn +∂z12 +� +· (ψm ∗ f) + +� +z23 +∂χn +∂z13 +− z13 +∂χn +∂z23 +� +· ∂ (ψm ∗ f) +∂z12 ++ ∂χn +∂z12 +· +� +z23 +∂ +∂z13 +− z13 +∂ +∂z23 +� +(ψm ∗ f) + 2z12 +∂χn +∂z12 +· (ψm ∗ f) . +(158) +Since Lwj, wj ∈ so3(R), are linear combinations of +∂ +∂α, +∂ +∂β, +∂ +∂γ, by (d) in Lemma 3.15 we have + + +� +wj∈so3(R) +qjLwj +� ∂χn +∂z12 +� + · (ψm ∗ f) = + + +� +wj∈so3(R) +qjLwjχn + + · ∂ (ψm ∗ f) +∂z12 +≡ 0. +(159) +Similarly, by (e) in Lemma 3.15, +� +z23 +∂ +∂z13 +− z13 +∂ +∂z23 +� � ∂χn +∂z12 +� +· (ψm ∗ f) = +� +z23 +∂χn +∂z13 +− z13 +∂χn +∂z23 +� +· ∂ (ψm ∗ f) +∂z12 +≡ 0. +(160) + +34 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +Therefore, +D12 (χn (ψm ∗ f)) = χnD12 (ψm ∗ f) + ∂χn +∂z12 + + +� +wj∈so3(R) +qjLwj (ψm ∗ f) + + ++ +� +z2 +12 + 1 +� +∂2χn +∂z12∂z12 +(ψm ∗ f) + 2 +� +z2 +12 + 1 +� ∂χn +∂z12 +∂ (ψm ∗ f) +∂z12 ++ ∂χn +∂z12 +· +� +z23 +∂ +∂z13 +− z13 +∂ +∂z23 +� +(ψm ∗ f) + 2z12 +∂χn +∂z12 +· (ψm ∗ f) +=: An + Bn + Cn + Dn + En + Fn. +(161) +In what follows, we shall prove that the functions Bn, Cn, Dn, En, Fn converge to 0 and An +converges to D12(ψm ∗ f) in L2(X), when n approaches ∞. Thanks to (f) in Lemma 3.15 and +the fact that qj are bounded functions on X, we have the pointwise estimate +|Bn(x)| ≤ + +M +sup +wj∈so3(R) +x∈X +|qj(x)| + + +� +wj∈so3(R) +��Lwj (ψm ∗ f) (x) +�� , +(162) +where the right hand side is an L2 function by (c) in Lemma 3.16. Then by (c) in Lemma 3.15, +we conclude by Lebesgue’s dominated convergence theorem that +lim +n→∞ ∥Bn∥L2(X) = 0. +(163) +Similarly, by (f) in Lemma 3.15, we have pointwisely +|Cn(x)| + |Fn(x)| ≤ 2M · |(ψm ∗ f) (x)| , +(164) +where the right hand side is an L2 function by (b) in Lemma 3.16. Then by (c) in Lemma 3.15, +we conclude by Lebesgue’s dominated convergence theorem that +lim +n→∞ ∥Cn∥L2(X) + lim +n→∞ ∥Fn∥L2(X) = 0. +(165) + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +35 +By Lemma 3.13, we have +� +z2 +12 + 1 +� ∂χn +∂z12 +∂ (ψm ∗ f) +∂z12 += ∂χn +∂z12 +· +� ∂ +∂z12 ++ z23 +∂ +∂z13 +� +(ψm ∗ f) − z23 +∂χn +∂z12 +∂ (ψm ∗ f) +∂z13 ++ z12 +∂χn +∂z12 +· +� +z12 +∂ +∂z12 +− z23 +∂ +∂z23 +� +(ψm ∗ f) + z12z23 +∂χn +∂z12 +∂ (ψm ∗ f) +∂z23 += ∂χn +∂z12 + + + + +� +v4,j∈U(sl3(R)) +are monomials +p4,jLv4,j (ψm ∗ f) + + + + − z23 +∂χn +∂z12 + + + + +� +v1,j∈U(sl3(R)) +are monomials +p1,jLv1,j (ψm ∗ f) + + + + ++ z12 +∂χn +∂z12 + + + + +� +v3,j∈U(sl3(R)) +are monomials +p3,jLv3,j (ψm ∗ f) + + + + + z12z23 +∂χn +∂z12 + + + + +� +v2,j∈U(sl3(R)) +are monomials +p2,jLv2,j (ψm ∗ f) + + + + . +(166) +Thanks to (f) in Lemma 3.15 and the fact that p1,j, p2,j, p3,j, p4,j are bounded functions on X, +|Dn(x)| ≤ + +M sup +1≤i≤4 +x∈X +|pi,j(x)| + + +4 +� +i=1 +� +vi,j∈U(sl3(R)) +are monomials +|Lvi,j (ψm ∗ f) (x)|. +(167) +Then by (c) in Lemma 3.15, we conclude by Lebesgue’s dominated convergence theorem that +lim +n→∞ ∥Dn∥L2(X) = 0. +(168) +Similarly, by Lemma 3.13, +∂χn +∂z12 +· +� +z23 +∂ +∂z13 +− z13 +∂ +∂z23 +� +(ψm ∗ f) = z23 +∂χn +∂z12 +· + + + + +� +v1,j∈U(sl3(R)) +are monomials +p4,jLv4,j (ψm ∗ f) + + + + +− z13 +∂χn +∂z12 +· + + + + +� +v2,j∈U(sl3(R)) +are monomials +p2,jLv2,j (ψm ∗ f) + + + + . +(169) +By (c), (f) in Lemma 3.15, and (c) in Lemma 3.16, we can conclude +lim +n→∞ ∥En∥L2(X) = 0. +(170) +Finally, we consider An. It is clear that +|An(x) − (D12 (ψm ∗ f)) (x)| = | (χn − 1) · (D12 (ψm ∗ f)) (x)| ≤ | (D12 (ψm ∗ f)) (x)|, +(171) +where the right hand side is an L2 function by (d) in Lemma 3.16. Then by (b) in Lemma 3.15, +lim +n→∞ ∥An − D12 (ψm ∗ f) ∥L2(X) = 0. +(172) + +36 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +As a conclusion, we complete the proof of Theorem 3.17. +Proof of Theorem 1.3. By Theorem 3.17, the graph of D12 on the domain C∞ +c (X) is dense +in the graph of D12 on the domain Dom(D12), with respect to the graph L2 × L2 norm. Then +D12 is essentially self-adjoint by Lemmas 3.1, 3.2. +Notice that the conjugations of the normalizer of A in SL3(R) (or equivalently the permuta- +tions of the indices 1, 2, 3) are isometries of L2(X), and such isometries transform D12 to D13 +and D23. Then, it is clear that D13 and D23 are essentially self-adjoint. +We complete the proof of Theorem 1.3. +Remark 3.18. In the proof of Theorem 3.17, one can see that our cutoff functions are specifi- +cally designed to treat the operator D12. Firstly, the first order derivative of our cutoff functions +with respect to the variable z12 contributes a decay of the order (z12 log z12)−1, which is better +than the more standard z−1 +12 decay. By stretching the cutoff functions in an inhomogeneous +way, one can balance a linear growth of the other variables z13, z23 exactly by the remaining +factor (log z12)−1, so that certain terms in D12 can be dealt with. Secondly, some other terms +in D12 force us to arrange the cutoff functions so that they are totally annihilated by those +differentials such as z23 +∂ +∂z13 −z13 +∂ +∂z23, in order to kill the wild terms such as +∂ +∂z12(ψm ∗f). Recall +that the singled out differential z23 +∂ +∂z13 − z13 +∂ +∂z23 has an intrinsic geometric explanation as the +rotation caused by an element in SO3(R). Because of these restrictions, the cutoff functions are +left with little wiggle room, and in particular when we perturb the cutoff functions for instance +by convolutions, they are no longer suitable for our task. Therefore, to deal with the cases D123 +and D213, one may want to decompose the operators in new geometric coordinates so that the +separation of terms is more transparent and suitable cutoff functions can be constructed. +Appendix A. Computations in the Euler-Iwasawa Coordinates +A.1. Explicit formulas for the generators of the left derivatives on SL3(R)/A. + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +37 +Lemma A.1. In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R), +− LX12 = +�1 +2 cos α cos β sin 2γ − sin α tan β sin2 γ − 1 +2 cos α sin β tan β sin 2γ +� ∂ +∂α ++ +� +−1 +2 sin α sin β sin 2γ − cos α sin2 β cos2 γ − cos α cos2 β sin2 γ +� ∂ +∂β ++ +� +− sin α sec β sin2 γ − 1 +2 cos α tan β sin 2γ + 1 +4 cos α sin 2β sin 2γ +� ∂ +∂γ ++ +��1 +2 sin α cos β sin 2γ + 1 +2 cos α sin 2β cos2 γ + 1 +2 cos α sin 2β +� +z12 ++ (− sin α sin β sin γ + cos α cos 2β cos γ) +� ∂ +∂z12 ++ +�� +sin α cos β sin 2γ + 1 +2 cos α sin 2β cos 2γ +� +z13 + (− sin α sin β sin γ ++ cos α cos 2β cos γ) z23 + +� +− sin α cos β cos 2γ + 1 +2 cos α sin 2β sin 2γ +�� +∂ +∂z13 ++ +��1 +2 sin α cos β sin 2γ − 1 +2 cos α sin 2β − 1 +2 cos α sin 2β sin2 γ +� +z23 ++ (sin α sin β cos γ + cos α cos 2β sin γ) +� ∂ +∂z23 +. +(173) +Proof of Lemma A.1. According to the definition, +(−LX12f)(α, β, γ, z12, z13, z23) = d +dt +� +f +� +�α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t) +������ +t=0 +, +(174) +where +� +�α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t) +� +are the Euler-Iwasawa coordinates of + + +1 +t +0 +0 +1 +0 +0 +0 +1 + + + + +cos β cos γ +− sin β +cos β sin γ +sin α sin γ + cos α cos γ sin β +cos α cos β +cos α sin β sin γ − cos γ sin α +cos γ sin α sin β − cos α sin γ +cos β sin α +cos α cos γ + sin α sin β sin γ + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + +=: + + +1 +t +0 +0 +1 +0 +0 +0 +1 + + + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + . +(175) +Applying the Gram-Schmidt process, we can obtain that, up to order 1 in t, + + +1 +t +0 +0 +1 +0 +0 +0 +1 + + + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + = + + + +�k11 +�k12 +�k13 +�k21 +�k22 +�k23 +�k31 +�k32 +�k33 + + + + + +1 +�z12 +�z13 +0 +1 +�z23 +0 +0 +1 + + + + +∆1 +0 +0 +0 +∆2 +0 +0 +0 +∆3 + + , (176) + +38 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +where +∆1 = 1 + t (k11k21) , ∆2 = 1 + t (k12k22) , ∆3 = 1 + t (k13k23) , +�k11 = k11 + t +� +k21 − k2 +11k21 +� +, �k21 = k21 + t +� +−k11k2 +21 +� +, �k31 = k31 + t (−k11k21k31) , +�k12 = k12 + t +� +k2 +13k22 − k11k12k21 +� +, +�k22 = k22 + t +� +−k12k2 +21 + k13k23k22 +� +, +�k32 = k32 + t (k12k23k33 − k11k22k31) , +�k13 = k13 + t +� +k2 +13k23 +� +, �k23 = k23 + t +� +−k13 + k13k2 +23 +� +, �k33 = k33 + t (k13k23k33) , +�z12 = (1 + t (k11k21 − k12k22)) z12 + t (k11k22 + k12k21) , +�z13 = (1 + t (k11k21 − k13k23)) z13 + t (k11k22 + k12k21) z23 + t (k11k23 + k13k21) , +�z23 = (1 + t (k12k22 − k13k23)) z23 + t (k12k23 + k13k22) . +(177) +Then for smooth functions f on X, we have by (98) that +(−LX12f)(α, β, γ, z12, z13, z23) = ∂f +∂α +d�α +dt (0) + ∂f +∂β +d�β +dt (0) + ∂f +∂γ +d�γ +dt (0) + ∂f +∂z12 +d�z12 +dt (0) ++ ∂f +∂z13 +d�z13 +dt (0) + ∂f +∂z23 +d�z23 +dt (0) += +�(k21k32 − k22k31)(k12k21 + k11k22) +k2 +22 + k2 +32 +� ∂f +∂α + +� +k11k12k21 − k2 +13k22 +� +1 − k2 +12 +� +∂f +∂β ++ +�−k13k21 − k11k12k13k22 +k2 +11 + k2 +13 +� ∂f +∂γ ++ ((k11k21 − k12k22) z12 + (k11k22 + k12k21)) ∂f +∂z12 ++ ((k11k21 − k13k23) z13 + (k11k22 + k12k21) z23 + (k11k23 + k13k21)) ∂f +∂z13 ++ ((k12k22 − k13k23) z23 + (k12k23 + k13k22)) ∂f +∂z23 +. +(178) +Substituting (97) into the above formula, we can derive (173). + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +39 +Lemma A.2. In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R), +− LX13 = +�1 +2 sin α cos β sin 2γ + cos α tan β sin2 γ − 1 +2 sin α sin β tan β sin 2γ +� ∂ +∂α ++ +�1 +2 cos α sin β sin 2γ − sin α sin2 β cos2 γ − sin α cos2 β sin2 γ +� ∂ +∂β ++ +�1 +4 sin α sin 2β sin 2γ − 1 +2 sin α tan β sin 2γ + cos α sec β sin2 γ +� ∂ +∂γ ++ +��1 +2 sin α sin 2β cos2 γ + 1 +2 sin α sin 2β − 1 +2 cos α cos β sin 2γ +� +z12 ++ (sin α cos 2β cos γ + cos α sin β sin γ) +� ∂ +∂z12 ++ +��1 +2 sin α sin 2β cos 2γ − cos α cos β sin 2γ +� +z13 + (sin α cos 2β cos γ ++ cos α sin β sin γ) z23 + +�1 +2 sin α sin 2β sin 2γ + cos α cos β cos 2γ +�� +∂ +∂z13 ++ +�� +−1 +2 sin α sin 2β − 1 +2 sin α sin 2β sin2 γ − 1 +2 cos α cos β sin 2γ +� +z23 ++ (sin α cos 2β sin γ − cos α sin β cos γ) +� ∂ +∂z23 +. +(179) +Proof of Lemma A.2. Similarly, consider the Iwasawa decomposition + + +1 +0 +t +0 +1 +0 +0 +0 +1 + + + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + = + + + +�k11 +�k12 +�k13 +�k21 +�k22 +�k23 +�k31 +�k32 +�k33 + + + + + +1 +�z12 +�z13 +0 +1 +�z23 +0 +0 +1 + + + + +∆1 +0 +0 +0 +∆2 +0 +0 +0 +∆3 + + , +(180) +where (kij) ∈ SO3(R) is parametrized by (97). Then up to order 1 in t, we have +∆1 = 1 + t (k11k31) , ∆2 = 1 + t (k12k32) , ∆3 = 1 + t (k13k33) , +�k11 = k11 + t +� +k31 − k2 +11k31 +� +, �k21 = k21 + t (−k11k31k21) , �k31 = k31 + t +� +−k11k2 +31 +� +, +�k12 = k12 + t +� +k32k2 +13 − k11k12k31 +� +, �k22 = k22 + t (−k12k31k21 + k13k32k23) , +�k32 = k32 + t +� +−k12k2 +31 + k13k33k32 +� +, �k13 = k13 + t +� +k2 +13k33 +� +, +�k23 = k23 + t (k13k23k33) , �k33 = k33 + t +� +−k13 + k13k2 +33 +� +, +�z12 = (1 + t (k11k31 − k12k32)) z12 + t (k11k32 + k12k31) , +�z13 = (1 + t (k11k31 − k13k33)) z13 + t (k11k32 + k12k31) z23 + t (k11k33 + k13k31) , +�z23 = (1 + t (k12k32 − k13k33)) z23 + t (k12k33 + k13k32) . +(181) + +40 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +Computation yields that +(−LX13f)(α, β, γ, z12, z13, z23) = d +dt +� +f +� +�α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t) +������ +t=0 += +�(k21k32 − k22k31)(k12k31 + k11k32) +k2 +22 + k2 +32 +� ∂f +∂α + +� +k11k12k31 − k2 +13k32 +� +1 − k2 +12 +� +∂f +∂β ++ +�−k13k31 − k11k12k13k32 +k2 +11 + k2 +13 +� ∂f +∂γ + ((k11k31 − k12k32) z12 + (k11k32 + k12k31)) ∂f +∂z12 ++ ((k11k31 − k13k33) z13 + (k11k32 + k12k31) z23 + (k11k33 + k13k31)) ∂f +∂z13 ++ ((k12k32 − k13k33) z23 + (k12k33 + k13k32)) ∂f +∂z23 +. +(182) +Substituting (97) into the above formula, we can derive (179). +Lemma A.3. In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R), +− LX23 = +� +cos 2α cos2 γ − cos2 α + 1 +2 sin 2α sin β sin 2γ +� ∂ +∂α ++ +�1 +4 sin 2α sin 2β cos 2γ − 1 +2 cos 2α cos β sin 2γ +� ∂ +∂β + +� +−1 +4 sin 2α cos2 β sin 2γ +� ∂ +∂γ ++ +��1 +2 sin 2α sin2 β cos2 γ − 1 +2 sin 2α sin2 γ − 1 +2 sin 2α cos2 β − 1 +2 cos 2α sin β sin 2γ +� +z12 ++ +�1 +2 sin 2α sin 2β cos γ − cos 2α cos β sin γ +�� +∂ +∂z12 ++ +��1 +2 sin 2α cos 2γ + 1 +2 sin 2α sin2 β cos 2γ − cos 2α sin β sin 2γ +� +z13 ++ +�1 +2 sin 2α sin 2β cos γ − cos 2α cos β sin γ +� +z23 ++ +�1 +2 sin 2α sin 2γ + 1 +2 sin 2α sin2 β sin 2γ + cos 2α sin β cos 2γ +�� +∂ +∂z13 ++ +��1 +2 sin 2α cos2 β + 1 +2 sin 2α cos2 γ − 1 +2 sin 2α sin2 β sin2 γ − 1 +2 cos 2α sin β sin 2γ +� +z23 ++ +�1 +2 sin 2α sin 2β sin γ + cos 2α cos β cos γ +�� +∂ +∂z23 +. +(183) +Proof of Lemma A.4. For (kij) ∈ SO3(R), consider the Iwasawa decomposition + + +1 +0 +0 +0 +1 +t +0 +0 +1 + + + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + = + + + +�k11 +�k12 +�k13 +�k21 +�k22 +�k23 +�k31 +�k32 +�k33 + + + + + +1 +�z12 +�z13 +0 +1 +�z23 +0 +0 +1 + + + + +∆1 +0 +0 +0 +∆2 +0 +0 +0 +∆3 + + . +(184) + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +41 +Up to order 1 in t, +∆1 = 1 + t (k21k31) , ∆2 = 1 + t (k22k32) , ∆3 = 1 + t (k23k33) , +�k11 = k11 + t (−k21k31k11) , �k21 = k21 + t +� +k31 − k2 +21k31 +� +, �k31 = k31 + t +� +−k21k2 +31 +� +, +�k12 = k12 + t (k22k33k13 − k21k32k11) , +�k22 = k22 + t +� +k32k2 +23 − k21k22k31 +� +, +�k32 = k32 + t +� +−k22k2 +31 + k23k33k32 +� +, +�k13 = k13 + t (k23k33k13) , �k23 = k23 + t +� +k2 +23k33 +� +, �k33 = k33 + t +� +−k23 + k23k2 +33 +� +, +�z12 = (1 + t (k21k31 − k22k32)) z12 + t (k21k32 + k22k31) , +�z13 = (1 + t (k21k31 − k23k33)) z13 + t (k21k32 + k22k31) z23 + t (k21k33 + k23k31) , +�z23 = (1 + t (k22k32 − k23k33)) z23 + t (k22k33 + k23k32) . +(185) +Then by (98), +(−LX23f)(α, β, γ, z12, z13, z23) = d +dt +� +f +� +�α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t) +������ +t=0 += +�−k2 +22k2 +31 − k2 +22k2 +32 − k2 +23k2 +32 +k2 +22 + k2 +32 +� ∂f +∂α + +� +k21k32k11 − k22k33k13 +� +1 − k2 +12 +� +∂f +∂β ++ +� +−k11k13k22k32 +k2 +11 + k2 +13 +� ∂f +∂γ + ((k21k31 − k22k32) z12 + (k21k32 + k22k31)) ∂f +∂z12 ++ ((k21k31 − k23k33) z13 + (k21k32 + k22k31) z23 + (k21k33 + k23k31)) ∂f +∂z13 ++ ((k22k32 − k23k33) z23 + (k22k33 + k23k32)) ∂f +∂z23 +. +(186) +Substituting (97) into the above formula, we can derive (183). +Lemma A.4. In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R), +LX12 − LX21 = (sin α tan β) ∂ +∂α + (cos α) ∂ +∂β + (sin α sec β) ∂ +∂γ . +(187) +Proof of Lemma A.4. For (kij) ∈ SO3(R), we have up to order 1 in t + + + +�k11 +�k12 +�k13 +�k21 +�k22 +�k23 +�k31 +�k32 +�k33 + + + = + + +1 +−t +0 +t +1 +0 +0 +0 +1 + + + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + = + + +k11 − tk21 +k12 − tk22 +k13 − tk23 +k21 + tk11 +k22 + tk12 +k23 + tk13 +k31 +k32 +k33 + + . (188) +Then by (98), +(LX12 − LX21)f = +� −k12k32 +k2 +22 + k2 +32 +� ∂f +∂α + +� +k22 +� +1 − k2 +12 +� +∂f +∂β + +�k13k21 − k11k23 +k2 +11 + k2 +13 +� ∂f +∂γ . +(189) +Substituting (97) into the above formula, we can derive (187). + +42 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +Lemma A.5. In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R), +LX13 − LX31 = − (cos α tan β) ∂ +∂α + (sin α) ∂ +∂β − (cos α sec β) ∂ +∂γ . +(190) +Proof of Lemma A.5. For (kij) ∈ SO3(R), we have up to order 1 in t + + + +�k11 +�k12 +�k13 +�k21 +�k22 +�k23 +�k31 +�k32 +�k33 + + + = + + +1 +0 +−t +0 +1 +0 +t +0 +1 + + + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + = + + +k11 − tk31 +k12 − tk32 +k13 − tk33 +k21 +k22 +k23 +k31 + tk11 +k32 + tk12 +k33 + tk13 + + . (191) +Then by (98), +(LX13 − LX31)f = +� k12k22 +k2 +22 + k2 +32 +� ∂f +∂α + +� +k32 +� +1 − k2 +12 +� +∂f +∂β + +�k13k31 − k11k33 +k2 +11 + k2 +13 +� ∂f +∂γ . +(192) +Substituting (97) into the above formula, we can derive (190). +Lemma A.6. In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R), +LX23 − LX32 = ∂ +∂α. +(193) +Proof of Lemma A.6. For (kij) ∈ SO3(R), we have up to order 1 in t + + + +�k11 +�k12 +�k13 +�k21 +�k22 +�k23 +�k31 +�k32 +�k33 + + + = + + +1 +0 +0 +0 +1 +−t +0 +t +1 + + + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + = + + +k11 +k12 +k13 +k21 − tk31 +k22 − tk32 +k23 − tk33 +k31 + tk21 +k32 + tk22 +k33 + tk23 + + . (194) +We derive (193) by (98). +A.2. Explicit formulas for the generators of the left-invariant differentials on SL3(R). +In this subsection, we compute the left-invariant differential operators R (Eij), 1 ≤ i ̸= j ≤ 3 +on SL3(R), where Eij is the 3 × 3 matrix unit with a 1 in the ith row and jth column. +Recall the Euler-Iwasawa coordinates (α, β, γ, z12, z13, z23, λ1, λ2) of SL3(R) attached to k0 +(see Definition 3.5). Then, +Lemma A.7. In each coordinate chart +� +(k0 · U EI) × R3 × R2, �Λk0 +� +, +R (E12) (α, β, γ, z12, z13, z23, λ1, λ2) = λ1 +λ2 +∂ +∂z12 +. +(195) +Proof of Lemma A.7. For matrix-valued functions g := � +1≤i,j≤3 gij(t)Eij, write g = O(2) if +dgij +dt (t) +���� +t=0 += 0, 1 ≤ i, j ≤ 3. +(196) + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +43 +Computation yields that for each k0 ∈ SO3(R), +k0 · + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + + +1 +t +0 +0 +1 +0 +0 +0 +1 + + += k0 · + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 + λ1 +λ2t +z13 +0 +1 +z23 +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + O(2). +(197) +For smooth functions f on SL3(R), we can derive that +(R (E12) f) (α, β, γ, z12, z13, z23, λ1, λ2) += d +dt +� +f +� +α, β, γ, z12 + λ1 +λ2 +t, z13, z23, λ1, λ2 +������ +t=0 += λ1 +λ2 +∂ +∂z12 +. +(198) +We complete the proof of Lemma A.7. +Lemma A.8. In each coordinate chart +� +(k0 · U EI) × R3 × R2, �Λk0 +� +, +R (E13) (α, β, γ, z12, z13, z23, λ1, λ2) = λ2 +1λ2 +∂ +∂z13 +. +(199) +Proof of Lemma A.8. The proof is similar. For each k0 ∈ SO3(R), computation yields +k0 · + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + + +1 +0 +t +0 +1 +0 +0 +0 +1 + + += k0 · + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 + λ2 +1λ2t +0 +1 +z23 +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + O(2). +(200) +Then, +R (E13) f = d +dt +� +f +� +α, β, γ, z12, z13 + λ2 +1λ2t, z23, λ1, λ2 +������ +t=0 += λ2 +1λ2 +∂f +∂z13 +. +(201) +We complete the proof of Lemma A.8. +Lemma A.9. In each coordinate chart +� +(k0 · U EI) × R3 × R2, �Λk0 +� +, +R (E23) (α, β, γ, z12, z13, z23, λ1, λ2) = λ1λ2 +2 +� ∂ +∂z23 ++ z12 +∂ +∂z13 +� +. +(202) + +44 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +Proof of Lemma A.9. The proof is similar. For each k0 ∈ SO3(R), computation yields +k0 · + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + + +1 +0 +0 +0 +1 +t +0 +0 +1 + + += k0 · + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 + λ1λ2 +2z12t +0 +1 +z23 + λ1λ2 +2t +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + O(2). +(203) +Then, +R (E13) f = d +dt +� +f +� +α, β, γ, z12, z13 + λ1λ2 +2z12t, z23 + λ1λ2 +2t, λ1, λ2 +������ +t=0 += λ1λ2 +2 +� ∂ +∂z23 ++ z12 +∂ +∂z13 +� +. +(204) +We complete the proof of Lemma A.9. +Lemma A.10. In each coordinate chart +� +(k0 · U EI) × R3 × R2, �Λk0 +� +, +R (E21) (α, β, γ, z12, z13, z23, λ1, λ2) = λ2 +λ1 +� +sec β sin γ ∂ +∂α + cos γ ∂ +∂β + tan β sin γ ∂ +∂γ ++ +� +z2 +12 + 1 +� +∂ +∂z12 ++ z23 +∂ +∂z13 +− z13 +∂ +∂z23 +� ++ λ2z12 +∂ +∂λ1 +− λ2 +2 +λ1 +z12 +∂ +∂λ2 +. +(205) +Proof of Lemma A.10. The proof is similar. For each k0 ∈ SO3(R), computation yields +k0 · K · Z · Λ(t) := + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + + +1 +0 +0 +t +1 +0 +0 +0 +1 + + = O(2)+ ++ k0 · + + + +�k11 +�k12 +k13 +�k21 +�k22 +k23 +�k31 +�k32 +k33 + + + + + +1 +z12 + λ2 +λ1 +� +z2 +12 + 1 +� +t +z13 + λ2 +λ1 z23t +0 +1 +z23 − λ2 +λ1 z13t +0 +0 +1 + + + + + +λ1 + λ2z12t +0 +0 +0 +λ2 − λ2 +2 +λ1 z12t +0 +0 +0 +λ−1 +1 λ−1 +2 + + + , +(206) +where +�k11 = k11 + λ1 +λ2 +k12t, +�k21 = k21 + λ1 +λ2 +k22t, +�k31 = k31 + λ1 +λ2 +k32t, +�k12 = k12 − λ1 +λ2 +k11t, +�k22 = k22 − λ1 +λ2 +k21t, +�k32 = k32 − λ1 +λ2 +k31t. +(207) +Denote by +� +�α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t), �λ1(t), �λ2(t) +� +the corresponding Euler-Iwasawa +coordinates of k0 · K · Z · Λ(t). By (98) we have up to order 1 in t, +�α = arctan +� +k32 − λ1 +λ2 k31t +k22 − λ1 +λ2 k31t +� +, �β = arctan + + + + +− +� +k12 − λ1 +λ2 k11t +� +� +1 − +� +k12 − λ1 +λ2 k11t +�2 + + + + , �γ = arctan +� +k13 +k11 + λ1 +λ2 k12t +� +. +(208) + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +45 +We can thus complete the proof of Lemma A.10 by direct computation. +Lemma A.11. In each coordinate chart +� +(k0 · U EI) × R3 × R2, �Λk0 +� +, +R (E31) (α, β, γ, z12, z13, z23, λ1, λ2) = λ−2 +1 λ−1 +2 +� +((sec β sin γ) z23 − (sec β cos γ) z12) ∂ +∂α ++ ((cos γ) z23 + (sin γ) z12) ∂ +∂β + ((tan β sin γ) z23 − (tan β cos γ) z12 − 1) ∂ +∂γ ++ +� +z23 + z2 +12z23 +� +∂ +∂z12 ++ +� +1 + z2 +13 + z2 +23 − z12z13z23 +� +∂ +∂z13 ++ +� +−z12 − z12z2 +23 +� +∂ +∂z23 +� ++ λ−1 +1 λ−1 +2 z13 +∂ +∂λ1 +− λ−2 +1 z12z23 +∂ +∂λ2 +. +(209) +Proof of Lemma A.11. Computation yields +K · Z · Λ(t) := + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + + +1 +0 +0 +0 +1 +0 +t +0 +1 + + += + + +k11 + λ−2 +1 λ−1 +2 +(k12z23 + k13) t +k12 + λ−2 +1 λ−1 +2 +(k11z23 + k13z12) t +k13 + λ−2 +1 λ−1 +2 +(k12z12 − k11) t +k21 + λ−2 +1 λ−1 +2 +(k22z23 + k23) t +k22 + λ−2 +1 λ−1 +2 +(k21z23 + k23z12) t +k23 + λ−2 +1 λ−1 +2 +(k22z12 − k21) t +k31 + λ−2 +1 λ−1 +2 +(k32z23 + k33) t +k32 + λ−2 +1 λ−1 +2 +(k31z23 + k33z12) t +k33 + λ−2 +1 λ−1 +2 +(k32z12 − k31) t + + +· + + +1 +z12 + λ−2 +1 λ−1 +2 +� +z23 + z2 +12z23 +� +t +z13 + λ−2 +1 λ−1 +2 +� +1 + z2 +13 + z2 +23 − z12z13z23 +� +t +0 +1 +z23 − λ−2 +1 λ−1 +2 +� +z12 + z12z2 +23 +� +t +0 +0 +1 + + +· + + +λ1 + λ−1 +1 λ−1 +2 z13t +0 +0 +0 +λ2 − λ−2 +1 z12z23t +0 +0 +0 +λ−1 +1 λ−1 +2 +− λ−3 +1 λ−2 +2 (z13 − z12z23)t + + . +(210) +Denote by +� +�α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t), �λ1(t), �λ2(t) +� +the corresponding Euler-Iwasawa +coordinates. By (98) we have up to order 1 in t, +�α(t) = arctan +�k32 + λ−2 +1 λ−1 +2 (k31z23 + k33z12) t +k22 + λ−2 +1 λ−1 +2 (k21z23 + k23z12) t +� +, +�β(t) = arctan + + +− +� +k12 + λ−2 +1 λ−1 +2 (k11z23 + k13z12) t +� +� +1 − +� +k12 + λ−2 +1 λ−1 +2 (k11z23 + k13z12) t +�2 + + , +�γ(t) = arctan +�k13 + λ−2 +1 λ−1 +2 (k12z12 − k11) t +k11 + λ−2 +1 λ−1 +2 (k12z23 + k13) t +� +. +(211) +Lemma A.11 follows from a direct computation. + +46 +HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG +Lemma A.12. In each coordinate chart +� +(k0 · U EI) × R3 × R2, �Λk0 +� +, +R (E32) (α, β, γ, z12, z13, z23, λ1, λ2) = λ−1 +1 λ−2 +2 +� +sec β cos γ ∂ +∂α − sin γ ∂ +∂β + tan β cos γ ∂ +∂γ ++ (z13 − z12z23) +∂ +∂z12 ++ z13z23 +∂ +∂z13 ++ +� +z2 +23 + 1 +� +∂ +∂z23 +� ++ λ−1 +1 λ−1 +2 z23 +∂ +∂λ2 +. +(212) +Proof of Lemma A.12. Computation yields +K · Z · Λ(t) := + + +k11 +k12 +k13 +k21 +k22 +k23 +k31 +k32 +k33 + + + + +1 +z12 +z13 +0 +1 +z23 +0 +0 +1 + + + + +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ−1 +1 λ−1 +2 + + + + +1 +0 +0 +0 +1 +0 +0 +t +1 + + += + + +k11 +k12 + λ−1 +1 λ−2 +2 k13t +k13 − λ−1 +1 λ−2 +2 k12t +k21 +k22 + λ−1 +1 λ−2 +2 k23t +k23 − λ−1 +1 λ−2 +2 k22t +k31 +k32 + λ−1 +1 λ−2 +2 k33t +k33 − λ−1 +1 λ−2 +2 k32t + + + + +1 +z12 + λ−1 +1 λ−2 +2 +(z13 − z12z23) t +z13 + λ−1 +1 λ−2 +2 z13z23t +0 +1 +z23 + λ−1 +1 λ−2 +2 +� +z2 +23 + 1 +� +t +0 +0 +1 + + +· + + +λ1 +0 +0 +0 +λ2 + λ−1 +1 λ−1 +2 z23t +0 +0 +0 +λ−1 +1 λ−1 +2 +− λ−2 +1 λ−3 +2 z23t + + . +(213) +Denote by +� +�α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t), �λ1(t), �λ2(t) +� +the corresponding Euler-Iwasawa +coordinates of K · Z · Λ(t). By (98) we have up to order 1 in t, +�α = arctan +�k32 + λ−1 +1 λ−2 +2 k33t +k22 + λ−1 +1 λ−2 +2 k23t +� +, �β = arctan + + +− +� +k12 + λ−1 +1 λ−2 +2 k13t +� +� +1 − +� +k12 + λ−1 +1 λ−2 +2 k13t +�2 + + , +�γ = arctan +�k13 − λ−1 +1 λ−2 +2 k12t +k11 +� +. +(214) +Lemma A.12 follows from a direct computation. +References +[AV] Andreotti, A., and Vesentini, E., Carleman estimates for the Laplace-Beltrami equation on complex man- +ifolds, Inst. Hautes ´Etudes Sci. Publ. Math. No. 25 (1965), 81-130. +[Ba] van den Ban, E. P., Invariant differential operators on a semisimple symmetric space and finite multiplic- +ities in a Plancherel formula, Ark. Mat. 25 (1987), no. 2, 175–187. +[BH] Bopp, N., and Harinck, P., Formule de Plancherel pour GL(n, C)/U(p, q), J. Reine Angew. Math. 428 +(1992), 45-95. +[BIK] Benoist, Y., Inoue, Y., and Kobayashi, T., Temperedness criterion of the tensor product of parabolic +induction for GLn, J. 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Groups 5 (2000), no. 2, +181-204. +[W] Wolf, J., Essential self-adjointness for the Dirac operator and its square, Indiana Univ. Math. J. 22 +(1972/73), p. 611-640. + +HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE +49 +H. Fang, School of Mathematical Sciences, Peking University, Beijing, Beijing, 100871, China. +(hlfang@pku.edu.cn) +X. Li, Beijing International Center for Mathematical Research, Beijing, Beijing, 100871, China. +(lixiaocheng@bicmr.pku.edu.cn) +Y. Zhang, School of Mathematical Sciences, Peking University, Beijing, Beijing, 100871, China. +(yunfengzhang108@gmail.com) + diff --git a/6tAyT4oBgHgl3EQfpvgE/content/tmp_files/load_file.txt b/6tAyT4oBgHgl3EQfpvgE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3de7a5067ed0be0a7938c6c94acc2ab23845d5a --- /dev/null +++ b/6tAyT4oBgHgl3EQfpvgE/content/tmp_files/load_file.txt @@ -0,0 +1,2367 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf,len=2366 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='00529v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='RT] 2 Jan 2023 HARMONIC ANALYSIS ON THE FOURFOLD COVER OF THE SPACE OF ORDERED TRIANGLES I: THE INVARIANT DIFFERENTIALS HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Denote by SLn(R) the group of n × n real matrices with determinant one, A the subgroup consisting of the diagonal matrices with positive entries, and SLn(R)/A the manifold of left cosets gA, g ∈ SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In this paper, we will be concerned with the harmonic analysis on the homogeneous space SLn(R)/A when n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In particular, we provide explicit generators and their relations for the algebra of the invariant differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Then we prove that some of the non-central generators are essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Structure of the Algebra of the Invariant Differentials 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Generators of the invariant differentials on SLn(R)/A 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Relations among the generators 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The center of the algebra of the invariant differentials 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Essential Self-Adjointness 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Essential self-adjointness of the central elements 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Reduction to the density of C∞ c (X) in Dom(∆) 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Coordinate charts induced by the Euler angles 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Presence of left derivatives in D12 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Proof of the density and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='3 28 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Computations in the Euler-Iwasawa Coordinates 36 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Explicit formulas for the generators of the left derivatives on SL3(R)/A 36 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Explicit formulas for the generators of the left-invariant differentials on SL3(R) 42 References 46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Introduction Denote by SLn(R) the group of n × n real matrices with determinant one, A the subgroup consisting of the diagonal matrices with positive entries, and SLn(R)/A the manifold of left cosets gA, g ∈ SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In this paper, we will be concerned with the harmonic analysis on the homogeneous space SLn(R)/A when n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In particular, we will restrict attention to the commutation relations and essential self-adjointness of the invariant differential operators 1 2 HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG on SL3(R)/A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' spectral decomposition of the invariant differential operators and its interaction with Plancherel theorems are left for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The space SL3(R)/A distinguishes itself among homogeneous spaces in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Firstly, it has a natural interpretation as the fourfold cover of the space of nondegenerate Schubert triangles in the plane ([Sc]), of which the compactification is well studied as of a homogeneous space of complexity 1 ([Sem], [Ti]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' One may wish to investigate the Plancherel type formulas (see [GGV], [GG] for the relation with the Radon transforms), via the Bernstein maps and the Maass-Selberg relations (see [SV], [DKKS] for the spherical case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Secondly, SL3(R)/A is one of the simplest examples of non-spherical homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' These spaces exhibit a very different nature compared with spherical ones, and very little of their harmonic theory is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' For instance, the finite multiplicity theorem for induction does not hold anymore ([KO]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' the algebra of invariant differential operators is a noncommutative algebra instead of a polynomial ring ([Kn]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' the representation theory of the ring of bi-invariant functions is mysterious as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' For our case, at least one knows, a priori by the systematic work of Benoist-Kobayashi ([BK1], [BK2], [BK3], [BK4], [BIK]), that the natural unitary representation of SLn(R) in L2 (SLn(R)/A) is tempered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' From the technical perspective, the investigation of the spectral theory of pseudo-Riemannian manifolds is challenging, for the traditional elliptic theory is not applicable to the Laplace- Beltrami operators with mixed signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' To remedy the situation, the analytic methods are always fused with the peculiar geometry of the underlying manifold related to the representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The considerable efforts have been made mainly on the symmetric spaces, such as the group manifolds ([HC1], [HC2]), the real hyperboloids ([RLN], [LNR1], [LNR2], [Sh], [St], [Ros], [Fa], [Sek], [Mo]), special symmetric spaces ([Ma], [Sa], [DP], [KD], [BH], [Ha]), and general symmetric spaces ([FJ], [OM], [OS], [Ba], [O], [De], [BS1], [BS2]), and, more recently, certain locally symmetric spaces and spherical varieties ([KK1], [KK2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In the present situation, SL3(R)/A has an inherent approachable geometry in spite of the rather involved analysis due to its non-sphericity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' For instance, the underlying manifold is diffeomorphic to SO3(R) × R3, so that an extensive calculation is possible, just as the hyperboloid case, where the usage of the spherical coordinates of the underlying product space Sp−1 × Sq−1 × R is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Before describing our results in more detail, we first set certain notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' To treat in a unified fashion, let G denote SL3(R), g the Lie algebra of G, and U(g) the universal enveloping algebra of the complexification of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Denote by C∞(G) the space of complex-valued smooth functions on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Then, the infinitesimal action R on C∞(G) induced by the right regular representation of G, maps U(g) into the algebra of algebraic differentials on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' More precisely, R acts on u = X1X2 · · · Xk ∈ U(g) by (Ruf) (g) := (R (X1 · · · Xk) f) (g) := ∂ ∂t1 ���� t1=0 · · ∂ ∂tk ���� tk=0 f(g exp (t1X1) · · · exp (tkXk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' (1) Here X1, X2, · · · , Xk ∈ g, and f ∈ C∞(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' the exponential map is given by exp(X) := γ(1), where γ : R → G is the one-parameter subgroup of G whose tangent vector at the identity is equal to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' It is easy to verify that Ru, u ∈ U(g), is a left G-invariant differential operators on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Denote by D(G) the algebra of the left G-invariant differential operators on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE 3 For a closed subgroup H ⊂ G, denote by D(G/H) the algebra of G-invariant differential operators on the homogeneous space G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Denote by π : G → G/H the natural projection, and h the Lie algebra of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Define DH(G) := {D ∈ D(G) | D(f ◦ Rh) ◦ R−1 h = Df, ∀h ∈ H and f ∈ C∞(G)}, (2) where Rh : g �→ gh is the right translation of G for h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Assuming G and H are reductive, we have the standard isomorphism (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='6 in Chapter 2 of [He]) DH(G)/ � DH(G) ∩ D(G)h � ∼= D(G/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' (3) It is induced by the map µ : DH(G) → D(G/H), such that for each D ∈ DH(G), µ(D) is the element of D(G/H) such that (µ(D)f) ◦ π = D(f ◦ π) for all smooth functions f on G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' (4) Denote by Eij the 3 × 3 matrix unit with a 1 in the ith row and jth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' For distinct i, j, k ∈ {1, 2, 3}, define differential operators on SL3(R)/A Dij = µ � 1 2 � σ∈S2 R � Eσ(i)σ(j)Eσ(j)σ(i) � � , (5) and Dijk = µ � 1 6 � σ∈S3 R � Eσ(i)σ(j)Eσ(j)σ(k)Eσ(k)σ(i) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' (6) Note that Dij = Dji and Dijk = Djki = Dkij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In the first part of the paper, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' D (SL3(R)/A) is the noncommutative associative algebra generated over C by {D12, D13, D23, D123, D213} with relations \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 [D123, D213] = 0, [Dij, Dik] = Dijk − Dikj, i, j, k ∈ {1, 2, 3} are distinct, [Dijk, Dij] = DjkDij − DijDik, i, j, k ∈ {1, 2, 3} are distinct, 2 (D123D213 + D213D123 − D12D23D31 − D13D32D21) = (D23 − D13 − D12)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' (7) The center of D (SL3(R)/A) is a polynomial ring in D123 + D213 and D12 + D23 + D13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In the theory of harmonic analysis on homogeneous spaces, one of the central problems is whether a symmetric invariant differential operator has a unique self-adjoint extension, as it would enable a simultaneous study of spectral decomposition of both the regular representation and the invariant differential operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' see [BS1], [BS2] for the case of reductive symmetric spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' There is considerable literature on essential self-adjointness for natural operators on a complete Riemannian or Hermitian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' See for instance [Ga1], [Ga2], [Ga3], [Roe], [Co], [Ri] for the Hodge-Laplace-Beltrami operator, [AV] for the ¯∂-Laplacian, [P], [W] for the Dirac operator, and [Ch] for certain first order differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' For differential operators on Rn, much more is known ([RS]), and we mention here that certain ellipticity ([IK]), or semi- boundedness together with temperedness ([Dev]), guarantees the essential self-adjointness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' 4 HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG For general homogeneous spaces, a classical result shows that the symmetric elements in the image of the center of the universal enveloping algebra are essentially self-adjoint ([Seg], [NS], or [Th]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' As it can be shown that the center of D (SL3(R)/A) equals the image of the center of U(sl3(R)), it follows Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Every symmetric differential operator in the center of D (SL3(R)/A) is es- sentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' For non-central elements, the most general result is due to Van den Ban [Ba], who established the essential self-adjointness of the symmetric invariant differential operators for semi-simple symmetric pairs, even if it is not a generalized Gelfand pair, semiboundedness is absent, or the underlying manifold is non-Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Beyond that, to the best of our knowledge, there is no general theory ensuring the essential self-adjointness in the pseudo-Riemannian setting, even for the Laplacian operators (see [KK1], [KK2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The major part of the paper is to devoted to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The differential operators D12, D13, D23 on SL3(R)/A are essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' We now briefly describe the basic ideas for the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' We exploit the universal enveloping algebra to extract the algebraic structure of D(SL3(R)/A), and the normal form theory in [FH] to determine the center of D(SL3(R)/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' To study the essential self-adjointness of symmetric operators, we modify the scheme of [Ba].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The elegant proof in [Ba] is to decompose the differential operator into a bounded sum of left derivatives so that the wild growth of the coefficients can be treated as bounded ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Unfortunately, in this non-spherical case, the left derivatives are too degenerate to span the whole space of invariant differentials in a mild way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' We make the observation that by choosing the cutoff functions and the mollifiers in a compatible way instead of separating the G˚arding type space as the operator core, one may gain extra decays to balance the extraordinary growth of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In fact, the chosen cutoff functions are annihilated by the wildest terms and contribute desired decays thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The organization of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' §2 is devoted to the algebraic structure of D(SL3(R)/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' We study the presence of left derivatives in the coordinate form of D12 in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='5, we establish the density of C∞ c (SL3(R)/A) in Dom(D12), and, as a consequence, prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The explicit formulas for the generators of the left derivatives and of the left invariant differentials are given in Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The authors appreciate greatly Professors N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Xu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Yu for many helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' This research is supported by National Key R& D Pro- gram of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' 2022YFA1006700).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The first author is partially supported by NSFC grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' 12201012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Structure of the Algebra of the Invariant Differentials 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Generators of the invariant differentials on SLn(R)/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Let S(g) be the symmetric algebra over g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Then for a basis {X1, · · · , Xn} of g, S(g) can be identified with the algebra of polynomials � (k1,··· ,kn)∈Nn ak1···knXk1 1 · · · Xkn n , ak1···kn ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' (8) HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE 5 We have the following symmetrizer map λ : S(g) → D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='1 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='3 in Chapter 2 of [He]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' There is a unique linear bijection λ from S(g) to D(G) such that λ(Xm) = R(Xm) for X ∈ g and m ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' More precisely, (λ(P)f)(g) := P � ∂ ∂t1 , · · · , ∂ ∂tn � f (g exp(t1X1 + · · · + tnXn)) ���� t1=···=tn=0 , (9) for P ∈ S(g) and f ∈ C∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' In particular (see Page 282 of [He]), λ(Y1 · · · Yk) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' � σ∈Sk R � Yσ(1) · · · Yσ(k) � , (10) where Y1, · · · , Yk ∈ g, and Sk is the symmetric group of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Denote by Eij the n × n matrix unit with a 1 in the ith row and jth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Define Xij := Eij, 1 ≤ i ̸= j ≤ n, Xll := Ell − Enn, 1 ≤ l ≤ n − 1, (11) which constitute a basis of sln(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' The algebra D(SLn(R)/A) is generated by � (µ ◦ λ) � Ei1i2Ei2i3 · · · Eik−1ikEiki1 � ���� 2 ≤ k ≤ n, 1 ≤ i1, i2, · · · , ik ≤ n i1, i2, · · · , ik are distinct � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' (12) Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' By (3), the invariant differential operators on SLn(R)/A are induced from the left SLn(R) and right A invariant differential operators on SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Then by (3), it suffices to prove that DA(SLn(R)) is generated by the elements in (12) and D(SLn(R))a, where a is the Lie algebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Take D ∈ DA(SLn(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' There is a polynomial PD such that λ(PD) = D by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' Arrange PD in the following lexicographic order, PD = ∞ � k=1 � 1≤i1≤i2≤···≤it≤···≤ik≤n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' 1≤jt≤n and (it,jt)̸=(n,n) for 1≤t≤k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'} +page_content=' if 1≤u 0 depending +only on the quantitative data. +2. The proof of main result +2.1. Interior regularity. Continuity of weak solutions for (1.1) was first obtained in [2]. +But for boundary continuity, we need the stronger interior H¨older regularity. We first +recall the following interior H¨older regularity result for (1.1) from [5, Theorem 1.3] or [6, +Theorem 1.1]. +Theorem 2.1 (Interior Regularity). Suppose u ∈ W k,2(B2k, Rm) is a solution of (1.1) with +(1.2) and (1.3). Then there exist α ∈ (0, 1), C > 0 and r0, depending only on k, m and +the data from (1.2) and (1.3), such that u is locally α-H¨older continuous and +oscBr(x)u ≤ Crα∥u∥W k,2(B2k) + +BOUNDARY REGULARITY +3 +for all x ∈ B 1 +4(0) and all 0 < r < r0. +We shall use the following version of Theorem 2.1 in our later proofs. There exists +R0 > 0 sufficiently small, such that, if x ∈ Ω and 0 < r < min{R0, dist(x, ∂Ω)/4}, then +u ∈ C0,α(Br(x0), Rm) for some α ∈ (0, 1) and +(2.1) +oscBτr(x)u ≲ τ α∥u∥W k,2(B4r(x0)) +for 0 < τ ≤ 1. +By [6, Theorem 1.1], one can indeed infer that (2.1) holds for all α ∈ (0, 1). But for our +purpose, the current estimate is sufficient. +2.2. Boundary Maximum Principle. Another ingredient for boundary regularity is a +boundary maximum principle, originally discovered by Qing [10] in his proof of boundary +regularity for weakly harmonic maps and was later adapted to the polyharmonic case in +Lamm-Wang [8]. +For x ∈ Ω and R > 0, denote by ΩR(x) = Ω ∩ BR(x). We shall prove the following +boundary maximal principle for solutions of (1.1). +Proposition 2.2 (Boundary Maximum Principle). There exists a constant C > 0 such that +for any x ∈ Ω and 0 < R < R0/4, for any q ∈ Rm, there holds +(2.2) +max +ΩR(x) |u − q| ≤ C +� +max +∂ΩR(x) |u − q| + ∥u∥W k,2(Ω4R(x)) +� +. +To prove Proposition 2.2, we need the following version of Courant-Lebesgue lemma, +which was essentially established in [8]. Since the formulation is slightly different from +there, for the convenience of the readers, we recall the proofs here. +Lemma 2.3. There exists C > 0 such that for any R > 0 and any x0 ∈ Ω, there exists +R1 ∈ (R, 2R) so that +osc∂BR1(x0)∩Ωu ≤ C∥u∥W k,2(Ω4R(x0)). +Proof. For x ∈ B2R(x0), set r = |x − x0| ∈ [0, 2R]. By Fubini’s theorem, we have +� +Ω2R(x0) +|∇u|2kdx ≥ +� 2R +R +dr +� +∂Br(x0)∩Ω +|∇Tu|2kdH2k−1 +≥ +�� 2R +R +1 +rdr +� +inf +R≤r≤2R +� +r +� +∂Br(x0)∩Ω +|∇Tu|2kdH2k−1 +� += ln 2 +inf +R≤r≤2R +� +r +� +∂Br(x0)∩Ω +|∇Tu|2kdH2k−1 +� +, +where ∇T denotes the gradient operator on ∂Br(x0) and dH2k−1 is the volume element +on ∂Br(x0). Then there exists R1 ∈ (R, 2R) such that +R1 +� +∂BR1(x0)∩Ω +|∇Tu|2kdH2k−1 ≤ +1 +ln 2 +� +Ω2R(x0) +|∇u|2kdx. +Hence u(R1, ·) ∈ W 1,2k(∂BR1(x0) ∩ Ω, Rm) and the Sobolev embedding theorem implies +that u(R1, ·) ∈ C +1 +2k (∂BR1(x0) ∩ Ω, Rm) and +osc∂BR1(x0)∩Ωu ≲ R1 +� +∂BR1(x0)∩Ω +|∇Tu|2kdH2k−1 ≲ ∥u∥W k,2(Ω4R(x0)). +□ + +4 +M.-L. LIU AND Y.-L. TIAN +Proof of Proposition 2.2. Denote M = max +ΩR(x) |u−q|, here q ∈ Rm is fixed. We may assume +that +(2.3) +M ≥ ∥u∥W k,2(ΩR(x)). +Choose x0 ∈ ΩR(x) such that +(2.4) +|u(x0) − q| ≥ 3 +4M. +Let r0 = dist(x0, ∂ΩR(x0)). Note that r0 ≤ R ≤ R0. Thus (2.1) implies that for any +r ∈ (0, r0 +4 ), we have +(2.5) +oscBr(x0)u ≤ C +� r +r0 +�α0 +∥u∥W k,2(ΩR(x)) ≤ CM +� r +r0 +�α0 +. +Pick r1 = r0/(4C)1/α in the above, and we obtain +oscBr1(x0)u ≤ 1 +4M +This together with (2.4) yields +(2.6) +inf +Br1(x0) |u − q| ≥ |u(x0) − q| − oscBr1(x0)u ≥ 1 +2M. +By Lemma 2.3, there exists r2 ∈ (r0, 2r0) such that +(2.7) +osc∂Br2(x0)∩ΩR(x)u ≤ C∥u∥W k,2(Ω4R(x)). +Note that ∂Br2(x0)∩∂ΩR(x) ̸= ∅. Using polar coordinates centered at x0, we estimate +inf +� +|u(r1, θ)−u(r2, θ)| : (ri, θ) ∈ ∂Bri(x0) ∩ ΩR(x), i = 1, 2 +� +≤C +� +S2k−1 dθ +� r2 +r1 +|ur|χ[r1,r2]×S2k−1(r, θ)dr +≤ C +r2k−1 +1 +� +S2k−1 dθ +� r2 +r1 +|ur|χ[r1,r2]×S2k−1(r, θ)r2k−1dr +≤ C +r2k−1 +1 +� +B2r0(x)∩ΩR(x) +|ur|dx +≤ C +r2k−1 +1 +|B2r0(x)| +2k−1 +2k +�� +ΩR(x) +|∇u|2kdx +� 1 +2k +≤C r2k−1 +0 +r2k−1 +1 +∥u∥W k,2(Ω4R(x)) ≤ C∥u∥W k,2(Ω4R(x)). +This implies that there exists θ0 ∈ ∂B1(x0) such that +(2.8) +|u(r1, θ0) − u(r2, θ0)| ≤ C∥u∥W k,2(Ω4R(x)). + +BOUNDARY REGULARITY +5 +Hence, by choosing an arbitrary x∗ ∈ ∂Br2(x0) ∩ ∂ΩR(x), we obtain from (2.6), (2.7) and +(2.8) that +M +2 ≤ +inf +Br1(x0) |u − q| ≤ |u(r1, θ0) − q| +≤|u(r1, θ0) − u(r2, θ0)| + |u(r2, θ0) − u(x∗)| + |u(x∗) − q| +≤C∥u∥W k,2(Ω4R(x)) + osc∂Br2(x0)∩ΩR(x)u + sup +∂ΩR(x) +|u − q| +≤C +� +sup +∂ΩR(x) +|u − q| + ∥u∥W k,2(Ω4R(x)) +� +. +The proof is complete. +□ +2.3. Proof of Theorem 1.1. Now we are ready to prove Theorem 1.1. +Proof of Theorem 1.1. Let x0 ∈ ∂Ω and take q = g(x0) = u(x0) in Proposition 2.2. Note +that +max +∂ΩR(x0) |u − u(x0)| ≤ +max +∂ΩR(x0)∩∂Ω |u − u(x0)| + osc∂ΩR(x0)∩Ωu += +max +∂ΩR(x0)∩∂Ω |g − g(x0)| + osc∂ΩR(x0)∩Ωu. +The first term tends to 0 as R → 0 since g ∈ C(∂Ω). The second term tends to 0 as +R → 0 by Lemma 2.3. This implies the continuity of u as desired. +□ +Remark 2.4. The proof above extends to solutions to the following inhomogeneous elliptic +system +(2.9) +∆ku = +k−1 +� +l=0 +∆l ⟨Vl, du⟩ + +k−2 +� +l=0 +∆lδ(wldu) + f +in Ω ⊂ R2k +with f ∈ Lp for some p > 1 and (1.2), (1.3). Indeed, by [6, Theorem 1.1], in this case, the +interior regularity estimate (2.1) becomes +(2.10) +oscBτr(x)u ≲ τ α � +∥u∥W k,2(B4r(x0)) + ∥f∥Lp(B4r(x0)) +� +for 0 < τ ≤ 1. +With this, the buondary maximal principle (2.2) remains valid with an extra term ∥f∥Lp(Ω4R(x)) +on the right hand side. The proof of Theorem 1.1 then works with obvious modifications. +Acknowledgements. The authors would like to Prof. Chang-Lin Xiang and Chang-Yu +Guo for posing this question to them and for many useful conservations. +References +[1] S.-Y.A. Chang, L. Wang and P.C. Yang, A regularity theory of biharmonic maps. Commun. +Pure Appl. Math. 52(9) (1999), 1113-1137. +[2] F.L. de Longueville and A. Gastel, Conservation laws for even order systems of polyharmonic +map type. Calc. Var. Partial Differential Equations 60, 138 (2021). +[3] A. Gastel and C. Scheven, Regularity of polyharmonic maps in the critical dimension. Comm. +Anal. Geom. 17 (2009), no. 2, 185-226. +[4] C.-Y. Guo and C.-L. Xiang, Regularity of solutions for a fourth order linear system via conser- +vation law. J. Lond. Math. Soc. (2) 101 (2020), no. 3, 907-922. +[5] C.-Y. Guo and C.-L. Xiang, Regularity of weak solutions to higher order elliptic systems in critical +dimensions. Tran. Amer. Math. Soc. 374 (2021), no. 5, 3579-3602. + +6 +M.-L. LIU AND Y.-L. TIAN +[6] C.-Y. Guo, C.-L. Xiang and G.-F. Zheng, Lp regularity theory for even order elliptic systems +with antisymmetric first order potentials., J. Math. Pures Appl. 165 (2022) 286-324. +[7] T. Lamm and T. Rivi`ere, Conservation laws for fourth order systems in four dimensions. Comm. +Partial Differential Equations 33 (2008), 245-262. +[8] T. Lamm and C.Y. Wang, Boundary regularity for polyharmonic maps in the critical dimension. +Adv. Calc. Var. 2 (2009), 1-16. +[9] F. M¨uller and A. Schikorra, Boundary regularity via Uhlenbeck-Rivi`ere decomposition. Analysis +(Munich) 29 (2009), 199-220. +[10] J. Qing, Boundary regularity of weakly harmonic maps from surfaces, J. Funct. Anal. 114 (1993) +458-466. +[11] T. Rivi`ere, Conservation laws for conformally invariant variational problems. Invent. Math. 168 +(2007), 1-22. +[12] C.Y. Wang, Stationary biharmonic maps from Rm into a Riemannian manifold. Comm. Pure Appl. +Math. 57 (2004), 419-444. +(Ming-Lun Liu) Research Center for Mathematics and Interdisciplinary Sciences, Shan- +dong University, Qingdao 266237, P. R. China and Frontiers Science Center for Nonlin- +ear Expectations, Ministry of Education, Qingdao, P. R. China +Email address: minglunliu2021@163.com +(Yao-Lan Tian) Center for Optics Research and Engineering, Shandong University, +Qingdao 266237, P. R. China +Email address: tianylbnu@126.com + diff --git a/9tAyT4oBgHgl3EQfqPjX/content/tmp_files/load_file.txt b/9tAyT4oBgHgl3EQfqPjX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a254b65d730114f46f245371bcdd9342a6d2b115 --- /dev/null +++ b/9tAyT4oBgHgl3EQfqPjX/content/tmp_files/load_file.txt @@ -0,0 +1,285 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf,len=284 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='00541v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='AP] 2 Jan 2023 BOUNDARY REGULARITY FOR AN EVEN ORDER ELLIPTIC SYSTEM IN THE CRITICAL DIMENSION MING-LUN LIU AND YAO-LAN TIAN* Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Given any continuous boundary data, we show that weak solutions are continuous up to boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Keywords: Polyharmonic maps, higher order elliptic system, Boudary continuity, Dirichlet prob- lem 2010 Mathematics Subject Classification: 35J48, 35B65, 35G35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Introduction In this paper, we consider the Dirichlet problem for the following even order elliptic system for u ∈ W k,2(Ω, Rm): (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) ∆ku = k−1 � l=0 ∆l ⟨Vl, du⟩ + k−2 � l=0 ∆lδ(wldu) in Ω ⊂ R2k with the following regularity assumptions on the coefficients: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2) wi ∈ W 2i+2−k,2 � Ω, Rm×m� for i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' , k − 2}, Vi ∈ W 2i+1−k,2 � Ω, Rm×m ⊗ ∧1R2k� for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' , k − 1}, and V0 = dη + F with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3) η ∈ W 2−k,2 (Ω, so(m)) and F ∈ W 2−k, 2k k+1,1 � Ω, Rm×m ⊗ ∧1R2k� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' This system was initially introduced by de Longueville and Gastel [2], aiming at a further extesion of the second order theory by Rivi`ere [11] (corresponding to the case k = 1) and the fourth order theory by Lamm-Rivi`ere [7] (corresponding to the case k = 2), addressing an open problem of Rivi`ere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' It includes the Euler-Lagrange equations of many interesting classes of geometric mappings such as the harmonic mappings, biharmonic mappings, polyharmonic mappings and so on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' see [1, 12, 11, 7, 3, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' A distinguished feature of this system is the criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' To see it, we consider the simpler case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Then system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) reduces to the second order Rivi`ere system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='4) ∆u = Ω′ · ∇u, Corresponding author: Yao-Lan Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Both authors are partially supported by the Young Scientist Program of the Ministry of Sci- ence and Technology of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' 2021YFA1002200), the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' 12101362) and the Natural Science Foundation of Shandong Province (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' ZR2022YQ01, ZR2021QA003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' LIU AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' TIAN where u ∈ W 1,2(Ω, Rm) and Ω′ ∈ L2(Ω, so(m) ⊗ Λ1R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='4) is merely in L1 by H¨older’s inequality and so standard Lp regularity theory for elliptic equations fails to apply here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' In the celebrated work [11], Rivi`ere succeeded in rewriting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='4) into an equivalent conservation law, from which the continuity of weak solutions follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The techniques were further extended to fourth order system in [7] and finally to general even order systems in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' In this paper, we shall consider the Dirichlet boundary value problem for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Recall that we say that u ∈ W k,2(Ω, Rm) has Dirichlet boundary value g ∈ Ck−1(Ω, Rm) if ∇αu = ∇αg on ∂Ω holds in the sense of traces for all 2k-dimensional multi-indices α with |α| ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Similarly, we say that u has Navier boundary value hi ∈ C(Ω, Rm), i = 0, · · · , k − 1, if for all i ∈ {0, · · · , k − 1} ∆iu = hi on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Now, we can state our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Fix k ∈ N and Ω ⊂ R2k a bounded smooth domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Suppose u ∈ W k,2(Ω, Rm) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' If either the Dirichlet bound- ary value g ∈ Ck−1(Ω, Rm) or the Navier boundary value hi for i = 0, · · · , k − 1, then u ∈ C(Ω, Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1 can be viewed as a natural extension of the corresponding boundary continuity results of M¨uller-Schikorra [9] for second order system, and Guo-Xiang [4] for fourth order system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' As a special case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1, we infer that every (extrinsic or intrinsic) polyharmonic mapping from the unit ball B2k ⊂ R2k into a closed manifold N ֒→ Rm is continuous up the boundary, under the Dirichlet boundary value condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' This partially extends the corrosponding boundary continuity reuslt of Lamm-Wang [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The approach to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1 is similar to that of Lamm and Wang [8], relying on interior H¨older regularity and a boundary maximal principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' As noticed in [4], this approach only requies zero order boundary assumption “u = g” or “u = h0” on ∂Ω for some continuous function g or h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' This observation extends to the general even order system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Our natations are rather standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' We write Br(x) for a ball centred at x with radius r in R2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The notation C denotes various constants that may be different from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' We sometimes write A ≲ B meaning that A ≤ CB for some constant C > 0 depending only on the quantitative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The proof of main result 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Interior regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Continuity of weak solutions for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) was first obtained in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' But for boundary continuity, we need the stronger interior H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' We first recall the following interior H¨older regularity result for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) from [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3] or [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1 (Interior Regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Suppose u ∈ W k,2(B2k, Rm) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Then there exist α ∈ (0, 1), C > 0 and r0, depending only on k, m and the data from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3), such that u is locally α-H¨older continuous and oscBr(x)u ≤ Crα∥u∥W k,2(B2k) BOUNDARY REGULARITY 3 for all x ∈ B 1 4(0) and all 0 < r < r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' We shall use the following version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1 in our later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' There exists R0 > 0 sufficiently small, such that, if x ∈ Ω and 0 < r < min{R0, dist(x, ∂Ω)/4}, then u ∈ C0,α(Br(x0), Rm) for some α ∈ (0, 1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) oscBτr(x)u ≲ τ α∥u∥W k,2(B4r(x0)) for 0 < τ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' By [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1], one can indeed infer that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) holds for all α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' But for our purpose, the current estimate is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Boundary Maximum Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Another ingredient for boundary regularity is a boundary maximum principle, originally discovered by Qing [10] in his proof of boundary regularity for weakly harmonic maps and was later adapted to the polyharmonic case in Lamm-Wang [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' For x ∈ Ω and R > 0, denote by ΩR(x) = Ω ∩ BR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' We shall prove the following boundary maximal principle for solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2 (Boundary Maximum Principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' There exists a constant C > 0 such that for any x ∈ Ω and 0 < R < R0/4, for any q ∈ Rm, there holds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2) max ΩR(x) |u − q| ≤ C � max ∂ΩR(x) |u − q| + ∥u∥W k,2(Ω4R(x)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' To prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2, we need the following version of Courant-Lebesgue lemma, which was essentially established in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Since the formulation is slightly different from there, for the convenience of the readers, we recall the proofs here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' There exists C > 0 such that for any R > 0 and any x0 ∈ Ω, there exists R1 ∈ (R, 2R) so that osc∂BR1(x0)∩Ωu ≤ C∥u∥W k,2(Ω4R(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' For x ∈ B2R(x0), set r = |x − x0| ∈ [0, 2R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' By Fubini’s theorem, we have � Ω2R(x0) |∇u|2kdx ≥ � 2R R dr � ∂Br(x0)∩Ω |∇Tu|2kdH2k−1 ≥ �� 2R R 1 rdr � inf R≤r≤2R � r � ∂Br(x0)∩Ω |∇Tu|2kdH2k−1 � = ln 2 inf R≤r≤2R � r � ∂Br(x0)∩Ω |∇Tu|2kdH2k−1 � , where ∇T denotes the gradient operator on ∂Br(x0) and dH2k−1 is the volume element on ∂Br(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Then there exists R1 ∈ (R, 2R) such that R1 � ∂BR1(x0)∩Ω |∇Tu|2kdH2k−1 ≤ 1 ln 2 � Ω2R(x0) |∇u|2kdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Hence u(R1, ·) ∈ W 1,2k(∂BR1(x0) ∩ Ω, Rm) and the Sobolev embedding theorem implies that u(R1, ·) ∈ C 1 2k (∂BR1(x0) ∩ Ω, Rm) and osc∂BR1(x0)∩Ωu ≲ R1 � ∂BR1(x0)∩Ω |∇Tu|2kdH2k−1 ≲ ∥u∥W k,2(Ω4R(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' □ 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' LIU AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' TIAN Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Denote M = max ΩR(x) |u−q|, here q ∈ Rm is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' We may assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3) M ≥ ∥u∥W k,2(ΩR(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Choose x0 ∈ ΩR(x) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='4) |u(x0) − q| ≥ 3 4M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Let r0 = dist(x0, ∂ΩR(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Note that r0 ≤ R ≤ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) implies that for any r ∈ (0, r0 4 ), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='5) oscBr(x0)u ≤ C � r r0 �α0 ∥u∥W k,2(ΩR(x)) ≤ CM � r r0 �α0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Pick r1 = r0/(4C)1/α in the above, and we obtain oscBr1(x0)u ≤ 1 4M This together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='4) yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='6) inf Br1(x0) |u − q| ≥ |u(x0) − q| − oscBr1(x0)u ≥ 1 2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3, there exists r2 ∈ (r0, 2r0) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='7) osc∂Br2(x0)∩ΩR(x)u ≤ C∥u∥W k,2(Ω4R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Note that ∂Br2(x0)∩∂ΩR(x) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Using polar coordinates centered at x0, we estimate inf � |u(r1, θ)−u(r2, θ)| : (ri, θ) ∈ ∂Bri(x0) ∩ ΩR(x), i = 1, 2 � ≤C � S2k−1 dθ � r2 r1 |ur|χ[r1,r2]×S2k−1(r, θ)dr ≤ C r2k−1 1 � S2k−1 dθ � r2 r1 |ur|χ[r1,r2]×S2k−1(r, θ)r2k−1dr ≤ C r2k−1 1 � B2r0(x)∩ΩR(x) |ur|dx ≤ C r2k−1 1 |B2r0(x)| 2k−1 2k �� ΩR(x) |∇u|2kdx � 1 2k ≤C r2k−1 0 r2k−1 1 ∥u∥W k,2(Ω4R(x)) ≤ C∥u∥W k,2(Ω4R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' This implies that there exists θ0 ∈ ∂B1(x0) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='8) |u(r1, θ0) − u(r2, θ0)| ≤ C∥u∥W k,2(Ω4R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' BOUNDARY REGULARITY 5 Hence, by choosing an arbitrary x∗ ∈ ∂Br2(x0) ∩ ∂ΩR(x), we obtain from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='8) that M 2 ≤ inf Br1(x0) |u − q| ≤ |u(r1, θ0) − q| ≤|u(r1, θ0) − u(r2, θ0)| + |u(r2, θ0) − u(x∗)| + |u(x∗) − q| ≤C∥u∥W k,2(Ω4R(x)) + osc∂Br2(x0)∩ΩR(x)u + sup ∂ΩR(x) |u − q| ≤C � sup ∂ΩR(x) |u − q| + ∥u∥W k,2(Ω4R(x)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Now we are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Let x0 ∈ ∂Ω and take q = g(x0) = u(x0) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Note that max ∂ΩR(x0) |u − u(x0)| ≤ max ∂ΩR(x0)∩∂Ω |u − u(x0)| + osc∂ΩR(x0)∩Ωu = max ∂ΩR(x0)∩∂Ω |g − g(x0)| + osc∂ΩR(x0)∩Ωu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The first term tends to 0 as R → 0 since g ∈ C(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The second term tends to 0 as R → 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' This implies the continuity of u as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The proof above extends to solutions to the following inhomogeneous elliptic system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='9) ∆ku = k−1 � l=0 ∆l ⟨Vl, du⟩ + k−2 � l=0 ∆lδ(wldu) + f in Ω ⊂ R2k with f ∈ Lp for some p > 1 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Indeed, by [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1], in this case, the interior regularity estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1) becomes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='10) oscBτr(x)u ≲ τ α � ∥u∥W k,2(B4r(x0)) + ∥f∥Lp(B4r(x0)) � for 0 < τ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' With this, the buondary maximal principle (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='2) remains valid with an extra term ∥f∥Lp(Ω4R(x)) on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='1 then works with obvious modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' The authors would like to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Chang-Lin Xiang and Chang-Yu Guo for posing this question to them and for many useful conservations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Chang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' Wang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' China and Frontiers Science Center for Nonlin- ear Expectations, Ministry of Education, Qingdao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' China Email address: minglunliu2021@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='com (Yao-Lan Tian) Center for Optics Research and Engineering, Shandong University, Qingdao 266237, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content=' China Email address: tianylbnu@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf'} diff --git a/BtE4T4oBgHgl3EQfeA0g/content/tmp_files/2301.05095v1.pdf.txt b/BtE4T4oBgHgl3EQfeA0g/content/tmp_files/2301.05095v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..634c2c12321b1cd69ecef205471db2d595919019 --- /dev/null +++ b/BtE4T4oBgHgl3EQfeA0g/content/tmp_files/2301.05095v1.pdf.txt @@ -0,0 +1,1577 @@ +Electron cooling in graphene enhanced by plasmon-hydron resonance +Xiaoqing Yu1, Alessandro Principi2, Klaas-Jan Tielrooij3,4, Mischa Bonn1 and Nikita Kavokine1,5 +1Max Planck Institute for Polymer Research, +Ackermannweg 10, Mainz 55128, Germany +2School of Physics and Astronomy, University of Manchester, M13 9PL Manchester, U.K. +3Catalan Institute of Nanoscience and Nanotechnology (ICN2), +BIST and CSIC, Campus UAB, Bellaterra, Barcelona, 08193, Spain +4Department of Applied Physics, TU Eindhoven, +Den Dolech 2, 5612 AZ, Eindhoven, The Netherlands and +5Center for Computational Quantum Physics, +Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA +Evidence is accumulating for the crucial role of a solid’s free electrons in the +dynamics of solid-liquid interfaces. +Liquids induce electronic polarization and +drive electric currents as they flow; electronic excitations, in turn, participate in +hydrodynamic friction. Yet, the underlying solid-liquid interactions have been +lacking a direct experimental probe. Here, we study the energy transfer across +liquid-graphene interfaces using ultrafast spectroscopy. The graphene electrons +are heated up quasi-instantaneously by a visible excitation pulse, and the time +evolution of the electronic temperature is then monitored with a terahertz pulse. +We observe that water accelerates the cooling of the graphene electrons, whereas +other polar liquids leave the cooling dynamics largely unaffected. A quantum +theory of solid-liquid heat transfer accounts for the water-specific cooling en- +hancement through a resonance between the graphene surface plasmon mode +and the so-called hydrons – water charge fluctuations –, particularly the water +libration modes, that allows for efficient energy transfer. +Our results provide +direct experimental evidence of a solid-liquid interaction mediated by collective +modes and support the theoretically proposed mechanism for quantum friction. +They further reveal a particularly large thermal boundary conductance for the +water-graphene interface and suggest strategies for enhancing the thermal con- +ductivity in graphene-based nanostructures. +arXiv:2301.05095v1 [cond-mat.mes-hall] 12 Jan 2023 + +2 +Free electrons in graphene exhibit rather unique dynamics in the terahertz (THz) frequency +range, including a highly non-linear response to photoexcitation by THz pulses [1, 2]. Graphene’s +distinctive dynamical properties on picosecond timescales have found several applications in, e.g., +ultrafast photodetectors, modulators, and receivers [3–5]. The THz frequency range acquires par- +ticular importance at room temperature T, where it corresponds to the typical frequency of thermal +fluctuations: kBT/ℏ ∼ 6 THz, with kB Boltzmann’s constant and ℏ Planck’s constant. One may +therefore expect non-trivial couplings between the graphene electrons and the thermal fluctuations +of their environment. These couplings have been intensively studied in the case of a solid environ- +ment: for instance, non-adiabatic effects have been shown to arise in the graphene electron-phonon +interaction [6], and plasmon-phonon coupling between graphene and a polar substrate has been +demonstrated [7–9]. More recently, it has been theoretically proposed that similar effects are at +play when graphene has a liquid environment: then, the interaction between the liquid’s charge +fluctuations – dubbed hydrons – and graphene’s electronic excitations tunes the hydrodynamic +friction at the carbon surface [10, 11]. This "quantum friction" mechanism holds the potential of +entirely new strategies for controlling liquid flows at nanometer scales [12, 13]. +Obtaining an experimental signature of the quantum friction mechanism would involve directly +visualizing momentum transfer between a solid and a liquid: that is, measuring a force. Force +measurements at solid-liquid interfaces suffer from a strong sensitivity to the surface state, coupled +with enormous technical challenges [14–16]. In this Article, we overcome this obstacle by measuring +energy transfer as a proxy for momentum transfer. Specifically, we use a femtosecond visible pulse to +introduce a quasi-instantaneous temperature difference between the graphene electrons and their +environment. The cooling rate of the electronic system is followed in real-time using terahertz +pulses. Such Optical Pump - Terahertz Probe (OPTP) spectroscopy is a well-established tool for +probing electron relaxation in 2D materials [17–21]. In high-quality graphene, it has been used to +identify the interaction of hot carriers with optical phonons [19, 20] and with substrate phonons +as the main electron cooling mechanisms [22]. +Here, we measure the electron relaxation time in the presence of different polar liquids to probe +the electron-liquid interaction, which we find to be significant compared to the electron - optical +phonon interaction only when the liquid is water. A complete theoretical analysis shows that this +specificity of water is explained by the strong coupling of its THz (libration) modes to the graphene +surface plasmon, with the electron-electron interactions in graphene playing a crucial role. + +3 +a +b +Liquid +Solid +Quantum friction +Liquid +Solid +Quantum heat transfer +c +FIG. 1. From friction to heat transfer. a. Artist’s view of the system under study: the interface +between a liquid and a graphene sheet. The liquid, at temperature T, may flow with an interfacial velocity +v, while the graphene electrons (depicted by the orange cloud) may be heated up to a temperature T + ∆T. +b. Schematic of the solid-liquid quantum friction mechanism: momentum is transferred directly through +quasiparticle tunneling at a rate γ between surface modes of the solid and the liquid (depicted by the blue +parabolas), at wavevector q and frequency ωq. The Bose distribution nB predicts a higher occupation of the +liquid mode (filling of the blue parabola) due to a flow-induced Doppler shift. c. Schematic, with the same +notations as in b, of solid-liquid quantum heat transfer. Here, the solid’s mode has a higher occupation +due to a higher temperature than the liquid. Energy and momentum transfer involve the same interaction +between surface modes. +Solid-liquid heat transfer +The energy transfer between a solid and a liquid is usually considered to be mediated by +molecular vibrations at the interface, as most of a solid’s heat capacity is contained in its phonon +modes [23]. Even if an optical excitation of the solid’s electrons is used to create the temperature +difference, the electrons are typically assumed to thermalize with phonons on a very short time +scale, so that the solid’s phonons ultimately mediate the energy transfer to the liquid’s vibrational +modes [24, 25]. However, if the electrons were to transfer energy to the liquid faster than to the +phonons, the interfacial thermal conductivity would contain a non-negligible contribution from +near-field radiative heat transfer [26, 27]. Such an electronic or "quantum" contribution to heat +transfer is in close analogy with the quantum contribution to hydrodynamic friction (Fig. +1). +Quantum hydrodynamic friction relies on momentum being transferred directly between the solid’s +and the liquid’s charge fluctuation modes. +In a simplified Fermi golden rule picture [10], the +corresponding friction force can be written as +FQ = +� +dqdω ℏq ∆γq(ω). +(1) + +4 +SiO2 cell +THz probe +Liquid +Graphene +Optical pump +∆E(t) +a +Te = 623 K +Pump-probe delay (ps) +Normalized ∆T +b +N2 +H2O +D2O +Methanol +Ethanol +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 + + +Te= 1241 K +Te = 1023 K +Te = 770 K +Te = 623 K +Cooling time (ps) +c +0 +1 +2 +3 +4 +5 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +N2 +H2O +Methanol +Ethanol + D2O +FIG. 2. Measurement of picosecond hot electron relaxation in graphene. a. Schematic of the +experimental setup. A graphene sample is placed in contact with a liquid inside a fused silica flow cell. An +optical excitation pulse impulsively heats up the graphene electrons, and the electron temperature dynamics +are then monitored with a THz probe. b. Normalized electron temperature as a function of time after +photoexcitation. The dotted lines represent raw data and the full lines are exponential fits. c. Electron +cooling time obtained through exponential fitting (see b) for the different liquids that have been placed in +the flow cell and different initial electron temperatures, set by the excitation laser fluence. Faster cooling +is observed in the presence of water and heavy water. Error bars represent 95% confidence intervals of the +exponential fits. +It is a sum over all the in-plane wavevectors q and frequencies ω of the elementary momentum +ℏq, multiplied by the net quasiparticle tunneling rate ∆γq(ω) between the solid’s and the liquid’s +modes at wavevector q and frequency ω. The quantum contribution to the solid-liquid energy +transfer rate then reads +QQ = +� +dqdω ℏω ∆γq(ω), +(2) +with the momentum quantum ℏq being replaced by the energy quantum ℏω. +Thus, quantum +friction and quantum energy transfer rely on the same solid-liquid interactions, contained in the +tunneling rates ∆γq(ω). In the same way that probing quantum friction requires it to dominate +over the surface roughness contribution, the quantum energy transfer needs to exceed the classical +phonon-based energy transfer in order to become measurable. We now show that this condition is +met upon optically exciting of a graphene-water interface, owing, in particular, to graphene’s weak +electron-phonon coupling [28]. +Time-resolved electron cooling +Our experimental setup is schematically represented in Fig. 2a. A monolayer graphene sample + +5 +was transferred onto a fused silica flow cell, filled with either nitrogen gas or a liquid of our choice +(SI Sec. +1.1). +In a typical experiment, the graphene electrons were excited using a ∼ 50 fs +laser pulse with 800 nm central wavelength. Then, the attenuation of a ∼ 1 ps THz probe pulse +(precisely, the modulation of the peak electric field) was monitored as a function of the pump- +probe delay (SI Sec. +1.2). +After absorption of the exciting pump pulse, the non-equilibrium +electron distribution typically thermalizes over a sub-100 fs timescale through electron-electron +scattering [29]: it can then be described as a Fermi-Dirac distribution at a given temperature. A +hotter electron distribution results in a lower THz photoconductivity, since hotter electrons are less +efficient at screening charged impurities [30, 31]. The pump-probe measurement thus gives access +to the electron temperature dynamics after photoexcitation (Fig. 2b). +Regardless of the medium that the graphene is in contact with, the electronic temperature T(t) +exhibits a relaxation that can be approximated by an exponential function : ∆T(t) = T(t) − T0 = +∆T0e−t/τ. This allows us to extract the cooling times τ for the different liquids and different initial +electronic temperatures (determined by the excitation laser fluence), displayed in Fig. 2c. We +observe that the cooling time is longer for an initially hotter electron distribution, in agreement +with previous reports [20]. Now, for all initial temperatures, we consistently observe the same +dependence of the cooling time on the sample’s liquid environment. +In the presence of water +(H2O) and heavy water (D2O), the graphene electrons cool faster than they do intrinsically, in an +inert nitrogen atmosphere. Conversely, methanol and ethanol have almost no effect on the electron +cooling time. Interestingly, we observe an isotope effect in the electron cooling process: there is +a difference in the cooling times in the presence of H2O and D2O that well exceeds experimental +uncertainties. +We are thus led to hypothesize, as anticipated above, that the liquid provides the electrons with +a supplementary cooling pathway, which, in the case of water, has an efficiency comparable to the +intrinsic cooling pathway. We then interpret the faster cooling as a signature of "quantum" electron- +liquid energy transfer. We assess the pertinence of this hypothesis by developing a complete theory +of quantum energy transfer at the solid-liquid interface. +Theoretical framework +In order to tackle the interaction between a classical liquid and an electronic system whose +behavior is intrinsically quantum, we describe the liquid in a formally quantum way. Following +ref. [10], we represent the liquid’s charge density as a free fluctuating field with prescribed corre- + +6 +lation functions. This naturally leads to a Fourier-space description of the solid-liquid interface +in terms of its collective modes, rather than the usual molecular scale interactions. Within this +description, the quantum solid-liquid energy transfer amounts to electron relaxation upon coupling +to a bosonic bath, a problem that has been extensively studied in condensed matter systems [32]. +Interestingly, in the case of graphene, many of these studies are carried out within a single-particle +Boltzmann formalism, which may incorporate multiple screening effects only in an ad hoc fash- +ion [20, 28, 33]. These effects turn out to be crucial for the solid-liquid system under consideration: +we have therefore developed an ab initio theory of solid-liquid heat transfer based on the non- +equilibrium Keldysh formalism [34], which has only very recently been considered for problems of +interfacial heat transfer [35]. Our computation, detailed in the SI Sec. 2.2, is closely analogous to +the one carried out for quantum friction in ref. [10]. The theoretical framework can formally apply +to fully non-equilibrium situations and take interactions into account to arbitrary order. However, +to obtain a closed-form result, we restrict ourselves to a two-temperature model, where the liquid +and the solid are assumed to be internally equilibrated at temperatures Tℓ and Te respectively. Fur- +thermore, we take electron-electron and electron-liquid Coulomb interactions into account at the +Random Phase Approximation (RPA) level. We these assumptions, we obtain the electron-liquid +energy transfer rate as +QQ = +1 +2π3 +� +dq +� +∞ +0 +dω ℏω[nB(ω, Te) − nB(ω, Tℓ)]Im [ge(q, ω)]Im [gℓ(q, ω)] +|1 − ge(q, ω)gℓ(q, ω)|2 , +(3) +consistently with the general form anticipated in Eq. (2). Here, nB(ω, T) = 1/(eℏω/T − 1) is the +Bose distribution and the ge,ℓ are surface response functions of the solid and the liquid, respectively. +These are analogues of the dielectric function for semi-infinite media, whose precise definition is +given in the SI, Sec. +2.3. +For the liquids under consideration, it will be sufficient to use the +long-wavelength-limit expression of the surface response function: +gℓ(q → 0, ω) = ϵℓ(ω) − 1 +ϵℓ(ω) + 1, +(4) +where ϵℓ(ω) is the liquid’s bulk dielectric function. For two-dimensional graphene, we show in the +SI (Sec. 2.3) that the surface response function can be expressed as +ge(q, ω) = − e2 +2ϵ0qχ(q, ω), +(5) +where χ(q, ω) is graphene’s charge susceptibility. We note that the result in Eq. (3) has been derived +for two solids separated by a vacuum gap in the framework of fluctuation-induced electromagnetic +phenomena [26, 36, 37]; we believe, however, that the framework we provide is better suited to the +solid-liquid system under consideration. + +7 +Coulomb +interaction +Surface +plasmon +Electron +cloud +Hindered rotation +(libration) +a +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Wavevector (nm-1) +0 +0.05 +0.1 +0.15 +0.2 +Frequency (eV) +Dirac cone +Plasmon-hydron +resonance +Te = 623 K +Cooling rate (1/ps) +Total rate (experiment) +Liquid contribution (theory) +N2 +H2O +D2O +Methanol +Ethanol +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Wavevector (nm-1) +0 +0.05 +0.1 +0.15 +0.2 +Frequency (eV) +Te = 623 K +Dirac cone +0.05 +0.1 +0.15 +0.2 +Frequency (eV) +0 +0.1 +0.2 +0.3 +0.4 +Surface excitation spectrum +H2O +D2O +Methanol +Ethanol +b +c +0.5 +0.6 +0.7 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +Cooling rate (1/ps) +N2 +H2O +D2O +Methanol +Ethanol +d +e +f +Plasmon +10 +20 +30 +40 +50 +Frequency (THz) +FIG. 3. Mechanism of electron-liquid heat transfer. a. Surface excitation spectra Im [gℓ(ω)] of the +different liquids under study obtained according to Eq. (4) from the experimentally-measured bulk dielectric +permittivities. The arrows indicate the libration modes of H2O and D2O. b. Graphene surface excitation +spectrum Im [ge(q, ω)], calculated at a chemical potential µ = 100 meV and temperature Te = 623 K. +The main feature is the collective plasmon mode. c. Theoretical prediction for the graphene-water energy +transfer rate resolved in frequency-wavevector space. The main contribution originates from a resonance +between the graphene plasmon mode and the water libration mode. d. Experimentally-measured electron +cooling rate in the presence of the various liquids. e. Theoretical prediction for the liquid contribution +to the electron cooling rate, reproducing the experimentally-observed trend in terms of the nature of the +liquid. The symbol size in the vertical direction represents the variation in the theoretical prediction when +the graphene chemical potential spans the range [100 meV, 180 meV]. f. Schematic of the water-mediated +electron cooling mechanism inferred from the combination of theoretical and experimental results. +The +cooling proceeds through the Coulomb interaction between the graphene plasmon mode and the hindered +molecular rotations (librations) in water. +Plasmon-hydron resonance +If the interaction with the liquid is the only mechanism for electron relaxation, our result in +Eq. (3) determines the time evolution of the electron temperature according to +C(Te)dTe(t) +dt += −QQ(Te, Tℓ), +(6) + +8 +where C(Te) is the graphene electronic heat capacity at temperature Te. This allows us to de- +fine the liquid contribution to the electron cooling rate as 1/τ = QQ(Te, Tℓ)/(C(Te) × (Te − Tℓ)), +which may be compared with the experimental results. The quantitative evaluation of τ requires +the surface response functions of graphene and of the various liquids. We compute the graphene +surface response function according to Eq. (5) by numerical integration [38], at the chemical po- +tential determined for our samples by Raman spectroscopy (SI Sec. +1.4). +For the liquids, we +use the expression in Eq. (4), with the bulk dielectric function determined by infrared absorption +spectroscopy (Fig. 3a and SI Sec. 1.3). +Our theoretical prediction for the various liquids’ contribution to the electron cooling rate is +shown in Fig. 3e. Quantitatively, we obtain cooling rates of the order of 1 ps−1, in excellent +agreement with the experimentally observed range (Fig. +3d) : our theory indicates that the +quantum electron-liquid cooling is a sufficiently efficient process to compete with intrinsic electron +relaxation mechanisms. Moreover, our theory reproduces the experimentally observed trend in +cooling rates, with a significant liquid contribution arising only for water and heavy water; the +dependence of the cooling rate on initial electron temperature is also well-reproduced (Fig. S7). +Finally, the theory reproduces the isotope effect, that is, the slightly slower cooling observed with +D2O as compared to H2O. +We may now exploit the theory to gain insight into the microscopic mechanism of the liquid- +mediated cooling process. In Eq. (3), the difference of Bose distributions decreases exponentially +at frequencies above Te/ℏ ∼ 100 meV. At frequencies below 100 meV, the graphene spectrum is +dominated by a plasmon mode, that corresponds to the collective oscillation of electrons in the +plane of the graphene layer [38] (Fig. 3b). In this same frequency range, water and heavy water +have a high spectral density due to their libration mode, that corresponds to hindered molecular +rotations [39] (Fig. 3a). As a result, the energy transfer rate resolved in frequency-momentum +space (the integrand in Eq. (3), plotted in Fig. 3c) has its main contribution from the spectral +region where the two modes overlap. We conclude that the particularly efficient electron-water +cooling is due to a resonance between the graphene plasmon mode and the water libration mode. +This conclusion is further supported by the isotope effect. Indeed, the libration of the heavier D2O +is at slightly lower frequency than that of the lighter H2O, and a higher frequency mode makes a +larger contribution to the cooling rate due to the factor ℏω in Eq. (3). Physically, the quasiparticle +tunneling rates are almost the same for the graphene-H2O and graphene-D2O systems, but in the +case of H2O each quasiparticle carries more energy. Overall, our experiments evidence a direct +interaction between the graphene plasmon and water libration, as shown schematically in Fig. 3f. + +9 +Cooling rate (1/ps) +Renormalized +No e-e interactions +Bare +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Wavevector (nm-1) +0 +0.05 +0.1 +0.15 +0.2 +Frequency (eV) +Te = 623 K +b +Full theory +First order +a +0 +0.1 +0.2 +0 +20 +40 +60 +Energy transfer rate (meV·Å2·s-1) +First order +Full theory +Frequency (eV) +c +H2O +D2O +Methanol +Ethanol +10-1 +100 +101 +0 +0.1 +0.2 +Frequency (eV) +FIG. 4. Strong plasmon-hydron coupling. a. Theoretical prediction for the graphene electron cooling +rate in contact with different liquids, within different treatments of interactions. The cooling rate is strongly +overestimated if no electron-electron interactions are taken into account (blue symbols), and underestimated +if the electron-liquid interactions are considered only to first order (orange symbols). b. Graphene surface +excitation spectrum Im [ge(q, ω)], calculated at a chemical potential µ = 180 meV and temperature Te = +623 K, renormalized by the presence of water according to Eq. (7). The white dashed lines are guides to +the eye showing the strongly-coupled plasmon-hydron mode. Inset: bare and renormalized graphene spectra +at fixed wavevector q0 = 0.15 nm−1. c. Comparison between the spectrally resolved energy transfer rates +obtained to first order and to arbitrary order in the solid-liquid interaction. Higher-order effects enhance +the energy transfer rate at low frequencies. +Interactions and strong coupling +The combination of theory and experiment allows us to identify the key physical ingredients +that are required to account for energy transfer at the water-graphene interface. First, our results +reveal that electron-electron interactions are crucial, since they produce the plasmon mode that +is instrumental to the energy transfer mechanism. Indeed, applying our theory to non-interacting +graphene would result in a strongly overestimated liquid contribution to the cooling rate (Fig. 4a). +This precludes single-particle Boltzmann approaches – such as those that have been used for the +electron-phonon interaction in graphene [20, 28] – for accurately describing the water-graphene +interaction. +Furthermore, the detailed examination of our theoretical result reveals that the efficiency of the +electron-water cooling is enhanced by the formation of a strongly-coupled plasmon-hydron mode. +Indeed, the result in Eq. (3) involves bare surface response functions, without any renormalization +due to the presence of the other medium. +However, the denominator |1 − gegℓ|2 accounts for +solid-liquid interactions to arbitrary order (at the RPA level) and contains the signature of any + +10 +potential strong coupling effects. We find that these effects are indeed important, as removing the +denominator in Eq. (3) (that is, treating the electron-liquid interactions only to first order) results +in under-estimation of the liquid-mediated cooling rate by about 30% (Fig. 4a) . In order to gain +physical insight into the nature of these higher order effects, we may compute the graphene surface +response function renormalized by the presence of water, which is given by (see SI Sec. 2.3) +˜ge(q, ω) = +ge(q, ω) +1 − ge(q, ω)gℓ(q, ω). +(7) +The renormalized surface excitation spectrum Im [˜ge(q, ω)] is plotted in Fig. 4b, for a chemical +potential µ = 180 meV. We observe that the graphene plasmon now splits into two modes, which +are both a mixture of the the bare plasmon and water libration. These are in fact analogous to +the coupled plasmon-phonon modes that have been predicted [7] and measured [8, 9] for graphene +on a polar substrate. It can be seen in the inset of Fig. 4b that coupling to the water modes also +increases the spectral density at low frequencies (below the plasmon peak), compared to the bare +graphene response function. This is in fact the higher-order effect that is mainly responsible for +the enhancement of the electron cooling rate. As shown in Fig. 4c, taking into account solid-liquid +interactions to arbitrary order mainly enhances the contribution of low frequencies to the energy +transfer. +Conclusions +We have carried out ultrafast measurements of electron relaxation in graphene, revealing signa- +tures of direct energy transfer between the graphene electrons and the surrounding liquid. These +results speak to the importance of electronic degrees of freedom in the dynamics of solid-liquid +interfaces, particularly interfaces between water and carbon-based materials. Despite conventional +theories and simulations that describe the interface in terms of atomic-scale Lennard-Jones poten- +tials [24, 25], or with electronic degrees of freedom in the Born-Oppenheimer approximation [40, 41], +here we demonstrate experimentally that the dynamics of the water-graphene interface need to be +considered at the level of collective modes in the terahertz frequency range. In particular, our semi- +quantitative theoretical analysis attributes the observed cooling dynamics to the strong coupling +between the graphene plasmon and water libration modes. +The experimental observation of such a collective mode interaction supports the proposed mech- +anism for quantum friction at the water-carbon interface, which is precisely based on momentum +transfer between collective modes [10]. The near-quantitative agreement between the experiment + +11 +and theory obtained for energy transfer suggests that a similar agreement should be achieved for +momentum transfer. We note, however, that quantum friction of water on graphene is typically +negligible compared to the classical surface roughness contribution, and it is only expected to play +a role in the presence of a phonon wind [12]. Quantum friction has been predicted to be much +more important for water on graphite due to the difference in plasmon dispersion between the two +materials [10]: the investigation of electron-water energy transfer in the case of carbon multilayers +will be the subject of future work. +Our results provide yet another example of the water-carbon interface outperforming other solid- +liquid systems [42]. Indeed, the electronic contribution to the graphene-water thermal boundary +conductance is as high as λ = 0.25 MW · m−2 · K−1, exceeding the value obtained with the +other investigated liquids by at least a factor of 2. +This even exceeds the thermal boundary +conductance obtained for the graphene-hBN interface, at which particularly fast "super-Planckian" +energy transfer was observed [33]. Our investigation thus suggests that the density of modes in the +terahertz frequency range is a key determinant for the thermal conductivity of graphene-containing +composite materials. +Acknowledgements +We acknowledge financial support from the MaxWater initiative of the Max Planck Society. We +thank Xiaoyu Jia and Hai Wang for carrying out preliminary experiments, and Maksim Grechko +and Detlev-Walter Scholdei for assisting with the FTIR measurements. X.Y. is grateful for support +from the China Scholarship Council. 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Kavokine +- +Contents +1 +Experimental methods +1 +1.1 +Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1.2 +OPTP measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1.3 +FTIR measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +1.4 +Raman measurements +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2 +Theoretical methods +5 +2.1 +Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.2 +General theory of electron-boson heat transfer . . . . . . . . . . . . . . . . . . . . . +7 +2.3 +Application to the graphene-liquid system . . . . . . . . . . . . . . . . . . . . . . . +8 +arXiv:2301.05095v1 [cond-mat.mes-hall] 12 Jan 2023 + +Figure S1: Schematic of the OPTP setup. +1 +Experimental methods +1.1 +Sample preparation +CVD-grown graphene samples supported on 1 mm-thick copper substrates were purchased from +Grolltex Inc. The MilliQ water (18.2 MΩ · cm) was used as obtained from the machine. Cellulose +acetate butyrate (CAB, average Mn ∼ 12000, Sigma-Aldrich), ammonium persulfate (APS, ACS +reagent, ≥ 98%, Honeywell FlukaTM) are used as received. CAB was dissolved in ethyl acetate +(Sigma-Aldrich), producing a 30 mg/mL solution. APS was dissolved in MilliQ water to prepare +1 M and 0.1 M solutions. The detachable fused silica flow cell was ordered from FireflySci, Inc. +The flow cell was cleaned by sonication in a hot acetone and ethanol baths for 10 minutes each +before using. +We transferred graphene onto the front substrate of the flow cell following a wet transfer +procedure [1, 2]. +First, we spin-coated graphene samples with CAB at 4000 rpm and baked +them at 180◦C for 3 minutes. Then, to remove unnecessary graphene on the backside of copper +substrates, the CAB-coated graphene samples were immersed into a 1 M solution of APS for 10 +minutes and subsequently rinsed with MilliQ water five times. The copper substrates were then +fully etched by 0.1 M APS solution for 2 hours, followed by a five times rinse with MilliQ water +to remove the attached ions. Then, the floating CAB-graphene monolayers were "fished" onto the +flow cell, and the CAB coating was removed by soaking in acetone for 2 hours and in isopropanol +for one hour. +1.2 +OPTP measurements +We probed electron relaxation in graphene using optical pump - terahertz probe (OPTP) spec- +troscopy. A schematic of the OPTP setup is shown in Fig. S1. The fundamental laser output was +generated by a regenerative Ti:sapphire amplifier system, which produces 5 W, 50 fs pulses at a +repetition rate of 1 kHz and a central wavelength of 800 nm. The generated pulses were then split +into three branches for THz generation, sampling, and optical excitation. A single-cycle THz pulse +of ∼ 1 ps duration was generated by pumping a 1 mm thick (110) ZnTe crystal with the 800 nm +fundamental pulses via optical rectification. +We photoexcited graphene to generate hot carriers by using 800 nm pulses with a diameter of +1 + +X +Polarizer +Delay stage +Beam splitter +P +Chopper +ZnTe 入/4 +Wollaston +sample +prism +White foam +ZnTe +Differential +detector0 +2 +4 +6 +8 +0.00 +0.02 +0.04 +0.06 +0.08 +Pump-probe delay (ps) +N2 +Fluence ( +2) + 5.86 + 4.32 + 2.85 + 1.50 + 0.89 + 0.29 +c +b +a +0.00 +0.02 +0.04 +0.06 +0.08 +0 +200 +400 +600 +800 +1000 +1200 +Peak value of +0 +1 +2 +3 +4 +5 +6 +0.00 +0.02 +0.04 +0.06 +0.08 + + +Peak value of +Fluence (μJ/cm2) +0 +200 +400 +600 +800 +1000 +1200 +T e-T l (K) +T e-T l (K) +∆E/E +∆E/E +∆E/E +μJ/cm +Figure S2: Electron temperature of the pumped graphene layer. a.The OPTP traces +of graphene in a nitrogen atmosphere with various excitation fluences. b. Peak value of ∆E/E +as a function of laser fluence and corresponding electron temperature. c. Increase in electron +temperature (Te) with respect to ambient temperature Tℓ as a function of ∆E/E: a linear relation +is observed. +5 mm to ensure a homogeneously photoexcited region. The transmitted THz wave was then recol- +limated and focused onto a ZnTe detection crystal together with an 800 nm sampling beam, where +the THz electrical field waveform was detected using the electro-optic sampling method [3]. The +THz pulse induces birefringence in the ZnTe detection crystal, and the polarization of the sampling +beam is thus changed. After passing through a quarter-wave plate, the sampling beam changes +from perfectly circular to slightly elliptical shape. The s and p components of this elliptically +polarized pulse are separated by a Wollaston prism, and the difference of these two components +is detected by a balance diode. The signal is collected by a lock-in amplifier that is phase-locked +to an optical chopper that modulates either the THz generation beam or the pump beam at a +frequency of 500 Hz. The ultrafast time evolution of the peak intensity of the THz field is tracked +by varying the time delay between optical pump and THz probe [3, 4]. The setup was purged with +dry nitrogen during the measurement to avoid the absorption of water vapor. +The raw data consists in time traces of the pump-induced transmission change at the peak of +the THz waveform (∆E), normalized by the peak value of the THz transmission without excitation +(E) (Fig. S2a). Assuming that a fraction γ = 1.6% of the pump pulse energy is absorbed by the +graphene electrons [5], the maximum electron temperature reached after photoexcitation can be +related to the pump laser fluence F according to γF = C(Te)Te, where C(Te) is the graphene heat +capacity at temperature Te. In the limit where the graphene Fermi energy µ is larger than kBTe +(as relevant for our samples), we may use the approximate expression [6, 7, 8] +C(Te) = αTe, +with +α = 2π +3 +k2 +Bµ +(ℏvF)2 , +(1) +where vF is graphene’s constant Fermi velocity. Then, +Te = T0 +� +1 + 2γF +αT 2 +0 +�1/2 +, +(2) +where T0 is ambient temperature. The peak value of ∆E/E after photoexcitation increases with +laser fluence. +Upon rescaling, we find that the plots of ∆E/E vs. +F and Te vs. +F collapse +upon each other (Fig. S2b), so that we may consider that ∆E/E is proportional to the electron +temperature within the range of temperatures probed in the experiment, as shown explicitly in +Fig. S2c. +The thickness of the liquid layer was set to 50 µm by the geometry of the flow cell. The liquids +were exchanged using a syringe and the spectroscopic measurement was always carried out at +2 + +0 +2 +4 +6 +8 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + no cover + with cover + 10 μm + 20 μm + 30 μm + 40 μm + 50 μm + 60 μm + +1.50 +1.55 +1.60 +1.65 +1.70 +1.75 +1.80 + Day 1 + Day 2 + Day 3 + + +no cover +with cover +10 μm +20 μm +30 μm +40 μm +50 μm +60 μm +Cooling time (ps) +Normalized ∆E/E +b +a +Pump-probe delay (ps) +Figure S3: Control experiments. a. The OPTP traces of graphene with varying water layer +thickness. b. Cooling times obtained by exponential fitting of the data in panel a. +the same spot of the graphene sample. To exclude the effect of beam dispersion in the different +liquids on the results, we repeated the measurement with different water layer thickness and using +different Teflon spacers between two fused silica windows (Fig. S3). +1.3 +FTIR measurements +We measured the dielectric functions of water, heavy water, ethanol and methanol using Fourier- +transform infrared (FTIR) spectroscopy. +We measured the transmitted and reflected infrared +intensities both for an empty cell (It,cell, Ir,cell) and for a cell filled with liquid (It,liquid, Ir,liquid) +thanks to an A510/Q-T Reflectance and Transmittance accessory placed in a commercial VERTEX +70 FTIR spectrometer (Fig. S4a). In order to avoid disassembling the cell when changing liquids, +we carried out the measurements inside a flow cell, made out of two-silicon wafers separated by a +10 µm Teflon spacer. We calculated the absorbance A(ω) according to +A(ω) = − log10 +� +It,solution(ω) +It,cell(ω) + Ir,cell(ω) − Ir,solution(ω) +� +. +(3) +The H2O and D2O show saturated absorption in the range of 3100-3600 and 2200-2700 cm−1, +respectively. We obtained the data in this frequency range by measuring the spectra without any +spacer between two CaF2 windows and then rescaled the spectra to overlap with the data with +spacer (Fig. S4b). The imaginary part k(ω) of the refractive index is related to the absorbance by +k(ω) = A(ω)ln(10) +4πωℓ , +(4) +where ℓ is the sample thickness. To accurately determine the thickness of the cell, we calculate +the absorbance of the empty cell without correction for multiple reflections, +A2(ω) = − log10 +� +It,cell(ω) +It,lamp(ω) +� +. +(5) +where It,lamp(ω) is the intensity of the lamp of the FTIR source (Fig. S5a). Fourier transformation +of this spectrum yields a peak at the time ∆t that light takes to travel twice through the cell (Fig. +S5b), so that ℓ = c∆t/2 = 10.29 µm. We then obtained the real part of the refractive index +through a numerical Kramers-Krönig transformation: +n(ω) = n∞ + 2 +π +� ∞ +0 +dω′ k(ω′) +ω′ − ω, +(6) +3 + +Figure S4: FTIR data analysis. a. Raw intensity data of empty cell and water-filled cell. b. +Absorbance of H2O and D2O, measured with spacer and without spacer (the latter is rescaled to +overlap with the former). +Figure S5: Determination of the cell thickness. a. Absorbance of empty cell. b. Fourier +transform of the data in panel a. +4 + +a 0.20 +celltransmission +cellreflection +0.15 +solutiontransmission +(a.u. +solutionreflection +0.05 +0.00 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +Frequency (cm-1) +6 +H,0 +b +:D20 +5 +-H,Owospacer(X5.0) +bsorbance +-D,Owospacer(X3.3) +H,O combined +3 +D,o combined +0 +1000 +2000 +3000 +4000 +Frequency (cm-1)a +1.2 +007 +0.06863ps +1.0. +0.06 +0.05 +Absorbance +itensity +0.04 +0.6 +0.03 +0.4 +0.02 +0.2. +0.01. +0.00 +0.0 +0 +2000 +4000 +6000 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +Frequency(cm") +Time (ps)a +b +Figure S6: Raman characterization of graphene sample. a. Spatial map of Raman G +band frequency for graphene sample in air. b. Distribution of the Raman G band frequency with +different liquids placed on the graphene surface. +where n∞ is the refractive index in the high frequency limit, which is obtained by the ATAGO +Digital Handheld Refractometer: PAL-RI. The measured values for H2O, D2O, methanol, ethanol +and isopropanol are 1.333, 1.3291, 1.3285, 1.3604, and 1.3706 respectively. +We then obtain the dielectric function ϵ(ω) = ϵ′(ω) + iϵ′′(ω) according to +� +ϵ′(ω) = n(ω)2 − k(ω)2 +ϵ′′(ω) = 2n(ω)k(ω) +. +(7) +1.4 +Raman measurements +We estimated the Fermi level µ in our liquid-covered graphene samples from the Raman G-band +frequency, according to the empirical equation [6] +|µ|(eV) = ωG − 1580 cm−1 +42 cm−1 +. +(8) +An example of a spatial map of the Raman G-band frequency is shown in Fig. S6a. The fre- +quency shows spatial inhomogeneities on the µm scale with an amplitude around 10 cm−1. The +corresponding distributions are shown in Fig. S6b. The average G-band frequency is essentially +independent of the nature of the liquid, which excludes a change in charge carrier density as a +possible mechanism for the liquid effect on the electron cooling rate. To take into account the +broadness of the distribution, in the theoretical analysis we considered chemical potentials in the +range µ = 100 − 180 meV. The theoretical prediction is independent of the electron or hole nature +of the charge carriers. +2 +Theoretical methods +In this section, we develop a description of energy transfer between the Dirac fermion charge +carriers in graphene and a liquid, treated as a bosonic bath, within the non-equilibrium Keldysh +framework of perturbation theory. For the sake of completeness, and in order to show consistency +with previous theoretical approaches, we apply the same description to energy transfer between +the graphene electrons and its optical phonon modes, showing that our formalism recovers the +results that were previously obtained within a Boltzmann equation approach [9]. +5 + +We use SI units throughout the text. We adopt the following convention for the n-dimensional +Fourier transform: +˜f(q) = +� +∞ +−∞ +dnr f(r)e−iq·r +and +f(r) = +1 +(2π)n +� +∞ +−∞ +dnq ˜f(q)eiq·r. +(9) +2.1 +Interaction Hamiltonian +2.1.1 +Electron-hydron interaction +In this section, r represents a vector in 3D space, and ρ a vector in 2D space. +The charge +fluctuations of the liquid in the z > 0 half-space couple to the graphene electrons via the Coulomb +potential V . In real space, the corresponding Hamiltonian is +Hew(t) = +� +drdr′nw(r, t)V (r − r′)ne(r, t), +(10) +where nw and ne are the liquid and graphene instantaneous charge density, respectively. +Let +c† +k,ν, ck,ν be the Dirac fermion creation and annihilation operators in the chiral basis (ν = ±1). A +2D Fourier transformation then yields +Hint = +� +dq +(2π)2 +e2 +2ϵ0qns(q, t) +� +k,ν,ν′ +⟨k + q, ν|eiqρeqz|k, ν′⟩c† +k+q,ν(t)ck,ν′(t), +(11) +with +ns(q) = +� +dρ +� +∞ +0 +dz e−iqρe−qznw(ρ, z, t). +(12) +As long as we consider wavevectors q such that q−1 is large compared to the extension of the +carbon pz orbitals perpendicular to the graphene plane, we may approximate +|⟨k + q, ν|eiqρeqz|k, ν′⟩|2 ≈ |⟨k + q, ν|eiqρ|k, ν′⟩|2 = 1 +2 +�1 + νν′ cos(φk+q − φq) +� . +(13) +2.1.2 +Electron-phonon interaction +Let d† +q,α, dq,α be the creation and annihilation operators of phonons in the mode α with frequency +ωα. The non-interacting electron-phonon system’s Hamiltonian is +H0 = +� +k,ν +Ek,νc† +k,νck,ν + +� +q,α +ℏωαd† +q,αdq,α, +(14) +where Ek,ν are the band energies, and � +k ≡ (1/ABZ) +� +BZ dk (ABZ is the area of the 2D Brillouin +zone). The electron-phonon interaction Hamiltonian has the general form [10] +Hep = +� +α +� +BZ +dq +(2π)2 +� +k,ν,ν′ +gνν′ +α,k,k+qc† +k+qck(d† +q,α + d−q,α), +(15) +Following [9], we consider the Γ point LO and TO phonons that scatter electrons within one valley, +and the K, K’ point LO phonons that scatter electrons between valleys. The electron-phonon +matrix elements read +|gνν′ +Γ,k,k+q|2 = g2 +Γ(1 ± νν′ cos(φk + φk+q − 2φq)), +(16) +where the + (−) sign is for LO (TO) phonons; and +|gνν′ +Γ,k,k+q|2 = g2 +K(1 ∓ νν′ cos(φk − φk+q)), +(17) +where the − (+) sign corresponds to scattering from K to K’ (from K’ to K); here, φv is the polar +angle of the vector v. The values of the coupling constants are gΓ = 0.55 eV·Å and gK = 0.85 eV·Å, +according to GW calculations [11]. +6 + +2.1.3 +General form +We find that for both types of interactions the Hamiltonian has the general form +Heb = +� +dq +(2π)2 nq(t)ϕq(t), +(18) +where nq is an electronic two-particle operator and ϕq is a free bosonic field. In the electron-phonon +case, we define +nq = +� +k,ν,ν′ +gνν′ +α,k,k+q +√ℏωα +c† +k+q,νck,ν′ +and +ϕq = +� +ℏωα(d† +q,α + d−q,α); +(19) +in the electron-hydron case +nq = +� +Vq +� +k,ν,ν′ +⟨k + q, ν|eiqρ|k, ν′⟩c† +k+q,νck,ν′ +and +ϕq = +� +Vqns(q), +(20) +where Vq ≡ e2/(2ϵ0q) is the 2D Fourier-transformed Coulomb potential. With these definitions, +both nq and φq have dimensionless correlation functions in frequency space. +2.2 +General theory of electron-boson heat transfer +2.2.1 +Non-equilibrium perturbation theory +We consider an initial state of the electron-boson system where the electrons are at a temperature +Te and the bosons at a temperature Tb. We wish to study the subsequent dynamics. In particular, +we are interested in the heat flux per unit surface from the electrons to the bosons: +Q(t) = − 1 +A +d +dt⟨Heb(t)⟩. +(21) +Since the system is under non-equilibrium conditions, this average value needs to be computed in +the Keldysh framework. In particular, we may define the Keldysh component of the electron-boson +correlation function: +χK +eb(q, t, t′) = − 1 +A +i +ℏ⟨{nq(t), ϕ−q(t′)}⟩. +(22) +Then, +Q(t) = −iℏ +2 +� +dq +(2π)2 +dχK +eb(q, t, t) +dt +. +(23) +Form this point on, the computation of the electron-boson correlation function follows the exact +same steps as in the theory of quantum friction [12], and we reproduce here only the main equations. +Diagramatically, the correlation function satisfies the following Dyson equation: +(24) +where the "bubble" represents the propagator of n (denoted χe), and the dashed line the propagator +of ϕ (denoted χb). When made explicit in terms of the R, A, K components, the Dyson equation +becomes +� +� +� +� +� +χK +eb = χR +e ⊗ χK +b + χK +e ⊗ χA +b + χR +e ⊗ χR +b ⊗ χK +eb + (χR +e ⊗ χK +b + χK +e ⊗ χA +b ) ⊗ χA +eb +χR,A +eb += χR,A +e +⊗ χR,A +b ++ χR,A +e +⊗ χR,A +b +⊗ χR,A +eb +, +(25) +where ⊗ represents time convolution. +While these equations are extremely general, they are +impractical to manipulate analytically, unless a number of assumptions are made. In order to +7 + +proceed, we will restrict ourselves to cooling dynamics that are slow enough for time-translation +invariance to hold when it comes to determining the cooling rate. This assumption is expected +to hold for small enough temperature differences, such that the cooling rate is approximately +temperature-independent. We will further assume that, in line with experimental observations, +that electron thermalization is much faster than electron-boson energy transfer, so that the electron +and boson propagators may be considered as equilibrium propagators, satisfying the fluctuation- +dissipation theorem: we work within a two-temperature model. We may then carry out Fourier +transforms in time, so that Eq. (23) becomes +Q = 1 +2 +� dqdω +(2π)3 ℏω χK +eb(q, ω). +(26) +The convolutions in Eq. (25) become products in Fourier space. Before proceeding, it is convenient +to flip the signs of all the correlation functions: we introduce, for all the labels, g ≡ −χ. Then, +after some algebra, we obtain an explicit expression for Q: +Q = +1 +2π3 +� +dq +� +∞ +0 +dω ℏω[nB(ω, Te) − nB(ω, Tb)]Im [gR +e (q, ω)]Im [gR +b (q, ω)] +|1 − gR +e (q, ω)gR +b (q, ω)|2 , +(27) +where nB(ω, T) ≡ 1/(eℏω/kBT − 1) is the Bose distribution at temperature T. We recover Eq. (3) +of the main text. +2.2.2 +Cooling rate +The cooling dynamics are governed by the equation +dE(Te) +dt += −Q(Te, Tb), +(28) +where E is the total energy per unit surface of the electronic system. We follow ref. [9] in de- +termining the electronic heat capacity (per unit surface) at constant density C(Te), such that +dtE = C(Te)dtTe. We may then define the instantaneous cooling rate +τ(Te, Tb) = C(Te)(Te − Tb) +Q(Te, Tb) +. +(29) +2.3 +Application to the graphene-liquid system +2.3.1 +Liquid-mediated cooling +We first consider electron cooling through the electron-hydron coupling. Using eqs. (12) and (20), +we find that +gR +b (q, t, t′) = − 1 +AVq +� +∞ +0 +dzdz′ e−q(z+z′) +� +− i +ℏθ(t − t′)⟨[ns(q, z, t), ns(−q, z′, t′)]⟩ +� +. +(30) +This is the microscopic definition of the liquid’s surface response function. In the long wavelength +limit, it can be expressed in terms of the liquid’s bulk dielectric function ϵ(ω) [12]: +gR +b (q, ω) = ϵ(ω) − 1 +ϵ(ω) + 1, +(31) +as stated in the main text. The electronic response function gR +e (q, ω) simply amount to (minus) +the density-density response function. Taking into account electron-electron interactions at the +RPA level [13], +gR +e (q, ω) = − +Vqχ0 +e(q, ω) +1 − Vqχ0e(q, ω). +(32) +8 + +600 +800 +1000 +1200 +Electron temperature (K) +1 +1.5 +2 +2.5 +Cooling time (ps) +Theory +Experiment +Figure S7: Dependence of water-mediated cooling time on initial electron temperature. The red +dots are experimental data for graphene in contact with water and the red dots correspond to the +prediction of Eq. (29) (with µ = 180 meV). +The non-interacting response function χ0 +e is given by [13] +χ0 +e(q, ω) = gsgv +� +dk +(2π)2 +� +ν,ν′ +|⟨k + q, ν|eiqρ|k, ν′⟩|2 nF(Eν +k, Te) − nF(Eν′ +k+q, Te) +Eν +k − Eν′ +k+q + ω + iδ +, +(33) +where gs = gv = 2 are the spin and valley degeneracies of graphene, respectively, Eν +k = νvF k are +the band energies in the Dirac fermion approximation, nF(E, T) = 1/(e(E−µ)/kBT +1) is the Fermi +distribution at chemical potential µ and temperature T, and δ → 0+. The integral is evaluated +numerically at non-zero temperature. +With all the above, we may compute theoretical predictions for the liquid-mediated cooling rate +by numerical integration according to Eq. (27). We considered a graphene chemical potential µ in +the range 100 − 180 meV (see section 1.4) and an electron temperature Te = 623 K, corresponding +to the lowest pump laser fluence. Our model is further able to reproduce the dependence of the +electron cooling time on Te, as shown in Fig. S7. +We note that Eq. (27) involves bare surface response functions, that contain no effect of the +presence of the neighboring medium, at least at the RPA level. Nevertheless, the physical response +function of graphene in the presence of water undergoes RPA renormalization according to +(34) +In this diagrammatic equation, when the propagators are interpreted as surface response functions, +the vertices reduce to unity, so that we obtain the renormalized graphene response function ˜ge as +˜ge(q, ω) = +ge(q, ω) +1 − ge(q, ω)gb(q, ω), +(35) +which is Eq. (7) of the main text. +2.3.2 +Phonon-mediated cooling +In the phonon case, the boson response function is proportional to the usual phonon propagator: +gR +b (q, ω) = +2ω2 +α +ω2α − ω2 . +(36) +9 + +The non-interacting electronic response function now involves the electron-phonon matrix elements: +gR +e (q, ω) = −gs +� +BZ +dk +(2π)2 +� +ν,ν′ +|gνν′ +α,k,k+q|2 +ℏωα +nF(Eν +k, Te) − nF(Eν′ +k+q, Te) +Eν +k − Eν′ +k+q + ω + iδ +. +(37) +We now show that we recover the results of ref. [9] for the electron-phonon cooling rate obtained in +a Boltzamann equation framework, if we neglect electron-electron interactions and treat electron- +phonon interactions to first order. Under these assumptions, Eq. (27) reduces to +Q = +1 +2π3 +� +dq +� +∞ +0 +dω ℏω[nB(ω, Te) − nB(ω, Tb)]Im [gR +e (q, ω)]Im [gR +b (q, ω)]. +(38) +We notice that +Im [gR +b (q, ω)] = πω2 +α[δ(ω − ωα) − δ(ω + ωα)] +(39) +and +Im [gR +e (q, ω)] = πgs +� +BZ +dk +(2π)2 +� +ν,ν′ +|gνν′ +α,k,k+q|2 +ℏωα +[nF(Eν +k, Te) − nF(Eν′ +k+q, Te)]δ(Eν +k − Eν′ +k+q + ω). (40) +Moreover, upon integration over k and q in Eq. (38), the angle-dependent parts of the electron- +phonon matrix elements vanish, and the intervalley phonons become formally identical to the +intravalley phonons: we may introduce the valley degeneracy and carry out integrations over a +single Dirac cone. Altogether, we obtain +Q = 2πgsgvωαg2 +α[nB(ωα, Te) − nB(ωα, Tb)] . . . +· · · +� +ν,ν′ +� dqdk +(2π)4 [nF(Eν +k, Te) − nF(Eν′ +q , Te)]δ(Eν +k − Eν′ +q + ωα). +(41) +If we introduce another delta function, according to +Q = 2πgsgvωαg2 +α[nB(ωα, Te) − nB(ωα, Tb)] . . . +· · · +� +ν,ν′ +� dqdk +(2π)4 +� +∞ +−∞ +dϵ[nF(ϵ − ωα, Te) − nF(ϵ, Te)]δ(Eν +k − ϵ + ωα)δ(ϵ − Eν′ +q ), +(42) +we recognize the graphene density of states, +ν(ϵ) = gsgv +� +ν +� +dk +(2π)2 δ(ϵ − Ek,ν) = 2|ϵ| +πv2 +F +. +(43) +Our result then simplifies according to +Q = 2πωαg2 +α +gsgv +[nB(ωα, Te) − nB(ωα, Tb)] +� +∞ +−∞ +dϵ[nF(ϵ − ωα, Te) − nF(ϵ, Te)]ν(ϵ)ν(ϵ − ωα), +(44) +which is Eq. (18) in the supplementary information of ref. [9]. +References +[1] Yogeswaran, N. et al. Piezoelectric graphene field effect transistor pressure sensors for tactile +sensing. Applied Physics Letters 113, 014102 (2018). +[2] Burwell, G., Smith, N. & Guy, O. Investigation of the utility of cellulose acetate butyrate +in minimal residue graphene transfer, lithography, and plasma treatments. Microelectronic +Engineering 146, 81–84 (2015). +10 + +[3] Ulbricht, R., Hendry, E., Shan, J., Heinz, T. F. & Bonn, M. Carrier dynamics in semicon- +ductors studied with time-resolved terahertz spectroscopy. Reviews of Modern Physics 83, +543–586 (2011). +[4] Lee, Y.-S. Principles of Terahertz Science and Technology (Springer US, 2009). +[5] Fu, S. et al. +Long-lived charge separation following pump-wavelength-dependent ultrafast +charge transfer in graphene/ws2 heterostructures. Science Advances 7, eabd9061 (2021). +[6] Shi, S. F. et al. +Controlling graphene ultrafast hot carrier response from metal-like to +semiconductor-like by electrostatic gating. Nano Letters 14, 1578–1582 (2014). +[7] Tielrooij, K. J. et al. Photoexcitation cascade and multiple hot-carrier generation in graphene. +Nature Physics 9, 248–252 (2013). +[8] Lui, C. H., Mak, K. F., Shan, J. & Heinz, T. F. Ultrafast photoluminescence from graphene. +Physical Review Letters 105, 127404 (2010). +[9] Pogna, E. A. et al. Hot-carrier cooling in high-quality graphene is intrinsically limited by +optical phonons. ACS Nano 15, 11285–11295 (2021). +[10] Neto, A. H. C. & Guinea, F. Electron-phonon coupling and raman spectroscopy in graphene. +Physical Review B 75, 045404 (2007). +[11] Sohier, T. et al. +Phonon-limited resistivity of graphene by first-principles calculations: +Electron-phonon interactions, strain-induced gauge field, and boltzmann equation. Physical +Review B 90, 125414 (2014). +[12] Kavokine, N., Bocquet, M.-L. & Bocquet, L. +Fluctuation-induced quantum friction in +nanoscale water flows. Nature 602, 84–90 (2022). +[13] Wunsch, B., Stauber, T., Sols, F. & Guinea, F. Dynamical polarization of graphene at finite +doping. New Journal of Physics 8, 318–318 (2006). +11 + diff --git a/BtE4T4oBgHgl3EQfeA0g/content/tmp_files/load_file.txt b/BtE4T4oBgHgl3EQfeA0g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..06292be57cef31faf4aa9f188d1cfa6db91ea8d8 --- /dev/null +++ b/BtE4T4oBgHgl3EQfeA0g/content/tmp_files/load_file.txt @@ -0,0 +1,1117 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf,len=1116 +page_content='Electron cooling in graphene enhanced by plasmon-hydron resonance Xiaoqing Yu1, Alessandro Principi2, Klaas-Jan Tielrooij3,4, Mischa Bonn1 and Nikita Kavokine1,5 1Max Planck Institute for Polymer Research, Ackermannweg 10, Mainz 55128, Germany 2School of Physics and Astronomy, University of Manchester, M13 9PL Manchester, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3Catalan Institute of Nanoscience and Nanotechnology (ICN2), BIST and CSIC, Campus UAB, Bellaterra, Barcelona, 08193, Spain 4Department of Applied Physics, TU Eindhoven, Den Dolech 2, 5612 AZ, Eindhoven, The Netherlands and 5Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA Evidence is accumulating for the crucial role of a solid’s free electrons in the dynamics of solid-liquid interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Liquids induce electronic polarization and drive electric currents as they flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' electronic excitations, in turn, participate in hydrodynamic friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Yet, the underlying solid-liquid interactions have been lacking a direct experimental probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Here, we study the energy transfer across liquid-graphene interfaces using ultrafast spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The graphene electrons are heated up quasi-instantaneously by a visible excitation pulse, and the time evolution of the electronic temperature is then monitored with a terahertz pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We observe that water accelerates the cooling of the graphene electrons, whereas other polar liquids leave the cooling dynamics largely unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' A quantum theory of solid-liquid heat transfer accounts for the water-specific cooling en- hancement through a resonance between the graphene surface plasmon mode and the so-called hydrons – water charge fluctuations –, particularly the water libration modes, that allows for efficient energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Our results provide direct experimental evidence of a solid-liquid interaction mediated by collective modes and support the theoretically proposed mechanism for quantum friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' They further reveal a particularly large thermal boundary conductance for the water-graphene interface and suggest strategies for enhancing the thermal con- ductivity in graphene-based nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05095v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='mes-hall] 12 Jan 2023 2 Free electrons in graphene exhibit rather unique dynamics in the terahertz (THz) frequency range, including a highly non-linear response to photoexcitation by THz pulses [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Graphene’s distinctive dynamical properties on picosecond timescales have found several applications in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=', ultrafast photodetectors, modulators, and receivers [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The THz frequency range acquires par- ticular importance at room temperature T, where it corresponds to the typical frequency of thermal fluctuations: kBT/ℏ ∼ 6 THz, with kB Boltzmann’s constant and ℏ Planck’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' One may therefore expect non-trivial couplings between the graphene electrons and the thermal fluctuations of their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' These couplings have been intensively studied in the case of a solid environ- ment: for instance, non-adiabatic effects have been shown to arise in the graphene electron-phonon interaction [6], and plasmon-phonon coupling between graphene and a polar substrate has been demonstrated [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' More recently, it has been theoretically proposed that similar effects are at play when graphene has a liquid environment: then, the interaction between the liquid’s charge fluctuations – dubbed hydrons – and graphene’s electronic excitations tunes the hydrodynamic friction at the carbon surface [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This "quantum friction" mechanism holds the potential of entirely new strategies for controlling liquid flows at nanometer scales [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Obtaining an experimental signature of the quantum friction mechanism would involve directly visualizing momentum transfer between a solid and a liquid: that is, measuring a force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Force measurements at solid-liquid interfaces suffer from a strong sensitivity to the surface state, coupled with enormous technical challenges [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In this Article, we overcome this obstacle by measuring energy transfer as a proxy for momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Specifically, we use a femtosecond visible pulse to introduce a quasi-instantaneous temperature difference between the graphene electrons and their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The cooling rate of the electronic system is followed in real-time using terahertz pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Such Optical Pump - Terahertz Probe (OPTP) spectroscopy is a well-established tool for probing electron relaxation in 2D materials [17–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In high-quality graphene, it has been used to identify the interaction of hot carriers with optical phonons [19, 20] and with substrate phonons as the main electron cooling mechanisms [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Here, we measure the electron relaxation time in the presence of different polar liquids to probe the electron-liquid interaction, which we find to be significant compared to the electron - optical phonon interaction only when the liquid is water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' A complete theoretical analysis shows that this specificity of water is explained by the strong coupling of its THz (libration) modes to the graphene surface plasmon, with the electron-electron interactions in graphene playing a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3 a b Liquid Solid Quantum friction Liquid Solid Quantum heat transfer c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' From friction to heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Artist’s view of the system under study: the interface between a liquid and a graphene sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The liquid, at temperature T, may flow with an interfacial velocity v, while the graphene electrons (depicted by the orange cloud) may be heated up to a temperature T + ∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Schematic of the solid-liquid quantum friction mechanism: momentum is transferred directly through quasiparticle tunneling at a rate γ between surface modes of the solid and the liquid (depicted by the blue parabolas), at wavevector q and frequency ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The Bose distribution nB predicts a higher occupation of the liquid mode (filling of the blue parabola) due to a flow-induced Doppler shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Schematic, with the same notations as in b, of solid-liquid quantum heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Here, the solid’s mode has a higher occupation due to a higher temperature than the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Energy and momentum transfer involve the same interaction between surface modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Solid-liquid heat transfer The energy transfer between a solid and a liquid is usually considered to be mediated by molecular vibrations at the interface, as most of a solid’s heat capacity is contained in its phonon modes [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Even if an optical excitation of the solid’s electrons is used to create the temperature difference, the electrons are typically assumed to thermalize with phonons on a very short time scale, so that the solid’s phonons ultimately mediate the energy transfer to the liquid’s vibrational modes [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' However, if the electrons were to transfer energy to the liquid faster than to the phonons, the interfacial thermal conductivity would contain a non-negligible contribution from near-field radiative heat transfer [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Such an electronic or "quantum" contribution to heat transfer is in close analogy with the quantum contribution to hydrodynamic friction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Quantum hydrodynamic friction relies on momentum being transferred directly between the solid’s and the liquid’s charge fluctuation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In a simplified Fermi golden rule picture [10], the corresponding friction force can be written as FQ = � dqdω ℏq ∆γq(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (1) 4 SiO2 cell THz probe Liquid Graphene Optical pump ∆E(t) a Te = 623 K Pump-probe delay (ps) Normalized ∆T b N2 H2O D2O Methanol Ethanol 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 Te= 1241 K Te = 1023 K Te = 770 K Te = 623 K Cooling time (ps) c 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='0 N2 H2O Methanol Ethanol D2O FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Measurement of picosecond hot electron relaxation in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Schematic of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' A graphene sample is placed in contact with a liquid inside a fused silica flow cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' An optical excitation pulse impulsively heats up the graphene electrons, and the electron temperature dynamics are then monitored with a THz probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Normalized electron temperature as a function of time after photoexcitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The dotted lines represent raw data and the full lines are exponential fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Electron cooling time obtained through exponential fitting (see b) for the different liquids that have been placed in the flow cell and different initial electron temperatures, set by the excitation laser fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Faster cooling is observed in the presence of water and heavy water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Error bars represent 95% confidence intervals of the exponential fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' It is a sum over all the in-plane wavevectors q and frequencies ω of the elementary momentum ℏq, multiplied by the net quasiparticle tunneling rate ∆γq(ω) between the solid’s and the liquid’s modes at wavevector q and frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The quantum contribution to the solid-liquid energy transfer rate then reads QQ = � dqdω ℏω ∆γq(ω), (2) with the momentum quantum ℏq being replaced by the energy quantum ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Thus, quantum friction and quantum energy transfer rely on the same solid-liquid interactions, contained in the tunneling rates ∆γq(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In the same way that probing quantum friction requires it to dominate over the surface roughness contribution, the quantum energy transfer needs to exceed the classical phonon-based energy transfer in order to become measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We now show that this condition is met upon optically exciting of a graphene-water interface, owing, in particular, to graphene’s weak electron-phonon coupling [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Time-resolved electron cooling Our experimental setup is schematically represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' A monolayer graphene sample 5 was transferred onto a fused silica flow cell, filled with either nitrogen gas or a liquid of our choice (SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In a typical experiment, the graphene electrons were excited using a ∼ 50 fs laser pulse with 800 nm central wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Then, the attenuation of a ∼ 1 ps THz probe pulse (precisely, the modulation of the peak electric field) was monitored as a function of the pump- probe delay (SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' After absorption of the exciting pump pulse, the non-equilibrium electron distribution typically thermalizes over a sub-100 fs timescale through electron-electron scattering [29]: it can then be described as a Fermi-Dirac distribution at a given temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' A hotter electron distribution results in a lower THz photoconductivity, since hotter electrons are less efficient at screening charged impurities [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The pump-probe measurement thus gives access to the electron temperature dynamics after photoexcitation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Regardless of the medium that the graphene is in contact with, the electronic temperature T(t) exhibits a relaxation that can be approximated by an exponential function : ∆T(t) = T(t) − T0 = ∆T0e−t/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This allows us to extract the cooling times τ for the different liquids and different initial electronic temperatures (determined by the excitation laser fluence), displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We observe that the cooling time is longer for an initially hotter electron distribution, in agreement with previous reports [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Now, for all initial temperatures, we consistently observe the same dependence of the cooling time on the sample’s liquid environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In the presence of water (H2O) and heavy water (D2O), the graphene electrons cool faster than they do intrinsically, in an inert nitrogen atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Conversely, methanol and ethanol have almost no effect on the electron cooling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Interestingly, we observe an isotope effect in the electron cooling process: there is a difference in the cooling times in the presence of H2O and D2O that well exceeds experimental uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We are thus led to hypothesize, as anticipated above, that the liquid provides the electrons with a supplementary cooling pathway, which, in the case of water, has an efficiency comparable to the intrinsic cooling pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We then interpret the faster cooling as a signature of "quantum" electron- liquid energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We assess the pertinence of this hypothesis by developing a complete theory of quantum energy transfer at the solid-liquid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Theoretical framework In order to tackle the interaction between a classical liquid and an electronic system whose behavior is intrinsically quantum, we describe the liquid in a formally quantum way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Following ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' [10], we represent the liquid’s charge density as a free fluctuating field with prescribed corre- 6 lation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This naturally leads to a Fourier-space description of the solid-liquid interface in terms of its collective modes, rather than the usual molecular scale interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Within this description, the quantum solid-liquid energy transfer amounts to electron relaxation upon coupling to a bosonic bath, a problem that has been extensively studied in condensed matter systems [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Interestingly, in the case of graphene, many of these studies are carried out within a single-particle Boltzmann formalism, which may incorporate multiple screening effects only in an ad hoc fash- ion [20, 28, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' These effects turn out to be crucial for the solid-liquid system under consideration: we have therefore developed an ab initio theory of solid-liquid heat transfer based on the non- equilibrium Keldysh formalism [34], which has only very recently been considered for problems of interfacial heat transfer [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Our computation, detailed in the SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2, is closely analogous to the one carried out for quantum friction in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The theoretical framework can formally apply to fully non-equilibrium situations and take interactions into account to arbitrary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' However, to obtain a closed-form result, we restrict ourselves to a two-temperature model, where the liquid and the solid are assumed to be internally equilibrated at temperatures Tℓ and Te respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Fur- thermore, we take electron-electron and electron-liquid Coulomb interactions into account at the Random Phase Approximation (RPA) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We these assumptions, we obtain the electron-liquid energy transfer rate as QQ = 1 2π3 � dq � +∞ 0 dω ℏω[nB(ω, Te) − nB(ω, Tℓ)]Im [ge(q, ω)]Im [gℓ(q, ω)] |1 − ge(q, ω)gℓ(q, ω)|2 , (3) consistently with the general form anticipated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Here, nB(ω, T) = 1/(eℏω/T − 1) is the Bose distribution and the ge,ℓ are surface response functions of the solid and the liquid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' These are analogues of the dielectric function for semi-infinite media, whose precise definition is given in the SI, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' For the liquids under consideration, it will be sufficient to use the long-wavelength-limit expression of the surface response function: gℓ(q → 0, ω) = ϵℓ(ω) − 1 ϵℓ(ω) + 1, (4) where ϵℓ(ω) is the liquid’s bulk dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' For two-dimensional graphene, we show in the SI (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3) that the surface response function can be expressed as ge(q, ω) = − e2 2ϵ0qχ(q, ω), (5) where χ(q, ω) is graphene’s charge susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We note that the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3) has been derived for two solids separated by a vacuum gap in the framework of fluctuation-induced electromagnetic phenomena [26, 36, 37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' we believe, however, that the framework we provide is better suited to the solid-liquid system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 7 Coulomb interaction Surface plasmon Electron cloud Hindered rotation (libration) a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='5 Wavevector (nm-1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 Frequency (eV) Dirac cone Plasmon-hydron resonance Te = 623 K Cooling rate (1/ps) Total rate (experiment) Liquid contribution (theory) N2 H2O D2O Methanol Ethanol 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='5 Wavevector (nm-1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 Frequency (eV) Te = 623 K Dirac cone 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 Frequency (eV) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 Surface excitation spectrum H2O D2O Methanol Ethanol b c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='8 1 Cooling rate (1/ps) N2 H2O D2O Methanol Ethanol d e f Plasmon 10 20 30 40 50 Frequency (THz) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Mechanism of electron-liquid heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Surface excitation spectra Im [gℓ(ω)] of the different liquids under study obtained according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (4) from the experimentally-measured bulk dielectric permittivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The arrows indicate the libration modes of H2O and D2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Graphene surface excitation spectrum Im [ge(q, ω)], calculated at a chemical potential µ = 100 meV and temperature Te = 623 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The main feature is the collective plasmon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Theoretical prediction for the graphene-water energy transfer rate resolved in frequency-wavevector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The main contribution originates from a resonance between the graphene plasmon mode and the water libration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Experimentally-measured electron cooling rate in the presence of the various liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Theoretical prediction for the liquid contribution to the electron cooling rate, reproducing the experimentally-observed trend in terms of the nature of the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The symbol size in the vertical direction represents the variation in the theoretical prediction when the graphene chemical potential spans the range [100 meV, 180 meV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Schematic of the water-mediated electron cooling mechanism inferred from the combination of theoretical and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The cooling proceeds through the Coulomb interaction between the graphene plasmon mode and the hindered molecular rotations (librations) in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Plasmon-hydron resonance If the interaction with the liquid is the only mechanism for electron relaxation, our result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3) determines the time evolution of the electron temperature according to C(Te)dTe(t) dt = −QQ(Te, Tℓ), (6) 8 where C(Te) is the graphene electronic heat capacity at temperature Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This allows us to de- fine the liquid contribution to the electron cooling rate as 1/τ = QQ(Te, Tℓ)/(C(Te) × (Te − Tℓ)), which may be compared with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The quantitative evaluation of τ requires the surface response functions of graphene and of the various liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We compute the graphene surface response function according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (5) by numerical integration [38], at the chemical po- tential determined for our samples by Raman spectroscopy (SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' For the liquids, we use the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (4), with the bulk dielectric function determined by infrared absorption spectroscopy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3a and SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Our theoretical prediction for the various liquids’ contribution to the electron cooling rate is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Quantitatively, we obtain cooling rates of the order of 1 ps−1, in excellent agreement with the experimentally observed range (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3d) : our theory indicates that the quantum electron-liquid cooling is a sufficiently efficient process to compete with intrinsic electron relaxation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Moreover, our theory reproduces the experimentally observed trend in cooling rates, with a significant liquid contribution arising only for water and heavy water;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' the dependence of the cooling rate on initial electron temperature is also well-reproduced (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Finally, the theory reproduces the isotope effect, that is, the slightly slower cooling observed with D2O as compared to H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We may now exploit the theory to gain insight into the microscopic mechanism of the liquid- mediated cooling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3), the difference of Bose distributions decreases exponentially at frequencies above Te/ℏ ∼ 100 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' At frequencies below 100 meV, the graphene spectrum is dominated by a plasmon mode, that corresponds to the collective oscillation of electrons in the plane of the graphene layer [38] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In this same frequency range, water and heavy water have a high spectral density due to their libration mode, that corresponds to hindered molecular rotations [39] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' As a result, the energy transfer rate resolved in frequency-momentum space (the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3), plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3c) has its main contribution from the spectral region where the two modes overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We conclude that the particularly efficient electron-water cooling is due to a resonance between the graphene plasmon mode and the water libration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This conclusion is further supported by the isotope effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Indeed, the libration of the heavier D2O is at slightly lower frequency than that of the lighter H2O, and a higher frequency mode makes a larger contribution to the cooling rate due to the factor ℏω in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Physically, the quasiparticle tunneling rates are almost the same for the graphene-H2O and graphene-D2O systems, but in the case of H2O each quasiparticle carries more energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Overall, our experiments evidence a direct interaction between the graphene plasmon and water libration, as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 9 Cooling rate (1/ps) Renormalized No e-e interactions Bare 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='5 Wavevector (nm-1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 Frequency (eV) Te = 623 K b Full theory First order a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 0 20 40 60 Energy transfer rate (meV·Å2·s-1) First order Full theory Frequency (eV) c H2O D2O Methanol Ethanol 10-1 100 101 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 Frequency (eV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Strong plasmon-hydron coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Theoretical prediction for the graphene electron cooling rate in contact with different liquids, within different treatments of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The cooling rate is strongly overestimated if no electron-electron interactions are taken into account (blue symbols), and underestimated if the electron-liquid interactions are considered only to first order (orange symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Graphene surface excitation spectrum Im [ge(q, ω)], calculated at a chemical potential µ = 180 meV and temperature Te = 623 K, renormalized by the presence of water according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The white dashed lines are guides to the eye showing the strongly-coupled plasmon-hydron mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Inset: bare and renormalized graphene spectra at fixed wavevector q0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='15 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Comparison between the spectrally resolved energy transfer rates obtained to first order and to arbitrary order in the solid-liquid interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Higher-order effects enhance the energy transfer rate at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Interactions and strong coupling The combination of theory and experiment allows us to identify the key physical ingredients that are required to account for energy transfer at the water-graphene interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' First, our results reveal that electron-electron interactions are crucial, since they produce the plasmon mode that is instrumental to the energy transfer mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Indeed, applying our theory to non-interacting graphene would result in a strongly overestimated liquid contribution to the cooling rate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This precludes single-particle Boltzmann approaches – such as those that have been used for the electron-phonon interaction in graphene [20, 28] – for accurately describing the water-graphene interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Furthermore, the detailed examination of our theoretical result reveals that the efficiency of the electron-water cooling is enhanced by the formation of a strongly-coupled plasmon-hydron mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Indeed, the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3) involves bare surface response functions, without any renormalization due to the presence of the other medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' However, the denominator |1 − gegℓ|2 accounts for solid-liquid interactions to arbitrary order (at the RPA level) and contains the signature of any 10 potential strong coupling effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We find that these effects are indeed important, as removing the denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3) (that is, treating the electron-liquid interactions only to first order) results in under-estimation of the liquid-mediated cooling rate by about 30% (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 4a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In order to gain physical insight into the nature of these higher order effects, we may compute the graphene surface response function renormalized by the presence of water, which is given by (see SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3) ˜ge(q, ω) = ge(q, ω) 1 − ge(q, ω)gℓ(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (7) The renormalized surface excitation spectrum Im [˜ge(q, ω)] is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 4b, for a chemical potential µ = 180 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We observe that the graphene plasmon now splits into two modes, which are both a mixture of the the bare plasmon and water libration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' These are in fact analogous to the coupled plasmon-phonon modes that have been predicted [7] and measured [8, 9] for graphene on a polar substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' It can be seen in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 4b that coupling to the water modes also increases the spectral density at low frequencies (below the plasmon peak), compared to the bare graphene response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This is in fact the higher-order effect that is mainly responsible for the enhancement of the electron cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 4c, taking into account solid-liquid interactions to arbitrary order mainly enhances the contribution of low frequencies to the energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Conclusions We have carried out ultrafast measurements of electron relaxation in graphene, revealing signa- tures of direct energy transfer between the graphene electrons and the surrounding liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' These results speak to the importance of electronic degrees of freedom in the dynamics of solid-liquid interfaces, particularly interfaces between water and carbon-based materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Despite conventional theories and simulations that describe the interface in terms of atomic-scale Lennard-Jones poten- tials [24, 25], or with electronic degrees of freedom in the Born-Oppenheimer approximation [40, 41], here we demonstrate experimentally that the dynamics of the water-graphene interface need to be considered at the level of collective modes in the terahertz frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In particular, our semi- quantitative theoretical analysis attributes the observed cooling dynamics to the strong coupling between the graphene plasmon and water libration modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The experimental observation of such a collective mode interaction supports the proposed mech- anism for quantum friction at the water-carbon interface, which is precisely based on momentum transfer between collective modes [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The near-quantitative agreement between the experiment 11 and theory obtained for energy transfer suggests that a similar agreement should be achieved for momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We note, however, that quantum friction of water on graphene is typically negligible compared to the classical surface roughness contribution, and it is only expected to play a role in the presence of a phonon wind [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Quantum friction has been predicted to be much more important for water on graphite due to the difference in plasmon dispersion between the two materials [10]: the investigation of electron-water energy transfer in the case of carbon multilayers will be the subject of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Our results provide yet another example of the water-carbon interface outperforming other solid- liquid systems [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Indeed, the electronic contribution to the graphene-water thermal boundary conductance is as high as λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='25 MW · m−2 · K−1, exceeding the value obtained with the other investigated liquids by at least a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This even exceeds the thermal boundary conductance obtained for the graphene-hBN interface, at which particularly fast "super-Planckian" energy transfer was observed [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Our investigation thus suggests that the density of modes in the terahertz frequency range is a key determinant for the thermal conductivity of graphene-containing composite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Acknowledgements We acknowledge financial support from the MaxWater initiative of the Max Planck Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We thank Xiaoyu Jia and Hai Wang for carrying out preliminary experiments, and Maksim Grechko and Detlev-Walter Scholdei for assisting with the FTIR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' is grateful for support from the China Scholarship Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 General theory of electron-boson heat transfer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3 Application to the graphene-liquid system .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 8 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05095v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='mes-hall] 12 Jan 2023 Figure S1: Schematic of the OPTP setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1 Experimental methods 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 Sample preparation CVD-grown graphene samples supported on 1 mm-thick copper substrates were purchased from Grolltex Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The MilliQ water (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 MΩ · cm) was used as obtained from the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Cellulose acetate butyrate (CAB, average Mn ∼ 12000, Sigma-Aldrich), ammonium persulfate (APS, ACS reagent, ≥ 98%, Honeywell FlukaTM) are used as received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' CAB was dissolved in ethyl acetate (Sigma-Aldrich), producing a 30 mg/mL solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' APS was dissolved in MilliQ water to prepare 1 M and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 M solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The detachable fused silica flow cell was ordered from FireflySci, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The flow cell was cleaned by sonication in a hot acetone and ethanol baths for 10 minutes each before using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We transferred graphene onto the front substrate of the flow cell following a wet transfer procedure [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' First, we spin-coated graphene samples with CAB at 4000 rpm and baked them at 180◦C for 3 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Then, to remove unnecessary graphene on the backside of copper substrates, the CAB-coated graphene samples were immersed into a 1 M solution of APS for 10 minutes and subsequently rinsed with MilliQ water five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The copper substrates were then fully etched by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 M APS solution for 2 hours, followed by a five times rinse with MilliQ water to remove the attached ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Then, the floating CAB-graphene monolayers were "fished" onto the flow cell, and the CAB coating was removed by soaking in acetone for 2 hours and in isopropanol for one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 OPTP measurements We probed electron relaxation in graphene using optical pump - terahertz probe (OPTP) spec- troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' A schematic of the OPTP setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The fundamental laser output was generated by a regenerative Ti:sapphire amplifier system, which produces 5 W, 50 fs pulses at a repetition rate of 1 kHz and a central wavelength of 800 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The generated pulses were then split into three branches for THz generation, sampling, and optical excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' A single-cycle THz pulse of ∼ 1 ps duration was generated by pumping a 1 mm thick (110) ZnTe crystal with the 800 nm fundamental pulses via optical rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We photoexcited graphene to generate hot carriers by using 800 nm pulses with a diameter of 1 X Polarizer Delay stage Beam splitter P Chopper ZnTe 入/4 Wollaston sample prism White foam ZnTe Differential detector0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='08 Pump-probe delay (ps) N2 Fluence ( 2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='29 c b a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='08 0 200 400 600 800 1000 1200 Peak value of 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='08 Peak value of Fluence (μJ/cm2) 0 200 400 600 800 1000 1200 T e-T l (K) T e-T l (K) ∆E/E ∆E/E ∆E/E μJ/cm Figure S2: Electron temperature of the pumped graphene layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='The OPTP traces of graphene in a nitrogen atmosphere with various excitation fluences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Peak value of ∆E/E as a function of laser fluence and corresponding electron temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Increase in electron temperature (Te) with respect to ambient temperature Tℓ as a function of ∆E/E: a linear relation is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 5 mm to ensure a homogeneously photoexcited region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The transmitted THz wave was then recol- limated and focused onto a ZnTe detection crystal together with an 800 nm sampling beam, where the THz electrical field waveform was detected using the electro-optic sampling method [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The THz pulse induces birefringence in the ZnTe detection crystal, and the polarization of the sampling beam is thus changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' After passing through a quarter-wave plate, the sampling beam changes from perfectly circular to slightly elliptical shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The s and p components of this elliptically polarized pulse are separated by a Wollaston prism, and the difference of these two components is detected by a balance diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The signal is collected by a lock-in amplifier that is phase-locked to an optical chopper that modulates either the THz generation beam or the pump beam at a frequency of 500 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The ultrafast time evolution of the peak intensity of the THz field is tracked by varying the time delay between optical pump and THz probe [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The setup was purged with dry nitrogen during the measurement to avoid the absorption of water vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The raw data consists in time traces of the pump-induced transmission change at the peak of the THz waveform (∆E), normalized by the peak value of the THz transmission without excitation (E) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Assuming that a fraction γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='6% of the pump pulse energy is absorbed by the graphene electrons [5], the maximum electron temperature reached after photoexcitation can be related to the pump laser fluence F according to γF = C(Te)Te, where C(Te) is the graphene heat capacity at temperature Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In the limit where the graphene Fermi energy µ is larger than kBTe (as relevant for our samples), we may use the approximate expression [6, 7, 8] C(Te) = αTe, with α = 2π 3 k2 Bµ (ℏvF)2 , (1) where vF is graphene’s constant Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Then, Te = T0 � 1 + 2γF αT 2 0 �1/2 , (2) where T0 is ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The peak value of ∆E/E after photoexcitation increases with laser fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Upon rescaling, we find that the plots of ∆E/E vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' F and Te vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' F collapse upon each other (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S2b), so that we may consider that ∆E/E is proportional to the electron temperature within the range of temperatures probed in the experiment, as shown explicitly in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The thickness of the liquid layer was set to 50 µm by the geometry of the flow cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The liquids were exchanged using a syringe and the spectroscopic measurement was always carried out at 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='0 no cover with cover 10 μm 20 μm 30 μm 40 μm 50 μm 60 μm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='80 Day 1 Day 2 Day 3 no cover with cover 10 μm 20 μm 30 μm 40 μm 50 μm 60 μm Cooling time (ps) Normalized ∆E/E b a Pump-probe delay (ps) Figure S3: Control experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The OPTP traces of graphene with varying water layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Cooling times obtained by exponential fitting of the data in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' the same spot of the graphene sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' To exclude the effect of beam dispersion in the different liquids on the results, we repeated the measurement with different water layer thickness and using different Teflon spacers between two fused silica windows (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3 FTIR measurements We measured the dielectric functions of water, heavy water, ethanol and methanol using Fourier- transform infrared (FTIR) spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We measured the transmitted and reflected infrared intensities both for an empty cell (It,cell, Ir,cell) and for a cell filled with liquid (It,liquid, Ir,liquid) thanks to an A510/Q-T Reflectance and Transmittance accessory placed in a commercial VERTEX 70 FTIR spectrometer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In order to avoid disassembling the cell when changing liquids, we carried out the measurements inside a flow cell, made out of two-silicon wafers separated by a 10 µm Teflon spacer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We calculated the absorbance A(ω) according to A(ω) = − log10 � It,solution(ω) It,cell(ω) + Ir,cell(ω) − Ir,solution(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3) The H2O and D2O show saturated absorption in the range of 3100-3600 and 2200-2700 cm−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We obtained the data in this frequency range by measuring the spectra without any spacer between two CaF2 windows and then rescaled the spectra to overlap with the data with spacer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The imaginary part k(ω) of the refractive index is related to the absorbance by k(ω) = A(ω)ln(10) 4πωℓ , (4) where ℓ is the sample thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' To accurately determine the thickness of the cell, we calculate the absorbance of the empty cell without correction for multiple reflections, A2(ω) = − log10 � It,cell(ω) It,lamp(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (5) where It,lamp(ω) is the intensity of the lamp of the FTIR source (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Fourier transformation of this spectrum yields a peak at the time ∆t that light takes to travel twice through the cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S5b), so that ℓ = c∆t/2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='29 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We then obtained the real part of the refractive index through a numerical Kramers-Krönig transformation: n(ω) = n∞ + 2 π � ∞ 0 dω′ k(ω′) ω′ − ω, (6) 3 Figure S4: FTIR data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Raw intensity data of empty cell and water-filled cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Absorbance of H2O and D2O, measured with spacer and without spacer (the latter is rescaled to overlap with the former).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Figure S5: Determination of the cell thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Absorbance of empty cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Fourier transform of the data in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 4 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='20 celltransmission cellreflection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='15 solutiontransmission (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' solutionreflection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='00 1000 2000 3000 4000 5000 6000 7000 Frequency (cm-1) 6 H,0 b :D20 5 H,Owospacer(X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='0) bsorbance D,Owospacer(X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3) H,O combined 3 D,o combined 0 1000 2000 3000 4000 Frequency (cm-1)a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='06863ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05 Absorbance itensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='0 0 2000 4000 6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='09 Frequency(cm") Time (ps)a b Figure S6: Raman characterization of graphene sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Spatial map of Raman G band frequency for graphene sample in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Distribution of the Raman G band frequency with different liquids placed on the graphene surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' where n∞ is the refractive index in the high frequency limit, which is obtained by the ATAGO Digital Handheld Refractometer: PAL-RI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The measured values for H2O, D2O, methanol, ethanol and isopropanol are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='333, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3291, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3285, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3604, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3706 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We then obtain the dielectric function ϵ(ω) = ϵ′(ω) + iϵ′′(ω) according to � ϵ′(ω) = n(ω)2 − k(ω)2 ϵ′′(ω) = 2n(ω)k(ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4 Raman measurements We estimated the Fermi level µ in our liquid-covered graphene samples from the Raman G-band frequency, according to the empirical equation [6] |µ|(eV) = ωG − 1580 cm−1 42 cm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (8) An example of a spatial map of the Raman G-band frequency is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The fre- quency shows spatial inhomogeneities on the µm scale with an amplitude around 10 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The corresponding distributions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The average G-band frequency is essentially independent of the nature of the liquid, which excludes a change in charge carrier density as a possible mechanism for the liquid effect on the electron cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' To take into account the broadness of the distribution, in the theoretical analysis we considered chemical potentials in the range µ = 100 − 180 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The theoretical prediction is independent of the electron or hole nature of the charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2 Theoretical methods In this section, we develop a description of energy transfer between the Dirac fermion charge carriers in graphene and a liquid, treated as a bosonic bath, within the non-equilibrium Keldysh framework of perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' For the sake of completeness, and in order to show consistency with previous theoretical approaches, we apply the same description to energy transfer between the graphene electrons and its optical phonon modes, showing that our formalism recovers the results that were previously obtained within a Boltzmann equation approach [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 5 We use SI units throughout the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We adopt the following convention for the n-dimensional Fourier transform: ˜f(q) = � +∞ −∞ dnr f(r)e−iq·r and f(r) = 1 (2π)n � +∞ −∞ dnq ˜f(q)eiq·r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 Interaction Hamiltonian 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 Electron-hydron interaction In this section, r represents a vector in 3D space, and ρ a vector in 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The charge fluctuations of the liquid in the z > 0 half-space couple to the graphene electrons via the Coulomb potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In real space, the corresponding Hamiltonian is Hew(t) = � drdr′nw(r, t)V (r − r′)ne(r, t), (10) where nw and ne are the liquid and graphene instantaneous charge density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Let c† k,ν, ck,ν be the Dirac fermion creation and annihilation operators in the chiral basis (ν = ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' A 2D Fourier transformation then yields Hint = � dq (2π)2 e2 2ϵ0qns(q, t) � k,ν,ν′ ⟨k + q, ν|eiqρeqz|k, ν′⟩c† k+q,ν(t)ck,ν′(t), (11) with ns(q) = � dρ � +∞ 0 dz e−iqρe−qznw(ρ, z, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (12) As long as we consider wavevectors q such that q−1 is large compared to the extension of the carbon pz orbitals perpendicular to the graphene plane, we may approximate |⟨k + q, ν|eiqρeqz|k, ν′⟩|2 ≈ |⟨k + q, ν|eiqρ|k, ν′⟩|2 = 1 2 �1 + νν′ cos(φk+q − φq) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (13) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 Electron-phonon interaction Let d† q,α, dq,α be the creation and annihilation operators of phonons in the mode α with frequency ωα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The non-interacting electron-phonon system’s Hamiltonian is H0 = � k,ν Ek,νc† k,νck,ν + � q,α ℏωαd† q,αdq,α, (14) where Ek,ν are the band energies, and � k ≡ (1/ABZ) � BZ dk (ABZ is the area of the 2D Brillouin zone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The electron-phonon interaction Hamiltonian has the general form [10] Hep = � α � BZ dq (2π)2 � k,ν,ν′ gνν′ α,k,k+qc† k+qck(d† q,α + d−q,α), (15) Following [9], we consider the Γ point LO and TO phonons that scatter electrons within one valley, and the K, K’ point LO phonons that scatter electrons between valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The electron-phonon matrix elements read |gνν′ Γ,k,k+q|2 = g2 Γ(1 ± νν′ cos(φk + φk+q − 2φq)), (16) where the + (−) sign is for LO (TO) phonons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' and |gνν′ Γ,k,k+q|2 = g2 K(1 ∓ νν′ cos(φk − φk+q)), (17) where the − (+) sign corresponds to scattering from K to K’ (from K’ to K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' here, φv is the polar angle of the vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The values of the coupling constants are gΓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='55 eV·Å and gK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='85 eV·Å, according to GW calculations [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3 General form We find that for both types of interactions the Hamiltonian has the general form Heb = � dq (2π)2 nq(t)ϕq(t), (18) where nq is an electronic two-particle operator and ϕq is a free bosonic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In the electron-phonon case, we define nq = � k,ν,ν′ gνν′ α,k,k+q √ℏωα c† k+q,νck,ν′ and ϕq = � ℏωα(d† q,α + d−q,α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (19) in the electron-hydron case nq = � Vq � k,ν,ν′ ⟨k + q, ν|eiqρ|k, ν′⟩c† k+q,νck,ν′ and ϕq = � Vqns(q), (20) where Vq ≡ e2/(2ϵ0q) is the 2D Fourier-transformed Coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' With these definitions, both nq and φq have dimensionless correlation functions in frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 General theory of electron-boson heat transfer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 Non-equilibrium perturbation theory We consider an initial state of the electron-boson system where the electrons are at a temperature Te and the bosons at a temperature Tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We wish to study the subsequent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In particular, we are interested in the heat flux per unit surface from the electrons to the bosons: Q(t) = − 1 A d dt⟨Heb(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (21) Since the system is under non-equilibrium conditions, this average value needs to be computed in the Keldysh framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In particular, we may define the Keldysh component of the electron-boson correlation function: χK eb(q, t, t′) = − 1 A i ℏ⟨{nq(t), ϕ−q(t′)}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (22) Then, Q(t) = −iℏ 2 � dq (2π)2 dχK eb(q, t, t) dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (23) Form this point on, the computation of the electron-boson correlation function follows the exact same steps as in the theory of quantum friction [12], and we reproduce here only the main equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Diagramatically, the correlation function satisfies the following Dyson equation: (24) where the "bubble" represents the propagator of n (denoted χe), and the dashed line the propagator of ϕ (denoted χb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' When made explicit in terms of the R, A, K components, the Dyson equation becomes � � � � � χK eb = χR e ⊗ χK b + χK e ⊗ χA b + χR e ⊗ χR b ⊗ χK eb + (χR e ⊗ χK b + χK e ⊗ χA b ) ⊗ χA eb χR,A eb = χR,A e ⊗ χR,A b + χR,A e ⊗ χR,A b ⊗ χR,A eb , (25) where ⊗ represents time convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' While these equations are extremely general, they are impractical to manipulate analytically, unless a number of assumptions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In order to 7 proceed, we will restrict ourselves to cooling dynamics that are slow enough for time-translation invariance to hold when it comes to determining the cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' This assumption is expected to hold for small enough temperature differences, such that the cooling rate is approximately temperature-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We will further assume that, in line with experimental observations, that electron thermalization is much faster than electron-boson energy transfer, so that the electron and boson propagators may be considered as equilibrium propagators, satisfying the fluctuation- dissipation theorem: we work within a two-temperature model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We may then carry out Fourier transforms in time, so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (23) becomes Q = 1 2 � dqdω (2π)3 ℏω χK eb(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (26) The convolutions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (25) become products in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Before proceeding, it is convenient to flip the signs of all the correlation functions: we introduce, for all the labels, g ≡ −χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Then, after some algebra, we obtain an explicit expression for Q: Q = 1 2π3 � dq � +∞ 0 dω ℏω[nB(ω, Te) − nB(ω, Tb)]Im [gR e (q, ω)]Im [gR b (q, ω)] |1 − gR e (q, ω)gR b (q, ω)|2 , (27) where nB(ω, T) ≡ 1/(eℏω/kBT − 1) is the Bose distribution at temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We recover Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (3) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 Cooling rate The cooling dynamics are governed by the equation dE(Te) dt = −Q(Te, Tb), (28) where E is the total energy per unit surface of the electronic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We follow ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' [9] in de- termining the electronic heat capacity (per unit surface) at constant density C(Te), such that dtE = C(Te)dtTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We may then define the instantaneous cooling rate τ(Te, Tb) = C(Te)(Te − Tb) Q(Te, Tb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (29) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3 Application to the graphene-liquid system 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='1 Liquid-mediated cooling We first consider electron cooling through the electron-hydron coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Using eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (12) and (20), we find that gR b (q, t, t′) = − 1 AVq � +∞ 0 dzdz′ e−q(z+z′) � − i ℏθ(t − t′)⟨[ns(q, z, t), ns(−q, z′, t′)]⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (30) This is the microscopic definition of the liquid’s surface response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' In the long wavelength limit, it can be expressed in terms of the liquid’s bulk dielectric function ϵ(ω) [12]: gR b (q, ω) = ϵ(ω) − 1 ϵ(ω) + 1, (31) as stated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The electronic response function gR e (q, ω) simply amount to (minus) the density-density response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Taking into account electron-electron interactions at the RPA level [13], gR e (q, ω) = − Vqχ0 e(q, ω) 1 − Vqχ0e(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (32) 8 600 800 1000 1200 Electron temperature (K) 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='5 Cooling time (ps) Theory Experiment Figure S7: Dependence of water-mediated cooling time on initial electron temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The red dots are experimental data for graphene in contact with water and the red dots correspond to the prediction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (29) (with µ = 180 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The non-interacting response function χ0 e is given by [13] χ0 e(q, ω) = gsgv � dk (2π)2 � ν,ν′ |⟨k + q, ν|eiqρ|k, ν′⟩|2 nF(Eν k, Te) − nF(Eν′ k+q, Te) Eν k − Eν′ k+q + ω + iδ , (33) where gs = gv = 2 are the spin and valley degeneracies of graphene, respectively, Eν k = νvF k are the band energies in the Dirac fermion approximation, nF(E, T) = 1/(e(E−µ)/kBT +1) is the Fermi distribution at chemical potential µ and temperature T, and δ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' The integral is evaluated numerically at non-zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' With all the above, we may compute theoretical predictions for the liquid-mediated cooling rate by numerical integration according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We considered a graphene chemical potential µ in the range 100 − 180 meV (see section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='4) and an electron temperature Te = 623 K, corresponding to the lowest pump laser fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Our model is further able to reproduce the dependence of the electron cooling time on Te, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' We note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (27) involves bare surface response functions, that contain no effect of the presence of the neighboring medium, at least at the RPA level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Nevertheless, the physical response function of graphene in the presence of water undergoes RPA renormalization according to (34) In this diagrammatic equation, when the propagators are interpreted as surface response functions, the vertices reduce to unity, so that we obtain the renormalized graphene response function ˜ge as ˜ge(q, ω) = ge(q, ω) 1 − ge(q, ω)gb(q, ω), (35) which is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (7) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content='2 Phonon-mediated cooling In the phonon case, the boson response function is proportional to the usual phonon propagator: gR b (q, ω) = 2ω2 α ω2α − ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (36) 9 The non-interacting electronic response function now involves the electron-phonon matrix elements: gR e (q, ω) = −gs � BZ dk (2π)2 � ν,ν′ |gνν′ α,k,k+q|2 ℏωα nF(Eν k, Te) − nF(Eν′ k+q, Te) Eν k − Eν′ k+q + ω + iδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (37) We now show that we recover the results of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' [9] for the electron-phonon cooling rate obtained in a Boltzamann equation framework, if we neglect electron-electron interactions and treat electron- phonon interactions to first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Under these assumptions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (27) reduces to Q = 1 2π3 � dq � +∞ 0 dω ℏω[nB(ω, Te) − nB(ω, Tb)]Im [gR e (q, ω)]Im [gR b (q, ω)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (38) We notice that Im [gR b (q, ω)] = πω2 α[δ(ω − ωα) − δ(ω + ωα)] (39) and Im [gR e (q, ω)] = πgs � BZ dk (2π)2 � ν,ν′ |gνν′ α,k,k+q|2 ℏωα [nF(Eν k, Te) − nF(Eν′ k+q, Te)]δ(Eν k − Eν′ k+q + ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (40) Moreover, upon integration over k and q in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (38), the angle-dependent parts of the electron- phonon matrix elements vanish, and the intervalley phonons become formally identical to the intravalley phonons: we may introduce the valley degeneracy and carry out integrations over a single Dirac cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Altogether, we obtain Q = 2πgsgvωαg2 α[nB(ωα, Te) − nB(ωα, Tb)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' · · � ν,ν′ � dqdk (2π)4 [nF(Eν k, Te) − nF(Eν′ q , Te)]δ(Eν k − Eν′ q + ωα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (41) If we introduce another delta function, according to Q = 2πgsgvωαg2 α[nB(ωα, Te) − nB(ωα, Tb)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' · · � ν,ν′ � dqdk (2π)4 � +∞ −∞ dϵ[nF(ϵ − ωα, Te) − nF(ϵ, Te)]δ(Eν k − ϵ + ωα)δ(ϵ − Eν′ q ), (42) we recognize the graphene density of states, ν(ϵ) = gsgv � ν � dk (2π)2 δ(ϵ − Ek,ν) = 2|ϵ| πv2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (43) Our result then simplifies according to Q = 2πωαg2 α gsgv [nB(ωα, Te) − nB(ωα, Tb)] � +∞ −∞ dϵ[nF(ϵ − ωα, Te) − nF(ϵ, Te)]ν(ϵ)ν(ϵ − ωα), (44) which is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' (18) in the supplementary information of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' References [1] Yogeswaran, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Piezoelectric graphene field effect transistor pressure sensors for tactile sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Applied Physics Letters 113, 014102 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' [2] Burwell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=', Smith, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' & Guy, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfeA0g/content/2301.05095v1.pdf'} +page_content=' Investigation of the utility of cellulose acetate 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b/BtE5T4oBgHgl3EQfTQ-D/content/tmp_files/load_file.txt @@ -0,0 +1,566 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf,len=565 +page_content='Using the profile of publishers to predict barriers across news articles Abdul Sittar1,2[0000−0003−0280−9594] and Dunja Mladeni´c1,2[0000−0002−0360−6505] 1 Joˇzef Stefan Institute, Slovenia, 2 Joˇzef Stefan International Postgraduate School, Slovenia, Jamova cesta 39 {abdul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='sittar, dunja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='mladenic}@ijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='si Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Detection of news propagation barriers, being economical, cultural, political, time zonal, or geographical, is still an open research issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We present an approach to barrier detection in news spreading by utilizing Wikipedia-concepts and metadata associated with each bar- rier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Solving this problem can not only convey the information about the coverage of an event but it can also show whether an event has been able to cross a specific barrier or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Experimental results on IPoNews dataset (dataset for information spreading over the news) reveals that simple classification models are able to detect barriers with high accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We believe that our approach can serve to provide useful insights which pave the way for the future development of a system for predicting information spreading barriers over the news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Keywords: news propagation news spreading barriers cultural bar rier economical barriers geographical barrier political barrier time zone barrier classification methods 1 Introduction The phenomenon of event-centric news spreading due to globalization has been exposed internationally [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' International events capture attention from all cor- ners of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' News agencies play their part to bring our attentions on some events and not on others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Varying nature of living styles, cultures, economic con- ditions, time zone, and geographical juxtaposition of countries present a signifi- cant role in process of publishing news related to different events [3, 6, 13, 19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For example, publishing about sports events could be dependent on culture, epi- demic events can reach firstly to neighboring countries due to geographic prox- imity and, news on a luxury product may be relevant for economically strong countries due to demand of wealthy people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We represent this differentiation along with different barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' These barriers include but are not limited to 1) Economic Barrier, 2) Cultural Barrier, 3) Political Barrier, 4) Geographical Bar- rier, and 5) Time Zone Barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Detection of the overpass of these barriers does Copyright © 2021 for this paper by its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Use permitted under Creative Commons License Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Sittar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' not only tell us the area where the broadcasting of an event reached, but it also shows us events-location relation as countries have different culture, economic conditions, geographical placement on the globe, political point of view, and time zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Following are the definitions of news crossing these barriers: Cultural Barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' If we identify the coverage of specific event-centric news by publishers that are surrounded by different cultures, then we can say that the news related to the event crossed cultural barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Political Barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' If news about a specific event is disseminated from publishers having different political alignment, we can say that the news related to that event crossed the political barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Geographical Barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We say that some news related to a specific event overpasses geographical barriers if that event gets attention by publishers of countries located in different geographical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Time Zone Barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We can claim that event-centric news has crossed the time zone barrier if it has been published by publishers located in different time zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Economic Barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' It can be asserted that a piece of event-centric news has crossed economic barriers if it is published in countries having different economic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' In this paper, we propose a methodology for detection of different barriers during information propagation in form of news that utilize data (IPoNews) [18] related to three contrasting events (earthquake, Global warming, and FIFA world cup) in different domains (natural disasters, climate changes, and sports) in 5 different languages: English, Slovene, Portuguese, German, and Spanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='1 Contributions Following are the main scientific contributions of this paper: – A novel methodology for barrier detection in news spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' – Experimental comparison of several simple classification models that can serve as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='2 Problem Statement Observing the spreading of news on a particular event over time, we want to predict whether a barrier (cultural, political, geographical, time zone, economi- cal) is likely to hamper information while information propagates over the news (binary classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 2 RELATED WORK Multiple barriers come across event-centric news specifically when the news is concerned about international or national events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' According to news flow theo- ries, multiple determinants impact international news spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' The economic Using the profile of publishers to predict barriers across news articles 3 power of a country is one of the factors that influence news spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Moreover, economic variations has different influence for different events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' protests, con- flicts, disasters) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' The magnitude of economic interactivity between countries can also impact the news flow [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Economic growth/income level shows the eco- nomic condition of a country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Multiple organizations are working on generating prosperity and welfare index on yearly basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Among them, “The Legatum Pros- perity Index” and “Human Development Index” are popular 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Geographical representation of entities and events has been utilized extensively in the past to detect local, global, and critical events [3, 13, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' It has been said that countries with close distance share culture and language up to a certain extent which can further unfold interesting facts about shared tendencies in informa- tion spreading [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' News agencies tend to follow the national context in which journalists op- erate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' One of the related examples is the SARS epidemic study which found that cross-national contextual values such as political and economic situations impact the news selection [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' It will be true to say that fake news is produced based on many factors and it is surrounded by a paramount factor that is polit- ical effect [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' A great amount of work regarding fake news dwells on different strategies and few studies considered political alignment to have a compelling effect on news spreading [4, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' [12] strongly proved it to be a major strategy in news agencies to control the news and change accordingly due to the involve- ment of journalists and political actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Countries that share common culture are expected to have heavier news flow about between them reporting on similar events [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Many quantitative studies found demographic, psychological, socio- cultural, source, system, and content-related aspects [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Many models have tried to explain cultural differences between societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Hofstede’s national culture di- mensions (HNCD) has been widely used and cited in different disciplines [7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' News classification for different kinds of problems is a well-known topic since the past and features used to classify varies depending upon the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' [17] used news content and user profile to classify the news whether it is fake or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' [2] calculated TF-IDF score and Word2Vec score of most frequent words and used them as features to classify into one of the five categories (state, econ- omy, entertainment, international, and sports).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Similarly, [14] performed part- of-speech (POS) tagging at sentences level and used them as features, and built supervised learning classifiers to classify news articles based on their location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Mostly classifier trained to utilize popular supervised learning methods such as Random Forest, Support Vector Machine (SVM), Naive Bayes, k-Nearest Neigh- bour (kNN), and Decision Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' In this work, we used the profile of each barrier for each news publisher (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='5) and most frequent 300 Wikipedia con- cepts from the dataset that appeared in the list of news articles related to three contrasting events (earthquake, Global Warming, and FIFA world cup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We also 1 http://hdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='undp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='org/en/content/human development index hdi 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='prosperity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com/ 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Sittar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' ≥ compared the results of popular classifiers such as SVM, Random Forest, Deci- sion Tree, Naive Bayes, and kNN (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 3 DATA DESCRIPTION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='1 Dataset We utilized dataset ”A dataset for information spreading over the news (IPoNews)” that consists of pairs of news articles that were labeled based on the level of their similarity, as described in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' This dataset was collected from Event Registry, a platform that identifies events by collecting related articles written in differ- ent languages from tens of thousands of news sources [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' The similarity score among cross-lingual news articles was calculated using concept-based similar- ity employing Wikifier service3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' [18] describes the criteria when information is considered to be propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Statistics of the data set are shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Statistics about dataset Dataset Domain Event type Articles per Language Total Articles 1 Sports FIFA World Cup Eng Spa Ger Slv Por 2682 983 762 711 10 216 2 Natural Disaster Earthquake 941 999 937 19 251 3147 3 Climate Changes Global Warming 996 298 545 8 97 1944 The dataset contains a list of pairs of news articles annotated with one of the labels such as ”information-Propagated”, ”Unsure”, or ”Information-Not- Propagated” (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' The information is considered to be propagated if the cosine similarity score of the two articles in the pair is above a predefined thresh- old ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='7 for Information-Propagated, < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='4 for Information-not-Propagated, otherwise Unsure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We restructured the original dataset to include only exam- ples labeled as spreading information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' In this way, we have pair of news articles where we observe information spreading from one to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Furthermore, for each example, instead of having a pair of articles, we kept only the article that was published earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' In this way, each example contains an article that spreads information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Articles with metadata from to weight Class from-publisher to-publisher from-pub-uri to-pub-uri Por44 Por43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='627 Unsure ClicRBS SAPO 24 jornald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='clicrbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='br 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='sapo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='pt English881 English880 1 Information-Propagated Sky News 247 Wall St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com 247wallst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com English258 English329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='313 Information-Not-Propagated Sify 4-traders sify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com 4-traders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com English793 English787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='238 Information-Not-Propagated Bioengineer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='org 7NEWS Sydney scienmag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com 7news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='au German237 German236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='979 Information-Propagated watson watson aargauerzeitung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='ch aargauerzeitung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='ch 3 http://wikifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='org/info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='html, https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com/abdulsittar/IPoNews Using the profile of publishers to predict barriers across news articles 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='2 Statistics after restructuring the data The original dataset describes in Section 3 contains pairs of articles along with the information on whether there was the propagation of information related to a specific event or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We used only examples labeled as propagating information 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Based on the available metadata for articles, we ignored articles that do not have metadata information in our database (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Table 3 shows the statistics for each barrier after filtering the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Statistics about barrier Dataset Domain Event type Articles for each barrier 1 Sports FIFA World Cup Time-Zone Cultural Political Geographical Economical 724 699 143 726 634 2 Natural Disaster Earthquake 1102 1113 227 1113 1010 3 Climate Changes Global Warming 586 445 108 487 463 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='3 Wikipedia Concepts as Features As our dataset already mention (see Section 3) if information in news is spread- ing from an article to another based on Wikipedia-concepts, we utilized the most frequent (top 300) Wikipedia-concepts as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Figure 1 portrays these Wikipedia-concepts for all three events in form of word clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Word clouds of most frequent words related to earthquake, FIFA World Cup and Global Warming events respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='4 Barriers Knowledge Barriers knowledge refers to a database that contains metadata about each bar- rier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Figure 3 shows schema of database and Table 4 presents barriers along with their characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Each barrier depends on one main information that is the country name of the headquarter of the news publishers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Since the utilized data 4 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='3950064 DEBpresident United Yor Wart overnmen States nameGermanname football SWar ummerWorld assoclation Unitednationa CUDFIFAASSO ation FIFAWorldYorKname War States IInchEarthFranceUnited United New Globa disambiguation6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Sittar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' set already contains headquarter of publishers therefore we fetched the coun- try associated with headquarters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For economical barrier, we fetched economical profile for each country using “”The Legatum Prosperity Index”” 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Cultural differences among different regions were collected using Hofstede’s national cul- ture dimensions (HNCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For time zone and geographical barrier, we stored general UTC-offset, latitude, and longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For political barrier we are using the political alignment of the newspaper/magazine that we determined based on Wikipedia infobox at their Wikipedia page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For instance, for Austrian newspa- per ”Der Standard” we find social liberalism as political alignment (See Figure 2), for British newspaper ”Daily Mail” we find right-wing as political alignment, for German ”Stern” magazine there is no information in its Wikipedia infobox on the political alignment thus we label political alignment as unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Three Wikipedia infobox for three different newspapers/magazines with political alignment 5 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='prosperity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com/ Der Standard DERSTANDARD Type Daily newspaper Owner(s) Oscar Bronner Publisher Oscar Bronner Martin Kotynek Founded 19 October 1988: 32 years ago Political Social liberalism alignment Headguarters Vienna Circulation 86,000 (2013) Website www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='derstandard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='de www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='derstandard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='at DailyMail DailumlailFREE MICHELIN SO MUCH FOR THE BONFIRE OF THE QUANGOS!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' aplasticheart DailyMail frontpageon 4August 2010 Type Dailynewspaper Format Tabloid Owner(s) DailyMail and General Trust Founder(s) AlfredHarmsworthandHarold Harmsworth Publisher DMGMedia Editor GeordieGreig Founded 4 May1896:124 years ago Political Right-wing[1]2][3] alignment Language English Headquarters Northcliffe House 2 Derry Street LondonW85TT Circulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='184(asofFebruary 2020)[4] ISSN 0307-7578 OCLC 16310567 number Website www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='dailymail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='ukStern ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' stern ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' stern KRERSMID Alein in turopa IXABE HRSECIEENE Sternmagazinecoveron18February2016 Editor FlorianGless,Anna-Beeke Gretemeier Categories Newsmagazine FrequencyWeekly Circulation390,000(2020) Year 1948 founded Firstissue 1August1948,72yearsago Company Gruner+Jahr Country Gemany Basedin Hamburg Language Geman Website www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='stern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='de ISSN 0039-1239Using the profile of publishers to predict barriers across news articles 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='5 Features for Individual Barrier We represented each barrier with a specific profile containing a list of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Table 4 depicts the list of features for each barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Economic and cultural bar- riers consist of a vector of length 11 and 6 features whereas geographical, time zone, and political only contain 1 or 2 features such as latitude-longitude, UTC- offset, and political alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Database Schema for Barriers 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='6 Dataset Annotation We queried the metadata information for each article and generated a CSV file for each barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We annotated each article based on that meta information to be used for model training and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For economic and cultural barriers, we calculated cosine similarity between vectors of economical values and vectors of cultural values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Score greater than the threshold value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='9 labeled as FALSE otherwise TRUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We set the lowest value as a threshold based on the fact that if two countries have a little gap concerning culture or economical values then there exists a barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For geographical barriers, we compared the latitude and longitude of the country of each publisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' If a country name or lat/lat appeared to be the same then we annotated it with FALSE otherwise TRUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=" Lastly, for Enterprise Conditions Social Capital Education EconomicQuality Marketaccessand Individualistic cuiture Living Conditions Economic infrastructure Power distance Profile Governance Natural Environment afety Fam Health nty Cultural Value's Has Has induigence vs restraint Headquarter Has Country Has Geographical Values (Lat/Lon) Has name Has Longitude Latitude Political barrier Time Zone Political UTC offset Alignment8 A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Sittar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Features of each barrier Barrier Features Economic Rank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Safety Security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Personal Freedom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Governance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Social Capital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Investment Environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Enterprise Conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Market Infrastructure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Economic Quality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Living Conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Health,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Natural Environment Cultural Power Distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Uncertainty Avoidance By Individuals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Individualistic Cultures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Masculinity Femininity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Long Term Orientation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Indulgence Restraint Geographical Latitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Longitude Time Zone UTC offset Political Political Alignment time-zone and political barriers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' we followed the same process that was for the geographical barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' if political alignment or UTC-offset appeared to be the same for a pair then it is annotated with FALSE otherwise TRUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Figure 4 depicts the class distribution for each barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We can notice unbalanced class distribution with majority of the examples being False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' This is especially true for Cultural and Political barrier with 91 percent of example being False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Thus in our evaluation we rely more on F1 measure than classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Class Distribution for Each Barrier 2000 True 2014 False 1500 1588 1599 1324 1000 948 670 500 478 408 203 42 0 I Barrier nical olitical Barrier Econom TimeUsing the profile of publishers to predict barriers across news articles 9 4 MATERIALS AND METHODS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='1 Problem Modeling For each barrier, we have a list of news articles where each article is associated with 300 Wikipedia-concepts and features related to that barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' The task is to predict the status S of each barrier B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' S = f (C, B) f is the learning function for barrier detection, C is donating here Wikipedia- concepts related to an article and B is the list of features related to a specific barrier (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='2 Methodology We utilized dataset IPoNews [18] and built a database on top of this dataset that includes barrier knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Figure 5 explains the overall process of model construction from news articles to results generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We created a list of in- stances using the most frequent Wikipedia-concepts based on news articles and joined them along with barrier knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' After performing the annotation (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='6), we trained popular classification models and generated the results on test data (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Steps for Model Construction 5 EXPERIMENTAL EVALUATION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='1 Baselines We used the following methods as baselines for all our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' – Uniform: Generates predictions uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' – Stratified: Generates predictions by respecting the training set’s class dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' – Most Frequent: Always predicts the most frequent label in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=" Barrier's Results knowledge Testset Newsarticles (IPoNews) Metadata Barriers'Annotation Wikipediaconcepts Model Construction Trainset10 A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Sittar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' sum sum sum sum 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='2 Classification Methods We trained popular classification models for each barrier such as SVM, kNN, Decision Tree, Random Forest, and Naive Bayes using Scikit-Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We applied a stratified 10-fold cross-validator to split the dataset for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For Random Forest, kNN, and Decision Tree, we varied the size of n-estimator, value of k, and max-leafs and chosen the one with the best score on test data respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Implementation of this methodology to barrier detection can be found on GitHub 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='3 Evaluation Metric Due to imbalance in the class distribution for all barriers, we used micro averaged precision and recall to evaluate our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 7 – Micro-Precision: The precision of average contributions from each class is calculated in micro-precision whereas the following question is answered by precision: What proportion of positive predictions was correct?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' It is defined as: TruePositivesum Micro − Precision = TruePositive + FalsePositive – Micro-Recall: Recall of average contributions from each class is calculated in micro-recall whereas the following question is answered by recall: What proportion of actual positives was predicted correctly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' It is defined as: TruePositivesum Micro − Recall = TruePositive + FalseNegative 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='4 Results and Analysis Table 5 shows the results of all the classifiers for each barrier along with baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Analysis of the experimental results show that overall all the machine learning models outperform the three baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For all the barriers, we can notice Micro- Recall is equal to Micro-Precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' The best performing baseline is the ”Most- frequent” with Micro-F1 for economic, cultural, geographical, time zone, and political barrier equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='90, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='58, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='70, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='90 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' The best performing models on all the barriers are Decision Tree, Random Forest, and kNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Looking at Micro-F1, we can see that on the Economic and Cultural barrier kNN achieved the best performance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='95 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' On Geographical barriers, kNN and Decision Tree performed the best achieving 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' On Time-Zone, the best performing classifier is Random Forest with Micro-F1 6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com/cleopatra-itn/BarrierDetection-Classification 7 https://peltarion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='com/knowledge-center/documentation/evaluation- view/classification-loss-metrics/micro-recall Using the profile of publishers to predict barriers across news articles 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' On Political barriers, SVM, kNN, and Random Forest achieve the best Micro-F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' In terms of classification accuracy, we can see that Random Forest outper- forms the baselines as well as the other four classifiers for the first four barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Notice that Random forest performs better than decision tree but takes more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Naive-Bayes achieves a little bit lower classification accuracy than the Deci- sion Tree for the first four barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' On the political barrier Naive-Bayes achieves the best classification accuracy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='98) but lower Micro-F1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 6 CONCLUSIONS AND FUTURE WORK It is highly important to detect the barriers while information propagates specif- ically through the news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' For journalists, marketers, and social scientists, the phe- nomenon of knowing which barrier appeared most frequently for what type of events, is significantly helpful to solve business and marketing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' In this regard, we proposed a simple methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Though its results are good enough for three types of events, we would like to enhance features as well as events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' We used only Wikipedia-concepts and meta information to detect barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' In the future, we would like to use DMoz categories provided by Event Registry [10], and transformation of the text of news articles as a feature for barrier detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Currently geographical and time zone barriers are calculated in a binary way ei- ther the same or different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' In the future, we would like to introduce the distance between countries and between time zones as labels instead of the currently used binary labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 7 ACKNOWLEDGMENTS The research described in this paper was supported by the Slovenian research agency under the project J2-1736 Causalify and co-financed by the Republic of Slovenia and the European Union’s Horizon 2020 research and innovation program under the Marie Sk-lodowska-Curie grant agreement No 812997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Sittar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Classifiers’ comparison with baselines Barrier Model CA Mic Pre Mic Rec Mic F1 Economic Uniform 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='49 Stratified 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='59 Most Frequent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='70 SVM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='69 kNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='75 Decision Tree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='73 Random Forest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='74 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' GeoInformatica pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' 1–31 (2020) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' : A brave new world for international news?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' exploring the determinants of the coverage of foreign news on us websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} +page_content=' International Communication Gazette 69(6), 539–551 (2007)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE5T4oBgHgl3EQfTQ-D/content/2301.05535v1.pdf'} diff --git a/CtAyT4oBgHgl3EQfR_f1/content/tmp_files/load_file.txt b/CtAyT4oBgHgl3EQfR_f1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..197be06d91fb9ce05da155fd1271986f351d67ef --- /dev/null +++ b/CtAyT4oBgHgl3EQfR_f1/content/tmp_files/load_file.txt @@ -0,0 +1,2932 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf,len=2931 +page_content='Semileptonic D Meson Decays D → P/V/Sℓ+νℓ with the SU(3) Flavor Symmetry/Breaking Ru-Min Wang1,†, Yue-Xin Liu1, Chong Hua1, Jin-Huan Sheng2,§, Yuan-Guo Xu1,♯ 1College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China 2School of Physics and Engineering, Henan University of Science and Technology, Luoyang, Henan 471000, China †ruminwang@sina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='com §jinhuanwuli@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='com ♯yuanguoxu@jxnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cn Many exclusive c → d/sℓ+νℓ (ℓ = e, µ, τ) transitions have been well measured, and they can be used to test the theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Motivated by this, we study the D → P/V/Sℓ+νℓ decays induced by the c → d/sℓ+νℓ transitions with the SU(3) flavor symmetry approach, where P denotes the pseudoscalar meson, V denotes the vector meson, and S denotes the scalar meson with a mass below 1 GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The different decay amplitudes of the D → Pℓ+νℓ, D → V ℓ+νℓ or D → Sℓ+νℓ decays can be related by using the SU(3) flavor symmetry and by considering the SU(3) flavor breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Using the present data of D → P/V/Sℓ+νℓ, we predict the not yet measured or not yet well measured processes in the D → P/V/Sℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' We find that the SU(3) flavor symmetry approach works well in the semileptonic D → P/V ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' For the D → Sℓ+νℓ decays, only the decay D+ s → f0(980)e+νe has been measured, the branching ratios of the D+ s → f0(980)e+νe and D → S(S → P1P2)ℓ+νℓ decays are used to constrain the nonperturbative parameters and then predict not yet measured D → Sℓ+νℓ decays, in addition, the two quark and the four quark scenarios for the light scalar mesons are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The SU(3) flavor symmetry predictions of the D → Sℓ+νℓ decays need to be further tested, and our predictions of the D → Sℓ+νℓ decays are useful for probing the structure of light scalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Our results in this work could be used to test the SU(3) flavor symmetry approach in the semileptonic D decays by the future experiments at BESIII, LHCb and BelleII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Introduction Semileptonic heavy meson decays dominated by tree-level exchange of W-bosons in the standard model have attracted a lot of attention in testing the stand model and in searching for the new physics beyond the stand model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Many semileptonic D → P/V ℓ+νℓ decays and one D → Sℓ+νℓ decay have been observed [1], and present experimental measurements give us an opportunity to additionally test theoretical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In theory, the description of semileptonic decays are relatively simple, and the weak and strong dynamics can be separated in these processes since leptons do not participate in the strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' All the strong dynamics in the initial and final hadrons is included in the hadronic form factors, which are important for testing the theoretical calculations of the involved strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The form factors of the D decays have been calculated, for examples, by quark model [2–7], QCD sum rules [8], light-cone sum rules [9–11], covariant light-front quark models [12–14], and lattice QCD [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The SU(3) flavor symmetry approach is independent of the detailed dynamics offering us an opportunity to relate different decay modes, nevertheless, it cannot determine the sizes of the amplitudes or the form factors by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='00079v1 [hep-ph] 31 Dec 2022 2 However, if experimental data are enough, one may use the data to extract the amplitudes or the form factors, which can be viewed as predictions based on symmetry, has a smaller dependency on estimated form factors, and can provide some very useful information about the decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The SU(3) flavor symmetry works well in the b-hadron decays [17–30], and the c-hadron decays [29–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Semileptonic decays of D mesons have been studied extensively in the standard model and its various extensions, for instance, in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [3, 46–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In this work, we will systematically study the D → P/V/Sℓ+νℓ decays with the SU(3) flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' We will firstly construct the amplitude relations between different decay modes of D → Pℓ+νℓ, D → V ℓ+νℓ or D → Sℓ+νℓ decays by the SU(3) flavor symmetry and the SU(3) flavor breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' We use the available data to extract the SU(3) flavor symmetry/breaking amplitudes and the form factors, and then predict the not yet measured modes for further tests in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The forward-backward asymmetries Aℓ F B, the lepton-side convexity parameters Cℓ F , the longitudinal polarizations of the final charged lepton P ℓ L, the transverse polarizations of the final charged lepton P ℓ T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions FL of the final vector mesons with two ways of integration have also been predicted in the D → P/V ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In addition, the q2 dependence of some differential observables for the D → P/V ℓ+νℓ decays are shown in figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' This paper will be organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' II, the theoretical framework in this work is presented, including the effective hamiltonian, the hadronic helicity amplitude relations, the observables and the form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The numerical results of the D → P/V/Sℓ+νℓ semileptonic decays will be given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Finally, we give the summary and conclusion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Theoretical Frame A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The effective Hamiltonian In the standard model, the four-fermion charged-current effective Hamiltonian below the electroweak scale for the decays D → Mℓ+νℓ (M = P, V, S) can be written as Heff(c → qℓ+νℓ) = GF √ 2 V ∗ cq¯qγµ(1 − γ5)c ¯νℓγµ(1 − γ5)ℓ, (1) with q = s, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The helicity amplitudes of the decays D → Mℓ+νℓ can be written as M(D → Mℓ+νℓ) = GF √ 2 Vcb � mm′ gmm′Lλℓλν m HλM m′ , (2) with Lλℓλν m = ϵα(m) ¯νℓγα(1 − γ5)ℓ, (3) HλM m′ = � � � ϵ∗ β(m′)⟨P/S(pP/S)|¯qγβ(1 − γ5)c|D(pD)⟩ ϵ∗ β(m′)⟨V (pV , ϵ∗)|¯qγβ(1 − γ5)c|D(pD)⟩ , (4) where the particle helicities λM = 0 for M = P/S, λM = 0, ±1 for M = V, λℓ = ± 1 2 and λν = + 1 2, as well as ϵµ(m) is the polarization vectors of the virtual W with m = 0, t, ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 3 The form factors of the D → P, D → S and D → V transitions are given by [2, 3, 13] � P(p) �� ¯dkγµc �� D(pD) � = f P + (q2)(p + pD)µ + � f P 0 (q2) − f P + (q2) � m2 D − m2 P q2 qµ, (5) � S(p) �� ¯dkγµγ5c �� D(pD) � = −i � f S +(q2)(p + pD)µ + � f S 0 (q2) − f S +(q2) � m2 D − m2 S q2 qµ � , (6) � V (p, ε∗) �� ¯dkγµ(1 − γ5)c �� D(pD) � = 2V V (q2) mD + mV ϵµναβε∗νpα Dpβ −i � ε∗ µ(mD + mV )AV 1 (q2) − (pD + p)µ(ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='pD) AV 2 (q2) mD + mV � +iqµ(ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='pD)2mV q2 [AV 3 (q2) − AV 0 (q2)], (7) where s = q2 (q = pD − pM), and ε∗ is the polarization of vector meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The hadronic helicity amplitudes can be written as H± = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (8) H0 = 2mDq|⃗pP | � q2 f P + (q2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (9) Ht = m2 Dq − m2 P � q2 f P 0 (q2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (10) for D → Pℓ+νℓ decays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' H± = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (11) H0 = i2mDq|⃗pS| � q2 f S +(q2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (12) Ht = im2 Dq − m2 S � q2 f S 0 (q2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (13) for D → Sℓ+νℓ decays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' and H± = (mDq + mV )A1(q2) ∓ 2mDq|⃗pV | (mDq + mV )V (q2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (14) H0 = 1 2mV � q2 � (m2 Dq − m2 V − q2)(mDq + mV )A1(q2) − 4m2 Dq|⃗pV |2 mDq + mV A2(q2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (15) Ht = 2mDq|⃗pV | � q2 A0(q2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (16) for D → V ℓ+νℓ decays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' where |⃗pM| ≡ � λ(m2 Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' m2 M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' q2)/2mDq with λ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' c) = a2 + b2 + c2 − 2ab − 2ac − 2bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Hadronic helicity amplitude relations by the SU(3) flavor symmetry Charmed mesons containing one heavy c quark are flavor SU(3) anti-triplets Di = � D0(c¯u), D+(c ¯d), D+ s (c¯s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (17) 4 Light pseudoscalar P and vector V meson octets and singlets under the SU(3) flavor symmetry of u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' s quarks are [57] P = � � � � π0 √ 2 + η8 √ 6 + η1 √ 3 π+ K+ π− − π0 √ 2 + η8 √ 6 + η1 √ 3 K0 K− K 0 − 2η8 √ 6 + η1 √ 3 � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (18) V = � � � � ρ0 √ 2 + ω8 √ 6 + ω1 √ 3 ρ+ K∗+ ρ− − ρ0 √ 2 + ω8 √ 6 + ω1 √ 3 K∗0 K∗− K ∗0 − 2ω8 √ 6 + ω1 √ 3 � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (19) where ω and φ mix in an ideal form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' and the η and η′ ( ω and φ) are mixtures of η1(ω1) = u¯u+d ¯d+s¯s √ 3 and η8(ω8) = u¯u+d ¯d−2s¯s √ 6 with the mixing angle θP (θV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' η and η′ (ω and φ) are given by � � η η′ � � = � � cosθP −sinθP sinθP cosθP � � � � η8 η1 � � , � � φ ω � � = � � cosθV −sinθV sinθV cosθV � � � � ω8 ω1 � � , (20) where θP = [−20◦, −10◦] and θV = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4◦ from Particle Data Group (PDG) [1] will be used in our numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The structures of the light scalar mesons are not fully understood yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Many suggestions are discussed, such as ordinary two quark states, four quark states, meson-meson bound states, molecular states, glueball states or hybrid states, for examples, in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [58–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In this work, we will consider the two quark and the four quark scenarios for the scalar mesons below or near 1 GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In the two quark picture, the light scalar mesons can be written as [67] S = � � � � a0 0 √ 2 + σ √ 2 a+ 0 K+ 0 a− 0 − a0 0 √ 2 + σ √ 2 K0 0 K− 0 K 0 0 f0 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (21) The two isoscalars f0(980) and f0(500) are obtained by the mixing of σ = u¯u+d ¯d √ 2 and f0 = s¯s � � f0(980) f0(500) � � = � � cosθS sinθS −sinθS cosθS � � � � f0 σ � � , (22) where the three possible ranges of the mixing angle, 25◦ < θS < 40◦, 140◦ < θS < 165◦ and − 30◦ < θS < 30◦ [58, 68] will be analyzed in our numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In the four quark picture, the light scalar mesons are given as [1, 69] σ = u¯ud ¯d, f0 = (u¯u + d ¯d)s¯s/ √ 2, a0 0 = (u¯u − d ¯d)s¯s/ √ 2, a+ 0 = u ¯ds¯s, a− 0 = d¯us¯s, K+ 0 = u¯sd ¯d, K0 0 = d¯su¯u, ¯K0 0 = s ¯du¯u, K+ 0 = s¯ud ¯d, (23) and the two isoscalars are expressed as � � f0(980) f0(500) � � = � � cosφS sinφS −sinφS cosφS � � � � f0 σ � � , (24) where the constrained mixing angle φS = (174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2)◦ [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 5 In terms of the SU(3) flavor symmetry, meson states and quark operators can be parameterized into SU(3) tensor forms, while the leptonic helicity amplitudes Lλℓ,λν m are invariant under the SU(3) flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' And the hadronic helicity amplitude relations of the D → Mℓ+νℓ(M = P, V, S) decays can be parameterized as H(D → Mℓ+νℓ) = cM 0 DiM i jHj, (25) where H2 ≡ V ∗ cd and H3 ≡ V ∗ cs are the CKM matrix elements, and cM 0 are the nonperturbative coefficients of the D → Mℓ+νℓ decays under the SU(3) flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Noted that the hadronic helicity amplitudes for the D → Sℓ+νℓ decays in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (25) are given in the two quark picture of the light scalar mesons, and ones in the four quark picture of the light scalar mesons will be given later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The SU(3) flavor breaking effects mainly come from different masses of u, d and s quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [70], the SU(3) breaking amplitudes of the D → Mℓ+νℓ decays can be give as ∆H(D → Mℓ+νℓ) = cM 1 DaW a i M i jHj + cM 2 DiM i aW a j Hj, (26) with W = � W i j � = � � � � 1 0 0 0 1 0 0 0 −2 � � � � , (27) where cM 1,2 are the nonperturbative SU(3) flavor breaking coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In the four quark picture of the light scalar mesons, the hadronic helicity amplitudes of the D → Sℓ+νℓ decays under the SU(3) flavor symmetry are H(D → Sℓ+νℓ)4q = c′S 0 DiSim jmHj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (28) And the corresponding SU(3) flavor breaking amplitudes of the D → Sℓ+νℓ decays are ∆H(D → Sℓ+νℓ)4q = c′S 1 DaW a i Sim jmHj + c′S 2 DiSim amW a j Hj + c′S 1 DiSim ja W a mHj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (29) In terms of the SU(3) flavor symmetry, the hadronic helicity amplitude relations for the D → Pℓ+νℓ, D → V ℓ+νℓ and D → Sℓ+νℓ decays are summarized in later Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' I, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IV and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' VIII, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Observables for the D → Mℓ+νℓ decays The double differential branching ratios of the D → Mℓ+νℓ decays are [56] dB(D → Mℓ+νℓ) dq2d(cos θ) = τDG2 F |Vcq|2λ1/2(q2 − m2 ℓ)2 64(2π)3M 3 D(s)q2 � (1 + cos2 θ)HU + 2 sin2 θHL + 2 cos θHP +m2 ℓ q2 (sin2 θHU + 2 cos2 θHL + 2HS − 4 cos θHSL) � , (30) where λ ≡ λ(m2 Dq, m2 M, q2), m2 ℓ ≤ q2 ≤ (mDq − mM)2, and HU = |H+|2 + |H−|2, HL = |H0|2, HP = |H+|2 − |H−|2, HS = |Ht|2, HSL = ℜ(H0H† t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (31) 6 The differential branching ratios integrated over cos θ are [56] dB(D(s) → Mℓ+νℓ) dq2 = τDG2 F |Vcq|2λ1/2(q2 − m2 ℓ)2 24(2π)3M 3 D(s)q2 Htotal, (32) with Htotal ≡ (HU + HL) � 1 + m2 ℓ 2q2 � + 3m2 ℓ 2q2 HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (33) The lepton flavor universality in D(s) → Mℓ+νℓ is defined in a manner identical Rµ/e as Rµ/e = � qmax qmin dB(D(s) → Mµ+νµ)/dq2 � qmax qmin dB(D(s) → Me+νe)/dq2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (34) The forward-backward asymmetries are defined as [56] Aℓ F B(q2) = � 0 −1 dcosθℓ dB(D→Mℓν) dq2dcosθℓ − � 1 0 dcosθℓ dB(D→Mℓν) dq2dcosθℓ � 0 −1 dcosθℓ dB(D→Mℓν) dq2dcosθℓ + � 1 0 dcosθℓ dB(D→Mℓν) dq2dcosθℓ (35) = 3 4 HP − 2m2 ℓ q2 HSL Htotal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (36) The lepton-side convexity parameters are given by [56] Cℓ F (q2) = 3 4 � 1 − m2 ℓ q2 � HU − 2HL Htotal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (37) The longitudinal polarizations of the final charged lepton ℓ are defined by [56] P ℓ L(q2) = (HU + HL) � 1 − m2 ℓ 2q2 � − 3m2 ℓ 2q2 HS Htotal , (38) and its transverse polarizations are P ℓ T (q2) = − 3πmℓ 8 � q2 HP + 2HSL Htotal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (39) The lepton spin asymmetry in the ℓ − ¯νℓ center of mass frame is defined by [71–74] Aλ(q2) = dB(D → Mℓ+νℓ)[λℓ = − 1 2]/dq2 − dB(D → Mℓ+νℓ)[λℓ = + 1 2]/dq2 dB(D → Mℓ+νℓ)[λℓ = − 1 2]/dq2 + dB(D → Mℓ+νℓ)[λℓ = + 1 2]/dq2 (40) = Htotal − 6m2 ℓ 2q2 HS Htotal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (41) For the D → V ℓ+νℓ decays, the longitudinal polarization fractions of the final vector mesons are given by [56] FL(q2) = HL � 1 + m2 ℓ 2q2 � + 3m2 ℓ 2q2 HS Htotal , (42) then its transverse polarization fraction FT (q2) = 1 − FL(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Noted that, for q2-integration of X(q2) = Aℓ F B, Cℓ F , P ℓ L, P ℓ T , Aλ and FL, following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [75], two ways of integration are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The normalized q2-integrated observables ⟨X⟩ are calculated by separately integrating the numerators and denominators with the same q2 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The “naively integrated” observables are obtained by X = 1 q2max − q2 min � q2 max q2 min dq2X(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (43) 7 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Form factors In order to obtain more precise observables, one also need considering the q2 dependence of the form factors for the D → Pℓ+νℓ, D → V ℓ+νℓ and D → Sℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The following cases will be considered in our analysis of D → P/V ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' C1: All form factors are treated as constants without the hadronic momentum-transfer q2 dependence, and different form factors are related by the SU(3) flavor symmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=', the SU(3) flavor breaking terms such as cM 1,2 and c′S 1,2,3 in later Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' I, IV and VIII are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' C2: With the SU(3) flavor symmetry, the modified pole model for the q2-dependence of Fi(q2) is used [76] Fi(q2) = Fi(0) � 1 − q2 m2 pole � � 1 − αi q4 m4 pole �, (44) where mpole = mD∗+ for c → dℓ+νℓ transitions and mpole = mD∗+ s for c → sℓ+νℓ transitions, and αi are free parameters and are different for f P + (q2), f P 0 (q2), V (q2), A1(q2) and A2(q2), we will take αi ∈ [−1, 1] in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' C3: With the SU(3) flavor symmetry, following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [2] Fi(q2) = Fi(0) � 1 − q2 m2 pole � � 1 − σ1i q2 m2 pole + σ2i q4 m4 pole � for f P + (q2) and V (q2), (45) Fi(q2) = Fi(0) � 1 − σ1i q2 m2 pole + σ2i q4 m4 pole � for f P 0 (q2), A1(q2) and A2(q2), (46) where σ1,2 for the D → π and D → K∗ transitions from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [2] will be used in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' C4: Considering the SU(3) flavor breaking terms such as cM 1,2 and c′S 1,2,3 in later Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' I, IV and VIII, the form factors in C3 case are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' As for the form factors of the D → Sℓ+νℓ decays, we find that the vector dominance model [77] and the double pole model [78] give the similar SU(3) flavor symmetry predictions for the branching ratios of the D → Sℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The following form factors from the vector dominance model will be used in the numerical results, Fi(q2) = Fi(0) � 1 − q2/m2 pole � for f S +(q2) and f S 0 (q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (47) After considering above q2 dependence, we only need to focus on the Fi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Since these form factors Fi(0) also preserve the SU(3) flavor symmetry, the same relations in Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' I, IV and VIII will be used for Fi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' If considering the form factors ratios f+(0)/f0(0) = 1 for D → P/Sℓ+νℓ decays, rV ≡ V (0)/A1(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07, r2 ≡ A2(0)/A1(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 in D0 → K∗−ℓ+νℓ decays from PDG [1] and the SU(3) flavor symmetry, there is only one free form factor f P,S + (0) and A1(0) for the D → P/Sℓ+νℓ and D → V ℓ+νℓ decays, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' As a result, the branching ratios only depend on one form factor f P + (0), f S +(0) or A1(0) and the CKM matrix element Vcq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 8 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Numerical results The theoretical input parameters and the experimental data within the 2σ errors from PDG [1] will be used in our numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' D → Pℓ+νℓ decays Considering both the SU(3) flavor symmetry and the SU(3) flavor breaking contributions, the hadronic helicity amplitudes for the D → Pℓ+νℓ decays are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' I, in which we keep the CKM matrix element Vcs and Vcd information for comparing conveniently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In addition, H(D+ s → π0ℓ+νℓ) are obtained by neutral meson mixing with δ2 = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='71) × 10−4 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' From Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' I, we can easily see the hadronic helicity amplitude relations of the D → Pℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' There are four nonperturbative parameters A1,2,3,4 in the D → Pℓ+νℓ decays with A1 ≡ cP 0 + cP 1 − 2cP 2 , A2 ≡ cP 0 − 2cP 1 − 2cP 2 , A3 ≡ cP 0 + cP 1 + cP 2 and A4 ≡ cP 0 − 2cP 1 + cP 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' If neglecting the SU(3) flavor breaking cP 1 and cP 2 terms, A1 = A2 = A3 = A4 = cP 0 , and then all hadronic helicity amplitudes are related by only one parameter cP 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Many decay modes of the D → Pe+νe, Pµ+νµ decays have been measured, and the experimental data with 2σ errors are listed in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' One can constrain the parameters Ai by the present experimental data within 2σ errors and then predict other not yet measured branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Four cases C1,2,3,4 will be considered in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The numerical results of B(D → Pℓ+νℓ) in the C1, C2, C3 and C4 cases are given in the third, forth, fifth and sixth columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' II, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' And our comments on the results are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Results in C1 case: From the third column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' II, one can see that the SU(3) flavor symmetry predictions of B(D → Pℓ+νℓ) in the C1 case are entirely consistent with all present experiential data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The not yet measured branching ratios of the D+ s → π0e+νe, D+ s → π0µ+νµ, D+ → η′µ+νµ and D+ s → K0µ+νµ decays are predicted TABLE I: The hadronic helicity amplitudes for the D → Pℓ+ν decays including both the SU(3) flavor symmetry and the SU(3) flavor breaking contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' A1 ≡ cP 0 + cP 1 − 2cP 2 , A2 ≡ cP 0 − 2cP 1 − 2cP 2 , A3 ≡ cP 0 + cP 1 + cP 2 , A4 ≡ cP 0 − 2cP 1 + cP 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' A1 = A2 = A3 = A4 = cP 0 if neglecting the SU(3) flavor breaking cP 1 and cP 2 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='Hadronic helicity amplitudes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='SU(3) flavor amplitudes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D0 → K−ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A1V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A1V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → ηℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− cosθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2/3 − sinθP / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → η′ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− sinθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2/3 + cosθP / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → π0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='−δ� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− cosθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2/3 − sinθP / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D0 → π−ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → π0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 A3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → ηℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cosθP / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6 − sinθP / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → η′ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='sinθP / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6 + cosθP / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → K0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='A4V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='TABLE II: Branching ratios of the D → Pℓ+ν decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' †Denotes that the corresponding experimental data from PDG [1] are not used to constrain Ai in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Branching ratios Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' data Ones in C1 Ones in C2 Ones in C3 Ones in C4 Previous ones B(D+ → K 0e+νe)(×10−2) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 B(D+ → π0e+νe)(×10−3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='40 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33† 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12† 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 B(D+ → ηe+νe)(×10−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 B(D+ → η′e+νe)(×10−4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 B(D0 → K−e+νe)(×10−2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='549 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='052 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 B(D0 → π−e+νe)(×10−3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='23 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='03† 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09† 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 B(D+ s → ηe+νe)(×10−2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16 B(D+ s → η′e+νe)(×10−3) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='43 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 B(D+ s → K0e+νe)(×10−3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='80 B(D+ s → π0e+νe)(×10−5) · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} 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2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 [76] B(D+ → K 0µ+νµ)(×10−2) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 B(D+ → π0µ+νµ)(×10−3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='31† 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12† 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 B(D+ → ηµ+νµ)(×10−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='21 [7] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 [79] B(D+ → η′µ+νµ)(×10−4) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 [7] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 [79] B(D0 → K−µ+νµ)(×10−2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 B(D0 → π−µ+νµ)(×10−3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='01† 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09† 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 B(D+ s → ηµ+νµ)(×10−2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16 B(D+ s → η′µ+νµ)(×10−2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 B(D+ s → K0µ+νµ)(×10−3) · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='9 [7] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='76 [79] B(D+ s → π0µ+νµ)(×10−5) · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 B(D+ s → π0τ +ντ)(×10−10) · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='21 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 ± 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 (27 ∼ 36) [76] Rµ/e(D+ → K 0ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='984 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='974 Rµ/e(D+ → π0ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='977 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='009 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='984 Rµ/e(D+ → ηℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='984 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='973 Rµ/e(D+ → η′ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='948 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='940 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='940 Rµ/e(D0 → K−ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='984 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='974 Rµ/e(D0 → π−ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='977 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='008 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='984 Rµ/e(D+ s → ηℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='987 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='976 Rµ/e(D+ s → η′ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='958 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 Rµ/e(D+ s → K0ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='992 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='978 Rµ/e(D+ s → π0ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='980 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='985 10 on the order of O(10−3 − 10−5), nevertheless, B(D+ s → π0τ +ντ) is predicted on the order of O(10−10) due to its narrow phase space and (q2 − m2 τ)2 suppression of the differential branching ratios in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Results in C2,3 cases: The numerical results in C2,3 cases are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The experimental upper limits of B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ) have not been used to constrain the predictions of B(D → Pℓ+νℓ), since the upper limits of the predictions of B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ) by the SU(3) flavor symmetry in C2,3 cases are slightly larger than their experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Other SU(3) flavor symmetry predictions are consistent with their experimental data within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Results in C4 case: As given in the sixth column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' II, if considering both the hadronic momentum- transfer q2 dependence of the form factors and the SU(3) flavor breaking contributions, all SU(3) flavor symmetry predictions are consistent with their experimental data within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' For some decays, the errors of the theoretical predictions are much smaller than ones of their experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The previous predictions for the not yet measured branching ratios are listed in the last column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' II, our predictions are in the same order of magnitude as previous ones for the D → Pe+νe, Pµ+νµ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' And our prediction of B(D+ s → π0τ +ντ) is one order smaller than previous one in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In addition, the lepton flavor universality parameters Rµ/e(D → Pℓ+νℓ) are also given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' II, since many terms are canceled in the ratios, these predictions are quite accurate, and all processes have similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' For the q2 dependence of the differential branching ratios of the D → Pℓ+νℓ decays with present experimental bounds, we only show the not yet measured processes D+ → η′µ+νµ, D+ s → K0µ+νµ, D+ s → π0µ+νµ and D+ s → π0τ +ντ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' We do not show dB(D+ s → π0e+νe)/dq2, since it is similar to dB(D+ s → π0µ+νµ)/dq2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 1 C 1 C 2 C 3 C 4 dB(D + s 0 + )/dq 2 ( x10 10 ) q 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 1: The q2 dependence of the differential branching ratios for some D → Pℓ+νℓ with present experimental bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' qB(D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0 文 →>,")qd Se 8(C) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' S 3 4 Q 00 0C Sc c (ε) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 0S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 3\' 001 5 3 4qB(D→>K^")/qd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content="0 0 xXoX S3(q) 3'S 3'3 3 003 3'4 0 +ix Q 0 (p) d." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content="1 s'o 20 0 0'42." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 1 gB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 2 个← 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content="1 s'o11 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 20 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 ( d ) ( c ) ( b ) ( a ) e in unit of 10 6 in unit of 10 2 dA FB (D + s K 0 l + l )/dq 2 q 2 e dC l F (D + s K 0 l + l )/dq 2 q 2 e dP l L (D + s K 0 l + l )/dq 2 q 2 e dP l T (D + s K 0 l + l )/dq 2 q 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 2: The differential forward-backward asymmetries, differential lepton-side convexity parameters, differential longitudinal lepton polarizations and differential transverse lepton polarizations for the D+ s → K0ℓ+νℓ decays in the C3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 1, one can see that present experimental measurements give quite strong bounds on the differential branching ratios of D+ → η′µ+νµ, D+ s → π0µ+νµ and D+ s → π0τ +ντ decays in the C1, C3 and C4 cases as well as D+ s → K0µ+νµ decays in the C1 and C3 cases, and all predictions of the four differential branching ratios in the C2 case have large error due to the form factor choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Comparing with dB(D+ s → π0µ+νµ)/dq2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 1 (c), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 1 (d), dB(D+ s → π0τ +ντ)/dq2 is suppressed about the order of O(10−4) by mτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The forward-backward asymmetries Aℓ F B, the lepton-side convexity parameters Cℓ F , the longitudinal polarizations of the final charged leptons P ℓ L and the transverse polarizations of the final charged leptons P ℓ T with two ways of integration for the D → Pℓ+νℓ decays could also be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' These predictions are very accurate, and they are similar to each other in the four C1,2,3,4 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' So we only give the predictions within the C3 case in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' III for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' From Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' III, one can see that the predictions are obviously different between two ways of q2 integration, and the slight difference in the same way of q2 integration is due to the different decay phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' For displaying the differences between the D → Pe+νe and D → Pµ+νµ decays, we take D+ s → K0e+νe and D+ s → K0µ+νµ as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The differential forward-backward asymmetries, the differential lepton-side convexity parameters, the differential longitudinal lepton polarizations and the differential transverse lepton polarizations of D+ s → K0e+νe and D+ s → K0µ+νµ decays within the C3 case are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' And one can see that differential observables between ℓ = e and ℓ = µ are obviously different, specially in the low and high q2 ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 12 TABLE III: Quantities ⟨X⟩ and X of the D → Pℓ+ν in C3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Decay modes ⟨Aℓ F B⟩ Ae F B(×10−6) Aµ,τ F B(×10−2) ⟨Cℓ F ⟩ Cℓ F ⟨P ℓ L⟩ P ℓ L ⟨P ℓ T ⟩ P e T (×10−3) P µ,τ T D+ → K 0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='087 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='254 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='239 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='768 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='273 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='442 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 D+ → π0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='083 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='054 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='252 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='780 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='260 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='730 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ → ηe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='087 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='476 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='239 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='768 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='273 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='490 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ → η′e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='093 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='222 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='753 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='290 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='890 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 D0 → K−e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='087 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='259 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='239 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='768 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='273 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='446 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 D0 → π−e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='083 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='077 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='252 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='779 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='260 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='751 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ s → ηe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='086 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='033 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='242 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='770 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='270 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='300 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 D+ s → η′e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='091 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='829 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='226 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='757 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='286 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='484 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 D+ s → K0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='085 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='814 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='245 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='773 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='267 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='118 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ s → π0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='082 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='850 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='254 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='781 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='258 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='634 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 D+ → K 0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='226 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='278 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='822 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='851 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='655 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='414 D+ → π0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='201 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='810 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='897 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='907 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='602 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='310 D+ → ηµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='227 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='490 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='819 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='846 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='657 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='419 D+ → η′µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='263 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='097 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='708 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='703 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='725 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='581 D0 → K−µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='226 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='285 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='822 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='850 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='656 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='414 D0 → π−µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='201 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='844 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='895 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='407 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='910 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='603 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='313 D+ s → ηµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='221 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='836 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='864 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='646 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='394 D+ s → η′µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='254 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='736 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='747 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='709 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='540 D+ s → K0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='215 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='701 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='856 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='879 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='632 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='367 D+ s → π0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='197 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='571 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='907 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='920 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='594 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='295 D+ s → π0τ +ντ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='281 −27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='429 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='105−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='211 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='212 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='868 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='873 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='447 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='437 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' D → V ℓ+νℓ decays The hadronic helicity amplitudes for the D → V ℓ+νℓ decays are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' There are four nonperturbative parameters B1,2,3,4 in the D → V ℓ+νℓ decay modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' If neglecting the SU(3) flavor breaking cV 1 and cV 2 terms, B1 = B2 = B3 = B4 = cV 0 , and then all hadronic helicity amplitudes of D → V ℓ+νℓ are related by only one parameter cV 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Among the D → V ℓ+νℓ decay modes, 13 branching ratios have been measured, and 2 branching ratios have been upper limited by the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The experimental data with 2σ errors are listed in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Now we use the listed experimental data to constrain the parameters Bi and then predict other not yet measured and not yet well measured branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The numerical results of B(D → V ℓ+νℓ) in the C1, C2, C3 and C4 cases are given in the third, forth, fifth and sixth columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The results in the C1, C2 and C3 cases are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Since the SU(3) flavor symmetry predictions of B(D+ → ωe+νe) and B(D0 → ρ−µ+νµ) are slightly larger than their experimental data within 2σ errors in the three cases, we 13 TABLE IV: The hadronic helicity amplitudes for D → V ℓ+ν decays including both the SU(3) flavor symmetry and the SU(3) flavor breaking contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' B1 = cV 0 +cV 1 −2cV 2 , B2 = cV 0 −2cV 1 −2cV 2 , B3 = cV 0 +cV 1 +cV 2 , B4 = cV 0 −2cV 1 +cV 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' If neglecting the SU(3) flavor breaking cV 1 and cV 2 terms, B1 = B2 = B3 = B4 = cV 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='Hadronic helicity amplitudes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='SU(3) IRA amplitudes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D0 → K∗−ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='B1V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='∗0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='B1V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → φℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− cosθV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2/3 − sinθV / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='B2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → ωℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− sinθV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2/3 + cosθV / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='B2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D0 → ρ−ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='B3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → ρ0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 B3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → φℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cosθV / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6 − sinθV / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='B3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → ωℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='sinθV / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6 + cosθV / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='B3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → K∗0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='B4V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='do not use them to constrain the nonperturbative parameter cV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' One can see that the prediction of B(D0 → ρ−µ+νµ) is agree with its experimental data within 3σ errors, nevertheless, the prediction of B(D+ → ωe+νe) still slightly larger than experimental data within 3σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' B(D+ s → K∗0µ+νµ) and B(D+ s → ωe+νe, ωµ+νµ) are predicted on the order of O(10−3) and O(10−5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' And they could be measured in BESIII, LHCb and BelleII experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In the C4 case, as given in the sixth column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' V, after considering both the hadronic momentum- transfer q2 dependence of the form factors and the SU(3) flavor breaking contributions, all SU(3) flavor symmetry predictions are consistent with their experimental data within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Among relevant not yet measured decays, B(D+ s → K∗0µ+νµ) is calculated in the SM using light-cone sum rules [79] and in the relativistic quark model [7], B(D+ s → K∗0µ+νµ) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32) × 10−3 [79] and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 × 10−3 [7], and our predictions of B(D+ s → K∗0µ+νµ) in the C1, C2, C3 and C4 cases are coincident with previous ones in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [7, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In addition, the lepton flavor universality parameters Rµ/e(D → V ℓ+νℓ) are also given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Since many terms are canceled in the ratios, these predictions of the lepton flavor universality parameters are quite accurate, and our predictions in all four cases are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' For the q2 dependence of the differential branching ratios of the D → V ℓ+νℓ decays with present experimental bounds, we only show the not yet measured processes D+ → φµ+νµ, D+ s → ωµ+νµ and D+ s → K∗0µ+νµ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The differential branching ratios of D+ → φe+νe (D+ s → ωe+νe) is similar to D+ → φµ+νµ (D+ s → ωµ+νµ), so we do not shown them in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 3, one can see that present experiment data give quite strong bounds on all differential branching ratios of D+ → φµ+νµ, D+ s → ωµ+νµ and D+ s → K∗0µ+νµ decays in the C1, C2 and C3 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The prediction of dB(D+ → φµ+νµ)/dq2 in the C4 case could be distinguished from ones in the C1,2,3 cases within the middle range of q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' And the error of dB(D+ s → K∗0µ+νµ)/dq2 in the C4 case is obviously larger than ones in C1,2,3 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The forward-backward asymmetries Aℓ F B, the lepton-side convexity parameters Cℓ F , the longitudinal polarizations P ℓ L, the transverse polarizations P ℓ T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions of the final vector mesons FL with two ways of integration have also been predicted in the four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Since many theoretical 14 TABLE V: Branching ratios of the D → V ℓ+ν within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' †The experimental data of B(D+ → ωe+νe) and B(D0 → ρ−µ+νµ) from PDG [1] are not used in the C1,2,3 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Branching ratios Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' data Ones in C1 Ones in C2 Ones in C3 Ones in C4 B(D+ → K ∗0e+νe)(×10−2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16 B(D+ → ρ0e+νe)(×10−3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 B(D+ → ωe+νe)(×10−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07† 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12† 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04† 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14 B(D+ → φe+νe)(×10−7) < 130 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='23 B(D0 → K∗−e+νe)(×10−2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 B(D0 → ρ−e+νe)(×10−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 B(D+ s → φe+νe)(×10−2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 B(D+ s → ωe+νe)(×10−5) < 200 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 B(D+ s → K∗0e+νe)(×10−3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56 B(D+ → K ∗0µ+νµ)(×10−2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 B(D+ → ρ0µ+νµ)(×10−3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14 B(D+ → ωµ+νµ)(×10−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 B(D+ → φµ+νµ)(×10−7) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='21 B(D0 → K∗−µ+νµ)(×10−2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 B(D0 → ρ−µ+νµ)(×10−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07† 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11† 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06† 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 B(D+ s → φµ+νµ)(×10−2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 B(D+ s → ωµ+νµ)(×10−5) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='36 B(D+ s → K∗0µ+νµ)(×10−3) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='53 Rµ/e(D+ → K ∗0ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='939 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='944 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='941 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='941 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 Rµ/e(D+ → ρ0ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='956 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 Rµ/e(D+ → ωℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='956 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 Rµ/e(D+ → φℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='923 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='928 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='925 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='925 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 Rµ/e(D0 → K∗−ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='939 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='944 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='941 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='941 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 Rµ/e(D0 → ρ−ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='956 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='952 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 Rµ/e(D+ s → φℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='937 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='942 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='939 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='939 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 Rµ/e(D+ s → ωℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='957 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='963 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='959 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='959 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 Rµ/e(D+ s → K∗0ℓ+νℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='949 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='955 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='951 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='951 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='001 uncertainties are canceled in the ratios, these predictions are very accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' These predictions are similar to each other in the four cases, and we only list the results in the C3 case in Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' VI-VII for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' One can see that the predictions are obviously different between two ways of q2 integration, and they are also quite different between D → V e+νe and D → V µ+νµ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The differential observables of D+ s → K∗0ℓ+νℓ decays in the C3 case are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' One can see that, in the low q2 ranges, the differential observables expect dFL(D+ s → K∗0ℓ+νℓ)/dq2 are obviously different between decays with ℓ = e and ℓ = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 15 C 3 C 4 dB(D + s K 0 + )/dq 2 ( x10 3 ) q 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 3: The q2 dependence of the differential branching ratios for some not yet measured D → V µ+νµ decays with present experimental bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 6 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 ( f ) ( e ) ( d ) ( c ) ( b ) ( a ) e dA FB (D + s K 0 l + l )/dq 2 q 2 e dC l F (D + s K 0 l + l )/dq 2 q 2 e dP l L (D + s K 0 l + l )/dq 2 q 2 e: in unit of 10 3 dP l T (D + s K 0 l + l )/dq 2 q 2 e dA (D + s K 0 l + l )/dq 2 q 2 e dF L (D + s K 0 l + l )/dq 2 q 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 4: The differential forward-backward asymmetries, differential lepton-side convexity parameters, differential longitudinal lepton polarizations and differential transverse lepton polarizations for the D+ s → K0ℓ+νℓ decays in the C3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=" S 0'4 a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0cqB(D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0 0 2 8←S 3(p) d S a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 0 H 0 X x"S 3(c) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 qB(D) S(0rx) "pbl( 7 e16 TABLE VI: The forward-backward asymmetries Aℓ F B, the lepton-side convexity parameters Cℓ F , the longitudinal polarizations P ℓ L of the D → V ℓ+ν decays in the C3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Decay modes ⟨Aℓ F B⟩ Aℓ F B ⟨Cℓ F ⟩ Cℓ F ⟨P ℓ L⟩ P ℓ L D+ → K ∗0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='125 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='190 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='020 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='786 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ → ρ0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='130 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='222 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='024 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='052 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='496 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='789 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ → ωe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='130 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='220 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='024 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='052 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='497 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='789 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ → φe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='121 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='164 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='017 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='037 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='784 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D0 → K∗−e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='125 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='191 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='020 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='786 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D0 → ρ−e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='130 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='221 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='024 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='052 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='497 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='789 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ s → φe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='122 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='176 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='043 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='016 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='500 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='786 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ s → ωe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='130 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='229 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='025 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='057 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='025 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='496 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='790 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ s → K∗0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='128 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='207 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='022 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='021 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='495 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='789 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 D+ → K ∗0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='284 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='226 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='466 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='021 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='395 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='514 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='886 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 D+ → ρ0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='292 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='252 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='491 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='405 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='524 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='903 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 D+ → ωµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='292 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='251 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='022 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='490 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='405 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='524 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='902 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 D+ → φµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='277 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='206 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='016 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='433 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='016 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='376 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='503 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='864 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 D0 → K∗−µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='284 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='226 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='466 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='021 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='395 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='514 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='886 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 D0 → ρ−µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='292 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='252 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='490 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='405 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='524 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='902 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 D+ s → φµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='277 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='213 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='459 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='391 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='514 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='882 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 D+ s → ωµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='291 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='257 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='024 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='509 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='029 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='414 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='531 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='913 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 D+ s → K∗0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='286 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='239 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='021 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='485 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='024 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='402 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='525 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='900 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 TABLE VII: The transverse polarizations P ℓ T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions of the final vector mesons FL of the D → V ℓ+ν decays in the C3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Decay modes ⟨P ℓ T ⟩ P e T (×10−3) P µ T ⟨Aλ⟩ Aλ ⟨FL⟩ FL D+ → K ∗0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='251 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='205 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='066 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='905 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='556 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 D+ → ρ0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='249 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='040 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='072 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='907 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 D+ → ωe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='249 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='073 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='907 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 D+ → φe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='254 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='003 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='417 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='061 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='902 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='556 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 D0 → K∗−e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='251 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='206 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='067 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='905 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='556 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 D0 → ρ−e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='249 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='045 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='073 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='907 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 D+ s → φe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='251 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='060 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='904 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 D+ s → ωe+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='247 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='953 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='071 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='908 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='020 D+ s → K∗0e+νe −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='248 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='004 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='066 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='905 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='553 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='016 D+ → K ∗0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='454 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='022 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='156 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='935 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='928 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='775 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='557 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 D+ → ρ0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='452 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='026 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='139 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='944 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='937 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='782 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 D+ → ωµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='452 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='026 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='140 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='944 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='937 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='782 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 D+ → φµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='455 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='175 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='924 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='915 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='763 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='557 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 D0 → K∗−µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='454 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='022 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='156 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='935 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='927 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='775 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='557 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 D0 → ρ−µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='452 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='026 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='140 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='944 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='937 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='782 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='018 D+ s → φµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='454 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='162 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='934 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='925 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='771 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='557 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 D+ s → ωµ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='452 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='131 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='943 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='788 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='019 D+ s → K∗0µ+νµ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='451 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='023 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='143 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='943 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='936 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='779 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='016 17 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' D → Sℓ+νℓ decays For D → Sℓ+νℓ decays, the two quark and the four quark scenarios for the scalar mesons below or near 1 GeV are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The hadronic helicity amplitudes for the D → Sℓ+νℓ decays are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' VIII, in which the CKM matrix element Vcs and Vcd information are kept for comparing conveniently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' There are four (five) nonperturbative parameters E1,2,3,4 (E′ 1,2,3,4,5) in the two quark (four quark) picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' After ignoring the SU(3) flavor breaking contributions, only one nonperturbative parameter E1 = E2 = E3 = E4 = cS 0 or E′ 1 = E′ 2 = E′ 3 = E′ 4 = E′ 5 = c′S 0 relates all decay amplitudes in the two quark or the four quark picture, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Unlike many measured decay modes in the D → Pℓ+νℓ and D → V ℓ+νℓ decays, among these D → Sℓ+νℓ decays, only D+ s → f0(980)e+νe decay has been measured, and its branching ratio with 2σ errors is [1] B(D+ s → f0(980)e+νe) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8) × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (48) In addition, the branching ratios of the D → P1P2ℓ+νℓ decays with the light scalar resonances can be obtained by using B(D → Sℓ+νℓ) and B(S → P1P2), and the detail analysis can been found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Five branching ratios TABLE VIII: The hadronic helicity amplitudes for D → Sℓ+ν decays including both the SU(3) flavor symmetry and the SU(3) flavor breaking contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In the two quark picture of the scalar mesons, E1 ≡ cS 0 + cS 1 − 2cS 2 , E2 ≡ cS 0 − 2cS 1 − 2cS 2 , E3 ≡ cS 0 + cS 1 + cS 2 , E4 ≡ cS 0 − 2cS 1 + cS 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' E1 = E2 = E3 = E4 = cS 0 if neglecting the SU(3) flavor breaking cS 1 and cS 2 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In the four quark picture of the scalar mesons, E′ 1 ≡ c′S 0 + c′S 1 − 2c′S 2 + c′S 3 , E′ 2 ≡ c′S 0 − 2c′S 1 − 2c′S 2 + c′S 3 , E′ 3 ≡ c′S 0 + c′S 1 + c′S 2 − 2c′S 3 , E′ 4 ≡ c′S 0 + c′S 1 + c′S 2 + c′S 3 , E′ 5 ≡ c′S 0 − 2c′S 1 + c′S 2 + c′S 3 , E′ 1 = E′ 2 = E′ 3 = E′ 4 = E′ 5 = c′S 0 if neglecting the SU(3) flavor breaking c′S 1 , c′S 2 and c′S 3 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='Hadronic helicity amplitudes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='ones for two-quark scenario ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='ones for four-quark scenario ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D0 → K− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E1V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E1V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → f0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → f0(980)ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cosθS E2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2cosφS E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → f0(500)ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='−sinθS E2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2sinφS E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D0 → a− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → a0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2E3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → f0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → σℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2E3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → f0(980)ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2sinθS E3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='( 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3cosφS + E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4sinφS)V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ → f0(500)ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2cosθS E3V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='(− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3sinφS + E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4cosφS)V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='H(D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='s → K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0ℓ+νℓ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E4V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='E′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='and two upper limits of B(D → Sℓ+νℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' S → P1P2) have been measured,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' and the data within 2σ errors are B(D+ s → f0(980)e+νe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' f0(980) → π+π−) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='63) × 10−3 [81], B(D+ s → f0(980)e+νe, f0(980) → π0π0) = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='9) × 10−4 [82], B(D0 → a0(980)−e+νe, a0(980)− → ηπ−) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='60) × 10−4 [1], B(D+ → a0(980)0e+νe, a0(980)0 → ηπ0) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4) × 10−4 [1], B(D+ → f0(500)e+νe, f0(500) → π+π−) = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0) × 10−4 [1], B(D+ → f0(980)e+νe, f0(980) → π+π−) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 × 10−5 [83], B(D+ s → f0(500)e+νe, f0(500) → π0π0) < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 × 10−4 [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (49) Two cases S1 and S2 will be considered in the D → Sℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In S1 case, only experimental datum of B(D+ s → f0(980)e+νe) is used to constrain one parameter cS 0 or c′S 0 and then predict other not yet measured branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The numerical results of B(D → Sℓ+ν) in S1 case are given in the 2-4th and 8th columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In the S2 case, the experimental data of both B(D+ s → f0(980)e+νe) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (48) and B(D → Sℓ+νℓ, S → P1P2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' (49) will be used to constrain the parameter cS 0 or c′S 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The predictions of B(D → Sℓ+ν) in S2 case are listed in the 5-7th and 9th columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Our comments on the results in the S1,2 cases are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Results in the two quark picture: In the two quark picture, the three possible ranges of the mixing angle, 25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS < 30◦ [58, 68] have been analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In S1 case, using the data of B(D+ s → f0(980)e+νe), many predictions of B(D → Sℓ+ν) are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' As given in the 2-4th columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IX, one can see that the predictions with 25◦ < θS < 40◦ are similar to ones with 140◦ < θS < 165◦, the predictions with −30◦ < θS < 30◦ are slightly different from the first two, and the errors of predictions are quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' After adding the experimental bounds of B(D → Sℓ+νℓ, S → P1P2), as given in the 5-7th columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IX, the three possible ranges of the mixing angle θS are obviously constrained, and they reduce to 25◦ < θS < 35◦, 144◦ < θS < 158◦ and 22◦ ≤ |θS| ≤ 30◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In addition, the error of every prediction become smaller by adding the experimental bounds of B(D → Sℓ+νℓ, S → P1P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Results in the four quark picture: The predictions in the four quark picture are listed in the 8-9th columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The majority of predictions in four quark picture are smaller than corresponding ones in two quark picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Strong coupling constants g′ 4 and g4 are appeared in S → P1P2 decays with the four quark picture of light scalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' At present, we only can determine �� g′ 4 g4 �� from the S → P1P2 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The results of involved decays with both g′ 4 g4 > 0 and g′ 4 g4 < 0 are given in the 9th column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IX, and one can see that, except B(D+ s → f0(500)e+νe) and B(D+ s → f0(980)µ+νµ), the other involved branching ratios are not obviously affected by the choice of g′ 4 g4 > 0 or g′ 4 g4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The errors of the branching ratio predictions are obviously reduced by the experimental bounds of B(D → Sℓ+νℓ, S → P1P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Comparing with previous predictions: Previous predictions are listed in the last column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' B(D+ s → f0(500)e+νe), B(D+ s → f0(500)µ+νµ) and B(D+ → f0(500)µ+νµ) are predicted for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Our predictions of B(D+ s → f0(980)µ+νµ), B(D+ → a0 0e+νe), B(D+ → f0(980)e+νe), B(D+ → f0(500)e+νe) and B(D+ → a0 0µ+νµ) are consistent with previous predictions in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [78, 84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Our other predictions are about one order smaller or one order larger than previous ones in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [67, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 19 TABLE IX: Branching ratios of D → Sℓ+ν decays within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' As given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' [80], g′ 4 and g4 are strong coupling constants obtained by the SU(3) flavor symmetry in S → P1P2 decays, adenotes the results with g′ 4 g4 > 0, and bdenotes ones with g′ 4 g4 < 0, †denotes the results with two quark picture, and ‡denotes the results with four quark picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Branching ratios ones for 2q state in S1 ones for 2q state in S2 ones for 4q ones for 4q Previous ones [25◦, 40◦] [140◦, 165◦] [−30◦, 30◦] [25◦, 35◦] [144◦, 158◦] 22◦ ≤ |θS| ≤ 30◦ state in S1 state in S2 B(D0 → K− 0 e+νe)(×10−3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='18 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='57 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='103 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='115† [67] B(D+ → K 0 0e+νe)(×10−3) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='66 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='55 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='99 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='48 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='74 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='88 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='78 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='77 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='68 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='85 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='36 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6† [67] B(D+ s → f0(980)e+νe)(×10−3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='61a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56b 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2† [78], 2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5† −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 [84] B(D+ s → f0(500)e+νe)(×10−3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='73 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='98 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='31a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17b B(D0 → K− 0 µ+νµ)(×10−3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='90 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='73 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='103 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='115† [67] B(D+ → K 0 0µ+νµ)(×10−3) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='46 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='81 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='87 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='88 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='65 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='69 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='43 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='59 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6† [67] B(D+ s → f0(980)µ+νµ)(×10−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='54a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='49b 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2† [78] B(D+ s → f0(500)µ+νµ)(×10−3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='21 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='66 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='53 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='01 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='29a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16b B(D0 → a− 0 e+νe)(×10−5) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='99 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='54 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='50 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='67 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='22 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='98 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='65 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='42 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5† [78], 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8+13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='7† −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 [86], 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0† [67] B(D+ → a0 0e+νe)(×10−5) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='62 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='62 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='67 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='35 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='19 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='81 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='71 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='97 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='49 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='68 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8† [78], 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8† −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='9 [86] 6∼8†[85], 5∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4‡[85] B(D+ → f0(980)e+νe)(×10−5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='92 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='48 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='59 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='35±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='80a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='35b 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='68† [78], 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3† [87] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5†[85], 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='9∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3‡[85] B(D+ → f0(500)e+νe)(×10−4) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='05 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='21 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='59 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='70 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='97 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='97±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='34a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='95±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='36b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='6†[85], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='88 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4‡[85] B(D+ s → K0 0e+νe)(×10−4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='73 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='41 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='99 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='35 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='35 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='51 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8† [67] B(D0 → a− 0 µ+νµ)(×10−5) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='25 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='91 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='61 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='37 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='51 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='57 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='83 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='99 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4† [78], 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='0† [67] B(D+ → a0 0µ+νµ)(×10−5) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='83 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='19 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='44 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='23 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='00 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='41 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='76 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='00 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='89 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='73 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='69 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='30 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='7† [78] B(D+ → f0(980)µ+νµ)(×10−5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='23 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='88 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='56 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='74±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='49a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='20±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='14b 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='67† [78] B(D+ → f0(500)µ+νµ)(×10−4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='69 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='96 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='71 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='84 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='32 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='42 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='54 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='52±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='49±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='12b B(D+ s → K0 0µ+νµ)(×10−4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='28 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='62 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='94 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='94 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='45 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='8† [67] 20 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Summary Many semileptonic D → P/V/Sℓ+νℓ decays have been measured, and these processes could be used to test the SU(3) flavor symmetry approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In terms of the SU(3) flavor symmetry and the SU(3) flavor breaking, the amplitude relations have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Then using the present data of B(D → P/V/Sℓ+νℓ), we have presented a theoretical analysis of the D → P/V/Sℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Our main results can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' D → Pℓ+νℓ decays: Our predictions with the SU(3) flavor symmetry in the C1 case and the predictions after adding SU(3) flavor breaking contributions in the C4 case are quite consistent with all present experimental data of B(D → Pℓ+νℓ) within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' In the C2 and C3 cases, our SU(3) flavor symmetry predictions are consistent with all present experimental data except B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ), which are slight larger than their experiential upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The not yet measured B(D+ s → π0e+νe), B(D+ → η′µ+νµ), B(D+ s → K0µ+νµ), B(D+ s → π0µ+νµ), B(D+ s → π0τ +ντ) and the lepton flavor universality parameters have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Moreover, the forward-backward asymmetries, the lepton-side convexity parameters, the longitudinal (transverse) polarizations of the final charged leptons with two ways of integration for the D → Pℓ+νℓ decays have been predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The q2 dependence of corresponding differential quantities of the D → Pℓ+νℓ decays in the C3 case have been displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' D → V ℓ+νℓ decays: As given in the C1, C2 and C3 cases, our SU(3) flavor symmetry predictions of B(D+ → ωe+νe) and B(D0 → ρ−µ+νµ) are slightly larger than its experimental upper limits, and other SU(3) flavor symmetry predictions are consistent with present data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' After considering the SU(3) flavor breaking effects, as given in the C4 case, all predictions are consistent with present data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The not yet measured or not yet well measured branching ratios of D+ → φe+νe, D+ s → ωe+νe, D+ → φµ+νµ, D+ s → ωµ+νµ, and D+ s → K∗0µ+νµ have been predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The q2 dependence of corresponding differential quantities of the D → V ℓ+νℓ decays in the C3 case have also been displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' D → Sℓ+νℓ decays: Among 18 D → Sℓ+νℓ decay modes, only B(D+ s → f0(980)e+νe) has been measured, and this experimental datum has been used to constrain the SU(3) flavor symmetry parameter and then predict other not yet measured branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' Furthermore, the relevant experimental bounds of B(D → Sℓ+νℓ, S → P1P2) have also been added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The two quark and the four quark scenarios for the light scalar mesons are considered, and the three possible ranges of the mixing angle θS in the two quark picture have been analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' The SU(3) flavor symmetry is approximate approach, and it can still provide very useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' We have found that the SU(3) flavor symmetry approach works well in the semileptonic D → P/V ℓ+νℓ decays, and the SU(3) flavor symmetry predictions of the D → Sℓ+νℓ decays need to be further tested, and our predictions of the D → Sℓ+νℓ decays are useful for probing the structure of light scalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' According to our predictions, some decay modes could be observed at BESIII, LHCb or BelleII in near future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The work was supported by the National Natural Science Foundation of China (12175088).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' 21 References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content=' L.' 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[arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} +page_content='5465 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQfR_f1/content/2301.00079v1.pdf'} diff --git a/DdAyT4oBgHgl3EQf4vob/content/tmp_files/2301.00790v1.pdf.txt b/DdAyT4oBgHgl3EQf4vob/content/tmp_files/2301.00790v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e24b41553c1fcdd01b401240b7210e42caac368a --- /dev/null +++ b/DdAyT4oBgHgl3EQf4vob/content/tmp_files/2301.00790v1.pdf.txt @@ -0,0 +1,2521 @@ +ROBUST MACHINE LEARNING PIPELINES FOR TRADING +MARKET-NEUTRAL STOCK PORTFOLIOS ∗ +THOMAS WONG† AND MAURICIO BARAHONA‡ +Abstract. The application of deep learning algorithms to financial data is difficult due to heavy +non-stationarities which can lead to over-fitted models that underperform under regime changes. +Using the Numerai tournament data set as a motivating example, we propose a machine learning +pipeline for trading market-neutral stock portfolios based on tabular data which is robust under +changes in market conditions. +We evaluate various machine-learning models, including Gradient +Boosting Decision Trees (GBDTs) and Neural Networks with and without simple feature engineer- +ing, as the building blocks for the pipeline. We find that GBDT models with dropout display high +performance, robustness and generalisability with relatively low complexity and reduced computa- +tional cost. We then show that online learning techniques can be used in post-prediction processing +to enhance the results. +In particular, dynamic feature neutralisation, an efficient procedure that +requires no retraining of models and can be applied post-prediction to any machine learning model, +improves robustness by reducing drawdown in volatile market conditions. Furthermore, we demon- +strate that the creation of model ensembles through dynamic model selection based on recent model +performance leads to improved performance over baseline by improving the Sharpe and Calmar ra- +tios. We also evaluate the robustness of our pipeline across different data splits and random seeds +with good reproducibility of results. +Key words. +Robust Machine Learning, Online Learning, Gradient Boosting Decision Trees, +Deep Learning, Stock Trading, Tabular Data +1. Introduction. As investors explore new ways to generate profit, machine +learning (ML) models are increasingly used as part of trading strategies, e.g., to pre- +dict the future return of stocks or stock portfolios. In particular, deep learning models +for time-series data, such as Recurrent Neural Networks (RNNs) and Convolutional +Neural Networks (CNNs), have been applied to the prediction of future stock re- +turns [1–3]. However, a major challenge for such methods is the highly stochastic, +non-stationary and non-ergodic nature of financial data, which violates the assump- +tions of many algorithms. Furthermore, deep learning models are over-parameterised, +with numbers of parameters orders of magnitude larger than typical sizes of time series +data. Therefore, deep models can be easily over-fitted to specific patterns in historical +market data not present in future market data, and the over-fitting worsens with the +more complicated neural network architectures, such as Long Short Term Memory +(LSTM) or Transformer networks. In addition, the continuous influx of data, coupled +with possible regime changes, requires costly updating and retraining of such models. +Therefore, such methods can lack reproducibility and robustness for the prediction of +future market data. +As pointed out in recent reviews [4,5], replication of ML studies is often difficult +due to several issues, including data leakage [5], program bugs [6], data and code +usability [7], and model representation and evaluation [4]. These problems and are +currently hindering the usage of ML in high-risk decision processes, such as healthcare +and finance. For trading applications in particular, these issues can have critical effects +on the validity of results. Data leakage, in the form of look-ahead bias or overlap +∗Funding: This work is supported by the Wellcome Trust under Grant 108908/B/15/Z and by +the EPSRC under grant EP/N014529/1. +†Department of Mathematics, +Imperial College London, +London SW7 2AZ, U.K (ming- +hei.wong15@imperial.ac.uk). +‡Department +of +Mathematics, +Imperial +College +London, +London +SW7 +2AZ, +U.K +(m.barahona@imperial.ac.uk,). +1 +arXiv:2301.00790v1 [q-fin.CP] 30 Dec 2022 + +2 +THOMAS WONG AND MAURICIO BARAHONA +of training/test sets [8], can inflate in-sample performance with poor performance +when deployed live. Furthermore, black-box ML models, such as neural networks, +can lack robustness as they are highly sensitive to small changes in parameters and +data, thus resulting in volatile predictions. The non-stationary data and the presence +of regime changes also mean that ML models need to be re-trained with the latest +financial data, a task that is not only computationally costly but also introduces +further uncertainty to the trading models. Yet most studies do not consider model +performance when trained on different segments of historical market data [1–3,9,10]. +Although reinforcement learning (RL) in online learning settings allows ML models +to adapt to changing environments, deep reinforcement learning models are complex +and require large computational resources [11]. Indeed, applying RL to stock trading +is difficult since the complexity of the action space increases exponentially with the +number of stocks in the portfolio. +The above issues suggest the need to further develop robust ML pipelines for +trading applications possibly based on simpler models that can still operate on non- +stationary, highly stochastic data under regime changes. Here we consider such a +pipeline based on tabular data, which allows the use of traditional ML models, such +as Gradient Boosting Decision Trees (GBDT) and other ensemble methods, to predict +trading stocks and stock indices [12, 13]. This approach also allows the integration +of additional sources of data, such as sentiment analysis of news articles to improve +the prediction accuracy of the direction of stock returns [14]. In particular, we find +that Gradient Boosting models, which are known to be robust to data perturbations, +outperform neural network models. Finally, we show that improved robustness of ML +models and adaptation to regime changes can be attained without the use of deep +reinforcement learning by employing: (i) dynamic feature neutralisation, a simple +approach that reduces the linear correlation to a subset of features evolving in time, +and (ii) dynamic model selection of optimal models from an ensemble based on recent +performance. These approaches robustly improve trading performances by reducing +volatility and drawdown during adversarial market regimes. +To exemplify the above issues, we consider a benchmark financial data platform +that is continuously updated and curated under the Numerai tournament of stock +portfolio prediction [15]. Numerai is a hedge fund that organises a data science com- +petition (as of Oct 2022) and provides free, open-source, high quality standardised +financial data to all participants. As discussed below in more detail, the data set is +given in the form of pre-processed temporal tabular data and the task is the predic- +tion of the relative performances of stocks within an evolving trading universe without +access to the identity of individual stocks. Unlike other financial research papers that +use proprietary data sets which can be difficult to validate [9,10], this open financial +data competition allows researchers to replicate findings transparently and allows us +to focus on establishing ML end-to-end pipelines to achieve consistent profits trad- +ing a market-neutral portfolio under changing market regimes. Our pipeline, shown +in Figure 1, is built upon simple, yet robust methodologies that avoid some of the +problems of over-fitting and high computational cost inherent to deep methods. The +robustness of the pipeline is enhanced since each step is implemented independently +avoiding data leakage, which is common in other methods such as neural networks, +where the pre-processing and the actual model often share data. +Key ingredients +are the post-prediction processing and feature engineering steps, which allow easy +adaptation of models towards regime changes without expensive retraining. +The paper is organised as follows. Section 2 introduces the Numerai datasets +used in this paper. Section +3 describes and discusses the different computational + +ROBUST ML MODELS IN FINANCE +3 +Fig. 1: Schematic of the Machine Learning pipeline. Starting with the Numerai +data set, we consider feature engineering methods to augment the dataset and train an +ML model (several are evaluated, including neural networks, but we settle for gradient +boosting trees) to obtain the raw predictions. These then go through post-prediction +processing (e.g., dynamic feature neutralisation) to provide normalised predictions, +which are then combined through model ensembling and dynamic model selection +methods to output the predictions that are submitted to the Numerai tournament. +methods, including online cross-validation, feature engineering and the different ML +models considered and evaluated for the pipeline. Section 4 presents the results from +our ML pipeline, including the impact of different design choices on the robustness of +trading performance. Performances of ML models under different market regimes are +discussed in Section 5. In Section 6, we introduce adaptations to our ML models +based on online learning approaches, which can work well under regime changes, +noting that these adaptations are generic and not limited to specific families of ML +models. Lastly, we discuss the results of the method, open directions and alternatives +and provide a study of the robustness of our ML pipeline in Section 9.4. +2. Numerai dataset and prediction task. Financial data are often available +in the form of time series. These time series can be treated directly using classic meth- +ods such as ARIMA models [16] and more recently through deep learning methods +such as Temporal Fusion Transformers [17]. However, such methods are easily over- +fitted and lead to expensive retraining for financial data, which are inherently affected +by regime changes and high stochasticity. Alternatively, one can use various feature +engineering methods to transform these time series into tabular form through a pro- +cess sometimes called ‘de-trending’ in the financial industry, where the characteristics +of a financial asset at a particular time point, including features from its history, are +represented by a single dimensional data row (i.e., a vector). In this representation, +the time dimension is not considered explicitly, as the state of the system is captured +through transformed features at each time point and the continuity of the temporal +dimension is not used. For example, we can summarise the time series of the return +of a stock with the mean and standard deviation over different look-back periods. +Grouping these data rows for different financial assets into a table at a given time +point we obtain a tabular dataset. If the features are informative, this representation +can be used for prediction tasks at each time point, and allow us to employ robust +and widely tested ML algorithms that are applicable to tabular data. The Numerai +competition is based on a curated tabular data set with high-quality features made +available to the research community. +Description of the dataset:. The Numerai dataset is a temporal tabular dataset. +A temporal tabular dataset is a collection of matrices {Xi}1≤i≤T collected over time +eras 1 to T. Each matrix Xi represents data available at era i with shape Ni × M, + +Machine +Dataset Creation +Feature +Post-Prediction +Model +Learning +(Numerai) +Engineering +Processing +Ensemble +Model Training4 +THOMAS WONG AND MAURICIO BARAHONA +where Ni is the number of data samples (number of stocks here) in era i and M +is the number of features describing the samples. Note that the features are fixed +throughout the eras, in the sense that the same computational formula is used to +compute the features in each week. On the other hand, the number of data samples +(stocks) Ni does not have to be constant across time. +In the Numerai dataset, the matrices Xi contain M obfuscated global stock mar- +ket features (computed by Numerai) for Ni stocks, which are updated weekly (i.e., the +eras are in our case weeks). It is important to remark that the dataset is obfuscated, +i.e., it is not possible for participants to know the identity of stocks or even which +stocks are present each week. Each data row is indexed by a hash index, known only +to Numerai, that maps the data rows to the stocks. As a result, it is not possible +for participants to concatenate different data rows to create a continuous history of +a stock. The matrix Xi thus provides a snapshot of the market at week i presented +as an unknown set of stocks described by a common set of features, such that the +features are computed consistently across all stocks in the week and also computed +consistently across different weeks. +The Numerai dataset starts on 2003-01-03 (Era 1). The tabular set has 1191 +features, which are already normalised into 5 equal-sized integer bins, from 0 to 4. +There are 28 target labels which are derived from stock returns using 14 proprietary +normalisation methods (nomi, jerome, janet, ben, alan, paul, george, william, arthur, +thomas, ralph, tyler, victor, waldo ) over 2 forward-looking periods (20 trading days, +60 trading days). The main target label to evaluate performance is target-nomi-v4- +20, i.e., forward 20 trading days return obtained by the nomi normalisation method. +Other targets are named similarly. The target labels are all scaled between 0 to 1, +where a smaller value represents a lower forward return, and are also grouped into +bins. For each normalisation method, the number of bins could be different, 5 to 7 bins +are created for each target with the bin sizes following a Gaussian-like distribution, +so that most stocks are within the central bin of 0.5 while only a small amount of +stocks are in the tail bins of 0 and 1. We transform the features and labels so that +both become zero-mean. (For features, we subtract 2 from the integer bins so that +the transformed bins are -2,-1,0,1,2. For the target labels, we subtract 0.5 so that the +new targets are in the range -0.5 to 0.5). +Prediction task:. The tournament task is to predict the stock rankings each week, +ordered from lowest to highest expected return. The scoring is based on Spearman’s +rank correlation of the predicted rankings with the main target label (target-nomi-v4- +20). Hence there is a single overall score each week regardless of the number of stocks +to predict each week. Participants are not scored on the accuracy of the ranking of +each stock individually. Numerai uses the predicted rankings to construct a market- +neutral portfolio which is traded every week (As of Sep 2022), i.e., the hedge fund +buys and short-sells the same dollar amount of stocks. Therefore the relative return +of stocks is more relevant than the absolute return, hence the prediction task is a +ranking problem instead of a forecast problem. +3. Methods. +3.1. Robustness in Machine Learning pipelines. In this paper, we aim to +design an ML pipeline focusing on its robustness. Table 1 details issues related to +robustness and reproducibility, as listed in a recent review [5], and how they are +addressed in this paper. By preventing look-ahead bias and other data leakage issues, +our pipeline can be robustly applied to live trading setups. +In addition to avoiding data leakage, the following design choices are used to im- + +ROBUST ML MODELS IN FINANCE +5 +Issues +affecting +robust- +ness of ML algorithms +How the issue is addressed here +‘No test set’ +A robust cross-validation scheme is used. +‘Pre-processing on train- +ing and test set’ +Numerai features are already standardised; hence +minimal pre-processing. +‘Feature +selection +on +training and test set’ +Feature Engineering is applied to each data row in- +dependently +‘Duplicates in datasets’ +A unique id for each data row reduces the chance of +duplicates in dataset +‘Model uses features that +are not legitimate’ +Only data provided by Numerai is used to train ML +models—no extra features from other resources, and +no cherry-picking of features. +‘Temporal leakage’ +We use Grouped Time-Series Cross-Validation with +no overlap between training/validation/test (Fig. 2). +Feature Engineering is applied to each data row in- +dependently, i.e., no data leakage between eras. +‘Non-independence +be- +tween training and test’ +Training and test samples are market data at different +periods without overlap. +‘Sampling bias in test dis- +tribution’ +The stocks trading each week are decided by Numerai +based on operational and risk considerations. +Table 1: Data analysis design. Some common issues regarding data leakage in +machine learning research [4,5] and how these issues are dealt with in this study. +prove the robustness and reliability of the results. Firstly, the impact of random seeds +is reduced by reporting results from average predictions over 10 different random seeds +for each machine learning method. Secondly, the metrics used for model evaluation +are the same as in the Numerai tournament to avoid researcher bias in discounting +unfavourable results. Finally, cross-validation is independent of the effects of random +seeds and other human selection, thus reducing the chance of overfitting models to a +particular data split. +For datasets that involve time, standard cross-validation schemes cannot be used +directly, as a random split of data eras could lead to the training set including data that +appears later in time than the validation and test sets, hence introducing look-ahead +bias. To avoid this problem, we use grouped time-series cross-validation, which splits +data eras according to their chronological order (Figure 2). Note that for financial +datasets, the target labels often involve future asset returns and are reported with a +lag. Therefore, we add a gap between the training and validation sets and similarly +between the validation and test sets. +3.2. Feature Engineering methods. Feature engineering is a crucial step in +enhancing the power of tabular methods for the analysis of time series data. Therefore, +we evaluate different feature engineering methods that can be applied to temporal +tabular data sets with numerical features, as the Numerai data set only contains +normalised numerical features. +New features can be created by applying polynomial transformations such as +multiplication and addition to the original features. Here we create new features by +multiplying two features that can be thought of as modelling the joint distribution + +6 +THOMAS WONG AND MAURICIO BARAHONA +Training Data +V alidation Data +Test Data +Time +gap +gap +Fig. 2: Illustration of data split using grouped time-series cross-validation +of feature pairs. When the number of features is large, we draw a random subset of +feature pairs to create new features to alleviate the computational cost. Note that +the computation of these features can be done in parallel for data from each era. +A simple way of data augmentation is to add randomness to the feature matrix +with different dropout methods, which are used extensively to reduce over-fitting of +neural network models [18]. Here we apply dropout by multiplying the original data +with a Boolean mask so that some numerical features are set to zero. The dropout is +characterised by its sparsity level (how many features are set to zero) and its sparsity +structure (how to choose the features set to zero). Since our tabular dataset has no +local spatial structure, we use a random Boolean matrix with uniform probability. +This encourages the machine learning methods to learn multiple feature relationships +and reduces reliance on a small set of important features. +For our dataset, we first augment the feature matrix by creating additional fea- +tures obtained by multiplying feature pairs, and then apply dropout with a random +Boolean mask on the augmented feature matrix. A grid search is used to find optimal +hyper-parameters for the feature engineering methods, in particular the number of +feature products and the sparsity level of dropout. +3.3. Machine Learning algorithms for tabular datasets. Numerous ma- +chine learning models have been proposed for tabular datasets, and different bench- +marking studies have shown conflicting views on their performance [18, 19]. +The +biggest disagreement in the literature is whether gradient-boosting decision trees or +neural networks are superior in regression and classification tasks of tabular datasets. +Whereas one paper claims gradient boosting models (XGBoost) outperformed deep +learning models in 8 out of 11 datasets and none of the deep learning models consis- +tently outperform others [19], another paper suggests that well-tuned multi-layer per- +ceptron (MLP) models with regularisation can outperform different gradient boosting +models such as XGBoost and CatBoost [18]. Both these studies, however, share the +same view that neural networks with complicated designs, such as attention layers and +other transformer layers, tend to generalise poorly with a strong drop in performance +when applied to data sets beyond their original study. +Importantly, the Numerai +data set is different from the data sets in the above benchmarking studies in that it +is growing instead of fixed. Hence the data distribution varies across time periods +due to market regime effects, and we do not have a homogeneous distribution across +cross-validation splits. With such a different problem setup, it is thus not possible to +use the above benchmarking studies to guide our choice of ML method. +In this study, we benchmark a wide range of machine learning models, including +different variants of gradient-boosting decision tree models and different neural net- +work models. The choice of ML models is based on the popularity of usage in data + +ROBUST ML MODELS IN FINANCE +7 +science competitions and code quality, as one of our aims, is the replicability of results. +We train all machine learning models with a single GPU, the standard setup for most +participants in data science competitions. Some brief details of the ML models used +are provided in the following. +Gradient Boosting Decision Trees. Boosting can be seen as a generalisation of +generalized additive models (GAM) where the additive components of smooth para- +metric functions can be replaced by any weak learners such as decision trees [20]. +Historically, various boosting algorithms have been proposed for different loss func- +tions. For example, AdaBoost [21] was proposed for binary classification problems +with exponential loss, whereas Gradient Boosting was first proposed by Friedman in +2001 [22] for any smooth loss functions. Algorithm 9.1 in the SI outlines the iterative +update equations of gradient boosting. +Of the various implementations of gradient boosting decision (GBDT) trees in +Python, we use LightGBM [23] in this paper. CatBoost [24] is not used here as the +Numerai dataset has no categorical features. XGBoost [25] is not used due to slower +computation and more memory consumption. Algorithm 9.2 in the SI shows how the +gradient boosting algorithm is implemented with decision trees being the weak learners +in LightGBM. LightGBM implements GBDT models with several computational and +numerical improvements from XGBoost and other implementations. In addition to +traditional gradient boosting decision trees (LightGBM-gbdt), we consider two other +implementations of GBDT models: +• Dropouts meet Multiple Additive Regression Trees (LightGBM-dart) ignores a +portion of trees when computing the gradient for subsequent trees [26], thus +avoiding over-specialisation where the later learned trees can only affect a +few data instances. This reduces the sensitivity of models towards decisions +made by the first few trees. +• Gradient-based One-Side Sampling (LightGBM-goss) reduces the number of +data instances used to build each tree: it keeps data instances with large +absolute gradients and randomly samples a subset of data with small absolute +gradients. +The approximation error of the gradient using LightGBM-goss +converges to the standard method when the number of data is large, and it +outperforms other data sampling (e.g., uniform sampling) in most cases. +For all LightGBM models, we use mean squared error (L2 loss) as the loss function +for the regression problems. The number of gradients boosting trees and learning rate +is optimised by hyper-parameter searches. To prevent the over-fitting of trees, the +maximum depth and number of leaves in each tree and the minimal number of data +samples in the leaves are tuned for each model. L1 and L2 regularisation are also +applied. Data and feature sub-sampling are used to reduce similarities between trees: +before building each tree, a random part of data is selected without re-sampling and +a random subset of features is chosen to build the tree. For LightGBM-dart models, +both the probability to apply dropout during the tree-building process and the portion +of trees to be dropped out are tuned. Early stopping is applied using the validation +dataset for LightGBM-gbdt models to further prevent the over-fitting of models. +Neural Networks. The most basic architecture of neural networks, multi-layer +perceptron (MLP), failed to outperform gradient boosting models in many benchmark +studies of tabular datasets [19]. +Recently, more complex network architectures have been proposed for tabular +data sets, as surveyed in +[27] These new architectures can be classified into two +major groups: +• Hybrid models that combine neural networks with other traditional ML meth- + +8 +THOMAS WONG AND MAURICIO BARAHONA +ods, e.g., decision trees. Neural Oblivious Decision Ensembles (NODE) [28] is +a generalisation of gradient boosting models into differentiable deep decision +trees allowing end-to-end training with gradient descent optimisers such as +PyTorch [29]. DeepGBM [30] combines two neural networks, CatNN to han- +dle sparse categorical features and GBDT2NN to distil tree structures from +a pre-trained GBDT model to handle numerical features. A major limitation +of these models is the large memory consumption, which makes them run out +of memory on the NVIDIA 3080ti GPU. Therefore we do not use them in our +benchmark analysis. +• Transformer-based models that use deep attention mechanisms to model com- +plex feature relationships. TabNet [31] uses sequential attention to perform +instance-wise feature selection at each decision step, enabling interpretability +and better learning. AutoInt [32] maps and models feature interactions in a +low-dimensional space with a multi-head self-attentive neural network with +residual connections. AutoInt runs out of memory on a single GPU and is +thus not used in our benchmark. Tabnet also had similar memory issues, +hence we down-sampled the data by keeping every fifth week of data (i.e., +20% of the original data) for the training/validation periods, so that Tab- +net could be trained on the single GPU used in this study. Our aim is to +compare performance under modest computational resources attainable by a +wide class of users. +In summary, our benchmark analysis includes two NN models: MLP and TabNet +implemented in PyTorch. +We use Adam [33] as the gradient optimiser, with the +learning rate automatically determined by PyTorch. We use mean squared error (L2 +loss) for the regression problems. +4. Evaluation of Machine Learning methods for the Numerai temporal +tabular data set. In this section, we study different ML methods applied to the +Numerai temporal tabular data set for the prediction of stock rankings aimed at +market-neutral stock portfolios. +Data Split. We use the latest version (v4) of the Numerai dataset. The training +period is fixed between 2003-01-03 (Era 1) to 2012-07-27 (Era 500), and the validation +period is fixed between 2012-12-21 (Era 521) and 2014-11-14 (Era 620). The test +period starts on 2015-05-15 (Era 646) and ends on 2022-09-23 (Era 1030). We apply a +1-year gap between training and validation periods to reduce the effect of recency bias +so that the performance of the validation period will better reflect future performance. +The gap between the validation period and test period is set to 26 weeks to allow for +sufficient time to deploy trained machine learning models. +Evaluation of performance. For each configuration of each ML method, we aver- +age over the predictions of different targets before scoring. The predictions are scored +in each era by calculating the correlation (Corr) between the rank-normalised pre- +dictions and the actual (binned) stock ranking. The mean and standard deviation +(volatility) of Corr are reported for both the validation and test periods. To measure +the downside risk of the model, we also compute the Maximum Drawdown, defined +as the largest drop suffered by an investor starting at any time during the valida- +tion/test period. As summary measures, we compute two standard ratios: (i) the +Sharpe ratio, defined as the ratio of the mean and standard deviation of Corr; and +(ii) the Calmar ratio, defined as the ratio of mean Corr over Maximum Drawdown. +Good performance is characterised by large values of both of these ratios, + +ROBUST ML MODELS IN FINANCE +9 +Model Training. We use Optuna [34] to perform the hyper-parameter search (see +section 9.3 in Supplementary Information) and select the hyper-parameters with the +highest Sharpe ratio for the main target (target-nomi-v4-20) in the validation period. +The optimised hyper-parameters for each ML method are so fixed, and we then train +10 models, starting the algorithms from 10 different random seeds. We report the +average prediction from these 10 models for evaluation. +Baseline Model. As a baseline, we consider a factor momentum model which is +created by linear combinations of signed features, where the sign of each feature is +determined by the sign of the 52-week moving average of correlations of that feature +with the target. This simple baseline linear model is then compared with the ML +models, which can capture non-linearity in the data. +Comparative results of the ML algorithms and Feature Engineering. Table 2 shows +the performance on the validation and test sets for the different algorithms. +We +concentrate on methods that achieve the highest mean Corr, and Sharpe and Calmar +ratios. +(a) Performance over the validation period (2012-12-21 to 2014-11-14) +Method +Mean +Volatility +Max Draw +Sharpe +Calmar +Factor Momentum (baseline) +0.0229 +0.0170 +0.0691 +1.3495 +0.3314 +MLP with FE +0.0423 +0.0208 +0.0241 +2.0338 +1.7552 +MLP without FE +0.0443 +0.0201 +0.0065 +2.2058 +6.8154 +TabNet without FE +0.0362 +0.0189 +0.0199 +1.9125 +1.8191 +LightGBM-gbdt with FE +0.0483 +0.0229 +0.0307 +2.1144 +1.5733 +LightGBM-gbdt without FE +0.0500 +0.0224 +0.0235 +2.2335 +2.1277 +LightGBM-dart with FE +0.0496 +0.0223 +0.0215 +2.2274 +2.3070 +LightGBM-dart without FE +0.0475 +0.0199 +0.0079 +2.3883 +6.0127 +LightGBM-goss with FE +0.0288 +0.0219 +0.0687 +1.3136 +0.4192 +LightGBM-goss without FE +0.0302 +0.0234 +0.0877 +1.2877 +0.3444 +(b) Performance over the test period (2015-05-15 to 2022-09-23) +Method +Mean +Volatility +Max Draw +Sharpe +Calmar +Factor Momentum (baseline) +0.0080 +0.0275 +0.7877 +0.2923 +0.0102 +MLP with FE +0.0237 +0.0330 +0.2912 +0.7189 +0.0814 +MLP without FE +0.0258 +0.0289 +0.1668 +0.8931 +0.1547 +TabNet without FE +0.0161 +0.0296 +0.5811 +0.5431 +0.0277 +LightGBM-gbdt with FE +0.0253 +0.0327 +0.3064 +0.7731 +0.0826 +LightGBM-gbdt without FE +0.0262 +0.0321 +0.2378 +0.8140 +0.1102 +LightGBM-dart with FE +0.0265 +0.0319 +0.2151 +0.8313 +0.1232 +LightGBM-dart without FE +0.0278 +0.0284 +0.1622 +0.9791 +0.1714 +LightGBM-goss with FE +0.0169 +0.0297 +0.5539 +0.5695 +0.0305 +LightGBM-goss without FE +0.0156 +0.0318 +0.7528 +0.4896 +0.0207 +Table 2: Performance of different machine learning methods with and without feature +engineering on the Numerai dataset for (a) validation period and (b) test period. +The three top methods according to Sharpe ratio and Maximum Drawdown over +the validation period are shown in italics in (a). The top method according to the +Sharpe ratio and Maximum Drawdown over the test period is shown in boldface in +(b). For TabNet, the pipeline with feature engineering cannot be run due to memory +constraints. + +10 +THOMAS WONG AND MAURICIO BARAHONA +Firstly, we see that almost all ML models performed substantially better than the +factor momentum model (baseline), in both validation and test periods. Whereas the +factor momentum model relies on linear relationships, the capability of ML models +to learn non-linear relationships, in addition to linear ones, adds to their robustness +and improved performance under different, often volatile, market regimes. +Secondly, we observe that Feature Engineering does not improve the performance +of ML models. Although, in principle, Feature Engineering allows GBDT-based meth- +ods to model feature interactions more easily, our results suggest that these interac- +tions are over-fitted during the training process. For neural network-based models, +feature engineering is not strictly necessary, as dropout is already embedded in net- +work architectures. +Thirdly, we note that all ML models scored better in the validation period than +the test period. This is expected, as it is well known that the performance of trading +models deteriorates over time due to overcrowding and regime changes (a phenomenon +known as alpha decay). Models that are over-fitted to recent training data will ex- +perience greater alpha decay than properly regularised models. +To select the ML +method, we consider the top models according to the Sharpe and Calmar ratios over +the validation period: a high Sharpe ratio ensures the model has good overall perfor- +mance, whereas a high Calmar ratio ensures good performance against the worst-case +scenario, thus capturing the tail risks of the trading model. +Indeed, we find that +LightGBM-dart without feature engineering generalises well to the test period further +into the future. +Finally, we note that LightGBM-gbdt has better generalisation to the test period +than neural network-based models (TabNet), suggesting over-fitting in these complex +deep NN models. This indicates that although over-parameterised models can learn +non-linear relationships in temporal tabular data sets, these relationships may be +difficult to generalise under non-stationary data environments. On the other hand, our +results suggest that, despite their relative simplicity, gradient Boosting models capture +non-linearity in a more robust and controlled manner, with early trees capturing linear +relationships and non-linear relationships captured by the later trees, thus reducing +the risk of catastrophic forgetting [35]. +In summary, we find that the best performing model in our set is LightGBM- +dart without feature engineering. In the rest of the paper, we will use this model to +illustrate how the pipeline can be further modified with online learning to account +for regime effects. To demonstrate the robustness of our pipeline, and how it can +be applied to improve the performance of any ML model, we will also report the +performance of two other models: a similar GBDT model (LightGBM-gbdt without +feature engineering) and a neural network model (MLP without feature engineering). +5. Dealing with regime effects in the ML pipeline. Financial data are +heavily influenced by regime changes. +Growth (‘Bull’) markets are characterised +by low volatility and positive expected return, whereas high volatility and negative +expected returns are characteristic of adverse (‘bear’) markets. +Switches between +regimes can be triggered by externalities, such as pandemics, economic recessions, +etc. From the perspective of the Numerai data set, such regime effects affect model +performance. Volatility is detrimental to long-term performance due to the negative +compounding of investment losses, a phenomenon known as ‘volatility tax’. Given +that hedge funds are leveraged, we consider consistent models with reasonably good +performance under different market regimes, rather than models that have excellent +performance in one market regime but fail in others. + +ROBUST ML MODELS IN FINANCE +11 +In this section, we focus on how to deal with regime effects when using ML +models for financial tabular temporal data sets. +Specifically, we consider feature +neutralisation, and reducing the dependence on the initial trees in gradient boosting +models. +Classification into high and low volatility regimes. To classify the financial market +into regimes, we consider an intrinsic measure derived directly from the Numerai +dataset. In particular, we first compute the Numerai Market Index (NMI), i.e., the +weekly performance of the baseline (linear) factor momentum portfolio, and we then +calculate the Numerai Realised Volatility Index (NRVIX), defined as the standard +deviation of NMI rolling over 52 weeks (Fig. 3). The eras are then classified into high +and low volatility, based on a threshold of NRVIX=0.025, the mean over the first 7 +years of data (2003-01-03 to 2010-02-26). According to this intrinsic characterisation, +low volatility regimes have stable linear relationships of features to stock returns, often +associated with a good performance by ML models. On the other hand, high Volatility +regimes correspond to unstable linear relationships of features to stock returns leading +to poor model performance. Figure 3 shows that high/low NRVIX regimes are well +aligned with macroeconomic events: high volatility regimes include the financial crisis +(2007-2009), the Euro crisis (2011-2012), and the Covid pandemic (2020), whereas low +volatility regimes correspond to benign market conditions with no significant macro +event risks, during which the factor momentum baseline portfolio had good returns. +2005-01-28 +2008-11-28 +2012-09-28 +2016-07-29 +2020-05-29 +0.050 +0.025 +0.000 +0.025 +0.050 +0.075 +0.100 +(a) NMI +2005-01-28 +2008-11-28 +2012-09-28 +2016-07-29 +2020-05-29 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +(b) NRVIX +Fig. 3: High and low volatility regimes in the Numerai data. (a) Numerai +Market Index (NMI) for the period between 2005-01-28 (Era 109) and 2022-09-23 +(Era 1016); (b) the computed Numerai Realised Volatility Index (NRVIX) used to +identify the high and low volatility regimes. The high volatility regime refers to weeks +where NRVIX is higher than 0.25 and the low volatility regime refers to weeks where +NRVIX is lower than 0.25. +5.1. Feature Neutralisation. Feature neutralisation is the general term to +denote the elimination of the effect of particular features in the model, thus reducing +the risk of over-relying on certain individual features. Because the predictive ability +of individual features is highly dependent on market regimes, this can lead to long +periods of drawdown when there is a regime change. It is therefore undesirable to +have ML models that could have heavy (linear)-dependence on certain features. +We start by evaluating here the feature neutralisation suggested by the Numerai +tournament. Numerai recommends that participants reduce model exposure to 420 + +12 +THOMAS WONG AND MAURICIO BARAHONA +‘risky features’ (out of the 1191 features). +This list of risky features can be used +for feature neutralisation by subtracting the linear correlation using the formula for +Feature Neutral Correlation (FNC). Specifically, given a week of data with n stocks, +let X ∈ Rn×420 be the matrix of risky features and y ∈ Rn the predicted rankings ob- +tained from a model. For a given neutralisation strength β, 0 ≤ β ≤ 1, the neutralised +predicted ranking ˆy is calculated as ˆy = y − β XX†y, where X† is the pseudo-inverse +of X. FNC is then calculated as the correlation of the neutralised predicted rankings. +Using this procedure, we reduce the linear dependencies of predictions on features. +(a) LightGBM-dart without FE +Regime +Feature Neutral +Mean +Volatility +Max Draw +Sharpe +Calmar +All +Yes +0.0215 +0.0182 +0.1153 +1.1806 +0.1865 +No +0.0278 +0.0284 +0.1622 +0.9791 +0.1714 +High Vol +Yes +0.0227 +0.0163 +0.0223 +1.3888 +1.0179 +No +0.0314 +0.0251 +0.0657 +1.2510 +0.4779 +Low Vol +Yes +0.0206 +0.0195 +0.1153 +1.0576 +0.1787 +No +0.0252 +0.0305 +0.1622 +0.8257 +0.1554 +(b) LightGBM-gbdt without FE +Regime +Feature Neutral +Mean +Volatility +Max Draw +Sharpe +Calmar +All +Yes +0.0204 +0.0211 +0.1998 +0.9665 +0.1021 +No +0.0262 +0.0321 +0.2378 +0.8140 +0.1102 +High Vol +Yes +0.0217 +0.0198 +0.0364 +1.0953 +0.5962 +No +0.0308 +0.0293 +0.1123 +1.0497 +0.2743 +Low Vol +Yes +0.0194 +0.0220 +0.1998 +0.8820 +0.0971 +No +0.0227 +0.0338 +0.2378 +0.6727 +0.0955 +(c) MLP without FE +Regime +Feature Neutral +Mean +Volatility +Max Draw +Sharpe +Calmar +All +Yes +0.0179 +0.0203 +0.2606 +0.8798 +0.0687 +No +0.0258 +0.0289 +0.1668 +0.8931 +0.1547 +High Vol +Yes +0.0196 +0.0193 +0.0326 +1.0191 +0.6012 +No +0.0298 +0.0276 +0.1247 +1.0802 +0.2390 +Low Vol +Yes +0.0165 +0.0210 +0.2606 +0.7875 +0.0633 +No +0.0228 +0.0296 +0.1668 +0.7721 +0.1367 +Table 3: The effect of feature neutralisation. Performance of different ML meth- +ods on the Numerai v4 dataset over the test period (2014-06-27 to 2022-09-23) with +and without feature neutralisation under different market regimes: the whole test +period (all), high volatility regime (high-vol), and low volatility regime (low-vol). +In Table 3, we compare the performance of the LightGBM-dart, LightGBM-gbdt +and MLP with and without feature neutralisation under different market regimes (all, +high volatility, low volatility). The neutralisation strength β is set to 1 throughout. +We find that the variance of models is consistently reduced by feature neutralisa- +tion, suggesting an overall reduction of risk. Further, feature neutralisation improves +the Sharpe and Calmar ratios of LightGBM-dart and LightGBM-gbdt under different +market regimes, but does not improve the performance of MLP models. +Importantly, this default feature neutralisation procedure suggested by Numerai + +ROBUST ML MODELS IN FINANCE +13 +is not optimal, and we will show in Section 6 how online learning approaches can be +used to improve the procedure. +5.2. Pruning initial trees in Gradient Boosting models. For gradient- +boosting tree models, we also consider a specific procedure consisting of pruning +initial trees during prediction to reduce feature dependencies. Specifically, we perform +a grid search over the number of initial trees to be pruned off in the trained LightGBM +models, and we cap the number of trees to be pruned to not more than half of the +trees to ensure our models do not degenerate. +Table 4 compares the performance of LightGBM-dart and LightGBM-gbdt models +pruning different numbers of initial trees before feature neutralisation. Pruning ini- +tial trees during prediction improves the Sharpe and Calmar ratios of both LightGBM +models, but LightGBM-gbdt models see a bigger improvement than LightGBM-dart +models. This is expected as LightGBM-dart models already employ a similar fun- +damental idea during training, i.e., the trained trees in LightGBM-dart models are +already optimised. Our numerics also suggest that there is a limit of trees to be pruned +such that there is little improvement in model performance once over a threshold of +around 100-250 trees. +(a) LightGBM-dart without FE +Prune Trees +Mean +Volatility +Max Draw +Sharpe +Calmar +0 +0.0278 +0.0284 +0.1622 +0.9791 +0.1714 +100 +0.0272 +0.0264 +0.1384 +1.0293 +0.1965 +250 +0.0264 +0.0255 +0.1299 +1.0336 +0.2032 +500 +0.0249 +0.0238 +0.1166 +1.0459 +0.2136 +(b) LightGBM-gbdt without FE +Prune Trees +Mean +Volatility +Max Draw +Sharpe +Calmar +0 +0.0262 +0.0321 +0.2378 +0.8140 +0.1102 +100 +0.0265 +0.0291 +0.1835 +0.9106 +0.1444 +250 +0.0253 +0.0259 +0.1490 +0.9769 +0.1698 +500 +0.0253 +0.0259 +0.1490 +0.9765 +0.1698 +Table 4: The effect of tree pruning. Strategy Performance of different LightGBM +models in the test period (2014-06-27 to 2022-09-23) when pruning different numbers +of initial trees. +5.3. Joint effect of feature neutralisation and tree pruning. We then +considered the joint effect of feature neutralisation and pruning initial trees. Table +5 compares the performance (FNC) of LightGBM-dart and LightGBM-gbdt models +pruning a different number of initial trees after feature neutralisation. The effect of +pruning on model performance for both LightGBM models after feature neutralisation +is at best modest. As FNC is a measure of the effect of non-linear relationships, this +suggests that in gradient boosting models, early weak learners (trees) mostly capture +linear relationships whereas most of the non-linear relationships are captured in the +later weak learners (trees). Therefore, pruning initial trees can be thought of as a +model-dependent feature neutralisation method. + +14 +THOMAS WONG AND MAURICIO BARAHONA +(a) LightGBM-dart without FE with Feature Neutralisation +Prune Trees +Mean +Volatility +Max Draw +Sharpe +Calmar +0 +0.0215 +0.0182 +0.1153 +1.1806 +0.1865 +100 +0.0208 +0.0174 +0.1079 +1.1998 +0.1928 +250 +0.0200 +0.0168 +0.1103 +1.1918 +0.1813 +500 +0.0183 +0.0156 +0.1044 +1.1748 +0.1753 +(b) LightGBM-gbdt without FE with Feature Neutralisation +Prune Trees +Mean +Volatility +Max Draw +Sharpe +Calmar +0 +0.0204 +0.0211 +0.1998 +0.9665 +0.1021 +100 +0.0206 +0.0200 +0.1912 +1.0293 +0.1077 +250 +0.0194 +0.0188 +0.2058 +1.0307 +0.0943 +500 +0.0193 +0.0188 +0.2063 +1.0301 +0.0936 +Table 5: The joint effect of feature neutralisation and tree pruning. Perfor- +mance of different LightGBM models after neutralisation in the test period (2014-06- +27 to 2022-09-23) when pruning different numbers of initial trees. +6. Online Learning to improve post-prediction processing and model +ensembles. As a further improvement to the ML pipeline, we apply online learning +approaches to both feature neutralisation and model ensembles to produce improved +versions called dynamic feature neutralisation and dynamic model selection. Dynamic +feature neutralisation acts by applying statistical rules to determine subsets of fea- +tures to neutralise predictions in each era. Dynamic model selection acts by updating +regularly the choice of model(s) from a model ensemble based on recent model per- +formance. +The aim of online learning is to derive an optimal procedure to select ML mod- +els and parameters as data arrives continuously. In a continuous time setting, the +Hamilton-Jacobi-Bellman (HJB) equation is solved to find the optimal determinis- +tic control for the decision problem [36]. The discrete-time equivalent, the Bellman +equation, is used in reinforcement learning to derive optimal policies of agents [37]. +For the Numerai tournament, we consider online learning in the discrete-time +setting, since data and predictions are required once per week. +For each week t +(1 ≤ t ≤ T), we have a state (data) process Xt, which contains all the infor- +mation we know about the environment (Numerai datasets and trained ML model +parameters) up to week t. Our task is then to derive a deterministic decision pro- +cess Dt(βt) described by parameters βt := βt(Xt), subject to the objective function +VT = maxDt +�T +t=1 q(Xt, Dt), where q(Xt, Dt) represents the utility at time instant t +given the data and decision process. +(Deep) Reinforcement learning algorithms are commonly used to solve online +learning problems. However, they are not used here due to the following reasons: +1. Limited data: Available data is not enough to train reinforcement learning +models, such as Deep Q Networks (DQN) [38], Proximal Policy Optimisation +(PPO) [39] and Soft Actor-Critic (SAC) [40]). Generating a large number of +samples is difficult here since we must avoid look-ahead bias. +2. Expanding action space: Most implementations of reinforcement learning al- + +ROBUST ML MODELS IN FINANCE +15 +gorithms, as found in Ray-RLlib [11], cannot adapt naturally to an expanding +action space. For the dynamic model selection problem, the number of po- +tential models is unbounded, as newer models can be trained with the latest +data available and added to the candidate list. Rule-based models, on the +other hand, can handle the issue of expanding action space easily. +3. Actions have negligible impact on environment: Highly successful reinforce- +ment learning algorithms are usually targetted at robotics and Atari games +[41], where agent actions can modify the environment. However, for the trad- +ing models considered here, the trading activities are assumed to have zero or +negligible market impact, and reinforcement learning algorithms thus reduce +to an online learning prediction problem. +4. Large, correlated feature sets for neutralisation: To improve feature neutral- +isation, we use a different subset of features to neutralise predictions in each +era. Yet the size of the set of risky features (420 features) makes it computa- +tionally infeasible to learn feature subsets through supervised ML methods or +reinforcement learning, as it is difficult to construct a robust reward function +for correlated features. Heuristic methods thus provide suitable alternatives +to learn interpretable and robust feature neutralisation schemes. +5. Model ensembling can be simplified in the Numerai problem: The model en- +semble step of the pipeline assigns portfolio weightings to different ML mod- +els. Although similar to a multi-armed bandit problem, in our problem ex- +ploration is not needed for the agent to learn the distribution of rewards from +different choices since the performance of all ML models up to the decision +time are known to the Numerai tournament participant. Hence there is less +need to employ trial-and-error as in multi-armed bandit algorithms. +As a consequence, instead of reinforcement learning algorithms, we use heuristics +which are shown to be effective in improving the robustness of the ML pipeline. These +heuristics can be interpreted as strong priors in Bayesian learning that greatly simplify +our problem. +6.1. Dynamic Feature neutralisation. In Section 5, the subset of ‘risky fea- +tures’ that are used to neutralise ML models is fixed throughout the whole validation +and test periods. As market conditions are variable, we suggest choosing a different +set of features to neutralise in each era to adapt our ML models without the need for +expensive re-training of models. Specifically, each week we update the set of features +to neutralise based on rolling statistical properties of features, as follows. For each +feature in the dataset, we calculate the correlation of the feature with the target (fea- +ture Corr) and then compute lagged moving average statistics, with a lag of 6 weeks +to account for the lagged reporting of future performance. The look-back window to +compute statistical properties of feature Corr is 52 weeks. We consider 5 different +criteria to select the subset of features to be neutralised: +1. ‘Fixed’: 420 features provided by the portfolio optimiser in Numerai, as in +Section 5 above +2. ‘Low Mean’: 420 features that are least correlated to the target recently +3. ‘High Mean’: 420 features that are most correlated to the target recently +4. ‘Low Volatility’: 420 features that have correlations least volatile recently +5. ‘High Volatility’: 420 features that have correlations most volatile recently +Table 6 compares the performance obtained by the different dynamic feature +neutralisation schemes on LightGBM-dart, LightGBM-gbdt and MLP models. All +Dynamic Feature Neutralisation methods perform better than using a fixed set of + +16 +THOMAS WONG AND MAURICIO BARAHONA +features but the ‘Low Mean’ neutralisation method has the best Sharpe and Calmar +ratios for all ML models, followed by neutralisation of ‘High Volatility’ features. The +worse performance of ‘High Mean’ and ’Low Volatility’ neutralisations suggests that a +large part of the model risks can be attributed to recently underperforming and high +volatility features. +(a) LightGBM-dart without FE +Dynamic Feature Neutral. +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0215 +0.0182 +0.1153 +1.1806 +0.1865 +Low Mean +0.0240 +0.0164 +0.0350 +1.4595 +0.6857 +High Mean +0.0218 +0.0185 +0.0986 +1.1783 +0.2211 +Low Vol +0.0244 +0.0200 +0.0538 +1.2220 +0.4535 +High Vol +0.0226 +0.0169 +0.0341 +1.3411 +0.6628 +(b) LightGBM-gbdt without FE +Dynamic Feature Neutral. +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0204 +0.0211 +0.1998 +0.9665 +0.1021 +Low Mean +0.0234 +0.0184 +0.0495 +1.2737 +0.4727 +High Mean +0.0199 +0.0212 +0.1469 +0.9381 +0.1355 +Low Vol +0.0224 +0.0228 +0.1852 +0.9797 +0.1210 +High Vol +0.0182 +0.1633 +0.0487 +1.1986 +0.4476 +(c) MLP without FE +Dynamic Feature Neutral. +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0179 +0.0203 +0.2606 +0.8798 +0.0687 +Low Mean +0.0211 +0.0185 +0.0806 +1.1387 +0.2618 +High Mean +0.0186 +0.0201 +0.1283 +0.9256 +0.1450 +Low Vol +0.0206 +0.0215 +0.0878 +0.9598 +0.2346 +High Vol +0.0191 +0.0172 +0.0730 +1.1150 +0.2616 +Table 6: The effect of Dynamic Feature Neutralisation. Performance of differ- +ent ML models in the test period (2014-06-27 to 2022-09-23) with different dynamic +feature neutralisation methods +Next we compared the performance obtained by different dynamic feature neu- +tralisations under different market regimes, as defined in Section +5. +The results +can be found in Tables 9 and 8 in the Supplementary Information. Neutralisation +by ‘Low Mean’ performs better than Neutralisation by ‘High Mean’ in low volatility +regimes, but not in high volatility regimes. Under high volatility regimes, neutralisa- +tion by ‘Low Volatility’ features in the models performs better than neutralisation by +‘Low Mean’. Under a low volatility regime, neutralisation by ‘Low Mean’ performs +significantly better than others. +Based on the above, we make the following observations: In a low volatility +regime, factors that are performing well recently continue to do so in the near future +as the feature correlation structure is more stable in low volatility regime. +This +works until there is a regime change. In a high volatility regime, the ML models +after neutralisation of ‘Low Volatility’ features have a much higher Mean Corr than +models obtained by other neutralisation methods. ‘Low Volatility’ represents features + +ROBUST ML MODELS IN FINANCE +17 +that have a low variance, and stable performance in the last 52 weeks. During volatile +regimes, these features will underperform. Models that neutralise these features can +then outperform when there is market stress. +6.2. Dynamic model selection. In practice, it is not possible to know the best +dynamic feature engineering methods in advance. Therefore, we propose an online +learning procedure to select the dynamic feature engineering method during the test +period consisting of two steps. The first step is to have a warm-up period to collect +data on model performances, during which all 5 feature neutralisation methods (fixed, +low mean, high mean, low vol, high vol) have equal weighting. The second step is to +allocate weights to the optimal model based on recent performance according to the +following criteria: +• ‘Average’: Using all five feature neutralisation methods with equal weighting +• ‘Momentum’: Using the feature neutralisation method with the highest Mean +Corr in the last 52 weeks +• ‘Sharpe’: Using the feature neutralisation method with the highest Sharpe +Ratio in the last 52 weeks +• ‘Calmar’: Using the feature neutralisation method with the highest Calmar +ratio in the last 52 weeks +In Table 7, we use these criteria to select the optimal dynamic feature engineering +method based on recent performance. As above, a lag of 6 weeks is applied to account +for data delays. +The online learning procedure can thus select the optimal dynamic feature engi- +neering method to outperform the ‘Average’ selection in most cases. For all three ML +models (LightGBM-dart/LightGBM-gbdt/MLP), the ‘Momentum’ selection method +has higher mean Corr and Calmar ratio than the‘Average’ (baseline) and ‘Sharpe’ +methods. This shows that the ‘Momentum’ method, a very simple model selection +method that chooses the recent best-performing model, can adapt a trained ML model +towards different market regimes efficiently. For LightGBM-dart and LightGBM-gbdt +models, the ‘Calmar’ selection method gives a higher Calmar ratio than the ‘Momen- +tum’ method but with a lower mean Corr. For MLP models, the ‘Calmar’ selection +method significantly under-performs other model selection methods, with a much +higher Max Drawdown. This suggests that selection based on historical drawdown is +not robust, especially under situations with regime changes. +In summary, the proposed online learning procedure to select optimal dynamic +feature engineering methods can significantly reduce trading risks and improve the +robustness of trading models, outperforming the baseline selection method that takes +a simple average of all available models. +7. Discussion. Motivated by the Numerai tournament, we have designed here +an ML pipeline that can be applied to tabular temporal data of stock prices to under- +pin strategies for trading of market-neutral stock portfolios. The various steps in the +ML pipeline are carefully designed for robustness against regime changes and to avoid +information leakage through time. We thus aim to obtain models with relatively low +complexity, so as to reduce the danger of over-fitting, and with high robustness to +changes in hyper-parameters and other choices in the algorithms. Another aim is to +Regarding the choice of ML models, we find that gradient-boosting decision tree +models are both more robust and interpretable than neural network-based models, +and they allow more consistent performance under different market regimes. +We also find that post-prediction processing, which is model-agnostic, is an effec- +tive means of adapting trained ML models towards new situations without the need + +18 +THOMAS WONG AND MAURICIO BARAHONA +(a) LightGBM-dart without FE +Model Selection +Mean +Volatility +Max Draw +Sharpe +Calmar +Average +0.0229 +0.0160 +0.0619 +1.4323 +0.3700 +Momentum +0.0246 +0.0180 +0.0533 +1.3654 +0.4615 +Sharpe +0.0234 +0.0165 +0.0533 +1.4148 +0.4390 +Calmar +0.0225 +0.0171 +0.0350 +1.3122 +0.6429 +(b) LightGBM-gbdt without FE +Model Selection +Mean +Volatility +Max Draw +Sharpe +Calmar +Average +0.0216 +0.0177 +0.0710 +1.2165 +0.3042 +Momentum +0.0228 +0.0201 +0.0729 +1.1342 +0.3128 +Sharpe +0.0224 +0.0187 +0.0729 +1.1966 +0.3073 +Calmar +0.0216 +0.0195 +0.0508 +1.1102 +0.4252 +(c) MLP without FE +Model Selection +Mean +Volatility +Max Draw +Sharpe +Calmar +Average +0.0195 +0.0175 +0.0918 +1.1149 +0.2124 +Momentum +0.0212 +0.0191 +0.0878 +1.1124 +0.2415 +Sharpe +0.0207 +0.0186 +0.0878 +1.1110 +0.2358 +Calmar +0.0187 +0.0201 +0.1973 +0.9309 +0.0948 +Table 7: The effect of dynamic model selection. Performance of different ML +models in the test period (2014-06-27 to 2022-09-23) with different online learning +procedures selecting the optimal dynamic feature neutralisation method. +to re-train ML models and introduce additional model uncertainty. Using dynamic +feature neutralisation produces models with different flavours in an interpretable way, +which also have better risk-adjusted performance than models with fixed feature neu- +tralisation. +Staking is commonly used in ML competitions to improve the robustness of mod- +els. The method suggested in this study, dynamic model selection can be applied to +online ML problems in guiding the selection of an optimal model(s) from a growing +model ensemble. We find that a simple design, such as equal-weighted models, has +a robust performance under different market regimes, but selecting the best model +based on recent performance provides an improvement compared to the baseline as +it switches to a lower-risk model during more volatile market regimes. It remains an +open research area into how reinforcement learning or other online learning methods +can be used to learn optimal staking weights between different ML models, given their +historical performance and correlations. +We also studied the robustness of our ML pipeline under different random seeds +and changes in data splits for cross-validation. The results are presented in Section +9.4 in the Supplementary Information, where we show that LightGBM dart mod- +els are robust against these changes. The statistical rules used in dynamic feature +neutralisation are also shown to perform better than features chosen at random. +In the following, we discuss some ideas for further work to improve the ML pipeline + +ROBUST ML MODELS IN FINANCE +19 +we designed. The diversity of models within a model ensemble is a key ingredient +for dynamic model selection and other model ensemble/staking methods. +A new +metric could be designed to study the impact of a new ML model on an existing +model ensemble. This metric could then be used to train new ML models that are +uncorrelated to existing ones. +The simple feature engineering methods used in our present study could not +improve the performance of ML models. +Identifying robust relationships between +features over different market regimes is difficult but generative models, such as Vari- +ational Autoencoders [42], could be used to create new features that summarise non- +linear relationships in existing features. +The Gradient Boosting models used in our pipeline are suitable for distributed +learning, where large datasets are split into smaller batches to train on different ma- +chines, often with various computational resource constraints. Data science compe- +titions like the Numerai tournament rely on community efforts of individual data +scientists to create a meta-model. This approach to crowd sourcing depends on the +assumption that a complicated ML model that needs to be trained with advanced +hardware can be approximated by combining a number of ML models (each trained +with fewer data or features). Studying the convergence of model performance would +be important for organising the data science competition as it decides how many +participants are needed to maintain a well-diverse pool of models to create the meta- +model. +Overall, our results suggest using simple, well-established ML models such as +gradient-boosting decision trees instead of specialised neural network models for this +tasks. +Rather than using a single neural network to perform feature engineering, +model training/inference and post-prediction transformations, the modularised de- +sign of the ML pipeline in this study offers increased robustness and transparency. +Researchers can add, modify or delete a component without affecting the rest of the +pipeline. Creating model ensembles improves model performances by reducing id- +iosyncratic variance from individual ML models. The simple model selection rules +based on recent performances provide a baseline that works well under different mar- +ket regimes, whereas various portfolio metrics such as Sharpe and Calmar ratios are +improved by using the recently best-performing models. +8. Data and Code Availability . The data and code used in this paper is +available at https://github.com/ThomasWong2022/numerai-benchmark. +REFERENCES +[1] J. Arosemena, N. Perez, D. Benitez, D. Riofrio, and R. Flores-Moyano, “Stock price analy- +sis with deep-learning models,” in 2021 IEEE Colombian Conference on Applications of +Computational Intelligence (ColCACI), 2021, pp. 1–6. +[2] S. Selvin, R. Vinayakumar, E. A. Gopalakrishnan, V. K. Menon, and K. P. 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Here we show +the performance of dynamic feature neutralisation for low and high volatility regimes. +(a) LightGBM-dart without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0206 +0.0195 +0.1153 +1.0576 +0.1787 +Low Mean +0.0255 +0.0175 +0.0350 +1.4578 +0.7286 +High Mean +0.0207 +0.0206 +0.0986 +1.0033 +0.2099 +Low Vol +0.0238 +0.0221 +0.0538 +1.0793 +0.4424 +High Vol +0.0235 +0.0180 +0.0341 +1.3069 +0.6891 +(b) LightGBM-gbdt without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0194 +0.0220 +0.1998 +0.8820 +0.0971 +Low Mean +0.0251 +0.0188 +0.0495 +1.3328 +0.5071 +High Mean +0.0184 +0.0228 +0.1469 +0.8053 +0.1253 +Low Vol +0.0214 +0.0247 +0.1852 +0.8657 +0.115 +High Vol +0.0225 +0.0188 +0.0487 +1.1939 +0.4620 +(c) MLP without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0165 +0.0210 +0.2606 +0.7875 +0.0633 +Low Mean +0.0215 +0.0187 +0.0496 +1.1496 +0.4335 +High Mean +0.0170 +0.0210 +0.1283 +0.8118 +0.1325 +Low Vol +0.0194 +0.0229 +0.0878 +0.8487 +0.2210 +High Vol +0.0194 +0.0177 +0.0730 +1.0990 +0.2658 +Table 8: Performance of ML models in the test period (2014-06-27 to 2022-09-23) +with different dynamic feature neutralisation methods in low volatility regime + +ROBUST ML MODELS IN FINANCE +23 +(a) LightGBM-dart without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0227 +0.0163 +0.0223 +1.3888 +1.0179 +Low Mean +0.0220 +0.0148 +0.0199 +1.4907 +1.1055 +High Mean +0.0233 +0.0151 +0.0206 +1.5372 +1.1311 +Low Vol +0.0252 +0.0168 +0.0330 +1.4980 +0.7636 +High Vol +0.0215 +0.0152 +0.0143 +1.4077 +1.5035 +(b) LightGBM-gbdt without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0217 +0.0198 +0.0364 +1.0953 +0.5962 +Low Mean +0.0212 +0.0176 +0.0380 +1.2039 +0.5579 +High Mean +0.0218 +0.0186 +0.0334 +1.1728 +0.6527 +Low Vol +0.0237 +0.0201 +0.0306 +1.1792 +0.7745 +High Vol +0.0209 +0.0173 +0.0308 +1.2068 +0.6786 +(c) MLP without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Fixed +0.0196 +0.0193 +0.0326 +1.0191 +0.6012 +Low Mean +0.0205 +0.0183 +0.0806 +1.1212 +0.2543 +High Mean +0.0170 +0.0210 +0.1283 +0.8118 +0.1325 +Low Vol +0.0222 +0.0194 +0.0397 +1.1442 +0.5592 +High Vol +0.0187 +0.0165 +0.0336 +1.1368 +0.5565 +Table 9: Performance of ML models in the test period (2014-06-27 to 2022-09-23) +with different dynamic feature neutralisation methods in high volatility regime +9.2. Pseudocode for algorithms in the text. For completeness, we present +here brief pseudocode for some of the main methods in the paper with the appropriate +references. +Algorithm 9.1 Gradient boosting algorithm [22,43] +Given N data samples (xi, yi), 1 ≤ i ≤ N with the aim to find an increasing better +estimate ˆf(x) of the minimising function f(x) which minimise the loss L(f) between +targets and predicted values. L(f) = � +i l(yi, f(xi)) where l is a given loss function +such as mean square losses for regression problems. Function f is restricted to the +class of additive models f(x) = �K +k=1 wkh(x, αk) where h(·, α) is a weak learner +with parameters α and wk are the weights. +Initialise f0(x) = arg minα0 +�N +i=1 l(yi, h(xi, α0)) +For k = 1 : K Compute the gradient residual using gik = − +� +∂l(yi,fk−1(xi)) +∂fk−1(xi) +� +Use the weak learner to compute αk which minimises �N +i=1(gik − h(xi, αk))2 +Update with learning rate λ fk(x) = fk−1(x) + λh(x, αk) +Return f(x) = fK(x) + +24 +THOMAS WONG AND MAURICIO BARAHONA +Algorithm 9.2 Gradient boosting tree algorithm implemented in LightGBM [22,23, +43] +Initialise f0(x) = arg minα0 +�N +i=1 l(yi, x, α0) +For k = 1 : +K For i += +1, 2, . . . N, +compute the gradient residual using +gik = − +� +∂l(yi,fk−1(xi)) +∂fk−1(xi) +� +Fit a decision tree to the targets gik giving terminal leaves Rkj, j = 1, 2, . . . Jk, where +Jk is the number of terminal leaves. +For j = 1, 2, . . . Jk, compute αjk = arg minα +� +xi∈Rkj l(yi, fk−1(xi) + α) +Update boosting trees with learning rate λ fk(x) = fk−1(x) + λ �Jk +j=1 αkjI(x ∈ Rkj) +Return fK(x) + +ROBUST ML MODELS IN FINANCE +25 +9.3. Hyper-parameter search space for different ML models. We ran all +experiments on a GPU cluster, each node of which contains a NVIDIA GeForce RTX +2080 Ti GPU, running with 4352 CUDA cores and 11GB memory. Hyper-parameter +search is performed using Optuna [34]. For each Feature Engineering/ML pipeline, +hyper-parameter search is ran for at most 8 hours or at most 100 configurations, +whichever came first. The default TPE sampler in Optuna is used to perform the +hyper-parameter search. In Figure 4 and 5, we list the Hyper-parameter search pa- +rameters defined in Optuna [34] for different ML models used in the main text to +train the models. +• Feature Engineering +– Numerai Basic Feature Engineering +∗ dropout pct: low:0.05, high:0.25, step:0.05, +∗ no product features: low:50, high:1000, step:50, +• ML Models +– LightGBM-gbdt +∗ n estimators: low:50, high:1000, step:50 +∗ learning rate: low:0.005, high:0.1, log:True +∗ min data in leaf: low:2500, high:40000, step:2500 +∗ lambda l1: low:0.01, high: 1, log:True +∗ lambda l2: low:0.01, high: 1, log:True +∗ feature fraction: low:0.1, high:1, step:0.05 +∗ bagging fraction: low:0.5, high:1, step:0.05 +∗ bagging freq: low:10, high:50, step:10 +– LightGBM-dart +∗ n estimators: low:50, high:1000, step:50 +∗ learning rate: low:0.005, high:0.1, log:True +∗ min data in leaf: low:2500, high:40000, step:2500 +∗ lambda l1: low:0.01, high: 1, log:True +∗ lambda l2: low:0.01, high: 1, log:True +∗ feature fraction: low:0.1, high:1, step:0.05 +∗ bagging fraction: low:0.5, high:1, step:0.05 +∗ bagging freq: low:10, high:50, step:10 +∗ drop rate: low:0.1, high:0.5, step:0.1 +∗ skip drop: low:0.1, high:0.8, step:0.1 +– LightGBM-goss +∗ n estimators: low:50, high:1000, step:50 +∗ learning rate: low:0.005, high:0.1, log:True +∗ min data in leaf: low:2500, high:40000, step:2500 +∗ lambda l1: low:0.01, high: 1, log:True +∗ lambda l2: low:0.01, high: 1, log:True +∗ feature fraction: low:0.1, high:1, step:0.05 +∗ bagging fraction: low:0.5, high:1, step:0.05 +∗ bagging freq: low:10, high:50, step:10 +∗ top rate: low:0.1, high:0.4, step:0.05 +∗ other rate: low:0.05, high:0.2, step:0.05 +Fig. 4: Hyper-parameter Space for ML models + +26 +THOMAS WONG AND MAURICIO BARAHONA +• Machine Learning +– MLP +∗ max epochs: low:10, high:100, step:5 +∗ patience: low:5, high:20, step:5 +∗ num layers: low:2, high:7, step:1 +∗ neurons: low:64, high:1024, step:64 +∗ neuron scale: low:0.3, high:1, log:True +∗ dropout: low:0.1, high:0.9, log:True +∗ batch size: low:10240, high:40960, step:10240 +– TabNet +∗ max epochs: low:10, high:100, step:5 +∗ patience: low:5, high:20, step:5 +∗ batch size: low:1024, high:4096, step:1024 +∗ num d: low:4, high:16, step:4 +∗ num a: low:4, high:16, step:4 +∗ num steps: low:1, high:3, step:1 +∗ num shared: low:1, high:3, step:1 +∗ num independent: low:1, high:3, step:1 +∗ gamma : low:1, high:2, step:0.1 +∗ momentum: low:0.01, high:0.4, step:0.01 +∗ lambda sparse: low:0.0001, high:0.01, log:True +Fig. 5: Hyper-parameter Space for ML models + +ROBUST ML MODELS IN FINANCE +27 +9.4. Robustness of ML pipeline. One of the aims in this work was to provide +a robust pipeline for tabular temporal data under regime changes. Here we present +additional results of the robustness of the method under different scenarios and sources +of variability. +Robustness under changes of random seeds in the learning algorithms. In Ta- +ble 10, we report the variability of the performance of the LightGBM-dart, LightGBM- +gbdt and MLP models trained starting from 10 different initial random seeds. The +performance is generally robust to the change in random seeds, with small variances +in the prediction of the mean Corr and volatility and moderate for the Maximum +Drawdown. +Model +Mean +Volatility +Max Draw +Sharpe +Calmar +LightGBM-dart without FE +mean +0.0254 +0.0266 +0.1567 +0.9593 +0.1639 +sd +0.0006 +0.0007 +0.0158 +0.0365 +0.0175 +LightGBM-gbdt without FE +mean +0.0253 +0.0312 +0.2338 +0.8104 +0.1100 +sd +0.0006 +0.0006 +0.0296 +0.0278 +0.0153 +MLP without FE +mean +0.0233 +0.0271 +0.1643 +0.8600 +0.1446 +sd +0.0009 +0.0011 +0.0248 +0.0365 +0.0219 +Table 10: Variability of the performance of ML models in the test period (2014-06- +27 to 2022-09-23). The mean and standard deviation of each portfolio metrics are +calculated over models with 10 different random seeds for each method +A general strategy to reduce the variance is to combine different ML models. +There are two ways to do so: (i) averaging over models, by calculating the average +performance of different models, and (ii) averaging over predictions, by calculating the +average predictions from each model and then scoring the average predictions against +the target. Table 11 shows that averaging over predictions gives higher mean Corr +and Sharpe/Calmar ratios than averaging over models. +Therefore, this averaging +method is used to compute model performances in Table 2 in the main text. +Model +Average +Mean +Volatility +Max Draw +Sharpe +Calmar +LightGBM-dart without FE +Over models +0.0254 +0.0266 +0.1567 +0.9593 +0.1639 +Over predictions +0.0278 +0.0284 +0.1622 +0.9791 +0.1714 +LightGBM-gbdt without FE +Over models +0.0253 +0.0312 +0.2338 +0.8104 +0.1100 +Over predictions +0.0262 +0.0321 +0.2378 +0.8140 +0.1102 +MLP without FE +Over models +0.0233 +0.0271 +0.1643 +0.8600 +0.1446 +Over predictions +0.0258 +0.0289 +0.1668 +0.8931 +0.1547 +Table 11: Performance of different ML methods on Numerai v4 dataset in the test +period (2014-06-27 to 2022-09-23) with different averaging methods +Robustness under different cross-validation data splits. As financial data are regime +dependent, an important measure of model robustness is to measure the performance +of ML models that have been trained using different cross-validation splits of the data +and compute how much the model performance changes over different test periods. +To ascertain the robustness of data splits, we have carried out 3 cross-validation +splits (CV 1, CV 2, CV 3) as shown in Table 12. The hyper-parameters are optimised +under CV 1, which is the cross-validation used to generate the model performances +in the main text. These hyper-parameters are fixed for the models trained under +the CV 2 and CV 3 splits. For ML methods that require early stopping, the data + +28 +THOMAS WONG AND MAURICIO BARAHONA +in the validation period (different for each split) are used to regularise the models. +Therefore, by reusing the optimised hyper-parameters across all splits, we evaluate +the robustness of the model performance to the optimisation of hyper-parameters. We +then compute the performance when applying the models to shifted cross-validation +datasets in the walk-forward CV 2 and CV 3 data splits. +Our results show good +consistency in performance across CV 2 and CV 3, with only a small deterioration of +the results as compared to CV 1 (over which the hyperparameters were optimised). +We also find that LightGBM-dart with FE, the ML method that has the highest +mean Corr in CV 1, has the greatest return and best Sharpe and Calmar ratios also +in other cross-validations, as seen in Table 13. +Train Start +Train End +Validation Start +Validation End +Enter Ensemble +CV 1 +2003-01-03 +2012-07-27 +2012-12-21 +2014-11-14 +2015-05-15 +CV 2 +2003-01-03 +2014-06-27 +2014-11-21 +2016-10-14 +2017-04-14 +CV 3 +2003-01-03 +2016-05-27 +2016-10-21 +2018-09-14 +2019-03-15 +Table 12: Various cross-validation schemes to train ML models on different parts of +the data. CV 1 is the cross-validation used for hyper-parameter optimisation and +training ML models in the main text. +(a) CV 1 (2015-05-15 to 2022-09-23) +Method +Mean +Volatility +Max Draw +Sharpe +Calmar +LightGBM-dart without FE +0.0278 +0.0284 +0.1622 +0.9791 +0.1714 +LightGBM-gbdt without FE +0.0262 +0.0321 +0.2378 +0.8140 +0.1102 +MLP without FE +0.0258 +0.0289 +0.1668 +0.8931 +0.1547 +(b) CV 2 (2017-04-14 to 2022-09-23) +Method +Mean +Volatility +Max Draw +Sharpe +Calmar +LightGBM-dart without FE +0.0250 +0.0278 +0.1817 +0.8990 +0.1376 +LightGBM-gbdt without FE +0.0231 +0.0324 +0.3227 +0.7104 +0.0716 +MLP without FE +0.0215 +0.0289 +0.2307 +0.7446 +0.0932 +(c) CV 3 (2019-03-15 to 2022-09-23) +Method +Mean +Volatility +Max Draw +Sharpe +Calmar +LightGBM-dart without FE +0.0264 +0.0297 +0.1380 +0.8140 +0.1913 +LightGBM-gbdt without FE +0.0261 +0.0336 +0.1584 +0.7772 +0.1648 +MLP without FE +0.0224 +0.0240 +0.1171 +0.9339 +0.1913 +Table 13: Performance of selected machine learning methods on the Numerai dataset +in the test period for various walk-forward cross-validation schemes, (a) CV 1, (b) +CV 2 and (c) CV 3 +Robustness under feature selection for dynamic feature neutralisation. A fixed +set of 420 features to be neutralised was given by the Numerai organisers based on +internal evaluations of parameters. In Section 6, we introduce several statistical rules +that allow us to select a varying subset of features to be neutralised in each era based +on empirical heuristic criteria motivated by financial modelling. + +ROBUST ML MODELS IN FINANCE +29 +To evaluate the robustness of the proposed statistical rules, we draw 100 subsets +of 420 features selected at random. and use each set to neutralise the raw predictions +from ML models. We then evaluate the performance of ML models based on each of +the random subsets. Using the procedure described in section 6.2 we then select the +optimal dynamic feature neutralisation method and compute the performance of the +top 10 models of the highest mean Corr, Sharpe and Calmar ratio over the test period. +The results are reported in Table 14 and should be compared to the performance of +the same models in Table 7, which were obtained with dynamic feature neutralisation +using the statistical rules defined in section 6.2. +The mean Corr of models obtained with random feature neutralisation for each +rule (Momentum/Sharpe/Calmar) are lower than those obtained using the statistical +rules in Table 7. On the other hand, the Sharpe ratio of models for models with +random feature neutralisation is slightly higher, as expected due to the variance re- +duction effect by averaging over 10 different models. For models selected based on the +Calmar rule, the models obtained with statistical rules have a much higher Calmar +ratio than random feature neutralisation. It suggests the statistical rules defined can +effectively reduce model risks by reducing linear exposure to undesirable features. +(a) LightGBM-dart without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Average +0.0214 +0.0147 +0.0482 +1.4547 +0.4440 +Momentum +0.0216 +0.0149 +0.0472 +1.4522 +0.4576 +Sharpe +0.0213 +0.0147 +0.0459 +1.4474 +0.4641 +Calmar +0.0214 +0.0148 +0.0453 +1.4504 +0.4724 +(b) LightGBM-gbdt without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Average +0.0203 +0.0167 +0.0664 +1.2140 +0.3057 +Momentum +0.0208 +0.0167 +0.0641 +1.2457 +0.3245 +Sharpe +0.0206 +0.0168 +0.0618 +1.2267 +0.3333 +Calmar +0.0216 +0.0195 +0.0508 +1.1102 +0.2743 +(c) MLP without FE +Feature Neutralisation +Mean +Volatility +Max Draw +Sharpe +Calmar +Average +0.0176 +0.0165 +0.0831 +1.0658 +0.2118 +Momentum +0.0179 +0.0165 +0.0790 +1.0842 +0.2266 +Sharpe +0.0177 +0.0164 +0.0762 +1.0751 +0.2323 +Calmar +0.0175 +0.0167 +0.0825 +1.0511 +0.2121 +Table 14: Performance of different ML models in the test period (2015-05-15 to 2022- +09-23) obtained with random feature neutralisation. These are averages obtained by +selecting the top 10 models under the different online learning procedures over the +test period. + diff --git a/DdAyT4oBgHgl3EQf4vob/content/tmp_files/load_file.txt b/DdAyT4oBgHgl3EQf4vob/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc0729aceb12aa054769ab3ba36deef93c4840bb --- /dev/null +++ b/DdAyT4oBgHgl3EQf4vob/content/tmp_files/load_file.txt @@ -0,0 +1,1756 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf,len=1755 +page_content='ROBUST MACHINE LEARNING PIPELINES FOR TRADING MARKET-NEUTRAL STOCK PORTFOLIOS ∗ THOMAS WONG† AND MAURICIO BARAHONA‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The application of deep learning algorithms to financial data is difficult due to heavy non-stationarities which can lead to over-fitted models that underperform under regime changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Using the Numerai tournament data set as a motivating example, we propose a machine learning pipeline for trading market-neutral stock portfolios based on tabular data which is robust under changes in market conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We evaluate various machine-learning models, including Gradient Boosting Decision Trees (GBDTs) and Neural Networks with and without simple feature engineer- ing, as the building blocks for the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We find that GBDT models with dropout display high performance, robustness and generalisability with relatively low complexity and reduced computa- tional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We then show that online learning techniques can be used in post-prediction processing to enhance the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In particular, dynamic feature neutralisation, an efficient procedure that requires no retraining of models and can be applied post-prediction to any machine learning model, improves robustness by reducing drawdown in volatile market conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Furthermore, we demon- strate that the creation of model ensembles through dynamic model selection based on recent model performance leads to improved performance over baseline by improving the Sharpe and Calmar ra- tios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We also evaluate the robustness of our pipeline across different data splits and random seeds with good reproducibility of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Robust Machine Learning, Online Learning, Gradient Boosting Decision Trees, Deep Learning, Stock Trading, Tabular Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As investors explore new ways to generate profit, machine learning (ML) models are increasingly used as part of trading strategies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', to pre- dict the future return of stocks or stock portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In particular, deep learning models for time-series data, such as Recurrent Neural Networks (RNNs) and Convolutional Neural Networks (CNNs), have been applied to the prediction of future stock re- turns [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' However, a major challenge for such methods is the highly stochastic, non-stationary and non-ergodic nature of financial data, which violates the assump- tions of many algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Furthermore, deep learning models are over-parameterised, with numbers of parameters orders of magnitude larger than typical sizes of time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore, deep models can be easily over-fitted to specific patterns in historical market data not present in future market data, and the over-fitting worsens with the more complicated neural network architectures, such as Long Short Term Memory (LSTM) or Transformer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In addition, the continuous influx of data, coupled with possible regime changes, requires costly updating and retraining of such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore, such methods can lack reproducibility and robustness for the prediction of future market data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As pointed out in recent reviews [4,5], replication of ML studies is often difficult due to several issues, including data leakage [5], program bugs [6], data and code usability [7], and model representation and evaluation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' These problems and are currently hindering the usage of ML in high-risk decision processes, such as healthcare and finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For trading applications in particular, these issues can have critical effects on the validity of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Data leakage, in the form of look-ahead bias or overlap ∗Funding: This work is supported by the Wellcome Trust under Grant 108908/B/15/Z and by the EPSRC under grant EP/N014529/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' †Department of Mathematics, Imperial College London, London SW7 2AZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='K (ming- hei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='wong15@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‡Department of Mathematics, Imperial College London, London SW7 2AZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='K (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='barahona@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='uk,).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='00790v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='CP] 30 Dec 2022 2 THOMAS WONG AND MAURICIO BARAHONA of training/test sets [8], can inflate in-sample performance with poor performance when deployed live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Furthermore, black-box ML models, such as neural networks, can lack robustness as they are highly sensitive to small changes in parameters and data, thus resulting in volatile predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The non-stationary data and the presence of regime changes also mean that ML models need to be re-trained with the latest financial data, a task that is not only computationally costly but also introduces further uncertainty to the trading models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Yet most studies do not consider model performance when trained on different segments of historical market data [1–3,9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Although reinforcement learning (RL) in online learning settings allows ML models to adapt to changing environments, deep reinforcement learning models are complex and require large computational resources [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Indeed, applying RL to stock trading is difficult since the complexity of the action space increases exponentially with the number of stocks in the portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The above issues suggest the need to further develop robust ML pipelines for trading applications possibly based on simpler models that can still operate on non- stationary, highly stochastic data under regime changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Here we consider such a pipeline based on tabular data, which allows the use of traditional ML models, such as Gradient Boosting Decision Trees (GBDT) and other ensemble methods, to predict trading stocks and stock indices [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This approach also allows the integration of additional sources of data, such as sentiment analysis of news articles to improve the prediction accuracy of the direction of stock returns [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In particular, we find that Gradient Boosting models, which are known to be robust to data perturbations, outperform neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Finally, we show that improved robustness of ML models and adaptation to regime changes can be attained without the use of deep reinforcement learning by employing: (i) dynamic feature neutralisation, a simple approach that reduces the linear correlation to a subset of features evolving in time, and (ii) dynamic model selection of optimal models from an ensemble based on recent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' These approaches robustly improve trading performances by reducing volatility and drawdown during adversarial market regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' To exemplify the above issues, we consider a benchmark financial data platform that is continuously updated and curated under the Numerai tournament of stock portfolio prediction [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Numerai is a hedge fund that organises a data science com- petition (as of Oct 2022) and provides free, open-source, high quality standardised financial data to all participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As discussed below in more detail, the data set is given in the form of pre-processed temporal tabular data and the task is the predic- tion of the relative performances of stocks within an evolving trading universe without access to the identity of individual stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Unlike other financial research papers that use proprietary data sets which can be difficult to validate [9,10], this open financial data competition allows researchers to replicate findings transparently and allows us to focus on establishing ML end-to-end pipelines to achieve consistent profits trad- ing a market-neutral portfolio under changing market regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Our pipeline, shown in Figure 1, is built upon simple, yet robust methodologies that avoid some of the problems of over-fitting and high computational cost inherent to deep methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The robustness of the pipeline is enhanced since each step is implemented independently avoiding data leakage, which is common in other methods such as neural networks, where the pre-processing and the actual model often share data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Key ingredients are the post-prediction processing and feature engineering steps, which allow easy adaptation of models towards regime changes without expensive retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Section 2 introduces the Numerai datasets used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Section 3 describes and discusses the different computational ROBUST ML MODELS IN FINANCE 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 1: Schematic of the Machine Learning pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Starting with the Numerai data set, we consider feature engineering methods to augment the dataset and train an ML model (several are evaluated, including neural networks, but we settle for gradient boosting trees) to obtain the raw predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' These then go through post-prediction processing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', dynamic feature neutralisation) to provide normalised predictions, which are then combined through model ensembling and dynamic model selection methods to output the predictions that are submitted to the Numerai tournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' methods, including online cross-validation, feature engineering and the different ML models considered and evaluated for the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Section 4 presents the results from our ML pipeline, including the impact of different design choices on the robustness of trading performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Performances of ML models under different market regimes are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In Section 6, we introduce adaptations to our ML models based on online learning approaches, which can work well under regime changes, noting that these adaptations are generic and not limited to specific families of ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Lastly, we discuss the results of the method, open directions and alternatives and provide a study of the robustness of our ML pipeline in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Numerai dataset and prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Financial data are often available in the form of time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' These time series can be treated directly using classic meth- ods such as ARIMA models [16] and more recently through deep learning methods such as Temporal Fusion Transformers [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' However, such methods are easily over- fitted and lead to expensive retraining for financial data, which are inherently affected by regime changes and high stochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Alternatively, one can use various feature engineering methods to transform these time series into tabular form through a pro- cess sometimes called ‘de-trending’ in the financial industry, where the characteristics of a financial asset at a particular time point, including features from its history, are represented by a single dimensional data row (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', a vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In this representation, the time dimension is not considered explicitly, as the state of the system is captured through transformed features at each time point and the continuity of the temporal dimension is not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For example, we can summarise the time series of the return of a stock with the mean and standard deviation over different look-back periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Grouping these data rows for different financial assets into a table at a given time point we obtain a tabular dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' If the features are informative, this representation can be used for prediction tasks at each time point, and allow us to employ robust and widely tested ML algorithms that are applicable to tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The Numerai competition is based on a curated tabular data set with high-quality features made available to the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Description of the dataset:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The Numerai dataset is a temporal tabular dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' A temporal tabular dataset is a collection of matrices {Xi}1≤i≤T collected over time eras 1 to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Each matrix Xi represents data available at era i with shape Ni × M, Machine Dataset Creation Feature Post-Prediction Model Learning (Numerai) Engineering Processing Ensemble Model Training4 THOMAS WONG AND MAURICIO BARAHONA where Ni is the number of data samples (number of stocks here) in era i and M is the number of features describing the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Note that the features are fixed throughout the eras, in the sense that the same computational formula is used to compute the features in each week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' On the other hand, the number of data samples (stocks) Ni does not have to be constant across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In the Numerai dataset, the matrices Xi contain M obfuscated global stock mar- ket features (computed by Numerai) for Ni stocks, which are updated weekly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', the eras are in our case weeks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' It is important to remark that the dataset is obfuscated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', it is not possible for participants to know the identity of stocks or even which stocks are present each week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Each data row is indexed by a hash index, known only to Numerai, that maps the data rows to the stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As a result, it is not possible for participants to concatenate different data rows to create a continuous history of a stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The matrix Xi thus provides a snapshot of the market at week i presented as an unknown set of stocks described by a common set of features, such that the features are computed consistently across all stocks in the week and also computed consistently across different weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The Numerai dataset starts on 2003-01-03 (Era 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The tabular set has 1191 features, which are already normalised into 5 equal-sized integer bins, from 0 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' There are 28 target labels which are derived from stock returns using 14 proprietary normalisation methods (nomi, jerome, janet, ben, alan, paul, george, william, arthur, thomas, ralph, tyler, victor, waldo ) over 2 forward-looking periods (20 trading days, 60 trading days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The main target label to evaluate performance is target-nomi-v4- 20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', forward 20 trading days return obtained by the nomi normalisation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Other targets are named similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The target labels are all scaled between 0 to 1, where a smaller value represents a lower forward return, and are also grouped into bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For each normalisation method, the number of bins could be different, 5 to 7 bins are created for each target with the bin sizes following a Gaussian-like distribution, so that most stocks are within the central bin of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5 while only a small amount of stocks are in the tail bins of 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We transform the features and labels so that both become zero-mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (For features, we subtract 2 from the integer bins so that the transformed bins are -2,-1,0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For the target labels, we subtract 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5 so that the new targets are in the range -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Prediction task:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The tournament task is to predict the stock rankings each week, ordered from lowest to highest expected return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The scoring is based on Spearman’s rank correlation of the predicted rankings with the main target label (target-nomi-v4- 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Hence there is a single overall score each week regardless of the number of stocks to predict each week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Participants are not scored on the accuracy of the ranking of each stock individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Numerai uses the predicted rankings to construct a market- neutral portfolio which is traded every week (As of Sep 2022), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', the hedge fund buys and short-sells the same dollar amount of stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore the relative return of stocks is more relevant than the absolute return, hence the prediction task is a ranking problem instead of a forecast problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Robustness in Machine Learning pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In this paper, we aim to design an ML pipeline focusing on its robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Table 1 details issues related to robustness and reproducibility, as listed in a recent review [5], and how they are addressed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' By preventing look-ahead bias and other data leakage issues, our pipeline can be robustly applied to live trading setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In addition to avoiding data leakage, the following design choices are used to im- ROBUST ML MODELS IN FINANCE 5 Issues affecting robust- ness of ML algorithms How the issue is addressed here ‘No test set’ A robust cross-validation scheme is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Pre-processing on train- ing and test set’ Numerai features are already standardised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' hence minimal pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Feature selection on training and test set’ Feature Engineering is applied to each data row in- dependently ‘Duplicates in datasets’ A unique id for each data row reduces the chance of duplicates in dataset ‘Model uses features that are not legitimate’ Only data provided by Numerai is used to train ML models—no extra features from other resources, and no cherry-picking of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Temporal leakage’ We use Grouped Time-Series Cross-Validation with no overlap between training/validation/test (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Feature Engineering is applied to each data row in- dependently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', no data leakage between eras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Non-independence be- tween training and test’ Training and test samples are market data at different periods without overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Sampling bias in test dis- tribution’ The stocks trading each week are decided by Numerai based on operational and risk considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Table 1: Data analysis design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Some common issues regarding data leakage in machine learning research [4,5] and how these issues are dealt with in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' prove the robustness and reliability of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Firstly, the impact of random seeds is reduced by reporting results from average predictions over 10 different random seeds for each machine learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Secondly, the metrics used for model evaluation are the same as in the Numerai tournament to avoid researcher bias in discounting unfavourable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Finally, cross-validation is independent of the effects of random seeds and other human selection, thus reducing the chance of overfitting models to a particular data split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For datasets that involve time, standard cross-validation schemes cannot be used directly, as a random split of data eras could lead to the training set including data that appears later in time than the validation and test sets, hence introducing look-ahead bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' To avoid this problem, we use grouped time-series cross-validation, which splits data eras according to their chronological order (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Note that for financial datasets, the target labels often involve future asset returns and are reported with a lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore, we add a gap between the training and validation sets and similarly between the validation and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Feature Engineering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Feature engineering is a crucial step in enhancing the power of tabular methods for the analysis of time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore, we evaluate different feature engineering methods that can be applied to temporal tabular data sets with numerical features, as the Numerai data set only contains normalised numerical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' New features can be created by applying polynomial transformations such as multiplication and addition to the original features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Here we create new features by multiplying two features that can be thought of as modelling the joint distribution 6 THOMAS WONG AND MAURICIO BARAHONA Training Data V alidation Data Test Data Time gap gap Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 2: Illustration of data split using grouped time-series cross-validation of feature pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' When the number of features is large, we draw a random subset of feature pairs to create new features to alleviate the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Note that the computation of these features can be done in parallel for data from each era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' A simple way of data augmentation is to add randomness to the feature matrix with different dropout methods, which are used extensively to reduce over-fitting of neural network models [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Here we apply dropout by multiplying the original data with a Boolean mask so that some numerical features are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The dropout is characterised by its sparsity level (how many features are set to zero) and its sparsity structure (how to choose the features set to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Since our tabular dataset has no local spatial structure, we use a random Boolean matrix with uniform probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This encourages the machine learning methods to learn multiple feature relationships and reduces reliance on a small set of important features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For our dataset, we first augment the feature matrix by creating additional fea- tures obtained by multiplying feature pairs, and then apply dropout with a random Boolean mask on the augmented feature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' A grid search is used to find optimal hyper-parameters for the feature engineering methods, in particular the number of feature products and the sparsity level of dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Machine Learning algorithms for tabular datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Numerous ma- chine learning models have been proposed for tabular datasets, and different bench- marking studies have shown conflicting views on their performance [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The biggest disagreement in the literature is whether gradient-boosting decision trees or neural networks are superior in regression and classification tasks of tabular datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Whereas one paper claims gradient boosting models (XGBoost) outperformed deep learning models in 8 out of 11 datasets and none of the deep learning models consis- tently outperform others [19], another paper suggests that well-tuned multi-layer per- ceptron (MLP) models with regularisation can outperform different gradient boosting models such as XGBoost and CatBoost [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Both these studies, however, share the same view that neural networks with complicated designs, such as attention layers and other transformer layers, tend to generalise poorly with a strong drop in performance when applied to data sets beyond their original study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Importantly, the Numerai data set is different from the data sets in the above benchmarking studies in that it is growing instead of fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Hence the data distribution varies across time periods due to market regime effects, and we do not have a homogeneous distribution across cross-validation splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' With such a different problem setup, it is thus not possible to use the above benchmarking studies to guide our choice of ML method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In this study, we benchmark a wide range of machine learning models, including different variants of gradient-boosting decision tree models and different neural net- work models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The choice of ML models is based on the popularity of usage in data ROBUST ML MODELS IN FINANCE 7 science competitions and code quality, as one of our aims, is the replicability of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We train all machine learning models with a single GPU, the standard setup for most participants in data science competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Some brief details of the ML models used are provided in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Gradient Boosting Decision Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Boosting can be seen as a generalisation of generalized additive models (GAM) where the additive components of smooth para- metric functions can be replaced by any weak learners such as decision trees [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Historically, various boosting algorithms have been proposed for different loss func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For example, AdaBoost [21] was proposed for binary classification problems with exponential loss, whereas Gradient Boosting was first proposed by Friedman in 2001 [22] for any smooth loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1 in the SI outlines the iterative update equations of gradient boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Of the various implementations of gradient boosting decision (GBDT) trees in Python, we use LightGBM [23] in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' CatBoost [24] is not used here as the Numerai dataset has no categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' XGBoost [25] is not used due to slower computation and more memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2 in the SI shows how the gradient boosting algorithm is implemented with decision trees being the weak learners in LightGBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' LightGBM implements GBDT models with several computational and numerical improvements from XGBoost and other implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In addition to traditional gradient boosting decision trees (LightGBM-gbdt), we consider two other implementations of GBDT models: Dropouts meet Multiple Additive Regression Trees (LightGBM-dart) ignores a portion of trees when computing the gradient for subsequent trees [26], thus avoiding over-specialisation where the later learned trees can only affect a few data instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This reduces the sensitivity of models towards decisions made by the first few trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Gradient-based One-Side Sampling (LightGBM-goss) reduces the number of data instances used to build each tree: it keeps data instances with large absolute gradients and randomly samples a subset of data with small absolute gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The approximation error of the gradient using LightGBM-goss converges to the standard method when the number of data is large, and it outperforms other data sampling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', uniform sampling) in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For all LightGBM models, we use mean squared error (L2 loss) as the loss function for the regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The number of gradients boosting trees and learning rate is optimised by hyper-parameter searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' To prevent the over-fitting of trees, the maximum depth and number of leaves in each tree and the minimal number of data samples in the leaves are tuned for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' L1 and L2 regularisation are also applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Data and feature sub-sampling are used to reduce similarities between trees: before building each tree, a random part of data is selected without re-sampling and a random subset of features is chosen to build the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For LightGBM-dart models, both the probability to apply dropout during the tree-building process and the portion of trees to be dropped out are tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Early stopping is applied using the validation dataset for LightGBM-gbdt models to further prevent the over-fitting of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The most basic architecture of neural networks, multi-layer perceptron (MLP), failed to outperform gradient boosting models in many benchmark studies of tabular datasets [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Recently, more complex network architectures have been proposed for tabular data sets, as surveyed in [27] These new architectures can be classified into two major groups: Hybrid models that combine neural networks with other traditional ML meth- 8 THOMAS WONG AND MAURICIO BARAHONA ods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Neural Oblivious Decision Ensembles (NODE) [28] is a generalisation of gradient boosting models into differentiable deep decision trees allowing end-to-end training with gradient descent optimisers such as PyTorch [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' DeepGBM [30] combines two neural networks, CatNN to han- dle sparse categorical features and GBDT2NN to distil tree structures from a pre-trained GBDT model to handle numerical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' A major limitation of these models is the large memory consumption, which makes them run out of memory on the NVIDIA 3080ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore we do not use them in our benchmark analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Transformer-based models that use deep attention mechanisms to model com- plex feature relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' TabNet [31] uses sequential attention to perform instance-wise feature selection at each decision step, enabling interpretability and better learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' AutoInt [32] maps and models feature interactions in a low-dimensional space with a multi-head self-attentive neural network with residual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' AutoInt runs out of memory on a single GPU and is thus not used in our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Tabnet also had similar memory issues, hence we down-sampled the data by keeping every fifth week of data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', 20% of the original data) for the training/validation periods, so that Tab- net could be trained on the single GPU used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Our aim is to compare performance under modest computational resources attainable by a wide class of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In summary, our benchmark analysis includes two NN models: MLP and TabNet implemented in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We use Adam [33] as the gradient optimiser, with the learning rate automatically determined by PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We use mean squared error (L2 loss) for the regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Evaluation of Machine Learning methods for the Numerai temporal tabular data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In this section, we study different ML methods applied to the Numerai temporal tabular data set for the prediction of stock rankings aimed at market-neutral stock portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Data Split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We use the latest version (v4) of the Numerai dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The training period is fixed between 2003-01-03 (Era 1) to 2012-07-27 (Era 500), and the validation period is fixed between 2012-12-21 (Era 521) and 2014-11-14 (Era 620).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The test period starts on 2015-05-15 (Era 646) and ends on 2022-09-23 (Era 1030).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We apply a 1-year gap between training and validation periods to reduce the effect of recency bias so that the performance of the validation period will better reflect future performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The gap between the validation period and test period is set to 26 weeks to allow for sufficient time to deploy trained machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Evaluation of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For each configuration of each ML method, we aver- age over the predictions of different targets before scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The predictions are scored in each era by calculating the correlation (Corr) between the rank-normalised pre- dictions and the actual (binned) stock ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The mean and standard deviation (volatility) of Corr are reported for both the validation and test periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' To measure the downside risk of the model, we also compute the Maximum Drawdown, defined as the largest drop suffered by an investor starting at any time during the valida- tion/test period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As summary measures, we compute two standard ratios: (i) the Sharpe ratio, defined as the ratio of the mean and standard deviation of Corr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' and (ii) the Calmar ratio, defined as the ratio of mean Corr over Maximum Drawdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Good performance is characterised by large values of both of these ratios, ROBUST ML MODELS IN FINANCE 9 Model Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We use Optuna [34] to perform the hyper-parameter search (see section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3 in Supplementary Information) and select the hyper-parameters with the highest Sharpe ratio for the main target (target-nomi-v4-20) in the validation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The optimised hyper-parameters for each ML method are so fixed, and we then train 10 models, starting the algorithms from 10 different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We report the average prediction from these 10 models for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Baseline Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As a baseline, we consider a factor momentum model which is created by linear combinations of signed features, where the sign of each feature is determined by the sign of the 52-week moving average of correlations of that feature with the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This simple baseline linear model is then compared with the ML models, which can capture non-linearity in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Comparative results of the ML algorithms and Feature Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Table 2 shows the performance on the validation and test sets for the different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We concentrate on methods that achieve the highest mean Corr, and Sharpe and Calmar ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (a) Performance over the validation period (2012-12-21 to 2014-11-14) Method Mean Volatility Max Draw Sharpe Calmar Factor Momentum (baseline) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0691 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3314 MLP with FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0241 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0338 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7552 MLP without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0443 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1547 TabNet without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0277 LightGBM-gbdt with FE 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5539 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0305 LightGBM-goss without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0207 Table 2: Performance of different machine learning methods with and without feature engineering on the Numerai dataset for (a) validation period and (b) test period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The three top methods according to Sharpe ratio and Maximum Drawdown over the validation period are shown in italics in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The top method according to the Sharpe ratio and Maximum Drawdown over the test period is shown in boldface in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For TabNet, the pipeline with feature engineering cannot be run due to memory constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 10 THOMAS WONG AND MAURICIO BARAHONA Firstly, we see that almost all ML models performed substantially better than the factor momentum model (baseline), in both validation and test periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Whereas the factor momentum model relies on linear relationships, the capability of ML models to learn non-linear relationships, in addition to linear ones, adds to their robustness and improved performance under different, often volatile, market regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Secondly, we observe that Feature Engineering does not improve the performance of ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Although, in principle, Feature Engineering allows GBDT-based meth- ods to model feature interactions more easily, our results suggest that these interac- tions are over-fitted during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For neural network-based models, feature engineering is not strictly necessary, as dropout is already embedded in net- work architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Thirdly, we note that all ML models scored better in the validation period than the test period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This is expected, as it is well known that the performance of trading models deteriorates over time due to overcrowding and regime changes (a phenomenon known as alpha decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Models that are over-fitted to recent training data will ex- perience greater alpha decay than properly regularised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' To select the ML method, we consider the top models according to the Sharpe and Calmar ratios over the validation period: a high Sharpe ratio ensures the model has good overall perfor- mance, whereas a high Calmar ratio ensures good performance against the worst-case scenario, thus capturing the tail risks of the trading model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Indeed, we find that LightGBM-dart without feature engineering generalises well to the test period further into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Finally, we note that LightGBM-gbdt has better generalisation to the test period than neural network-based models (TabNet), suggesting over-fitting in these complex deep NN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This indicates that although over-parameterised models can learn non-linear relationships in temporal tabular data sets, these relationships may be difficult to generalise under non-stationary data environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' On the other hand, our results suggest that, despite their relative simplicity, gradient Boosting models capture non-linearity in a more robust and controlled manner, with early trees capturing linear relationships and non-linear relationships captured by the later trees, thus reducing the risk of catastrophic forgetting [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In summary, we find that the best performing model in our set is LightGBM- dart without feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In the rest of the paper, we will use this model to illustrate how the pipeline can be further modified with online learning to account for regime effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' To demonstrate the robustness of our pipeline, and how it can be applied to improve the performance of any ML model, we will also report the performance of two other models: a similar GBDT model (LightGBM-gbdt without feature engineering) and a neural network model (MLP without feature engineering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Dealing with regime effects in the ML pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Financial data are heavily influenced by regime changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Growth (‘Bull’) markets are characterised by low volatility and positive expected return, whereas high volatility and negative expected returns are characteristic of adverse (‘bear’) markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Switches between regimes can be triggered by externalities, such as pandemics, economic recessions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' From the perspective of the Numerai data set, such regime effects affect model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Volatility is detrimental to long-term performance due to the negative compounding of investment losses, a phenomenon known as ‘volatility tax’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Given that hedge funds are leveraged, we consider consistent models with reasonably good performance under different market regimes, rather than models that have excellent performance in one market regime but fail in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ROBUST ML MODELS IN FINANCE 11 In this section, we focus on how to deal with regime effects when using ML models for financial tabular temporal data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Specifically, we consider feature neutralisation, and reducing the dependence on the initial trees in gradient boosting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Classification into high and low volatility regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' To classify the financial market into regimes, we consider an intrinsic measure derived directly from the Numerai dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In particular, we first compute the Numerai Market Index (NMI), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', the weekly performance of the baseline (linear) factor momentum portfolio, and we then calculate the Numerai Realised Volatility Index (NRVIX), defined as the standard deviation of NMI rolling over 52 weeks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The eras are then classified into high and low volatility, based on a threshold of NRVIX=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='025, the mean over the first 7 years of data (2003-01-03 to 2010-02-26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' According to this intrinsic characterisation, low volatility regimes have stable linear relationships of features to stock returns, often associated with a good performance by ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' On the other hand, high Volatility regimes correspond to unstable linear relationships of features to stock returns leading to poor model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Figure 3 shows that high/low NRVIX regimes are well aligned with macroeconomic events: high volatility regimes include the financial crisis (2007-2009), the Euro crisis (2011-2012), and the Covid pandemic (2020), whereas low volatility regimes correspond to benign market conditions with no significant macro event risks, during which the factor momentum baseline portfolio had good returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 2005-01-28 2008-11-28 2012-09-28 2016-07-29 2020-05-29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='100 (a) NMI 2005-01-28 2008-11-28 2012-09-28 2016-07-29 2020-05-29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='040 (b) NRVIX Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 3: High and low volatility regimes in the Numerai data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (a) Numerai Market Index (NMI) for the period between 2005-01-28 (Era 109) and 2022-09-23 (Era 1016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (b) the computed Numerai Realised Volatility Index (NRVIX) used to identify the high and low volatility regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The high volatility regime refers to weeks where NRVIX is higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='25 and the low volatility regime refers to weeks where NRVIX is lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Feature Neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Feature neutralisation is the general term to denote the elimination of the effect of particular features in the model, thus reducing the risk of over-relying on certain individual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Because the predictive ability of individual features is highly dependent on market regimes, this can lead to long periods of drawdown when there is a regime change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' It is therefore undesirable to have ML models that could have heavy (linear)-dependence on certain features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We start by evaluating here the feature neutralisation suggested by the Numerai tournament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Numerai recommends that participants reduce model exposure to 420 12 THOMAS WONG AND MAURICIO BARAHONA ‘risky features’ (out of the 1191 features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This list of risky features can be used for feature neutralisation by subtracting the linear correlation using the formula for Feature Neutral Correlation (FNC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Specifically, given a week of data with n stocks, let X ∈ Rn×420 be the matrix of risky features and y ∈ Rn the predicted rankings ob- tained from a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For a given neutralisation strength β, 0 ≤ β ≤ 1, the neutralised predicted ranking ˆy is calculated as ˆy = y − β XX†y, where X† is the pseudo-inverse of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' FNC is then calculated as the correlation of the neutralised predicted rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Using this procedure, we reduce the linear dependencies of predictions on features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (a) LightGBM-dart without FE Regime Feature Neutral Mean Volatility Max Draw Sharpe Calmar All Yes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1153 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1865 No 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1714 High Vol Yes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0223 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3888 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0179 No 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0314 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0633 No 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1367 Table 3: The effect of feature neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Performance of different ML meth- ods on the Numerai v4 dataset over the test period (2014-06-27 to 2022-09-23) with and without feature neutralisation under different market regimes: the whole test period (all), high volatility regime (high-vol), and low volatility regime (low-vol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In Table 3, we compare the performance of the LightGBM-dart, LightGBM-gbdt and MLP with and without feature neutralisation under different market regimes (all, high volatility, low volatility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The neutralisation strength β is set to 1 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We find that the variance of models is consistently reduced by feature neutralisa- tion, suggesting an overall reduction of risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Further, feature neutralisation improves the Sharpe and Calmar ratios of LightGBM-dart and LightGBM-gbdt under different market regimes, but does not improve the performance of MLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Importantly, this default feature neutralisation procedure suggested by Numerai ROBUST ML MODELS IN FINANCE 13 is not optimal, and we will show in Section 6 how online learning approaches can be used to improve the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Pruning initial trees in Gradient Boosting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For gradient- boosting tree models, we also consider a specific procedure consisting of pruning initial trees during prediction to reduce feature dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Specifically, we perform a grid search over the number of initial trees to be pruned off in the trained LightGBM models, and we cap the number of trees to be pruned to not more than half of the trees to ensure our models do not degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Table 4 compares the performance of LightGBM-dart and LightGBM-gbdt models pruning different numbers of initial trees before feature neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Pruning ini- tial trees during prediction improves the Sharpe and Calmar ratios of both LightGBM models, but LightGBM-gbdt models see a bigger improvement than LightGBM-dart models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This is expected as LightGBM-dart models already employ a similar fun- damental idea during training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=', the trained trees in LightGBM-dart models are already optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Our numerics also suggest that there is a limit of trees to be pruned such that there is little improvement in model performance once over a threshold of around 100-250 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (a) LightGBM-dart without FE Prune Trees Mean Volatility Max Draw Sharpe Calmar 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0278 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Joint effect of feature neutralisation and tree pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We then considered the joint effect of feature neutralisation and pruning initial trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Table 5 compares the performance (FNC) of LightGBM-dart and LightGBM-gbdt models pruning a different number of initial trees after feature neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The effect of pruning on model performance for both LightGBM models after feature neutralisation is at best modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As FNC is a measure of the effect of non-linear relationships, this suggests that in gradient boosting models, early weak learners (trees) mostly capture linear relationships whereas most of the non-linear relationships are captured in the later weak learners (trees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore, pruning initial trees can be thought of as a model-dependent feature neutralisation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} 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joint effect of feature neutralisation and tree pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Perfor- mance of different LightGBM models after neutralisation in the test period (2014-06- 27 to 2022-09-23) when pruning different numbers of initial trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Online Learning to improve post-prediction processing and model ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As a further improvement to the ML pipeline, we apply online learning approaches to both feature neutralisation and model ensembles to produce improved versions called dynamic feature neutralisation and dynamic model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Dynamic feature neutralisation acts by applying statistical rules to determine subsets of fea- tures to neutralise predictions in each era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Dynamic model selection acts by updating regularly the choice of model(s) from a model ensemble based on recent model per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The aim of online learning is to derive an optimal procedure to select ML mod- els and parameters as data arrives continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In a continuous time setting, the Hamilton-Jacobi-Bellman (HJB) equation is solved to find the optimal determinis- tic control for the decision problem [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The discrete-time equivalent, the Bellman equation, is used in reinforcement learning to derive optimal policies of agents [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For the Numerai tournament, we consider online learning in the discrete-time setting, since data and predictions are required once per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For each week t (1 ≤ t ≤ T), we have a state (data) process Xt, which contains all the infor- mation we know about the environment (Numerai datasets and trained ML model parameters) up to week t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Our task is then to derive a deterministic decision pro- cess Dt(βt) described by parameters βt := βt(Xt), subject to the objective function VT = maxDt �T t=1 q(Xt, Dt), where q(Xt, Dt) represents the utility at time instant t given the data and decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (Deep) Reinforcement learning algorithms are commonly used to solve online learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' However, they are not used here due to the following reasons: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Limited data: Available data is not enough to train reinforcement learning models, such as Deep Q Networks (DQN) [38], Proximal Policy Optimisation (PPO) [39] and Soft Actor-Critic (SAC) [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Generating a large number of samples is difficult here since we must avoid look-ahead bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Expanding action space: Most implementations of reinforcement learning al- ROBUST ML MODELS IN FINANCE 15 gorithms, as found in Ray-RLlib [11], cannot adapt naturally to an expanding action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For the dynamic model selection problem, the number of po- tential models is unbounded, as newer models can be trained with the latest data available and added to the candidate list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Rule-based models, on the other hand, can handle the issue of expanding action space easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Actions have negligible impact on environment: Highly successful reinforce- ment learning algorithms are usually targetted at robotics and Atari games [41], where agent actions can modify the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' However, for the trad- ing models considered here, the trading activities are assumed to have zero or negligible market impact, and reinforcement learning algorithms thus reduce to an online learning prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Large, correlated feature sets for neutralisation: To improve feature neutral- isation, we use a different subset of features to neutralise predictions in each era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Yet the size of the set of risky features (420 features) makes it computa- tionally infeasible to learn feature subsets through supervised ML methods or reinforcement learning, as it is difficult to construct a robust reward function for correlated features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Heuristic methods thus provide suitable alternatives to learn interpretable and robust feature neutralisation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Model ensembling can be simplified in the Numerai problem: The model en- semble step of the pipeline assigns portfolio weightings to different ML mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Although similar to a multi-armed bandit problem, in our problem ex- ploration is not needed for the agent to learn the distribution of rewards from different choices since the performance of all ML models up to the decision time are known to the Numerai tournament participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Hence there is less need to employ trial-and-error as in multi-armed bandit algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As a consequence, instead of reinforcement learning algorithms, we use heuristics which are shown to be effective in improving the robustness of the ML pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' These heuristics can be interpreted as strong priors in Bayesian learning that greatly simplify our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Dynamic Feature neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In Section 5, the subset of ‘risky fea- tures’ that are used to neutralise ML models is fixed throughout the whole validation and test periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As market conditions are variable, we suggest choosing a different set of features to neutralise in each era to adapt our ML models without the need for expensive re-training of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Specifically, each week we update the set of features to neutralise based on rolling statistical properties of features, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For each feature in the dataset, we calculate the correlation of the feature with the target (fea- ture Corr) and then compute lagged moving average statistics, with a lag of 6 weeks to account for the lagged reporting of future performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The look-back window to compute statistical properties of feature Corr is 52 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We consider 5 different criteria to select the subset of features to be neutralised: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Fixed’: 420 features provided by the portfolio optimiser in Numerai, as in Section 5 above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Low Mean’: 420 features that are least correlated to the target recently 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘High Mean’: 420 features that are most correlated to the target recently 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Low Volatility’: 420 features that have correlations least volatile recently 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘High Volatility’: 420 features that have correlations most volatile recently Table 6 compares the performance obtained by the different dynamic feature neutralisation schemes on LightGBM-dart, LightGBM-gbdt and MLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' All Dynamic Feature Neutralisation methods perform better than using a fixed set of 16 THOMAS WONG AND MAURICIO BARAHONA features but the ‘Low Mean’ neutralisation method has the best Sharpe and Calmar ratios for all ML models, followed by neutralisation of ‘High Volatility’ features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The worse performance of ‘High Mean’ and ’Low Volatility’ neutralisations suggests that a large part of the model risks can be attributed to recently underperforming and high volatility features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (a) LightGBM-dart without FE Dynamic Feature Neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1153 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1865 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0350 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='6857 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0986 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2211 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0538 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4535 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0341 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='6628 (b) LightGBM-gbdt without FE Dynamic Feature Neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1021 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0495 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4727 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1355 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1210 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0487 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4476 (c) MLP without FE Dynamic Feature Neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0687 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0806 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1387 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2618 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1450 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2346 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0730 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2616 Table 6: The effect of Dynamic Feature Neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Performance of differ- ent ML models in the test period (2014-06-27 to 2022-09-23) with different dynamic feature neutralisation methods Next we compared the performance obtained by different dynamic feature neu- tralisations under different market regimes, as defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The results can be found in Tables 9 and 8 in the Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Neutralisation by ‘Low Mean’ performs better than Neutralisation by ‘High Mean’ in low volatility regimes, but not in high volatility regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Under high volatility regimes, neutralisa- tion by ‘Low Volatility’ features in the models performs better than neutralisation by ‘Low Mean’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Under a low volatility regime, neutralisation by ‘Low Mean’ performs significantly better than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Based on the above, we make the following observations: In a low volatility regime, factors that are performing well recently continue to do so in the near future as the feature correlation structure is more stable in low volatility regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This works until there is a regime change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In a high volatility regime, the ML models after neutralisation of ‘Low Volatility’ features have a much higher Mean Corr than models obtained by other neutralisation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ‘Low Volatility’ represents features ROBUST ML MODELS IN FINANCE 17 that have a low variance, and stable performance in the last 52 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' During volatile regimes, these features will underperform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Models that neutralise these features can then outperform when there is market stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Dynamic model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In practice, it is not possible to know the best dynamic feature engineering methods in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore, we propose an online learning procedure to select the dynamic feature engineering method during the test period consisting of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The first step is to have a warm-up period to collect data on model performances, during which all 5 feature neutralisation methods (fixed, low mean, high mean, low vol, high vol) have equal weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The second step is to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='allocate weights to the optimal model based on recent performance according to the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='following criteria: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='‘Average’: Using all five feature neutralisation methods with equal weighting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='‘Momentum’: Using the feature neutralisation method with the highest Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='Corr in the last 52 weeks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='‘Sharpe’: Using the feature neutralisation method with the highest Sharpe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='Ratio in the last 52 weeks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='‘Calmar’: Using the feature neutralisation method with the highest Calmar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='ratio in the last 52 weeks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='In Table 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' we use these criteria to select the optimal dynamic feature engineering method based on recent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As above, a lag of 6 weeks is applied to account for data delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The online learning procedure can thus select the optimal dynamic feature engi- neering method to outperform the ‘Average’ selection in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For all three ML models (LightGBM-dart/LightGBM-gbdt/MLP), the ‘Momentum’ selection method has higher mean Corr and Calmar ratio than the‘Average’ (baseline) and ‘Sharpe’ methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This shows that the ‘Momentum’ method, a very simple model selection method that chooses the recent best-performing model, can adapt a trained ML model towards different market regimes efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For LightGBM-dart and LightGBM-gbdt models, the ‘Calmar’ selection method gives a higher Calmar ratio than the ‘Momen- tum’ method but with a lower mean Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For MLP models, the ‘Calmar’ selection method significantly under-performs other model selection methods, with a much higher Max Drawdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This suggests that selection based on historical drawdown is not robust, especially under situations with regime changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In summary, the proposed online learning procedure to select optimal dynamic feature engineering methods can significantly reduce trading risks and improve the robustness of trading models, outperforming the baseline selection method that takes a simple average of all available models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Motivated by the Numerai tournament, we have designed here an ML pipeline that can be applied to tabular temporal data of stock prices to under- pin strategies for trading of market-neutral stock portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The various steps in the ML pipeline are carefully designed for robustness against regime changes and to avoid information leakage through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We thus aim to obtain models with relatively low complexity, so as to reduce the danger of over-fitting, and with high robustness to changes in hyper-parameters and other choices in the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Another aim is to Regarding the choice of ML models, we find that gradient-boosting decision tree models are both more robust and interpretable than neural network-based models, and they allow more consistent performance under different market regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We also find that post-prediction processing, which is model-agnostic, is an effec- tive means of adapting trained ML models towards new situations without the need 18 THOMAS WONG AND MAURICIO BARAHONA (a) LightGBM-dart without FE Model Selection Mean Volatility Max Draw Sharpe Calmar Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0619 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3700 Momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0533 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4615 Sharpe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0533 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4390 Calmar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0350 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='6429 (b) LightGBM-gbdt without FE Model Selection Mean Volatility Max Draw Sharpe Calmar Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0710 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3042 Momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0729 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3128 Sharpe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0729 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3073 Calmar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0508 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4252 (c) MLP without FE Model Selection Mean Volatility Max Draw Sharpe Calmar Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0918 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2124 Momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0878 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2415 Sharpe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0878 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2358 Calmar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0948 Table 7: The effect of dynamic model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Performance of different ML models in the test period (2014-06-27 to 2022-09-23) with different online learning procedures selecting the optimal dynamic feature neutralisation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' to re-train ML models and introduce additional model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Using dynamic feature neutralisation produces models with different flavours in an interpretable way, which also have better risk-adjusted performance than models with fixed feature neu- tralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Staking is commonly used in ML competitions to improve the robustness of mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The method suggested in this study, dynamic model selection can be applied to online ML problems in guiding the selection of an optimal model(s) from a growing model ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We find that a simple design, such as equal-weighted models, has a robust performance under different market regimes, but selecting the best model based on recent performance provides an improvement compared to the baseline as it switches to a lower-risk model during more volatile market regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' It remains an open research area into how reinforcement learning or other online learning methods can be used to learn optimal staking weights between different ML models, given their historical performance and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We also studied the robustness of our ML pipeline under different random seeds and changes in data splits for cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The results are presented in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4 in the Supplementary Information, where we show that LightGBM dart mod- els are robust against these changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The statistical rules used in dynamic feature neutralisation are also shown to perform better than features chosen at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In the following, we discuss some ideas for further work to improve the ML pipeline ROBUST ML MODELS IN FINANCE 19 we designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The diversity of models within a model ensemble is a key ingredient for dynamic model selection and other model ensemble/staking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' A new metric could be designed to study the impact of a new ML model on an existing model ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This metric could then be used to train new ML models that are uncorrelated to existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The simple feature engineering methods used in our present study could not improve the performance of ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Identifying robust relationships between features over different market regimes is difficult but generative models, such as Vari- ational Autoencoders [42], could be used to create new features that summarise non- linear relationships in existing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The Gradient Boosting models used in our pipeline are suitable for distributed learning, where large datasets are split into smaller batches to train on different ma- chines, often with various computational resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Data science compe- titions like the Numerai tournament rely on community efforts of individual data scientists to create a meta-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' This approach to crowd sourcing depends on the assumption that a complicated ML model that needs to be trained with advanced hardware can be approximated by combining a number of ML models (each trained with fewer data or features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Studying the convergence of model performance would be important for organising the data science competition as it decides how many participants are needed to maintain a well-diverse pool of models to create the meta- model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Overall, our results suggest using simple, well-established ML models such as gradient-boosting decision trees instead of specialised neural network models for this tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Rather than using a single neural network to perform feature engineering, model training/inference and post-prediction transformations, the modularised de- sign of the ML pipeline in this study offers increased robustness and transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Researchers can add, modify or delete a component without affecting the rest of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Creating model ensembles improves model performances by reducing id- iosyncratic variance from individual ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The simple model selection rules based on recent performances provide a baseline that works well under different mar- ket regimes, whereas various portfolio metrics such as Sharpe and Calmar ratios are improved by using the recently best-performing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 8.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='org/abs/1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='02438 [42] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Kingma and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Welling, “Auto-encoding variational bayes,” in 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, Y.' metadata={'source': 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+page_content='6114 [43] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Buhlmann and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Hothorn, “Boosting algorithms: Regularization, prediction and model fitting,” Statistical Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 4, nov 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1214%2F07-sts242 22 THOMAS WONG AND MAURICIO BARAHONA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Additional results for dynamic feature neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Here we show the performance of dynamic feature neutralisation for low and high volatility regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (a) LightGBM-dart without FE Feature Neutralisation Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1153 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1787 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0350 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7286 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0986 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2099 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0538 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4424 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0341 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='6891 (b) LightGBM-gbdt without FE Feature Neutralisation Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0971 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0495 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5071 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1253 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='115 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0487 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4620 (c) MLP without FE Feature Neutralisation Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0633 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0496 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4335 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1325 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2210 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0730 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2658 Table 8: Performance of ML models in the test period (2014-06-27 to 2022-09-23) with different dynamic feature neutralisation methods in low volatility regime ROBUST ML MODELS IN FINANCE 23 (a) LightGBM-dart without FE Feature Neutralisation Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0223 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3888 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0179 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0199 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4907 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1055 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0206 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5372 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1311 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0330 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7636 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0143 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4077 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5035 (b) LightGBM-gbdt without FE Feature Neutralisation Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0364 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5962 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0380 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5579 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0334 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='6527 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0306 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7745 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0308 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='6786 (c) MLP without FE Feature Neutralisation Mean Volatility Max Draw Sharpe Calmar Fixed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0326 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='6012 Low Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0806 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2543 High Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1325 Low Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0397 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5592 High Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0336 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5565 Table 9: Performance of ML models in the test period (2014-06-27 to 2022-09-23) with different dynamic feature neutralisation methods in high volatility regime 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Pseudocode for algorithms in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For completeness, we present here brief pseudocode for some of the main methods in the paper with the appropriate references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1 Gradient boosting algorithm [22,43] Given N data samples (xi, yi), 1 ≤ i ≤ N with the aim to find an increasing better estimate ˆf(x) of the minimising function f(x) which minimise the loss L(f) between targets and predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' L(f) = � i l(yi, f(xi)) where l is a given loss function such as mean square losses for regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Function f is restricted to the class of additive models f(x) = �K k=1 wkh(x, αk) where h(·, α) is a weak learner with parameters α and wk are the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Initialise f0(x) = arg minα0 �N i=1 l(yi, h(xi, α0)) For k = 1 : K Compute the gradient residual using gik = − � ∂l(yi,fk−1(xi)) ∂fk−1(xi) � Use the weak learner to compute αk which minimises �N i=1(gik − h(xi, αk))2 Update with learning rate λ fk(x) = fk−1(x) + λh(x, αk) Return f(x) = fK(x) 24 THOMAS WONG AND MAURICIO BARAHONA Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2 Gradient boosting tree algorithm implemented in LightGBM [22,23, 43] Initialise f0(x) = arg minα0 �N i=1 l(yi, x, α0) For k = 1 : K For i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' N, compute the gradient residual using gik = − � ∂l(yi,fk−1(xi)) ∂fk−1(xi) � Fit a decision tree to the targets gik giving terminal leaves Rkj, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Jk, where Jk is the number of terminal leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Jk, compute αjk = arg minα � xi∈Rkj l(yi, fk−1(xi) + α) Update boosting trees with learning rate λ fk(x) = fk−1(x) + λ �Jk j=1 αkjI(x ∈ Rkj) Return fK(x) ROBUST ML MODELS IN FINANCE 25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Hyper-parameter search space for different ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We ran all experiments on a GPU cluster, each node of which contains a NVIDIA GeForce RTX 2080 Ti GPU, running with 4352 CUDA cores and 11GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Hyper-parameter search is performed using Optuna [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For each Feature Engineering/ML pipeline, hyper-parameter search is ran for at most 8 hours or at most 100 configurations, whichever came first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The default TPE sampler in Optuna is used to perform the hyper-parameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In Figure 4 and 5, we list the Hyper-parameter search pa- rameters defined in Optuna [34] for different ML models used in the main text to train the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Feature Engineering – Numerai Basic Feature Engineering ∗ dropout pct: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='25, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05, ∗ no product features: low:50, high:1000, step:50, ML Models – LightGBM-gbdt ∗ n estimators: low:50, high:1000, step:50 ∗ learning rate: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='005, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, log:True ∗ min data in leaf: low:2500, high:40000, step:2500 ∗ lambda l1: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01, high: 1, log:True ∗ lambda l2: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01, high: 1, log:True ∗ feature fraction: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, high:1, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05 ∗ bagging fraction: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5, high:1, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05 ∗ bagging freq: low:10, high:50, step:10 – LightGBM-dart ∗ n estimators: low:50, high:1000, step:50 ∗ learning rate: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='005, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, log:True ∗ min data in leaf: low:2500, high:40000, step:2500 ∗ lambda l1: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01, high: 1, log:True ∗ lambda l2: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01, high: 1, log:True ∗ feature fraction: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, high:1, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05 ∗ bagging fraction: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5, high:1, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05 ∗ bagging freq: low:10, high:50, step:10 ∗ drop rate: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1 ∗ skip drop: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1 – LightGBM-goss ∗ n estimators: low:50, high:1000, step:50 ∗ learning rate: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='005, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, log:True ∗ min data in leaf: low:2500, high:40000, step:2500 ∗ lambda l1: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01, high: 1, log:True ∗ lambda l2: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01, high: 1, log:True ∗ feature fraction: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, high:1, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05 ∗ bagging fraction: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='5, high:1, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05 ∗ bagging freq: low:10, high:50, step:10 ∗ top rate: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05 ∗ other rate: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='05 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 4: Hyper-parameter Space for ML models 26 THOMAS WONG AND MAURICIO BARAHONA Machine Learning – MLP ∗ max epochs: low:10, high:100, step:5 ∗ patience: low:5, high:20, step:5 ∗ num layers: low:2, high:7, step:1 ∗ neurons: low:64, high:1024, step:64 ∗ neuron scale: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3, high:1, log:True ∗ dropout: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9, log:True ∗ batch size: low:10240, high:40960, step:10240 – TabNet ∗ max epochs: low:10, high:100, step:5 ∗ patience: low:5, high:20, step:5 ∗ batch size: low:1024, high:4096, step:1024 ∗ num d: low:4, high:16, step:4 ∗ num a: low:4, high:16, step:4 ∗ num steps: low:1, high:3, step:1 ∗ num shared: low:1, high:3, step:1 ∗ num independent: low:1, high:3, step:1 ∗ gamma : low:1, high:2, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1 ∗ momentum: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4, step:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01 ∗ lambda sparse: low:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0001, high:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='01, log:True Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' 5: Hyper-parameter Space for ML models ROBUST ML MODELS IN FINANCE 27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Robustness of ML pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' One of the aims in this work was to provide a robust pipeline for tabular temporal data under regime changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Here we present additional results of the robustness of the method under different scenarios and sources of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Robustness under changes of random seeds in the learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In Ta- ble 10, we report the variability of the performance of the LightGBM-dart, LightGBM- gbdt and MLP models trained starting from 10 different initial random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The performance is generally robust to the change in random seeds, with small variances in the prediction of the mean Corr and volatility and moderate for the Maximum Drawdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Model Mean Volatility Max Draw Sharpe Calmar LightGBM-dart without FE mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1639 sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0175 LightGBM-gbdt without FE mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2338 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1100 sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0153 MLP without FE mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1446 sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0219 Table 10: Variability of the performance of ML models in the test period (2014-06- 27 to 2022-09-23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The mean and standard deviation of each portfolio metrics are calculated over models with 10 different random seeds for each method A general strategy to reduce the variance is to combine different ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' There are two ways to do so: (i) averaging over models, by calculating the average performance of different models, and (ii) averaging over predictions, by calculating the average predictions from each model and then scoring the average predictions against the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Table 11 shows that averaging over predictions gives higher mean Corr and Sharpe/Calmar ratios than averaging over models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore, this averaging method is used to compute model performances in Table 2 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Model Average Mean Volatility Max Draw Sharpe Calmar LightGBM-dart without FE Over models 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1639 Over predictions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1714 LightGBM-gbdt without FE Over models 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2338 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1100 Over predictions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1102 MLP without FE Over models 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1446 Over predictions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1547 Table 11: Performance of different ML methods on Numerai v4 dataset in the test period (2014-06-27 to 2022-09-23) with different averaging methods Robustness under different cross-validation data splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' As financial data are regime dependent, an important measure of model robustness is to measure the performance of ML models that have been trained using different cross-validation splits of the data and compute how much the model performance changes over different test periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' To ascertain the robustness of data splits, we have carried out 3 cross-validation splits (CV 1, CV 2, CV 3) as shown in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The hyper-parameters are optimised under CV 1, which is the cross-validation used to generate the model performances in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' These hyper-parameters are fixed for the models trained under the CV 2 and CV 3 splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For ML methods that require early stopping, the data 28 THOMAS WONG AND MAURICIO BARAHONA in the validation period (different for each split) are used to regularise the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Therefore, by reusing the optimised hyper-parameters across all splits, we evaluate the robustness of the model performance to the optimisation of hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We then compute the performance when applying the models to shifted cross-validation datasets in the walk-forward CV 2 and CV 3 data splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Our results show good consistency in performance across CV 2 and CV 3, with only a small deterioration of the results as compared to CV 1 (over which the hyperparameters were optimised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We also find that LightGBM-dart with FE, the ML method that has the highest mean Corr in CV 1, has the greatest return and best Sharpe and Calmar ratios also in other cross-validations, as seen in Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Train Start Train End Validation Start Validation End Enter Ensemble CV 1 2003-01-03 2012-07-27 2012-12-21 2014-11-14 2015-05-15 CV 2 2003-01-03 2014-06-27 2014-11-21 2016-10-14 2017-04-14 CV 3 2003-01-03 2016-05-27 2016-10-21 2018-09-14 2019-03-15 Table 12: Various cross-validation schemes to train ML models on different parts of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' CV 1 is the cross-validation used for hyper-parameter optimisation and training ML models in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (a) CV 1 (2015-05-15 to 2022-09-23) Method Mean Volatility Max Draw Sharpe Calmar LightGBM-dart without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1714 LightGBM-gbdt without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1102 MLP without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1547 (b) CV 2 (2017-04-14 to 2022-09-23) Method Mean Volatility Max Draw Sharpe Calmar LightGBM-dart without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1376 LightGBM-gbdt without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='3227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0716 MLP without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0932 (c) CV 3 (2019-03-15 to 2022-09-23) Method Mean Volatility Max Draw Sharpe Calmar LightGBM-dart without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='8140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1913 LightGBM-gbdt without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1584 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='7772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1648 MLP without FE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='9339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='1913 Table 13: Performance of selected machine learning methods on the Numerai dataset in the test period for various walk-forward cross-validation schemes, (a) CV 1, (b) CV 2 and (c) CV 3 Robustness under feature selection for dynamic feature neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' A fixed set of 420 features to be neutralised was given by the Numerai organisers based on internal evaluations of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' In Section 6, we introduce several statistical rules that allow us to select a varying subset of features to be neutralised in each era based on empirical heuristic criteria motivated by financial modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' ROBUST ML MODELS IN FINANCE 29 To evaluate the robustness of the proposed statistical rules, we draw 100 subsets of 420 features selected at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' and use each set to neutralise the raw predictions from ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' We then evaluate the performance of ML models based on each of the random subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' Using the procedure described in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2 we then select the optimal dynamic feature neutralisation method and compute the performance of the top 10 models of the highest mean Corr, Sharpe and Calmar ratio over the test period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The results are reported in Table 14 and should be compared to the performance of the same models in Table 7, which were obtained with dynamic feature neutralisation using the statistical rules defined in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' The mean Corr of models obtained with random feature neutralisation for each rule (Momentum/Sharpe/Calmar) are lower than those obtained using the statistical rules in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' On the other hand, the Sharpe ratio of models for models with random feature neutralisation is slightly higher, as expected due to the variance re- duction effect by averaging over 10 different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' For models selected based on the Calmar rule, the models obtained with statistical rules have a much higher Calmar ratio than random feature neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' It suggests the statistical rules defined can effectively reduce model risks by reducing linear exposure to undesirable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' (a) LightGBM-dart without FE Feature Neutralisation Mean Volatility Max Draw Sharpe Calmar Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='0482 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='4440 Momentum 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content='2121 Table 14: Performance of different ML models in the test period (2015-05-15 to 2022- 09-23) obtained with random feature neutralisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} +page_content=' These are averages obtained by selecting the top 10 models under the different online learning procedures over the test period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQf4vob/content/2301.00790v1.pdf'} diff --git a/F9FKT4oBgHgl3EQfbS5P/content/tmp_files/2301.11811v1.pdf.txt b/F9FKT4oBgHgl3EQfbS5P/content/tmp_files/2301.11811v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..177df94215d32c2ca10e47b29cad397b1db93118 --- /dev/null +++ b/F9FKT4oBgHgl3EQfbS5P/content/tmp_files/2301.11811v1.pdf.txt @@ -0,0 +1,944 @@ +Indonesian Journal of Electrical Engineering and Computer Science +Vol. 28, No. 1, October 2022, pp. 328~338 +ISSN: 2502-4752, DOI: 10.11591/ijeecs.v28.i1.pp328-338 + 328 + + +Journal homepage: http://ijeecs.iaescore.com +A systematic review of structural equation modeling in +augmented reality applications + + +Vinh The Nguyen1, Chuyen Thi Hong Nguyen2 +1Faculty of Information Technology, TNU-University of Information and Communication Technology, Thai Nguyen, Vietnam +2Faculty of Primary Education, Thai Nguyen University of Education, Thai Nguyen, Vietnam + + +Article Info + +ABSTRACT +Article history: +Received Mar 26, 2022 +Revised Jun 21, 2022 +Accepted Jul 14, 2022 + + +The purpose of this study is to present a comprehensive review of the use of +structural equation modeling (SEM) in augmented reality (AR) studies in the +context of the COVID-19 pandemic. IEEE Xplore Scopus, Wiley Online +Library, Emerald Insight, and ScienceDirect are the main five data sources +for data collection from Jan 2020 to May 2021. The preferred reporting +items for systematic reviews and meta-analyses (PRISMA) approach was +used to conduct the analysis. At the final stage, 53 relevant publications were +included for analysis. Variables such as the number of participants in the +study, +original +or +derived +hypothesized +model, +latent +variables, +direct/indirect contact with users, country, limitation/suggestion, and +keywords were extracted. The results showed that a variety of external +factors were used to construct the SEM models rather than using the +parsimonious ones. The reports showed a fair balance between the direct and +indirect methods to contact participants. Despite the COVID-19 pandemic, +few publications addressed the issue of data collection and evaluation +methods, whereas video demonstrations of the augmented reality (AR) apps +were utilized. The current work influences new AR researchers who are +searching for a theory-based research model in their studies. +Keywords: +Augmented reality +COVID-19 +External factors +Structural equation modeling +Theory-based research +This is an open access article under the CC BY-SA license. + +Corresponding Author: +Vinh The Nguyen +Faculty of Information Technology, TNU-University of Information and Communication Technology +Z115 Street, Quyet Thang Commune, Thai Nguyen, Vietnam +Email: vinhnt@ictu.edu.vn + + +1. +INTRODUCTION +Augmented reality (AR) is a technology that has attracted a lot of attention in various domains [1]- +[3]. Unlike virtual reality (VR) which allows users to be totally immersed in a virtual environment, AR +enriches the real world with virtual artifacts [4]. The primary value of AR is that it allows digital objects to +be blended more seamlessly into a person’s perception of the real world than simply displaying data on a +screen. Market research [5] anticipates that AR’s market will reach USD 88.4 billion, growing 31.5% from +2021 to 2026. In addition, in response to the COVID-19 pandemic, more companies and organizations have +adopted remote work and are utilizing augmented reality technology [6]. What that means is that a huge +number of AR applications are being developed, especially in electrical engineering and computer science +[1]-[3], [7], [8]. +Assessment is one of the key factors in ensuring the success of an AR application, especially when it +is involved with end-users. However, literature work reported that only a few studies afforded time for this +type of evaluation (only 8% of published papers) [9]. One plausible explanation was that AR +researchers/developers had to devote their time to solving technical issues [10]. Moreover, the lack of +methods or theory-driven research on evaluating AR apps, considering end users’ involvement, contributed + +cC +BY +SAIndonesian J Elec Eng & Comp Sci +ISSN: 2502-4752 + + +A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) +329 +to the scarcity of AR evaluation [11]. In addition, after the COVID-19 outbreak, many conferences (e.g., +ISMAR) encouraged researchers to find alternative means of evaluating AR apps rather than canceling the +submissions due to social distancing. There has been no study addressing this issue so far, thus it remains a +gap in the literature. To close this gap, this paper-based on prior AR studies–provided an overview of theory- +based methods that can effectively be used for AR assessment. Among many other end-user evaluation +methods, the scope of the current study focused on structural equation modeling (SEM), a model commonly +used in behavioral science. SEM is a comprehensive statistical method that examines relationships between +observed and latent factors [12]. It has been widely used in confirmatory factory analysis in many topics and +fields [13]-[15]. +A number of review studies on SEM applications have been conducted in various research domains, +including ecology [16], social science [17], psychological research [18], and strategic management [19]. It +indicated that a review study would be valuable for new researchers to quickly acquire knowledge in the field +effectively. Yet, it also implies that it would be important to look at SEM from AR’s perspective since AR is +one of the emerging trends in the digital transformation era. However, there is no study of SEM for AR +applications other than previously mentioned review studies. Thus, the current research is unique on its own +by the AR’s topic and the outcomes of this study can be used as a referencguidene for researchers in similar +studies, particularly in electrical engineering and computer science. More specifically, the present study tries +to answer to following research questions: i) What are the preferred theory-driven models being used in prior +AR studies amid the COVID-19 pandemic? ii) What are the dimensions or variables being investigated by +AR researchers so far? iii) How do researchers of prior AR studies communicate with end-users for +evaluation? vi) How many participants are typically involved in a study? Would this number still be +considered appropriate from the literature? v) What are the main drawbacks of tR studies? Do they suffer +from the COVID-19 pandemic? + + +2. +METHOD +This study involves a review of SEM in AR applications; thus, the preferred reporting items for +systematic reviews and meta-analyses (PRISMA) statement was applied [20]. The PRISMA statement aims +to assist scholars in improving the reporting of scientific reviews and meta-analyses. It is an evidence-based +minimum set of elements for systematic review reports that are intended to assist systematic reviewers in +clearly explaining why the review was conducted and what the authors performed. It has previously been +used to target comparable research objectives [21], [22]. + +2.1. Source selection +IEEE Xplore, Scopus, Wiley Online Library, Emerald Insight, and ScienceDirect databases were +used to build the corpus, encompassing titles, abstracts, and keywords. These five databases are regarded as +essential and dependable sources of high-quality articles in the fields of computer science and engineering +[21], [23]. Although, some other indexing databases are available (i.e., Scholar) but they are out of scope in +the current study. + +2.2. Search criteria +To add articles to our corpus, both of the following related criteria need to be fulfilled, i) Structural +equation modeling search term: at least one SEM-related term must appear in an article’s title, abstract, or +author keywords (i.e., structural equation modeling, SEM, planned behavior, theory of planned behaviour +(TPB), motivational model, Michaelis–Menten (MM), reasoned action, theory of reasoned action (TRA), +social cognitive, SCT, diffusion of innovation, IDT); and ii) Augmented Reality search term: terms include +augmented reality, AR. Using the aforementioned criteria, 16 articles were discovered in IEEE Xplore, 107 +articles in Scopus, 197 papers in Wiley Online Library, 68 papers in Emerald Insight, and 695 papers in +ScienceDirect. The corpus was collected between June 3, 2021, and June 12, 2021. + +2.3. Eligibility assessment for the final analysis corpus +To determine the acceptability of the obtained papers, the first researcher personally reviewed the +entry criteria mentioned below by reviewing the titles and abstracts of the obtained publications. When a +clear judgment could not be reached, other aspects of the publication, particularly the method and data +acquisition descriptions, were discussed in conjunction with the second author. Only items that meet the +following criteria are retained in the corpus: i) Peer-reviewed: The paper was peer-reviewed in the two +indexing databases. This is due to the trustworthiness of peer-reviewed journals and the rigorous peer-review +processes, only articles in these databases are considered for this review; ii) Topic relevant: The topic of an +article is pertinent to the applications of SEM in AR; iii) Language: Publication was reported in English; and +vi) Duration: Paper was published between Jan 2020 and May 2021. + + + + + ISSN: 2502-4752 +Indonesian J Elec Eng & Comp Sci, Vol. 28, No. 1, October 2022: 328-338 +330 +If the article meets any of the following criteria, it will be excluded from the corpus: i) Books and +cover page, abstract only, poster; ii) The paper was not written in English; iii) Application of SEM is not for +AR; and vi) Paper was published before Jan 2020 and after May 2021. +Figure 1 depicts the flow of information through the different phases of the systematic review +utilizing PRISMA approach. 1,083 records were found in all data sources. Duplications were removed based +on the titles. Each paper was screened individually to remove items that are out of scope. Then 230 records +were excluded. As such, 309 candidates left for full-text retrieval. Of these remaining items, 9 records cannot +be retrieved due to access restrictions. The authors examined each report for eligibility and removed 247 +studies. In the end, 53 items were included in this research. The remaining papers were examined +individually to extract interesting variables such as the number of participants, original or derived +hypothesized model, latent variables, direct/indirect contact with the user, country of origin, +limitation/suggestion (if any), and keywords. + + + + +Figure 1. The flow diagram represents the movement of information through the various stages of a +systematic review + + +2.4. Data coding and analysis +To extract the data, all articles were loaded into NVivo software, and a coding scheme was created. +NVivo is a program that facilitates qualitative analytical method research. This tool enables researchers to +organize, analyze and explore unstructured or qualitative data, including interviews, reviews, articles, social +media, and web content. Codes included authors, journal name, year of publication, countries of authorship, +title, abstract, author keywords, method, objectives, findings and limitations on how SEM was used. + + +3. +RESULTS AND DISCUSSION +3.1. What are the preferred theory-based driven models being used in prior AR studies amid the +COVID-19 pandemic? +Figure 2 depicts the distribution of papers over hypothesized models. Most publications fall into the +SEM category (accounted for 58.49%), followed by eTAM and TAM with 20.76% and 11.32% respectively. +Although the UTAUT model was developed recently, the result shows less popularity of adopting this model +(only 3.77%), which is the same as the SOR model. +Technology acceptance model (TAM): originally developed by Davis [24], TAM is known as a +theory of information systems that describes how consumers come to accept and use technology. Real system +usage is the point at which people interact with technology. People utilize technology because of their +behavioral intentions. In this survey, 6 articles (11.32%) used original TAM for their research. +Extended technology acceptance model (eTAM): In this category, 11 publications (20.75%) +extended TAM with external variables such as perceived task-technology fit [25]-[28]–which asserted that + +Identification ofnew studies viadatabases +Recordsidentifiedfromdatabases: +Recordsremovedbeforescreening: +N=1.083 +N=544 +Records screened: +Records excluded: +N=539 +N=230 +Reports soughtfor retrieval +Reports not retrieved: +N=309 +N=9 +Reports excluded:247 +Reports assessed for eligibility: +NotinvolveSEM(N=27) +N=300 +NotinvolveAR(N=127) +ReviewOnly(N=93) +Reports of included studies: +N=53Indonesian J Elec Eng & Comp Sci +ISSN: 2502-4752 + + +A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) +331 +the technology must be utilized and a good fit with the tasks it supports to have positive impacts on +individual performance, perceived visual design/appeal [25]-[31] which assumed that beauty is important, +and it impacts decisions that should not be influenced by aesthetics, perceived enjoyment [32]-[35]-which +refers to the hedonic value of new technology and expresses how pleasurable a person finds its use. + + + + +Figure 2. Models distribution across prior studies + + +The unified theory of acceptance and use of technology (UTAUT): Venkatesh et al. [36] developed +the UTAUT after reviewing and consolidating the components of eight previous models used to describe +information system user behavior. In this review, several external variables were incorporated into the +existing UTAUT model (eUTAUT) such as innovativeness, reward, trust, enjoyment, hedonic motivation, +habit, and gamification [37], [38]. +Stimulus-organism-response (SOR): Mehrabian-stimulus Russell's model [39] depicts the +occurrence of a person's response to environmental stimuli. Qin et al. [40] decomposed stimulus into two +external factors (i.e., Interactivity, Virtuality), Organism into 4 variables (i.e., Hedonic, Utilitarian, +Informativeness, and Ease of Use), and Response into 2 factors including Attitude and Behavioral Intention. +Similarly in the scope of this review, Qin et al. [40] also included (critical mass, social interaction, +information timelines, content richness) into stimulus, (attachment, conformity) into Organism, and (visiting +intention, continue intention) into Response. +Structural equation modeling (SEM): This category contains the largest portion of the papers +included in our investigation (58.49%). Authors in this group mainly adapted constructs, measures in the +literature to form hypothesis. As such, PLS-SEM was utilized as an analytical method to conduct +confirmatory factor analysis and path analysis. Confirmatory factor analysis, which originates in +psychometrics, aims to quantify underlying psychological characteristics such as attitude and satisfaction. +Path analysis, on the other hand, has its origins in biometrics and is intended to discover the causal link +between variables by drawing a path diagram [41]. + + +3.2. What are dimensions or variables being investigated by AR researchers so far? +Figure 3 depicts 77 unique constructs/latent variables from hypothesized models. There are 184 +unique constructs found in this study. Behavioral intention, usefulness, ease of use, attitude, user behavior, +and enjoyment are the most frequent items used in the hypothesized models. + + + + +Figure 3. Wordcloud depicts 77 unique constructs from all hypothesized models + +Count of SEM +35 +31 +30 +25 +20 +15 +11 +10 +6 +5 +2 +0 +eTAM +eUTAUT +SEM +SOR +TAMHedonic +EaseOf Use +Novelty ConceptualUnderstanding +Performance Anxiety Quality +Technology +SocialInteraction +Enjoyment +Responses +ntention +Interactivity +UseBehavior +Knowledge Gain +Immersion +Voluntariness TaskSatisfaction Aesthetics +Reievance +Sublective Norms +Control +Behavioral Intentionsli-Efiacy +TaskTechnologyFit +Embedding +Environmental Motivation +Trust Fit +Learning +Playfulness +Behaviora +Game +Involvement +Value +BenefitJob +Attitude +ExperienceVisual +Presence +Effort Richness +Perceived +Informativeness Soclal ActualUsage +Engagement +Simulated +Purchase Intention +Behavior +System +Augmentation +information +Education +Usefulness +Expectancy +Service +Image Entertainment + + + ISSN: 2502-4752 +Indonesian J Elec Eng & Comp Sci, Vol. 28, No. 1, October 2022: 328-338 +332 +Figure 4 captures the top 14 dominant keywords in the collection of papers in this study. Aside from +“augmented reality”, TAM is the most popular term that the authors used for indexing their papers. In total, +this study extracted 319 keywords with 230 unique terms, indicating that there is a high variation of +topics/techniques used. However, in terms of their broad contents, the major theme of these collected papers +can be categorized as the “social marketing” theme as they were mainly focused on “Intention to Purchase” +or “Intention to Visit”. + + + + +Figure 4. Frequency of keywords extracted from publications + + +3.3. How do researchers of prior AR studies communicate with end-users for evaluation? +Table 1 reports the communication channels used to gather data from respondents. Results showed +that there is a fair balance between the direct (45.28%) and indirect (50.94%) methods. Here, the indirect +method means that the research teams did not contact participants directly (e.g., lab setting, or field study). +Instead, they contact users via online channels (e.g., social network, email, discussion group). On the other +hand, the direct method requires subjects to be at the site of the study for the experiment. + + +Table 1. Communication channels to collect data from respondents +Communication channel +Count +Percentage +Indirect +27 +50.94 +Direct +24 +45.28 +Direct and Indirect +2 +3.77 +Total +53 +100 + + +Figure 5 depicts the spatial locations of authors researching AR utilizing the SEM method across the +globe. It can be observed that most publications were conducted in the United States although this country +was suffered heavily from the COVID-19 pandemic. However, 8 out of 10 papers utilized the indirect +research method to recruit and gather data, meaning that the study was conducted remotely, and opinions +were collected through online tools. + + + +'SocialMarketing':AugmentedRealityappearsmostoften. +AugmentedReality +TechnologyAcceptance Model +BehavioralIntentions +Pokemon Go +VirtualReality +Social Marketing +GeneralizedStructuredComponentAnalysis +Presence +TAM +TechnologyAdoption +Interactivity +User Experience +MobileAugmented Reality +A-Frame +MobileAugmented RealityApplications +0 +5 +10 +15 +20 +25 +30 +35 +40 +Social MarketingIndonesian J Elec Eng & Comp Sci +ISSN: 2502-4752 + + +A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) +333 + + +Figure 5. Spatial locations of authors researching on AR utilizing SEM in 2020-2021 + + +3.4. How many participants are typically involved in a study? Would this number still be considered +appropriate from the literature? +Figure 6 shows the distribution of sample size across peer-reviewed papers. The whisker plot +indicates that on average the sample size (the number of participants) who took part in the studies was +approximately 300 subjects considering 4 extreme values (or outliers). The minimum sample size is 9 and the +maximum is 1,566. The median indicates that most papers recruited around 200 users for their studies. When +the four extreme values were not considered, the average sample size for direct communication with +participant was 142 (median=113, range=340, min=24, max=364), and indirect method was 286 +(median=302, range=710, min=9, max=719). + + + + +Figure 6. Distribution of sample size in the peer-reviewed papers + + +Sample size is a debating subject in the literature. As such, the determination of sample size varies +from study to study. Some researchers advocate a minimum sample size of 100–200 per a study, an +acceptable sample size can range between 300 and 500, or with criteria such as acceptable of five cases per +free parameter, moderate of ten cases per free parameter [12], and ideal of 20 instances per free parameter in +the model. Kock and Hadaya [42] proposed a technique for determining an adequate sample size based on +“inverse square root” and “gamma-exponential” approaches which were adapted by Nikhashemi et al. [43] +included in this study. To some extent, Figure 6 reflects the balance of sample size recommendation in the +literature. Interestingly, the median sample size calculated in this study (Median=200) was aligned with the +findings based on reviews of studies in different research areas, including operations management, education +and psychology. + +3.5. What are the main drawbacks of the AR studies? Do they suffer from the COVID-19 pandemic +Table 2 reports the frequency of limitations addressed by authors in the collected publications. The +most common flaw that needs to be examined further in future studies is the failure to incorporate additional +external components (39.62%) in the postulated model, followed by convenience sampling (35.85%), multi- +level analysis (32.08%) and limited to one region (30.19%). In terms of convenience sampling drawback, + +United States +10 +Germany +n +Taiwan +4 +Greece +4 +UnitedKingdom +China. +SouthKorea +Indonesia +2 +Thailand +2 +Romania +2 +Vietnam +2 +Australia +2 +France +1 +Italy +1 +Spain +1 +HongKong +1 +Ireland +1 +Turkey +Netherlands +1 +India +1 +Oman +1 +Portugal +1 +Malaysia +1 +Powered by Bing +Tom,WikipediaDistribution of sample size across publications +1800 +1600 +·1566 +1400 +1200 +:1183 +:1192 +1000 +800 +719 +600 +400 +412 +X298.8113208 +200 +200 +68 + + + ISSN: 2502-4752 +Indonesian J Elec Eng & Comp Sci, Vol. 28, No. 1, October 2022: 328-338 +334 +many authors acknowledged that they used the non-probability method to acquire sample data through their +networks of interest. As such, their reports/findings cannot be generalized to the population. + + +Table 2. Frequency of limitations addressed by the authors in the collected publications +Limitations +References +Not consider other factors (21) +[25], [26], [30], [31], [32], [38], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], +[55], [56], [57] +Convenience sampling (19) +[32], [35], [37], [40], [43], [45], [46], [50], [52], [53], [55], [57], [58], [59], [60], [61], [62], [63], +[64] +Multi levels analysis (17) +[25], [37], [44], [45], [46], [48], [49], [51], [55], [56], [59], [62], [65], [66], [67], [68], [69] +Limited to one region (16) +[30], [32], [33], [37], [38], [46], [47], [48], [49], [54], [56], [58], [59], [63], [68], [70] +Tailored to a specific AR product +(14) +[45], [46], [47], [52], [53], [54], [57], [61], [62], [64], [65], [67], [68], [70] +Small Sample Size (10) +[30], [32], [33], [38], [40], [47], [50], [54], [60], [71] +Short term effect (10) +[29], [31], [38], [43], [45], [58], [63], [65], [69], [72] +Not specified (9) +[34], [41], [50], [73], [74], [75], [76], [77], [78] +Only Intention Model (6) +[31], [51], [52], [56], [58], [79] +Lack of AR features (6) +[25], [29], [32], [48], [63], [71] +Lack of functions (4) +[25], [26], [29], [32] +Self-Administered Survey (3) +[58], [66], [79] +Use Videos for demonstrations (3) +[25], [26], [65] +Technical challenges (2) +[27], [28] +Standardized tools (2) +[29], [52] +Single Analysis technique (2) +[33], [48] +Lab setting (2) +[55], [64] +Not consider privacy concerns (2) +[25], [60] +Others (8) +[25], [26], [29], [32], [56], [59], [58], [70] + + +Along with convenience sampling, limited study to one region is another shortcoming that is often +mentioned with non-probability method limitation. Unlike convenience sampling drawback that subjects may +come from different parts of the world, the regional issue was arising where the study was intentionally +designed for a specific region through a case study or in the lab setting [55], [64]. A large portion of the +published work was carried out with the help of pre-existing AR products. This evaluation includes examples +such as IKEA Place, YouCam Makeup, and Pokémon Go. Participants were asked if they had any experience +with these AR apps, and if so, they were encouraged to take part in the survey. Furthermore, the authors' +capacity to extend the study to additional products/services was limited because they did not have control or +flexibility over the AR apps. +The results show that though the sample size was a sufficiently addressed problem by the +researchers, the proportion of this limitation was just 18.87%. Without considering publications that did not +report limitations in their work (i.e., not specified (9)), 77.27% (34/44 papers) of the research group justified +their sample size using an analytical tool/method, a sample size recommendation in the literature, and the use +of PLS-SEM, which can work with small sample sizes. As a result, sample estimation was deemed sufficient. +Another issue worth mentioning is the short-term effect addressed by 10 author groups (18.87%). The short- +term impact was explained by the fact that the experiments were only conducted for a limited period. As a +result, the theorized models can only explain variables impacting user behavior at that point in time. The +authors emphasized that because technology has evolved drastically over the years, the question of whether +their proposed models stand up remained unresolved. In addition, people's perspectives shift throughout time +as they gain experience [36], as a consequence, long-term research was suggested to validate the models. +In terms of the indirect method to conduct an experiment with users, four studies administered their +AR applications through video demonstrations [25], [26], [31], [65]. In this regard, instead of asking +participants to download or use the AR apps directly, the authors created videos demonstrating the features of +their studied AR apps. Based on the evidence of previous studies using video depictions of AR prototypes +[80], [81], these authors argued that the technology itself was not available for participants to interact with at +the time, and the purpose of the hypothesized models was to examine the influential factors that affect +behavioral intention before releasing the actual AR product to the market. As such in this category, studies in +[26], [29], [52] recommended that there is a need to have a tool or new evaluation method to overcome the +current issue. +In summary, compared with previous studies [16]-[19], this study has some similarities and +differences as: First, it is the selection of model, our report also shows similar results, that is, many different +types of models and variables are applied to the research. There has not yet been a general consensus set to +guide new researchers to follow. The difference is that the variables in this study revolve around technology + +Indonesian J Elec Eng & Comp Sci +ISSN: 2502-4752 + + +A systematic review of structural equation modeling in augmented reality applications (Vinh The Nguyen) +335 +rather than ecology, social science, psychology, and management. Second is the issue of limitations. While +similar studies only listed restrictions that exist in articles, our study quantified these limitations by specific +numbers and arranges them in descending order. As such, interested researchers can rely on it to cover the +information more broadly. The third consideration is the study’s time span. This investigation was carried out +in the context of digital transformation and the influence of COVID-19. Many new factors emerge and exert +effect that have received little consideration in prior research (see Figure 3). Summarizing these factors will +help researchers have more options instead of reading different articles. And finally, by synthesizing how the +experiments were carried out during the pandemic, not only new researchers can adapt prior evaluation +approach in the current situations but also improve them in the subsequent studies. + + +4. +CONCLUSION +This paper presented a systematic review of the use of SEM in AR studies during the COVID-19 +pandemic. The PRISMA model was adapted as a guideline for doing the research. Five data sources were +used for data retrieval. After a series of preprocessing steps, 53 publications were included in the study. The +results showed that authors used a variety of external factors to form the generative hypothesized models +(SEM), followed by the extension of TAM. The diversity of external factors indicated that there is no +consensus among AR scholars for using common factors influencing AR adoption, thus opening a huge +potential research gap for the AR community. Interestingly, United States was the most active country in +conducting AR studies during the Covid-19 pandemic, however 80% of its studies were conducted through +indirect communication channels. Hence, they were not affected by the pandemic. A large portion of AR +studies focused on understanding factors influencing user behavioral toward using third-party AR apps. As +such, participants were required to download and use the apps then answer the survey questionnaires. Sample +size, in this regard, cannot be excused due to social distancing. Only few studies examined user behavioral +through developed AR apps and the corresponding authors suggested that there is a need to have an +alternative approach to conduct user study rather than the traditional face-to-face fashion. Watching two +separate videos (one with AR and one without AR) was currently be used as an alternative method to +alleviate the issue but not a plausible approach in the long run. Therefore, this research gap remains open and +needs to be addressed in further studies. Thus, the outcomes of this study can be used as a reference guideline +for researchers in similar studies where there is a lack of theoretical framework for assessment, particular in +electrical engineering and computer science. + + +ACKNOWLEDGEMENTS +This research is supported by project T2022-07-09 undertaken at the TNU–University of +Information and Communication Technology, Thai Nguyen, Vietnam. + + +REFERENCES +[1] +K. Awang, S. N. W. Shamsuddin, I. Ismail, N. A. Rawi, and M. M. Amin, “The usability analysis of using augmented reality for +linus students,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 13, no. 1, pp. 58–64, 2019, doi: +10.11591/ijeecs.v13.i1.pp58-64. +[2] +A. Ihsan, N. Fadillah, and C. R. 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Tsai, “Students’ context-specific epistemic justifications, prior knowledge, engagement, and +socioscientific reasoning in a mobile augmented reality learning environment,” Journal of Science Education and Technology, +vol. 29, no. 3, pp. 399–408, 2020, doi: 10.1007/s10956-020-09825-9. +[79] D. Harborth and S. Pape, “Investigating privacy concerns related to mobile augmented reality Apps – A vignette based online +experiment,” Computers in Human Behavior, vol. 122, p. 106833, Sep. 2021, doi: 10.1016/j.chb.2021.106833. +[80] A. C. Haugstvedt and J. Krogstie, “Mobile augmented reality for cultural heritage: A technology acceptance study,” ISMAR 2012 +- 11th IEEE International Symposium on Mixed and Augmented Reality 2012, Science and Technology Papers, pp. 247–255, +2012, doi: 10.1109/ISMAR.2012.6402563. +[81] J. Cheon, S. Lee, S. M. Crooks, and J. Song, “An investigation of mobile learning readiness in higher education based on the +theory of planned behavior,” Computers & Education, vol. 59, no. 3, pp. 1054–1064, Nov. 2012, doi: +10.1016/j.compedu.2012.04.015. + + +BIOGRAPHIES OF AUTHORS + + +Dr. Vinh The Nguyen + + + + is currently a lecturer at the Faculty of Information +Technology, University of Information and Communication Technology. He is also a senior +visiting lecturer at FPT University Greenwich, Hanoi branch. He graduated with a master's +degree in information systems management from Oklahoma State University, USA (under +scholarship 322). He completed his PhD program under Project 911 in 2020 at Texas Tech +University, USA. His main research interests are Computer Vision, Computer Visualization, +and Computer in Human Behavior. He has authored or coauthored more than 35 publications +with 10 H-index and more than 250 citations. He can be contacted at email: +vinhnt@ictu.edu.vn. + + + +Chuyen Thi Hong Nguyen + + + + is currently a lecturer at the Faculty of Primary +Education, Thai Nguyen University of Education, Vietnam. She graduated with a master's +degree in Theory and History of Education from Hanoi University of Education, Vietnam +(2008). She completed her PhD program in 2016 at The Vietnam Institute of educational +Sciences, Vietnam. Her main research interests are method teaching, assessment in primary +education, computational thinking, learning style, and augmented reality in education. She can +be contacted at email: chuyennh@tnue.edu.vn. + + + + + +pp \ No newline at end of file diff --git a/FtE3T4oBgHgl3EQftQtg/content/tmp_files/2301.04674v1.pdf.txt b/FtE3T4oBgHgl3EQftQtg/content/tmp_files/2301.04674v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5f387552a96bbd20b2012df36a5d811c825bb38 --- /dev/null +++ b/FtE3T4oBgHgl3EQftQtg/content/tmp_files/2301.04674v1.pdf.txt @@ -0,0 +1,1488 @@ +Prepared for submission to JHEP +IPARCOS-23-002 +Late vacuum choice and slow roll approximation in +gravitational particle production during reheating +Jose A. R. Cembranos,a Luis J. Garay,a Álvaro Parra-Lópeza and Jose M. Sánchez +Velázquezb +aDepartamento de Física Teórica and IPARCOS, Facultad de Ciencias Físicas, +Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain +bInstituto de Física Teórica UAM/CSIC, c/ Nicolás Cabrera 13-15, +Cantoblanco, 28049, Madrid, Spain +E-mail: cembra@fis.ucm.es, luisj.garay@ucm.es, alvaparr@ucm.es, +jm.sanchez.velazquez@csic.es +Abstract: In the transition between inflation and reheating, the curvature scalar typically +undergoes oscillations which have significant impact on the density of gravitationally +produced particles. The commonly used adiabatic vacuum prescription for the extraction +of produced particle spectra becomes a non-reliable definition of vacuum in the regimes for +which this oscillatory behavior is important. In this work, we study particle production for +a scalar field non-minimally coupled to gravity, taking into account the complete dynamics +of spacetime during inflation and reheating. We derive an approximation for the solution +to the mode equation during the slow-roll of the inflaton and analyze the importance of +Ricci scalar oscillations in the resulting spectra. Additionally, we propose a prescription for +the vacuum that allows to safely extrapolate the result to the present, given that the test +field interacts only gravitationally. Lastly, we calculate the abundance of dark matter this +mechanism yields and compare it to observations. +Keywords: Cosmology of Theories beyond the SM, Effective Field Theories, Classical +Theories of Gravity +arXiv:2301.04674v1 [gr-qc] 11 Jan 2023 + +Contents +1 +Introduction +1 +2 +Dynamics of a scalar field in flat FLRW cosmologies +3 +3 +Background dynamics +5 +3.1 +Inflationary era - Slow-roll approximation +6 +3.2 +Late reheating +7 +4 +Particle production +8 +4.1 +Solution to the mode equation +8 +4.2 +Choice of reference vacua +9 +4.3 +Slow-roll approximation for the solution to the mode equation +11 +4.4 +Adiabaticity and oscillations +16 +5 +Spectra of particles and total density +17 +6 +Conclusions +21 +A Parameters +23 +1 +Introduction +The theory of quantum fields in curved spacetimes accommodates a plethora of unexpected +phenomena such as Hawking radiation [1], the Unruh effect [2], or entanglement across +horizons [3–6], that have changed our perspective on the interplay between quantum fields +and gravity. Gravitational particle production due to the spacetime dynamics [7, 8] is one of +these phenomena and can be particularly important during the early stages of the universe, +since it may be able to explain the dark matter abundance, as it has been extensively +discussed in the literature. The rapidly evolving spacetime during inflation [9–11] and +the consequent transient to reheating [12–16] produce a significant abundance of particles +for any field that is coupled to the geometry. Since this is the only requirement for this +process to occur, it is of particular interest to analyze it from the perspective of dark +matter production mechanisms. Because of the absence of interactions with other fields, the +abundance of dark matter produced in the early universe due to the expansion of spacetime +is not diluted as a consequence of thermalization with other fields. It remains then as a relic +abundance, so that this mechanism alone can in fact explain current observations. This +has been mostly explored for scalar fields that are non-minimally coupled to gravity in a +myriad of works, such as [17–20] for supermassive dark matter candidates (WIMPZillas), +or, more recently, in references [21–24], where the importance of the oscillatory behavior of +– 1 – + +the background geometry was incorporated. On the other hand, gravitational production +of more general fields, such as fermion and vector fields, has also been analyzed in [25, 26]. +Usually, the dark matter candidate is regarded as a spectator field [27, 28] which does not +source gravity, and with no direct coupling to the inflationary fields. However, it is generally +non-minimally coupled to the geometry via the curvature scalar, and interactions with +other fields are disregarded. In all these works, it is customary to make use of the adiabatic +prescription to define the vacuum state of the dark matter field in order to calculate the +gravitational production. This definition seems to hold after a few oscillations of the +inflaton in the reheating stage, but only in the case of very large masses of the dark matter +candidate. In the regime of low masses, however, this vacuum provides a correct prediction +only when considering very late times, after many oscillations have occured. Importantly, +this oscillating behavior influences gravitational production [24]. It is worth mentioning +that the type of dark matter produced in this way is adiabatic [23, 29], and therefore the +observational constraints on isocurvature perturbations [30] do not have to be considered. +In this work, we study the gravitational production of a massive scalar field ϕ described +by a Klein-Gordon action that includes a non-minimal coupling to the Ricci curvature scalar +R through a term of the form ξRϕ2. The strength of this coupling is determined by the +parameter ξ. In an attempt to accommodate the arguments put forward in refs. [24, 31–33] +concerning vacuum instability, overproduction, and quantum cosmology analyses, we restrict +ourselves to the range 1/6 ≤ ξ ≤ 1 for the coupling constant ξ. As inflationary model, we +consider a single inflaton field φ that slowly rolls down a quadratic potential and starts +oscillating around its minimum, leading then to a reheating phase. The dynamics for the +inflaton is analytically solved at the onset of inflation, while the transition to the reheating +epoch is modeled numerically. Our scalar field is assumed to be in the Bunch-Davies +vacuum state when inflation starts. In order to extract the gravitational production, the +Klein-Gordon equation of the field ϕ is solved from that point in time until well inside +the reheating era. This moment effectively corresponds to the time where the dynamics +enters the adiabatic regime and particle production becomes negligible. Moreover, one also +needs to provide a definition of vacuum for this instant, for which the adiabatic prescription +is usually adopted. We discuss its validity and introduce as well an averaged vacuum +that produces the same density of particles but allows to obtain the correct result much +earlier than the time at which adiabaticity is reached. This is particularly helpful when +considering masses way below the inflaton mass for our scalar field, which are of great +interest concerning dark matter candidates. Also, we stress the importance of taking into +account the first few hundreds of oscillations of the inflaton in the final prediction and +present the results in the form of spectra and total density of produced particles for different +values of the scalar field mass m and its coupling ξ to the Ricci scalar. +The remainder of this paper is organized as follows. In section 2, we introduce the field +that is coupled to the expanding geometry, and work out the formalities of Bogoliubov- +like particle production in this context. In order to determine the complete form of the +mode equation, we need to provide the background dynamics coming from the particular +inflationary model in consideration, which we do in section 3. With all these ingredients, we +explore the gravitational production for the scalar field in section 4, analyzing the solution +– 2 – + +to the mode equation in the different regimes and studying the influence of the oscillations +of the curvature scalar in the final result. Moreover, we discuss the importance of the +vacuum choice when obtaining the numer density of produced particles. Lastly, we present +our results in the form of spectra and total density of particles in section 5 and elaborate +our conclusions in section 6. +Notation. We set MP = +√ +G, ℏ = c = kB = 1, and use the metric signature (−, +, +, +). +Furthermore, greek indices µ, ν run from 0 to 3, while latin indices i, j run from 1 to 3. +2 +Dynamics of a scalar field in flat FLRW cosmologies +We will consider a massive scalar field ϕ non-minimally coupled to gravity in a Friedmann- +Lemaître-Robertson-Walker (FLRW) spacetime with vanishing spatial curvature [34–40]. +We will not consider here any coupling of the derivatives of the scalar field (see [41]). +The dynamics of our scalar field is encoded in the action +S = −1 +2 +� +d4x√−g +� +∂µϕ∂µϕ + +� +m2 + ξR +� +ϕ2� +, +(2.1) +where g is the determinant of the metric, m is the bare mass of the field, and ξ is the +coupling to the Ricci curvature scalar R. The geometry is determined by the spatially flat +FLRW line element +ds2 = a2(η) +� +−dη2 + dx2 + dy2 + dz2� +, +(2.2) +where we have considered Cartesian coordinates for the flat spatial sections, and η is the +conformal time, related to cosmological time by a(η)dη = dt. +It is convenient to work with the auxiliary field +χ(η, x) = a(η)ϕ(η, x), +(2.3) +whose equation of motion can be obtained from the action (2.1), +χ′′(η, x) − +� +∆ + a′′(η) +a(η) − a2(η) +� +m2 + ξR +�� +χ(η, x) = 0, +(2.4) +where ∆ is the Laplace operator, the prime denotes derivative with respect to conformal +time, and R = 6a′′/a3. +We can use the eigenfunctions of the Laplace operator, which in our case are Fourier +modes, as a basis of functions to expand the scalar field χ, +χ(η, x) = +� +d3k +(2π)2/3 +�akvk(η) + a∗ +−kv∗ +k(η) +� eikx, +(2.5) +where the coefficients ak, a∗ +k become creation and annihilation operators upon quantization +of the field, with the standard commutation relations [42–45]. The time-dependent mode +functions vk(η) and v∗ +k(η) satisfy a harmonic oscillator equation +v′′ +k(η) + ω2 +k(η)vk(η) = 0, +(2.6) +– 3 – + +with k = +√ +k2 and a time-dependent frequency +ω2 +k(η) = k2 + a2(η) +� +m2 + (ξ − 1/6)R(η) +� +. +(2.7) +The solutions to (2.6) have to fulfill the normalization condition +vkv′ ∗ +k − v′ +kv∗ +k = i, +(2.8) +so that they are compatible with the standard commutation relations of creation and +annihilation operators. +For a given evolution of the background geometry, encoded in the scale factor a(η) and +the Ricci scalar R(η), both (2.8) and (2.6) are sufficient to determine vk(η), v∗ +k(η), which is +a basis of the space of solutions of the mode equations. Since any other solution can be +expressed as a linear combination of vk(η) and v∗ +k(η), any two sets of solutions vk(η) and +uk(η) must be related by uk = αkvk + βkv∗ +k, where normalization (2.8) on the temporal +modes implies the relation |αk|2 − |βk|2 = 1 for the complex coefficients αk and βk, which +are known as Bogoliubov coefficients [42]. Note that the expansion (2.5) can be carried out +using either basis of solutions. +Upon quantization of the field, both sets of coefficients ak and bk (associated with the +basis vk and uk, respectively) and their complex conjugates become operators that give rise +to two different definitions on quanta and vacua [43], +ˆak |0a⟩ = 0 +and +ˆbk |0b⟩ = 0, +∀ k. +(2.9) +These two quantizations are related by the Bogoliubov transformation ˆbk = α∗ +kˆak − β∗ +kˆa† +k. +The mean number density of b-particles in the a-vacuum, which will be, in general, a +non-vacuum state according to the ˆbk operators, is given by +⟨0a| ˆnb +k |0a⟩ = |βk|2. +(2.10) +Integrating over all modes, we find the total mean density +� d3k |βk|2, which will remain +finite as long as |βk|2 → 0 faster than k−3 for increasing k. +Let us now associate each basis of solutions to two observers living at different times +ta < tb. If spacetime is static, the frequency (2.7) is constant, so that the solution to (2.6) +takes the same form at all times. As a consequence, observers at different times have +the same notion of particle, and therefore βk = 0. However, if geometry undergoes an +expansion, two observers living at different times (before and after the expansion) have +different notions of vacuum. Thus, βk ̸= 0 and therefore nb ̸= 0, which can be understood +as the number density of particles produced out of the original vacuum state due to the +expansion of spacetime. +For the problem at hand, the goal is to extract the number of produced particles after +the evolution of the universe during inflation and reheating, once these stages have finished. +Then, as long as the test particle is not (strongly) interacting, this will be related to the +abundance one observer would measure today only by the expansion dilution. Hence, we +will take the Bunch-Davies vacuum as initial state, as defined by the solution of the mode +– 4 – + +equation at very early times. In our case, we will take the geometry to approach de Sitter +spacetime at the beginning of inflation. On the other hand, the notion of vacuum for an +inertial observer after reheating will be different. If the evolution of spacetime is sufficiently +adiabatic after this phase, we can assume this is the same vacuum we observe nowadays. +Therefore, the corresponding operators will measure the number of particles created in the +evolution. +The specific form of the scale factor and the Ricci scalar will be determined by the +specific inflationary model under consideration, which we describe in the next section. +3 +Background dynamics +We will describe the early epoch of the universe with a chaotic inflationary model consisting +of a single scalar field φ with a quadratic potential of the form V (φ) = 1 +2m2 +φφ2, where +mφ denotes the inflaton mass. The equation of motion for the inflaton is, if we assume +homogeneity and isotropy, +0 = ¨φ + 3H(t) ˙φ + ∂φV (φ), +(3.1) +where H(t) ≡ ˙a(t)/a(t) is the Hubble parameter. Note that in this context it is customary +to work with cosmological time t. We will assume that the inflaton contribution to the +total energy-momentum tensor is dominant when deriving the corresponding Friedmann +equation, +H2 = +4π +3M2 +P +� ˙φ2 + 2V (φ) +� +. +(3.2) +We will also need the Ricci curvature scalar in order to properly describe the frequency of +the mode equation (2.6), which in terms of the inflaton field reads +R = 8π +M2 +P +� +4V (φ) − ˙φ2� +. +(3.3) +Equation (3.1), together with (3.2), has no analytic solution in general. However, +one can find approximations for certain regimes. When this is not possible, we must rely +on numerical computation. We analyze two different regions which, in conformal time, +correspond to +η = +� +� +� +ηi ≤ η < η∗, +Slow-roll approximation, +η∗ ≤ η ≤ ηf, +Numerical solution. +(3.4) +For the inflationary period, we can use the well-known slow-roll approximation to obtain a +solution to the inflaton equation of motion, as we describe in subsection (3.1). However, +during the transition between inflation and reheating, the dynamics of the inflaton has to +be obtained numerically. Both the inflaton field φ and the Ricci scalar R start to oscillate +with decreasing amplitude, as can be observed in figure 1, where φ(η) and R(η) are depicted +for an interval of time during the transition phase. This is the epoch in which most of the +particles are produced and the inflaton dynamics is solved until a numerically accessible time +ηf is reached, when production becomes negligible. For late times, deep in the reheating +era, we can also use an analytic approximation for the solution of the inflaton equation of +– 5 – + +-3 +-2 +-1 +0 +1 +2 +3 +0 +2 +4 +6 +-2 +-1 +0 +1 +2 +3 +4 +0 +5 +10 +15 +20 +1 +2 +3 +-0.25 +0 +0.25 +0.5 +Figure 1. Inflaton field φ(η) (left panel) and curvature scalar R(η) (right panel) as functions of +conformal time. The range of time corresponds to the end of inflation and the beginning of reheating. +The parameters used for all figures in this article are given in Appendix A. +motion, given in subsection 3.2, which —although not used in our calculations— will be +used to make some remarks in section 4. +3.1 +Inflationary era - Slow-roll approximation +We will choose the inflationary period to start at the negative, initial time ti. Inflation +requires that the inflaton field changes slowly in comparison to the potential. Within the +slow-roll approximation [46, 47], we can neglect the derivative of the field in favor of the +potential, namely ˙φ2 ≪ |V (φ)|. When this condition is satisfied, the field slowly rolls over +until it falls to a minimum and starts oscillating. At this point, inflation ends. With this +assumption, we can approximately write (3.2) during the slow roll as +H ≃ +� +8π +3M2 +P +V (φ). +(3.5) +A slowly-varying inflaton implies that H ∼ constant for this regime. Hence, the expansion +of spacetime is said to be quasi-exponential, as it resembles the pure de Sitter solution. +Usually, one also assumes a small rate of change for the (already slow) velocity of φ, such +that |¨φ| ≪ 3H| ˙φ|. This allows the slow-roll condition to be maintained long enough to solve +the flatness and horizon problems. With these assumptions, equation (3.1) becomes easily +solvable, +˙φ ≃ −∂φV (φ) +3H +≃ −∂φV (φ) +MP +� +24πV (φ). +(3.6) +– 6 – + +For the particular potential V (φ) = 1 +2m2 +φφ2, the solution to (3.6) is +φSR(t) = φ0 − MP +√ +12πmφt, +(3.7) +where t < 0 corresponds to the inflationary period. Note that t = 0 and φ0 are the ending +time of inflation and the value of the field at this instant, respectively. From here, it is +straightforward to obtain an explicit expression for the Ricci scalar, introducing the solution +into (3.3). +The scale factor is obtained by integrating the Hubble rate, and in the slow-roll +approximation it reads +aSR(t) ≃ a0e +−� φ(t) +φ0 +dφ 8π +M2 +P +V (φ) +∂φV (φ) , +(3.8) +which for the quadratic potential becomes +aSR(t) = a0e +− 2π +M2 +P [φ2 +SR(t)−φ2 +0]. +(3.9) +Lastly, we need the relation between cosmological and conformal time in order to write +both a(η) and R(η). This relation can be obtained numerically from η = η0 + +� t +0 dt/a(t). +These are the necessary ingredients for determining the frequency of the mode equation in +this region, under the slow-roll approximation. +This regime is valid as long as the slow-roll parameter, ϵH = − ˙H/H2, is much smaller +than one. When this no longer holds, at, say, t > t∗ with t∗ < 0, the equation of motion (3.1) +has to be solved numerically. The field begins to exit the inflationary regime and t = η = 0 +marks both the end of inflation and the beginning of reheating. At this point, the scale +factor reaches the value a0, which merely sets the scale and hence we take it to be a0 = 1. +3.2 +Late reheating +For late times, well into the reheating epoch (η∗ ≪ η ≲ ηf), one can find an approximate +solution to (3.1) [24]. We do not use it for obtaining our results, but it will be important +for the discussion in subsection 4.2. In this approximation, the Hubble rate reads +H(t) ≃ 2 +3t +� +1 − sin (2mφt − 2ϕ) +2mφt ++ O(m−2 +φ t−2) +�−1 +, +(3.10) +whereas the inflaton field is given by the expression +φ = Φ0 +t sin mφt +� +1 − cos 2mφt +2mφt ++ O(m−2 +φ t−2) +� +, +(3.11) +with Φ0 ≡ MP /( +√ +3πmφ). This solution is valid as long as mφt ≫ 1, condition which is +fulfilled during reheating, since, as we will see, the scale factor behaves as that of a matter +dominated universe. Indeed, we can integrate H(t) in order to approximately obtain the +scale factor a(t), +a(t) = Ct2/3 � +1 + O(m−2 +φ t−2) +� +. +(3.12) +– 7 – + +The constant C is determined by requiring that the value of the scale factor at late times +coincides with the one obtained from the numerical simulation in the previous region. One +can now integrate the scale factor in order to obtain t(η) = (Cη/3)3. +Now that we have a solution for the inflaton field and the scale factor valid for late +times, we can obtain the Ricci scalar from (3.3) by taking the solution for φ(t) to first order +in (mφt)−1. We end up with +R = 8 +3t2 +� +�2 sin2 mφt − +� +cos mφt − sin mφt +mφt +�2 ++ O(m−3 +φ t−3) +� +� . +(3.13) +With this, we are able to describe the frequency of the mode equation until very late +times, for which the approximations derived in this subsection behave even better. The +density of produced particles will be calculated at a sufficiently large time ηf, such that the +particle production is negligible from that point in time onwards. (3.13). +4 +Particle production +Once we have determined the behavior of the background geometry during inflation and +reheating, we can solve the mode equation in order to extract the Bogoliubov coefficients +after the evolution. +4.1 +Solution to the mode equation +In order to compute the gravitational production once reheating has ended, we need to +solve equation (2.6) from the onset of inflation at ti until a time tf well inside the adiabatic +regime at the end of reheating, with the frequency of the oscillator determined by the +background geometry described in the previous section. In a similar way as we did for the +background dynamics in section 3, the mode equation is solved in the regions +η = +� +� +� +ηi ≤ η ≤ η∗, +Slow-roll approximation, +η∗ ≤ η ≤ ηf, +Numerical solution. +(4.1) +Let us start with the slow-roll era. In a de Sitter geometry, the Hubble rate is exactly +constant, H0, the Ricci scalar is R = 12H0, and the scale factor reads a(η) = 1/(1 − H0η). +Therefore, the frequency (2.7) takes the form +ω2 +k,dS = k2 + +µ2 +(η − η0)2 , +with +µ2 = m2/H2 +0 + 12(ξ − 1/6), +(4.2) +where H0 = H(ηi) = 1/η0 is the Hubble rate at the beginning of inflation. The solution to +equation (2.6) in this simplified scenario which assimptotically at η → −∞ behaves as a +positive frequency plane wave is given by +vk,dS(η) = +� +π|η − η0|/2 eiπνH(1) +ν +(k|η − η0|) , +ν = +� +1/4 − µ2. +(4.3) +This is the so-called Bunch-Davies solution [42]. Note that there is a critical value µ2 = 1/4 +for which ν = 0, which separates the regimes of real and imaginary ν. In particular, for +– 8 – + +m2/H2 +0 ≪ 1, we can approximately write µ2 ≈ 12 (ξ − 1/6), and therefore µ2 = 1/4 for +ξ = 3/16. At this point, there is no gravitational pair production in a de Sitter geometry +[41], and this fact will be important for the analysis in section 4. +However, our background geometry is not exactly de Sitter, but given by the inflaton +dynamics derived in section 3. Within the slow-roll approximation, valid from the start of +inflation at ηi until η∗, the mode equation to solve is +v′′ +k(η) + ω2 +k,SR(η)vk(η) = 0, +(4.4) +where the scale factor and the Ricci scalar in ωk,SR(η) correspond to the analysis in +subsection 3.1. Nevertheless, in the slow-roll regime, and for a certain range in k, m, +and ξ, we can approximate the solution satisfying Bunch-Davies initial conditions by (see +subsection 4.3 for details) +vk,SR(η) ≃ +� +π|τk|/2eiπνH(1) +ν +(k|τk|) , +τk = ωk,SR(η) +ωk,dS(η) (η − η∗,k) + η∗,k − η0, +(4.5) +where η∗,k marks the limit of validity of the approximation. From this point on, equation (2.6) +has to be solved numerically, independently of the background dynamics being numerical or +analytical, taking as initial condition solution (4.5) and its derivative at η∗,k. The frequency +one has to use in this case is that in (2.7). +4.2 +Choice of reference vacua +The solution vk(η) to the mode equation is associated with a particular choice of vacuum: +the one that behaves as a plane wave at η → −∞. The procedure in subsection 4.1 allows +us to evaluate vk(ηf). However, in order to obtain the Bogoliubov coefficient βk, we also +need uk(ηf), which is the solution to the mode equation associated with the vacuum at this +point in time. Then, from the Bogoliubov coefficients αk and βk, we will be able to extract +the number density of produced particles at ηf. This time is chosen such that particle +production becomes negligible for later times, condition that is fulfilled in the adiabatic +regime, i.e., when +����� +ω′ +k(ηf) +ω2 +k(ηf) +����� ≪ 1. +(4.6) +The value of ηf highly depends on the parameters of the scalar field, and in particular, it +becomes larger as the mass m decreases. This is why, for certain regions in parameter space, +it may be convenient to use the late-time approximation for the background dynamics +described in 3.2, instead of solving numerically the equation of motion of the inflaton field. +It is worth mentioning that at the same time, a smaller coupling ξ to the curvature implies +that the Ricci scalar oscillations, which are the main source of non-adiabaticity, are less +important, therefore resulting in an earlier ηf at which (4.6) holds true. +As long as the background is not static, the meaning of vacuum will change in time. +Nevertheless, if the evolution is adiabatic enough, namely condition (4.6) is fulfilled, one +can use the so-called adiabatic prescription to define the instantaneous vacuum at a given +– 9 – + +instant ηf, +uk(ηf) = +1 +� +ωk(ηf) +, +u′ +k(ηf) = − +1 +√ωk +� +iωk(ηf) + 1 +2 +ω′ +k(ηf) +ωk(ηf) +� +. +(4.7) +In fact, it is this feature that allows us to extrapolate the results obtained at ηf to the +present when considering fields that interact only gravitationally [22, 24]. +When the mass of the field ϕ is above mφ, particle production is governed by the mass +term of the frequency (2.7), namely +ω2 +k(η) ≃ k2 + a2(η)m2. +(4.8) +Since the scale factor at late times behaves as a(η) ∼ η2, condition (4.6) is fulfilled after few +oscillations of the inflaton. In other words, in this case we have that ηf is small enough that +we do not need to invoke the late-time solution for the background, since everything can be +calculated numerically in an efficient way. This is not the case for masses smaller than the +inflaton, for which production stabilizes after many, many oscillations. As a consequence, if +we want to use the adiabatic vacuum description, we need to go up to a very large ηf, and +therefore we need to use the analytic approximation for the inflaton dynamics described +in (3.11). +Alternatively, we can take a different definition for the vacuum that allows us to +calculate the number density of produced particles at ¯η ≪ ηf, even for m ≪ mφ. Because +the oscillating term in (2.7) becomes negligible at sufficiently large (numerically accessible) ¯η, +we can define the frequency +ω(avg) 2 +k +(η) = k2 + a2(η) +� +m2 + (ξ − 1/6) ⟨R⟩ (η) +� +, +(4.9) +where the Ricci scalar oscillations are averaged. We can take this frequency to calculate the +averaged vacuum +u(avg) +k +(¯η) = +1 +� +ω(avg) +k +(¯η) +, +u(avg) ′ +k +(¯η) = − +1 +� +ω(avg) +k +(η) +� +iω(avg) +k +(¯η) + 1 +2 +ω(avg) ′ +k +(¯η) +ω(avg) +k +(¯η) +� +. +(4.10) +This prescription of vacuum is such that the spectrum of produced particles obtained at ¯η +essentially concides with the one given by the adiabatic vacuum at the time where we reach +the adiabatic regime, ηf, namely +n(avg) +k +��� +η=¯η ≃ n(ad) +k +��� +η=ηf +. +(4.11) +The larger discrepancies will reside in low wavenumbers, for which k ∼ a2(η) ⟨R⟩, but this +region of momentum space is supressed in the total density of produced particles by a factor +k2 (for details see next subsection), since +n(m, ξ) = +� +d3k +(2π)3 ⟨0| ˆnk |0⟩ = +� +dk +2π2 k2|βk|2. +(4.12) +– 10 – + +As a consequence, no differences are appreciated at the chosen ¯η. +This procedure has a limitation: It is valid up to the smallest mass m for which +the dynamics presented here remain the same until ηf. If reheating ends before ηf for +a particular mass, the result provided by the averaged vacuum will not be the particles +produced after this period is over. Nevertheless, a simple estimation shows that masses +above the order of m ∼ 10−30 eV would reach adiabaticity early enough. This is many +orders of magnitude below the mass of fuzzy cold dark matter, and hence all the interesting +range of masses lie within the regime of validity of our method. +4.3 +Slow-roll approximation for the solution to the mode equation +During inflation, spacetime expands quasi-exponentially. More specifically, the number of +e-folds +a(t0) +a(ti) = eN +(4.13) +is required to be such that N ≈ 50 − 60 [9–11]. +Because eq. (2.6) cannot be solved +analytically, even considering a slowly rolling inflaton field, one would need to use numerical +methods in order to find a solution. However, the large amount of e-folds to cover makes +it more interesting and feasible to rely on an analytic approximation, such as (4.5). We +dedicate this subsection to formally develop the approximation and to test its validity. For +notational convenience, in the calculations that follow we will write η − η0 as η, and drop +the mode index k. Let us start by defining the following small parameters for given values +of k, m and ξ which will be useful in the following. +• First, we have +ϵ(m, ξ) = max +η∈I1 +�����1 − ωSR(η; m, ξ) +ωdS(η; m, ξ) +�����, +with +I1 = (−∞, η1), +(4.14) +where η1 is chosen such that ϵ ≪ 1. Then, we can define f(η; m, ξ) by +ωSR +ωdS += 1 + ϵf. +(4.15) +By construction, |f(η)| ≤ 1 for η ∈ I1. Moreover, f′(η) ≥ 0. +• It will also be convenient to define +σ(m, ξ) = max +η∈I2 +���f′(η; m, ξ)η +���, +with +I2 = (−∞, η2), +(4.16) +and choose η2 such that σ ≤ ϵ. Then, we introduce g(η; m, ξ) as +f′(η) = σg(η) +η +, +(4.17) +for which again we have that |gk(η)| ≤ 1 for η ∈ I2. +– 11 – + +• Similarly, we define +ρ(m, ξ) = max +η∈I3 +����� +ω′ +dS(η) +ωdS(η)η +�����, +with +I3 = (−∞, η3), +(4.18) +and choose η3 such that ρ ≤ ϵ. +Now, we take η∗ = min(η1, η2, η3) and I = (−∞, η∗), where I is the interval for which +the three parameters ϵ, σ, ρ are small. Note that η∗ < 0 since inflation ends at η = 0. +• We also need |η∗/η0| > 1. +The task is to solve equation (4.4), for which we define a new time coordinate ζ within +the interval I, +dζ = ωSR(η) +ωdS(η) dη = [1 + ϵf(η)] dη. +(4.19) +After integration until η ∈ I and taking the absolute value, this becomes +|(ζ − ζ∗) − (η − η∗)| = ϵ +����� +� η∗ +η +f(t)dt +����� = O(ϵ)(η − η∗). +(4.20) +Then, choosing ζ∗ = η∗, this can be expressed as +ζ = η [1 + O(ϵ)] . +(4.21) +We change time coordinates η → ζ in the mode equation, which takes the form +¨w(ζ) + ω2 +dS [η(ζ)] w(ζ) + ϵf′ [η(ζ)] ω2 +dS [η(ζ)] +ω2 +SR [η(ζ)] ˙w(ζ) = 0, +(4.22) +where w(ζ) = v [η(ζ)] and the dot denotes here derivative with respect to ζ. +Let us analyze the last term. With this aim, we introduce the dimensionless time +¯ζ = ζ/η0. Then, in terms of ¯ζ, the equation above has the same form except for the last +term that acquires an extra factor. Using the definition of f′ and σ above, the coefficient of +this term is +ϵf′ ω2 +dS +ω2 +SR +η0 = ϵσg(1 + ϵf)η0 +η = O(ϵ2)η0 +η +(4.23) +If we choose η∗ such that |η∗/η0| > 1, as mentioned above, this coefficient is of order O(ϵ2). +Furthermore, the frequency in the second term of (4.22) is +ω2 +dS(η(ζ)) = ω2 +dS (ζ [1 + O(ϵ)]) +(4.24) += ω2 +dS(ζ) +� +�1 + 2ω′ +dS +ωdS +����� +ζ +· ζ O(ϵ) +� +� +(4.25) += ω2 +dS (ζ) +� +1 + O(ϵ2) +� +, +(4.26) +provided that |ζ ω′ +dS(ζ)/ωdS(ζ)| ≤ ρ = O(ϵ). This is satisfied for ζ = η [1 + O(ϵ)] < η∗, i.e., +for η < η∗. Thus, the equation for w can finally be written as +¨w(ζ) + ω2 +dS(ζ)w(ζ) = O(ϵ2 +k). +(4.27) +– 12 – + +We can perturbatively solve the differential equation by writting w = w0 + ϵw1 + O(ϵ2). +The solution to order ϵ0 is nothing but the de Sitter modes (4.3), +w0(ζ) = +� +π|ζ| eiπνH(1) +ν +(k|ζ|) , +ν = +� +1/4 − µ2, +(4.28) +and as a consequence, wk,0 behaves asymptotically (ζ → −∞) as a plane wave. On the +other hand, the coefficients of the solution to order ϵ1 will satisfy the same original equation +but with the initial conditions that w1(−∞) = 0 and therefore w1 is identically zero. We +can then write w as +w(ζ) = w0(ζ) +� +1 + O(ϵ2) +� += +� +π|ζ| eiπνH(1) +ν +(k|ζ|) +� +1 + O(ϵ2) +� +. +(4.29) +In order to undo the coordinate transformation ζ → η while keeping the error up to +O(ϵ2), we need to consider the O(ϵ1) terms in ζ = η [1 + O(ϵ)]. For this, we note that +����� (ζ − η∗) − ωSR(η) +ωdS(η) (η − η∗) +����� = +����� (η − η∗) + ϵ +� η +η∗ +f(t)dt − [1 + ϵf(η)] (η − η∗) +����� += ϵ +����� +� η +η∗ +f(t)dt − +� η +η∗ f(η)dt +����� +≤ ϵ +� η +η∗ +|f(t) − f(η)|dt += ϵ +� η +η∗ +����f′(η)(t − η) + 1 +2!f′′(η)(t − η)2 + · · · +���� dt +≤ ϵ +����� +1 +2f′(η)(η − η∗)2 +���� + +���� +1 +3!f′′ (η − η∗)3 +���� + · · · +� +. +(4.30) +This means that, as long as the terms in curly brackets are of order O(ϵ), we can write +ζ = η∗ + +�ωSR(η) +ωdS(η) + O(ϵ2) +� +(η − η∗) = η∗ + ωSR(η) +ωdS(η) (η − η∗) +� +1 + O(ϵ2) +� +. +(4.31) +The first term is equal to +1 +2σ +����g(η)η − η∗ +η +���� = O(ϵ). +(4.32) +The next terms are of the form f(n) (η − η∗)n+1 /n!, which numerically can be seen to be +smaller than the first one. +Therefore, undoing the translation of η to η − η0 that we did at the beginning of this +calculation, the solution to the mode equation can be written as (4.5) up to terms of order +O(ϵ2). With fixed ξ, and choosing η∗ independent of k, the error ϵk increases with increasing +m and decreasing k. +When we numerically solve the mode equation (2.6) from η∗, the error in the initial +condition coming from the slow-roll solution (4.5) carries through as +vk(η) = vk,0(η) +� +1 + O(ϵ2 +k) +� +, +(4.33) +– 13 – + +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0 +Figure 2. Maximum of the errors squared as function of the wave number k and the field mass m, +for ξ = 0.2 (left) and ξ = 0.8 (right). We take η∗ = −500mφ for all values of k, m and ξ. +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +0.01 +0.1 +1 +10 +100 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0 +Figure 3. Maximum of the errors squared times k2 as function of the wave number k and the field +mass m, for ξ = 0.2 (left) and ξ = 0.8 (right). We take η∗ = −500mφ for all values of k, m and ξ. +such that vk(η) → vk,SR(η) as η → η∗. Therefore, we have for the total density defined +in (4.12) that +n(m, ξ) = +� ∞ +0 +dk +2π2 k2|βk|2 = n0 +� +1 + 1 +n0 +� ∞ +0 +dk +2π2 k2|βk,0|2O(ϵ2 +k) +� +, +(4.34) +where n0 = +� ∞ +0 +dk +2π2 k2|βk,0|2. Although the error ϵk increases as k decreases, the factor k2 +compensates this increase for low k. Essentially, although ϵ2 +k increases for k < mφ, the +quantity k2ϵ2 +k remains small, whereas |βk,0|2 is roughly of the same order. More explicitly, +for the calculations in this paper, we take η∗ = −500mφ, for which the maximum of the +– 14 – + +-800 +-600 +-400 +10-5 +10-4 +0.001 +0.010 +-800 +-600 +-400 +10-8 +10-6 +10-4 +0.010 +Figure 4. Relative error in the absolute value (left panel) and the phase (right panel) of the +numerical solution to the exact mode equation (2.6) compared to the analytical approximation (4.5), +for wavenumbers ranging from k = 0.01mφ to k = 100mφ, and m = mφ, ξ = 0.5. Here, we take +ηdS = −1000/mφ and η∗ = −500/mφ. +three small parameters squared, ϵ2 +k, σ2 +k, ρ2 +k, as function of mass and wavenumber, for two +different choices of coupling ξ, is shown in figure 2. For m ≤ mφ and k ≥ 0.1mφ, the +error is of order O(0.01) or smaller for the various values of ξ considered, and thus the +approximation is controlled in this regime. At the same time, we can observe in figure 3 that +k2ϵ2 +k decreases as we move to the low-part of the momentum range. This guarantees that +this region of the spectrum is robust against errors in the mode equation approximation we +used. +On the other hand, from figure 3 we observe that the quantity k2ϵ2 +k grows with k for +k > mφ, since the decrease in ϵ2 +k (c.f. figure 2) can not compensate the power k2. However, +gravitational production for high-momentum particles is very small, namely |βk|2 ≈ 0 for +k ≫ mφ. As a consequence, n(m, ξ) ≈ n0 approximates well the total number density of +particles produced, since the weight of wavenumbers k ≫ mφ is very small when compared +to the rest of the spectrum. +Furthermore, we can test the validity of (4.5) when compared to the numerical solution +of (2.6) by putting ourselves in the following scenario: Let us assume that the geometry +can be approximated by a de Sitter spacetime during the early stages of inflation, such +that the solution (4.3) is valid for a region ηi ≤ η < ηdS. At ηdS, slow-roll starts to matter, +and deviations from the de Sitter solution vk,dS(η) occur. In this scenario, we explore two +different paths to continue continuing solving the equation: +1. We assume slow-roll inflation is a good description for the background dynamics in +the region ηdS ≤ η < η∗, and take as solution the approximation (4.5). +2. We solve numerically the exact equation of motion for the inflaton, eq. (3.1), obtaining +the frequency corresponding to (2.6), equation which we again solve numerically. This +– 15 – + +solution, vk(η), will be valid even for η ≥ η∗. +In figure 4, we compare the analytical slow-roll solution with the exact numerical solution +by plotting the relative difference between their absolute values, +∆rAbs [vk,SR(η)] ≡ +����� +Abs [vk(η)] − Abs [vk,SR(η)] +Abs [vk(η)] +�����, +(4.35) +as well as their phase difference, +∆rArg [vk,SR(η)] ≡ +����� +Arg [vk(η)] − Arg [vk,SR(η)] +π +�����. +(4.36) +We do so for different wavenumbers, ranging from k = 0.01mφ to k = 100mφ, denoted by +the different shapes in figure 4. We have taken ηdS = −1000/mφ as start of the slow-roll +and η∗ = −500/mφ as the time when the slow-roll approximation breaks down. For k = mφ, +the relative error is very small, of order ∼ 10−4 at η∗. For wavenumbers larger than the +mass of the inflaton, k > mφ, the approximation is still good, although it worsens. On +the other hand, the error for k = 0.01mφ starts becoming significant, and gets worse for +k < 0.01mφ. However, the corresponding region of the spectrum of produced particles is +highly suppressed, as discussed above, and therefore the contribution to the total density of +particles is negligible. Similarly, particle production is very small for wavenumbers larger +than k > 100mφ, and therefore the range of interest in k is under control. Hence, we can +assume the approximation is valid in the region ηdS ≤ η < η∗. +Note that if this solution behaves well in this region, it has to become an even better +approximation before ηdS, since the further towards the past we go, the more de Sitter-like +is the geometry. Thus, eq. (4.5) can be taken as well as a solution to the mode equation in +the region ηi ≤ η < ηdS. Under this approximations, eq. (2.6) can be solved analytically +from the start of inflation, ηi, until η∗, for which the slow-roll approximation starts to fail. +From there, the mode equation is solved numerically. +4.4 +Adiabaticity and oscillations +In order to illustrate the importance of the choice of vacuum, we studied the evolution +of spectra when calculated using prescription (4.7) before the dynamics has entered the +adiabatic regime. As an example, we plotted in figure 5 the spectra of particles with mass +m = 10−3mφ obtained at two different times. The dots correspond to η = 40/mφ, whereas +the solid lines denote η = ηf = 100/mφ. For this particular choice of mass, the latter time +lies within the adiabatic regime, and this is the reason why the non-adiabatic dots relax to +their final value as we approach this limit. As expected, the effect is less noticeable the +lower the coupling to the geometry is, as it is the main source of non-adiabaticity in the +frequency. +At the same time, we also characterized the importance of the first oscillations of the +curvature scalar in the final spectrum of produced particles, obtained with the averaged +vacuum defined in eq. (4.10). As can be seen in figure 6, even after several oscillations of +– 16 – + +0.01 +0.1 +1 +10 +100 +0 +0.2 +0.4 +0.6 +0.8 +Figure 5. Spectra of produced particles of mass m = 10−3mφ and different values of ξ, obtained with +the adiabatic prescription of the vacuum. The dots correspond to η = 40/mφ, before the adiabatic +regime has been reached for this value of the mass. The solid lines correspond to η = ηf = 100/mφ, +when most of the particles have been produced. +R(η) (for example, at η = 2/mφ), the production changes greatly if one compares with the +obtained spectra at ¯η. Even when looking only at the total number of produced particles in +eq. (4.12), differences are still significant. We observe that the spectrum does not stabilize +until η ≃ 5/mφ, which for our model means after hundreds of oscillations of the curvature +scalar R(η). With this, we want to stress that obtaning the particle production after one or +two oscillations does not account for the whole process. +5 +Spectra of particles and total density +Let us finally give the results for the spectra of produced particles as function of the +parameters of the field, the mass m, and the coupling to the curvature ξ. +We explore first the regime of masses below the inflaton mass. Represented by the solid +line in figure 7, we have masses m ≤ 10−4mφ. For these values, the mass contribution to +the frequency becomes negligible, and the dynamics is entirely given by the coupling to the +geometry. The spectra lie on top of each other, with very small differences in the low values +of k ∼ a(η)m. We observe, however, slight differences in the shape of the spectrum when +increasing the mass, especially for small wavenumbers, as the rest of the curves in figure +7 show. We can choose a mass in this regime, m = 10−1mφ, and explore the influence of +the coupling ξ in the final result. This is shown in figure 8, where one observes increasing +production of particles with larger values of the coupling. Lastly, let us come to the mass +of the inflaton, whose corresponding spectra are shown in figure 9. In such a case, it is +– 17 – + +0.01 +0.1 +1 +10 +100 +0 +2 +4 +6 +Figure 6. Spectra for m = 10−4mφ and ξ = 1, obtained with the averaged vacuum prescription, +for different instants of time. The spectrum stabilises after very many oscillations of the curvature +scalar. +0.001 +0.01 +0.1 +1 +10 +100 +0 +0.1 +0.2 +Figure 7. +Spectrum of particles for masses below the mass of the inflaton, with ξ = 0.26. +For very small masses (m ≤ 10−4mφ), production is dominated by curvature. +In the region +10−3 ≤ m ≤ 10−1mφ, differences in production due to the mass can be noticed, especially for low +values of k/mφ ≃ 0.1 − 1. +harder to characterize the behavior with ξ. It is clear, nevertheless, that particle production +decreases as the mass of particles becomes larger. +– 18 – + +0.01 +0.1 +1 +10 +100 +0 +0.2 +0.4 +0.6 +Figure 8. Spectra for m = 10−1mφ and several values of the coupling ξ. Particle production +increases when the curvature term becomes more important, and the maximum of the spectrum is +shifted towards higher values of k. +0.01 +0.1 +1 +10 +100 +0 +0.01 +0.02 +Figure 9. Spectra of particles with the mass of the inflaton, for different values of the coupling. In +this particular case, increasing the coupling does not translate directly into an increase of particle +production. This can be more clearly seen by examining the total density of particles. +It is easier to characterize particle production in this regime using the total number +density of particles (4.12), which we show in figure 10 as function of the two parameters of +the field, m and ξ. Here, one clearly sees that the prediction is independent of the value of +– 19 – + +Figure 10. Logarithm of the total density of produced particles for different values of m and ξ. In +order to give the mass and density in units of GeV, we took mφ = 1.2 × 1013 GeV for the mass of +the inflaton. We explore a wide range of masses in the left panel while we focus on a smaller region +close to the mass of the inflaton on the right panel in order to appreciate the dependence of the +total density with the coupling ξ. +the mass as long as it is below m ∼ 10−2mφ, in particular for a sufficiently high value of +the coupling, ξ ≳ 0.2. In this case, the mass is completely negligible when compared to +the dynamics of the curvature scalar. Only when the coupling to the curvature is close +to ξ ∼ 1/6, the production of particles is still sensible to m, up to m ∼ 10−7mφ. For this +value, even in the conformal case, the relevant wavenumbers, k ∼ a(η)m, are too suppressed +to make a difference. In all these regime of low masses, the number of produced particles +increases with larger coupling ξ. Closer to the mass of the inflaton, 10−2mφ < m < mφ, +the fact that a heavier particle translates into a lower production becomes apparent. Lastly, +in the region around the mass of the inflaton, m ∼ mφ, the behavior with the coupling is +different, and production may even decrease when raising the value of ξ. In fact, there +appears to exist a critical value ξc ≃ 0.22 which separates two qualitatively different regimes. +As we commented previously, this value is related to the parameter µ2 = 1/4 of the Hankel +functions, which were a good approximation of the mode functions of our problem. For +m < mφ, the number density drops very rapidly if ξ < ξc. For m ∼ mφ, ξc is the value below +which production decreases with ξ, and above which it increases. This is also illustrated +in figure 9, where production for ξ = 1/6 is larger than for ξ = 0.26, and from there it +increases again with the coupling. Moreover, we observe the expected strong suppression in +the number density of produced particles for masses above the mass of the inflaton. We +can confirm this behaviour by calculating the spectra for even higher masses, provided we +select a negative enough η∗ — and therefore leading to a very heavy computation — in +this case, as explained in 4.3. Note that we took mφ = 1.2 × 1013 GeV for the mass of the +inflaton, and as a consequence, the density in figure 10 is given in units of GeV3. +– 20 – + +10-3 +101 +105 +109 +1013 +1/6 +0.2 +0.4 +0.6 +0.8 +1.0 +10-3 +101 +105 +109 +1013 +1/6 +0.2 +0.4 +0.6 +0.8 +1.0 +-7.5 +-3.0 +1.5 +6.0 +10.5 +15.0 +Figure 11. Logarithm of the predicted abundance of dark matter today for different values of m +and ξ, and a reheating temperature of Treh = 1015 GeV (left) and Treh = 1013 GeV (right). In order +to give the mass and density in units of GeV, we took mφ = 1.2 × 1013 GeV for the mass of the +inflaton. +Finally, one can consider these gravitationally produced scalar particles as dark matter. +In this case, it is interesting to compare the resulting abundance with observations. Assuming +that the scalar field is non-interacting, the evolution of the particle density showed in figure 10 +is only due to the expansion of the universe. Then, the predicted abundance can be written +in terms of the background radiation temperature [24] as +Ω(m, ξ) = +8π +3M2 +P H2 +today +gtoday +∗S +grh +∗S +�Ttoday +Trh +�3 +m n(m, ξ), +(5.1) +where Ttoday and Trh are the radiation temperature today and at the end of reheating, +respectively, and gtoday +∗S +and grh +∗S are the corresponding relativistic degrees of freedom. This is +represented in figure 11 for two different reheating temperatures, together with the observed +abundance given by the dashed line. We observe that the proposed mechanism can explain +observations if the dark matter candidate is light enough (m ≲ 108 GeV), independently of +the value of the coupling ξ for the range that we considered. In addition, heavier particles +can also reach the observed dark matter abundance since their production is strongly +suppressed above the inflaton mass. +6 +Conclusions +Gravitational particle production is a very interesting process due to its universality. It +only requires the studied field to interact with gravity. Even without a direct coupling to +the inflaton, as it is the case of spectator fields such as the one we have studied, it can +– 21 – + +give rise to a significant abundance for the species considered after the heavy expansion of +spacetime during the early stages of the universe. However, predictions need for a definition +of vacuum after reheating, since the non-static geometry leads to certain ambiguity in the +meaning of particle. +In this manuscript, we studied the production of massive, scalar particles whose +dynamics is described by a non-minimally coupled to gravity action. However, the discussion +on the validity of the definition of vacuum is pertinent when considering any other field +as well. First, we have provided a method for solving in a complete form the background +dynamics, governed by a single scalar inflaton field. For this, we did not have to assume a +de Sitter geometry of spacetime, which would significantly change the amount of particles +produced. Although we make a choice of potential, this procedure can be extended to other +cases as well. We provided an analytic approximation to the solution of the slow-roll mode +equation where the error is well under control in our parameter region of interest. More +importantly, we showed that, for masses smaller than the inflaton mass, the commonly +used adiabatic prescription for the vacuum determines correctly the production of particles +after reheating only when calculated at very late times. Moreover, we define an alternative +vacuum choice that allows one to obtain the right abundance when calculating particle +production at a much earlier time. This allowed us to explore the contribution of the +first oscillations to the total number of produced particles, revealing that the spectra only +stabilizes after hundreds of periods. Lastly, after all these considerations have been taken +into account, we analyzed both the spectra and the total density of particles for different +values of the mass of the field and its coupling to the curvature scalar. When regarded as +dark matter, the production of the spectator field can be directly related to the abundance +that would be observed today if one assumes no couplings to any other fields also after +reheating. In particular, we find agreement with the observed dark matter abundance for a +certain range of masses and couplings of the spectator field. Moreover, this analysis can be +used to constrain the values of the field parameters by demanding that the predicted dark +matter abundance does not exceed observations. +Acknowledgements +This work was partially supported by the MICINN (Ministerio de Ciencia e Innovación, +Spain) projects PID2019-107394GB-I00/AEI/10.13039/501100011033 (AEI/FEDER, UE) +and PID2020-118159GBC44. Additionally, Á.P.-L. is supported by the MIU (Ministerio +de Universidades, Spain) fellowship FPU20/0560. Finally, JARC acknowledges support by +Institut Pascal at Université Paris-Saclay during the Paris-Saclay Astroparticle Symposium +2022, with the support of the P2IO Laboratory of Excellence (program “Investissements +d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I axis of +the Graduate School of Physics of Université Paris-Saclay, as well as IJCLab, CEA, APPEC, +IAS, OSUPS, and the IN2P3 master projet UCMN. +– 22 – + +A +Parameters +In the majority of the analyses, we have left all the quantities expressed in terms of the mass +of the inflaton, mφ, which sets up the scale of the problem. When it has been necessary to +assume a numerical value for such a mass, we have taken mφ = 1.2 × 1013 GeV. Accordingly, +the Planck mass MP has the value MP = 1.02 × 106mφ. +The initial value for the inflaton field, under the slow-roll assumption, is taken to +be φSR(ti) = φi = 3MP . When inflation ends, at t = 0, the field value is φSR(t = 0) = +φ0 = 0.5MP . The slow-roll approximation can then be used to extract ti ≃ −15.35/mφ as +the time when inflation starts. Equation of motion (3.1) can also be solved numerically +taking as initial conditions the same as for slow-roll, φ(ti) = φi, and the derivative of the +approximate solution at this point, φ′(ti) = φ′ +SR(ti). Both solutions will be very close up to +t∗, where the slow-roll approximation starts to break down. Then, φ(t = 0) slightly deviates +from φ0. The scale factor is chosen such that a(t = 0) = a0 = 1. Slow-roll is a assumed to +be a good approximation until η∗ = −500/mφ. +Unless the contrary is expressly stated, particle production is calculated using the +averaged vacuum prescription at ¯η = 16.33/mφ. The range of masses explored is 10−7mφ ≤ +m ≤ 100.5mφ, although for obtaining figure 11 it is assumed that production is the same +for m ≤ 10−7mφ. On the other hand, the coupling ξ is such that 1/6 ≤ ξ ≤ 1. +References +[1] S.W. Hawking, Particle creation by black holes, Comm. 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Hu, Nonequilibrium Quantum Field Theory, Cambridge Monographs +on Mathematical Physics, Cambridge University Press (2008), 10.1017/CBO9780511535123. +[45] L.E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and +Gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press (8, +2009), 10.1017/CBO9780511813924. +[46] V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, Oxford +(2005), 10.1017/CBO9780511790553. +[47] S. Weinberg, Cosmology, Oxford University Press, Oxford (2008). +– 25 – + diff --git a/GdE1T4oBgHgl3EQfrAWz/content/tmp_files/2301.03350v1.pdf.txt b/GdE1T4oBgHgl3EQfrAWz/content/tmp_files/2301.03350v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2371f058f0cdabdc13d970dbb7abd972729416a0 --- /dev/null +++ b/GdE1T4oBgHgl3EQfrAWz/content/tmp_files/2301.03350v1.pdf.txt @@ -0,0 +1,785 @@ +mRpostman: An IMAP Client for R +Allan V. C. Quadros +Department of Statistics +Kansas State University +Manhattan, KS 66506, United States +Abstract +Internet Message Access Protocol (IMAP) clients are a common feature in +several programming languages. Despite having some packages for electronic +messages retrieval, the R language, until recently, lacked a broader solution, +capable of coping with different IMAP servers and providing a wide spec- +trum of features. mRpostman covers most of the IMAP 4rev1 functionalities +by implementing tools for message searching, selective fetching of message +attributes, mailbox management, attachment extraction, and several other +IMAP features that can be executed in virtually any mail provider. By doing +so, it enables users to perform data analysis based on e-mail content. The +goal of this article is to showcase the toolkit provided with the mRpostman +package, to describe its key features and provide some application examples. +Keywords: +IMAP, e-mail, R +1. Motivation and significance +The acknowledgement of the R programming language[1] as having re- +markable statistical capabilities is much due to the excellence brought by +its statistical and data analysis packages. This reputation also stands on +the capabilities of a myriad of utility packages, which extends the use of the +language by facilitating the integration of the steps involved in data collec- +tion, analysis, and communication. With that in mind, and considering the +amount of data transmitted daily through e-mail, mRpostman was conceived +to fill the absence of an Internet Message Access Protocol (IMAP) client in +the R statistical environment; therefore, providing an appropriate toolkit for +electronic messages retrieval, and paving the way for e-mail data analysis in +R. +Email address: quadros@k-state.edu (Allan V. C. Quadros) +Preprint submitted to SoftwareX +January 10, 2023 +arXiv:2301.03350v1 [cs.NI] 11 Dec 2022 + +The Comprehensive R Archive Network (CRAN) has at least seven pack- +ages for sending emails (Table 1). Whereas some of these packages aim to +provide a plain Simple Mail Transport Protocol (SMTP) client for R (e.g. +sendmailR and emayili), others focus on more sophisticated implementations, +using Application Program Interfaces (API), or providing seamless integra- +tion between SMTP and other R features such as rmarkdown[2]. However, +despite the surplus of available clients in R, the SMTP protocol is not suit- +able for receiving e-mails. It only allows clients to communicate with servers +to deliver their messages. +For the purpose of message retrieval, there are the Post Office Protocol 3 +(POP3) and the Internet Message Access Protocol (IMAP). In comparison +with IMAP, POP3 is a very limited protocol, working as a simple interface +for clients to download e-mails from servers. IMAP, on the other hand, is +a much more complex protocol, and can be considered as the evolution of +POP3, with a very different and broader set of functionalities. In contrast to +POP3, all the messages are kept on the IMAP server and not locally. This +means that a user can access the same mail account using parallel connections +from different clients[3]. Besides the mail folders structure and management, +the capacity of issuing sophisticated search queries also contribute to the +level of complexity of the IMAP protocol. +Amid CRAN packages for e-mail communication, only gmailr and edeR +have IMAP capabilities (Table 1). However, those capabilities are restricted +to Gmail accounts and few IMAP functionalities. Although gmailr supports +both protocols, the package is more SMTP-focused, which explains its low +number of IMAP features. Therefore, R was clearly lacking a broader IMAP +client solution. It was in that mainstay that mRpostman was conceived. +2 + +Features +protocol +mail +providers +search +queries +message +fetch +attachment +extrac- +tion +mailbox +manage- +ment +active +develop- +ment +sendmailR[4] +SMTP +- +- +- +- +- +- +mailR[5] +SMTP +- +- +- +- +- +- +mail[6] +SMTP +- +- +- +- +- +- +blatr[7] +SMTP +- +- +- +- +- +- +gmailr[8] +SMTP/IMAP Gmail +no +limited +limited +no +yes +blastula[9] +SMTP +- +- +- +- +- +- +emayili[10] +SMTP +- +- +- +- +- +- +edeR[11] +IMAP +Gmail +no +limited +no +no +no +mRpostman +IMAP +all +yes +yes +yes +yes +yes +Table 1: Comparison of the current available CRAN packages for e-mail communica- +tion. The following attributes are evaluated: protocol - the supported protocol (SMTP +or IMAP); mail providers - if the IMAP protocol is supported, which mail providers are +supported by the package; Features - which type of IMAP features are available in the +package; active development - if the package is currently under active development. If the +package does not provide IMAP support, the remaining fields do not apply. +In this article, we present a brief view of the main functionalities of the +package and its applications. +2. Software description +mRpostman is conceived to be an easy-to-use session-based IMAP client +for R. The package implements intuitive methods for executing the major- +ity of the IMAP commands described in the Request for Comments 35011, +such as mailbox management, and selectively search and fetch of message at- +tributes. The package also implements complementary functions for decoding +quoted-printable and base 64 content, following the MIME specification2. +All these methods and functions play an important role in facilitating e- +mail data analysis. We shall not overlook the amount of data analyses daily +performed on e-mail content. The package has proved to be very useful as an +1The RFC 3501[12] is a formal document from the Internet Engineering Task Force +(IETF) specifying standards for the IMAP, Version 4rev1 (IMAP4rev1). +2The RFC 2047[13] specifies rules for encoding and decoding non-ASCII characters in +electronic messages. +3 + +additional feature in this workflow by, for instance, enabling the possibility +of automating the attachments retrieval step. Also, by fetching other mes- +sage contents, users are able to apply statistical techniques for analysing the +frequency of e-mails with regard to some message aspect, running sentiment +analysis on e-mail content, etc. +Since mRpostman works as a session-based IMAP client, one can think +of the provided methods following a natural order in which the steps shall be +organised in the event of an IMAP session (Fig. 1). For instance, if the goal +is to search messages within a specific period of time and/or containing a +specific word, first we need to configure the connection to the IMAP server; +then, choose a mail folder where the search is to be performed; and execute +the single criteria (left) or the custom multi-criteria search (right). If the +user intends to fetch the matched message(s) or its parts, additional fetch +steps can be chained to the described schema. +con <- configure imap() +con$select folder() +con$fetch *() +con$search *() +con$search() +a connection +object is configured +a mailbox +is +selected +a mailbox +is +selected +return message ids +return message ids +Fig. 1: Basic schema for fetching the full content of a message or its parts after a search +query. +mRpostman is flexible in the sense that the aforementioned steps can be +used either under the tidy framework, with pipes[14], or via the conventional +base R approach. +4 + +3. Software architeture +The software was designed following the object-oriented framework from +the R6 package[15]. A class called ImapCon is implemented to retain and +organize the necessary IMAP connection parameters. All the methods that +derive from this class will serve one of the two following purposes: to issue a +request toward the IMAP server (request methods) or re-configure an existing +IMAP connection (reset methods). +In order to execute IMAP commands, this package makes extensive use +of the curl[16] R package3. All mRpostman’s request methods are built on +top of the so-called curl handles. Under the hood, a curl handle consists +of a C pointer variable that gathers the necessary parameters to execute a +request to the server. As a matter of fact, the handle itself does not issue +any command, but is used as a parameter inside a curl’s fetch function. This +last object is the one that actually triggers the request to the server, ranging +from mail folder selection to search queries, or message fetch requests. +The object-oriented framework combined with the use of one curl handle +per session enables mRpostman to elegantly run as a session based IMAP +client, without demanding a connection reconfiguration between commands. +For example, if a mail folder is selected on the current session, all requests +using the same connection token will be performed on the selected folder, +unless the user re-selects a different one. +3.1. Software functionalities +3.1.1. Configuring an IMAP connection +As we demonstrated in Fig. 1, the first step for using mRpostman is to +configure an IMAP connection. It consists of creating a connection token +object of class ImapCon that will retain all the relevant information to issue +requests toward the server. +configure imap is the function used to configure and create a new IMAP +connection. +The mandatory arguments are three character strings: url, +username, and password for plain authentication; or url, username, and +xoauth2 bearer for OAuth2.0 authentication4. +The following example illustrates how to configure a connection to a Mi- +crosoft Exchange IMAP 4 server; more specifically, to an Office 365 Outlook +account using plain authentication. +library("mRpostman") +3The curl package is a binding for the libcurl[17] C library. +4Please refer to the “IMAP OAuth2.0 authentication in mRpostman” vignette in [18]. +5 + +con <- configure_imap(url = "imaps://outlook.office365.com", +username = "user@agency.gov", +password = rstudioapi::askForPassword()) +We opted for using an Outlook Office 365 account as an example in order +to highlight the difference between mRpostman and the other two CRAN +packages which, although also capable of receiving e-mails, are restricted to +Gmail accounts and fewer IMAP functionalities. Although mRpostman is +able to theoretically connect to any mail provider5, the Outlook Office 365 +service is broadly used by universities and companies. This enriches the range +of data analyses applications of this package, thus justifying our choice. +In a hypothetical situation where the user needs to simultaneously con- +nect to more than one e-mail account (in different providers or not) in the +same R session, it can be easily attained by creating and configuring multiple +connection tokens, such as con1, con2, and so on. +3.1.2. Selecting a mail folder +Mailboxes are structured as folders in the IMAP protocol. This allows us +to replicate many of the operations done in a local folder such as creating, +renaming or deleting folders. As messages are kept inside the mail folders, +users need to select one of them whenever they intend to execute a search, +fetch or other message-related operation, as presented in Fig. 1. +In this sense, the select folder method is one of the key features of +this package. It selects a mail folder for the current IMAP section. The +mandatory argument is a character string containing the name of the folder +to be selected. +Supposing that we want to select the "INBOX" folder and considering that +we are going to use the same connection object (con) that has been previously +created, the command would be: +con$select_folder(name = "INBOX") +Further details on other important mailbox management features are pro- +vided in [18]. +3.1.3. Message search +The IMAP protocol is designed to allow the execution of single or multi- +criteria queries on the mailboxes. This package implements a vast range of +5Besides Outlook Office 365, the package has been already successfully tested with +Gmail, Yahoo, Yandex, AOL, and Hotmail accounts. +6 + +IMAP search commands, which consist of a critical feature for performing +data analysis on email content. +As of its version 1.0.0, mRpostman has five types of single-criterion +search methods implemented: by date; string; flag, size; and span of time +(WITHIN extension)6. +The custom-search, on the other hand, enables the +execution of multi-criteria queries by allowing the combination of two or +more types of search. However, in this article, we will focus on the single- +criterion search-by-string type. +The search string method searches messages that contain a specific +string or expression. One or more specific sections of a message, such as the +TEXT section or the TO header field, for example, must be specified. +In the following code snippet, we search for messages from senders whose +mail domain is "@ksu.edu". +ids <- con$search_string(expr = "@ksu.edu", where = "FROM") +The resulting object is a vector containing the matched unique ids (UID) +or the message sequence numbers7 such as presented below: +[1] +60 145 147 159 332 333 336 338 341 428 +Further details on the other single-search methods and the custom-search +method available in this package are provided in [18]. +3.1.4. Message fetch +After executing a search query, users may be interested in fetching the +full content or some part of the messages indicated in the search results. In +this regard, mRpostman implements six types of fetch features: +fetch body Fetches the message body (message’s full content), or an +specified MIME level, which can refer to the text or the attachments if there +are any. +fetch header Fetches the message header, which comprises all the com- +ponents of the HEADER section of a message. Besides the traditional ones +(from, to, cc, subject), it may include several more fields. +fetch metadata Fetches the message metadata, which consists of some +message’s attributes such as the internal date, and the envelope (from, to, +cc, and subject fields). +6The WITHIN extension is not supported by all IMAP servers. A call to the list - +server capabilities method will present all the IMAP extensions supported by the +mail provider[18]. +7More details on the message identification methodology deployed by the IMAP pro- +tocol are provided in [19, 12, 18]. +7 + +fetch text Fetches the message text section, which can comprise attach- +ment MIME levels if applicable. +Each of these methods can be seamlessly integrated into a previous search +operation so that the returned ids are used as input for the fetch method. +Above all, these methods consist of a powerful source of information for +performing data analysis on e-mail content. Here, we mimic the extraction +of the TEXT portion of a message. Although there is a fetch text method, +the recommended approach is to use fetch body(..., mime level = 1L) +because the former may collect attachment parts along with the message +text. +out <- ids %>% +fetch_body(mime_level = 1L) +Once the messages are fetched, the text can be cleaned and decoded with +the clean msg text helper function. A subsequent call to the writeLines +base R function produces a clean printing of the fetched text: +cleaned_text <- clean_msg_text(msg_list = out) +writeLines(cleaned_text[[1]]) +Receipt Number: XXXXXXX +Customer: Vieira de Castro Quadros, Allan +Kansas State University +Current Date: 04/15/2020 +Description +Amount +-------------------------------------------------------------------------------- +HOUSING & DINING +$30.00 +User Number: XXXXXXXXX +Total +$30.00 +Payments Received +Amount +-------------------------------------------------------------------------------- +07 CREDIT CARD PAYMENTS +$30.00 +Visa XXXXXXXXXXXX8437 +Authorization # XXXXXX +Total +$30.00 +Thank you for the payment. +Besides other applications, the exported function clean msg text can be +used to decode hexadecimal and base 64 characters in the text and other +parts of the message. In some locales such as French, German or Portuguese +speaking countries, message parts may contain non-ASCII characters. SMTP +servers, then, encode it using the RFC 2047 specifications when sending the +e-mail. In these cases, clean msg text is capable of correctly decoding the +non-ASCII characters. +8 + +3.1.5. Attachment extraction +In its pretension to be an IMAP client for R, mRpostman provides meth- +ods that enable users to list and download message payloads. This feature +can be particularly critical for automating the analysis of attachment data +files, for instance. +Attachments can be downloaded using two different approaches in this +package: extending the fetch text/body operation by adding an attach- +ment extraction step at the end of the workflow with get attachments; or +directly fetching attachment parts via the fetch attachments method. In +this article, we focus on the first type of attachment methods, adding a step +to our previous workflow. +The get attachments method extracts attachment files from the fetched +messages and saves these files to the disk. In the following code excerpt, we +extract attachments in a unique pipeline that gathers fetching and search +steps. +con$search_string(expr = "@ksu.edu", where = "FROM") %>% +con$fetch_text() %>% +con$get_attachments() +During the execution, the software locally saves the extracted attach- +ments into sub-folders inside the user’s working directory. These sub-folders +are named following the messages’ ids. +The attachments are placed into +their respective messages’ sub-folders as demonstrated in Fig. 2. Note that +the parent levels are named after the informed username and the selected +mail folder. +For more information on the other attachment-related methods, the reader +should refer to the documentation in [18]. +4. Illustrative Examples +To demonstrate the capabilities of the proposed software, we explore two +use cases of this package in support of data analysis tasks: a simple study +of the frequency of e-mails grouped by senders; and a sentiment analysis +run on a set of e-mails received during a period. The R scripts needed for +reproducing these examples are provided in the appendixes. Although the +results cannot be exactly reproduced once it reflects the author’s mailbox +contents, they can be easily adapted to the reader’s context. +9 + +. (working directory) +user@company.com +INBOX +141 +final.zip +prob plot.svg +staa2072.pdf +144 +app.R +image001.png +recording.mp4 +Fig. 2: Local directory tree for the extracted attachment files +4.1. Frequency analysis of e-mail data +In the first example, we run a simple analysis of the e-mail frequency with +regard to senders. This can be especially useful in professional fields, such +as marketing and customer service offices. A period of analysis was defined, +and a search-by-date is performed using the search period method. Then, +senders’ information for the returned ids are fetched via fetch metadata, +using the ENVELOPE attribute. After some basic manipulation with regular +expressions, the data is ready to be plotted as shown in Fig. 3. +10 + +omitted@tbs−education.fr +omitted@lsbu.ac.uk +omitted@gmail.com +cortana@microsoft.com +no−reply@researchgatemail.net +E−mail Frequency (by sender) +count +0 +2 +4 +6 +8 +10 +12 +14 +ResearchGate +Cortana +Claudio Piga +Chen, Daqing +MANTOVANI Andrea +Period: 01−Nov to 01−Dec−2020 +Fig. 3: An example of e-mail frequency analysis grouped by sender +The same kind of analysis can be replicated for the messages’ subjects +with only a few modifications in the regular expressions code chunks. Con- +sidering that some companies/users deal with subject-standardized e-mails, +this approach can be useful to analyze the frequency of e-mails with regard +to different categories of subjects. +4.2. Sentiment analysis on e-mail data +For the sentiment analysis example, we also define a period of analysis and +run a search period query. Then, we retrieve the text part of the messages +by fetching the first MIME level with fetch body(..., mime level = 1L). +The texts go trough a first cleaning step with a call to the clean msg text +function. +After further cleaning procedures, we use a lexicon[20] via the +syuzhet package[21] to evaluate the sentiment of each e-mail. The output +below is a subset of the resulting data frame. The last two columns indicate, +respectively, the counts of negative and positive words for each message. +The other columns provide counts related to detailed emotions, which are +not necessarily positive nor negative. +anger anticipation disgust fear joy sadness surprise trust negative positive +body91 +1 +5 +1 +1 +2 +2 +0 +9 +1 +13 +body92 +0 +1 +0 +0 +1 +0 +0 +3 +0 +1 +body93 +0 +3 +0 +2 +0 +1 +2 +2 +1 +3 +body94 +0 +1 +0 +1 +0 +0 +1 +4 +1 +4 +body95 +0 +5 +0 +0 +3 +0 +2 +8 +0 +13 +body96 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +body97 +4 +20 +4 +11 +13 +11 +4 +25 +16 +51 +11 + +body98 +0 +3 +0 +0 +2 +0 +1 +4 +0 +6 +body99 +3 +9 +1 +6 +1 +5 +2 +16 +14 +24 +body100 +4 +15 +1 +13 +6 +7 +6 +15 +16 +31 +5. Impact +As we have demonstrated, mRpostman clearly fills an existent gap of a +broad, complete, and, at the same time, easy-to-use IMAP client for the +R language. The package has consolidated itself as an important tool for +collecting massive e-mail content, thus contributing to data analysis tasks in +R. +Although all sort of users have been taking advantage of this package, +we are inclined to think that its use has been prevailing amid companies. +We have received a considerable number of feedback from enterprise users +who deploy mRpostman as an additional feature for automatically produc- +ing daily reports based on attachment data files. +Besides this, there are +important applications for marketing and post-sales departments, for exam- +ple. They can also deploy this package to collect e-mail data for analyzing +e-mail frequency, or performing sentiment analysis, as we have demonstrated +in Section 4. +6. Conclusions +mRpostman aims to provide an easy-to-use IMAP client for R. Its design +allows the efficient, elegant, and intuitive execution of several IMAP com- +mands on a wide range of mail providers. Consequently, users cannot only +manage their mailboxes but also conduct e-mail data analysis from inside R. +Finally, because IMAP is such a complex protocol, this package is in con- +stant development, which means that new features are to be implemented in +future versions. +7. Conflict of Interest +No conflict of interest exists: We wish to confirm that there are no known +conflicts of interest associated with this publication and there has been no +significant financial support for this work that could have influenced its out- +come. +Acknowledgements +The author would like to acknowledge the Department of Statistics at +Kansas State University (K-State) for the assistantship provided for his doc- +torate studies. He wants to especially thank Dr. Christopher Vahl and Dr. +12 + +Michael Higgins for the academic support. The author also acknowledges the +academic guidance of Dr. George von Borries at the University of Brasilia +(UnB). The contents of this article are the responsibility of the author and +do not reflect the views of K-State or UnB. +Appendix A. Code for example 1 +library(mRpostman) +con <- configure_imap( +url="imaps://outlook.office365.com", +username="user@company.com", +password=rstudioapi::askForPassword() +) +con$select_folder(name = "INBOX") +meta_res <- con$search_period(since_date_char = "01-Nov-2020", +before_date_char = "01-Dec-2020") %>% +con$fetch_metadata(attribute = "ENVELOPE") +# cleaning +# step 1 +clean_meta <- lapply(meta_res, function(x){ +regmatches(x, regexpr(pattern = "\\(\\(.*\"(.*?)\"\\)\\)", x, perl = TRUE)) +}) +# step 2 +# cleaning Ccs +senders1 <- lapply(clean_meta, function(x){ +gsub(")) NIL .*$|)) .*$|))$", "", x) +}) +# step 3 +senders1 <- lapply(senders1, function(x){ +gsub(’^\\(\\(|\"+’, "", x) +}) +# splitting +name <- c() +email <- c() +for (i in seq_along(senders1)){ +# i = 1 +out <- unlist(strsplit(senders1[[i]], " NIL ")) +name <- c(name, out[1]) +email <- c(email, gsub(" ", "@", out[2])) +} +df <- data.frame(name, email) +df$name <- decode_mime_header(string = as.character(df$name)) +df2 <- as.data.frame(table(df$email)) +colnames(df2) <- c("email", "count") +df2 <- df2[order(-df2[,2]), ][1:5,] +df2$name <- unique(df$name[df$email %in% df2$email]) +par(mar=c(5,13,4,1)+.1) +pal_cols <- c(’#3B4992FF’, ’#EE0000FF’, ’#008B45FF’, ’#631879FF’, ’#008280FF’) +barplot(rev(df2$count), +main = "E-mail Frequency (by sender)", +xlab = "count", +names.arg = rev(df2$email), +las = 1, +col = pal_cols, +horiz = TRUE) +mysubtitle <- "Period: 01-Nov to 01-Dec-2020" +legend(x = "bottomright", legend = df2$name, fill = rev(pal_cols), bty = "n", y.intersp = 1) +mtext(side=3, line=0.3, at=-0.07, adj=0, cex=0.9, mysubtitle) +13 + +Appendix B. Code for example2 +library(mRpostman) +con <- configure_imap(url="imaps://outlook.office365.com", +username="user@company.com", +password=rstudioapi::askForPassword(), +timeout_ms = 20000 +) +con$select_folder("INBOX") +ids <- con$search_period(since_date_char = "10-Oct-2020", before_date_char = "20-Dec-2020") +fetch_res2 <- ids %>% +con$fetch_body(mime_level = 1L) +cleaned_text_list <- clean_msg_text(msg_list = fetch_res2) +cleaned_text_list[[4]] +for (i in seq_along(cleaned_text_list)) { +clean_text <- gsub("\r\n", " ", cleaned_text_list[[i]]) +clean_text <- unlist(strsplit(clean_text, " ")) +words <- clean_text[!grepl("\\d|_|http|www|nbsp|@|(?<=[[:lower:]])(?=[[:upper:]])", +clean_text, perl = TRUE)] +words <- tolower(gsub("\\W+", "", words)) +words <- gsub(’[^a-zA-Z|[:blank:]]’, "", words) +cleaned_text_list[[i]] <- paste(words, collapse = " ") +} +cleaned_text_df <- do.call(’rbind’, cleaned_text_list) +library(syuzhet) +email_sentiment_df <-get_nrc_sentiment(cleaned_text_df) +rownames(email_sentiment_df) <- rownames(cleaned_text_df) +head(email_sentiment_df,10) +References +[1] R Core Team, R: A Language and Environment for Statistical Comput- +ing, R Foundation for Statistical Computing, Vienna, Austria (2020). +URL https://www.R-project.org/ +[2] J. Allaire, Y. Xie, J. McPherson, J. Luraschi, K. Ushey, A. Atkins, +H. Wickham, J. Cheng, W. Chang, R. Iannone, rmarkdown: Dynamic +Documents for R, r package version 2.5 (2020). +URL https://github.com/rstudio/rmarkdown +[3] P. Heinlein, P. Hartleben, The Book of IMAP: Building a Mail Server +with Courier and Cyrus, No Starch Press, 2008. +[4] O. Mersmann, sendmailR: send email using R, r package version 1.2-1 +(2014). +URL https://CRAN.R-project.org/package=sendmailR +[5] R. Premraj, mailR: A Utility to Send Emails from R, r package version +0.4.1 (2015). +URL https://CRAN.R-project.org/package=mailR +14 + +[6] L. Himmelmann, mail: Sending Email Notifications from R, r package +version 1.0 (2011). +URL https://CRAN.R-project.org/package=mail +[7] S. M. Bache, blatr: Send Emails Using ’Blat’ for Windows, r package +version 1.0.1 (2015). +URL https://CRAN.R-project.org/package=blatr +[8] J. Hester, gmailr: Access the ’Gmail’ ’RESTful’ API, r package version +1.0.0 (2019). +URL https://CRAN.R-project.org/package=gmailr +[9] R. Iannone, J. Cheng, blastula: Easily Send HTML Email Messages, r +package version 0.3.2 (2020). +URL https://CRAN.R-project.org/package=blastula +[10] A. B. Collier, emayili: Send Email Messages, r package version 0.4.4 +(2020). +URL https://CRAN.R-project.org/package=emayili +[11] A. B. Collier, edeR: Email Data Extraction Using R, r package version +1.0.0 (2014). +URL https://CRAN.R-project.org/package=edeR +[12] M. Crispin, Internet message access protocol - version 4rev1, request for +Comments 3501 (RFC 3501), Internet Engineering Task Force (IETF) +(2003). +URL https://tools.ietf.org/html/rfc3501 +[13] K. Moore, Multipurpose Internet Mail Extensions (MIME), part three: +Message header extensions for non-ascii text, request for Comments 2047 +(RFC 2047), Internet Engineering Task Force (IETF) (1996). +URL https://tools.ietf.org/html/rfc2047 +[14] S. M. Bache, H. Wickham, magrittr: A Forward-Pipe Operator for R, r +package version 1.5 (2014). +URL https://CRAN.R-project.org/package=magrittr +[15] W. Chang, R6: Encapsulated Classes with Reference Semantics, r pack- +age version 2.5.0 (2020). +URL https://CRAN.R-project.org/package=R6 +[16] J. Ooms, curl: A Modern and Flexible Web Client for R, r package +version 4.3 (2020). +URL https://CRAN.R-project.org/package=curl +15 + +[17] D. Stenberg, libcurl - the multiprotocol file transfer library, version +7.69.1 (2020). +URL https://curl.haxx.se/ +[18] A. Quadros, mRpostman: An IMAP Client for R, r package version +1.0.0 (2020). +URL https://allanvc.github.io/ +[19] P. Resnick, Internet message format, request for Comments 5322 (RFC +5322), Internet Engineering Task Force (IETF) (2008). +URL https://tools.ietf.org/html/rfc5322 +[20] S. Mohammad, P. Turney, Emotions evoked by common words and +phrases: +Using mechanical turk to create an emotion lexicon, in: +CAAGET ’10: Proceedings of the NAACL HLT 2010 Workshop on +Computational Approaches to Analysis and Generation of Emotion in +Text, Los Angeles, California, 2010, p. 26–34, june, 2010. +URL http://saifmohammad.com/WebPages/lexicons.html +[21] M. L. Jockers, Syuzhet: Extract Sentiment and Plot Arcs from Text, r +package version 1.0.4 (2015). +URL https://github.com/mjockers/syuzhet +16 + diff --git a/GdE1T4oBgHgl3EQfrAWz/content/tmp_files/load_file.txt b/GdE1T4oBgHgl3EQfrAWz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a08f1360f7de04a41597a1c5d835f3ee9fdb5824 --- /dev/null +++ b/GdE1T4oBgHgl3EQfrAWz/content/tmp_files/load_file.txt @@ -0,0 +1,392 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf,len=391 +page_content='mRpostman: An IMAP Client for R Allan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Quadros Department of Statistics Kansas State University Manhattan, KS 66506, United States Abstract Internet Message Access Protocol (IMAP) clients are a common feature in several programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Despite having some packages for electronic messages retrieval, the R language, until recently, lacked a broader solution, capable of coping with different IMAP servers and providing a wide spec- trum of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' mRpostman covers most of the IMAP 4rev1 functionalities by implementing tools for message searching, selective fetching of message attributes, mailbox management, attachment extraction, and several other IMAP features that can be executed in virtually any mail provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' By doing so, it enables users to perform data analysis based on e-mail content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The goal of this article is to showcase the toolkit provided with the mRpostman package, to describe its key features and provide some application examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Keywords: IMAP, e-mail, R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Motivation and significance The acknowledgement of the R programming language[1] as having re- markable statistical capabilities is much due to the excellence brought by its statistical and data analysis packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' This reputation also stands on the capabilities of a myriad of utility packages, which extends the use of the language by facilitating the integration of the steps involved in data collec- tion, analysis, and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' With that in mind, and considering the amount of data transmitted daily through e-mail, mRpostman was conceived to fill the absence of an Internet Message Access Protocol (IMAP) client in the R statistical environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' therefore, providing an appropriate toolkit for electronic messages retrieval, and paving the way for e-mail data analysis in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Email address: quadros@k-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='edu (Allan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Quadros) Preprint submitted to SoftwareX January 10, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='03350v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='NI] 11 Dec 2022 The Comprehensive R Archive Network (CRAN) has at least seven pack- ages for sending emails (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Whereas some of these packages aim to provide a plain Simple Mail Transport Protocol (SMTP) client for R (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' sendmailR and emayili), others focus on more sophisticated implementations, using Application Program Interfaces (API), or providing seamless integra- tion between SMTP and other R features such as rmarkdown[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' However, despite the surplus of available clients in R, the SMTP protocol is not suit- able for receiving e-mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' It only allows clients to communicate with servers to deliver their messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' For the purpose of message retrieval, there are the Post Office Protocol 3 (POP3) and the Internet Message Access Protocol (IMAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In comparison with IMAP, POP3 is a very limited protocol, working as a simple interface for clients to download e-mails from servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' IMAP, on the other hand, is a much more complex protocol, and can be considered as the evolution of POP3, with a very different and broader set of functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In contrast to POP3, all the messages are kept on the IMAP server and not locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' This means that a user can access the same mail account using parallel connections from different clients[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Besides the mail folders structure and management, the capacity of issuing sophisticated search queries also contribute to the level of complexity of the IMAP protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Amid CRAN packages for e-mail communication, only gmailr and edeR have IMAP capabilities (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' However, those capabilities are restricted to Gmail accounts and few IMAP functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Although gmailr supports both protocols, the package is more SMTP-focused, which explains its low number of IMAP features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Therefore, R was clearly lacking a broader IMAP client solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' It was in that mainstay that mRpostman was conceived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 2 Features protocol mail providers search queries message fetch attachment extrac- tion mailbox manage- ment active develop- ment sendmailR[4] SMTP mailR[5] SMTP mail[6] SMTP blatr[7] SMTP gmailr[8] SMTP/IMAP Gmail no limited limited no yes blastula[9] SMTP emayili[10] SMTP edeR[11] IMAP Gmail no limited no no no mRpostman IMAP all yes yes yes yes yes Table 1: Comparison of the current available CRAN packages for e-mail communica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The following attributes are evaluated: protocol - the supported protocol (SMTP or IMAP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' mail providers - if the IMAP protocol is supported, which mail providers are supported by the package;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Features - which type of IMAP features are available in the package;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' active development - if the package is currently under active development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' If the package does not provide IMAP support, the remaining fields do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In this article, we present a brief view of the main functionalities of the package and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Software description mRpostman is conceived to be an easy-to-use session-based IMAP client for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The package implements intuitive methods for executing the major- ity of the IMAP commands described in the Request for Comments 35011, such as mailbox management, and selectively search and fetch of message at- tributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The package also implements complementary functions for decoding quoted-printable and base 64 content, following the MIME specification2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' All these methods and functions play an important role in facilitating e- mail data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' We shall not overlook the amount of data analyses daily performed on e-mail content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The package has proved to be very useful as an 1The RFC 3501[12] is a formal document from the Internet Engineering Task Force (IETF) specifying standards for the IMAP, Version 4rev1 (IMAP4rev1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 2The RFC 2047[13] specifies rules for encoding and decoding non-ASCII characters in electronic messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 3 additional feature in this workflow by, for instance, enabling the possibility of automating the attachments retrieval step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Also, by fetching other mes- sage contents, users are able to apply statistical techniques for analysing the frequency of e-mails with regard to some message aspect, running sentiment analysis on e-mail content, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Since mRpostman works as a session-based IMAP client, one can think of the provided methods following a natural order in which the steps shall be organised in the event of an IMAP session (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' For instance, if the goal is to search messages within a specific period of time and/or containing a specific word, first we need to configure the connection to the IMAP server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' then, choose a mail folder where the search is to be performed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' and execute the single criteria (left) or the custom multi-criteria search (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' If the user intends to fetch the matched message(s) or its parts, additional fetch steps can be chained to the described schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' con <- configure imap() con$select folder() con$fetch *() con$search *() con$search() a connection object is configured a mailbox is selected a mailbox is selected return message ids return message ids Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 1: Basic schema for fetching the full content of a message or its parts after a search query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' mRpostman is flexible in the sense that the aforementioned steps can be used either under the tidy framework, with pipes[14], or via the conventional base R approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Software architeture The software was designed following the object-oriented framework from the R6 package[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' A class called ImapCon is implemented to retain and organize the necessary IMAP connection parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' All the methods that derive from this class will serve one of the two following purposes: to issue a request toward the IMAP server (request methods) or re-configure an existing IMAP connection (reset methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In order to execute IMAP commands, this package makes extensive use of the curl[16] R package3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' All mRpostman’s request methods are built on top of the so-called curl handles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Under the hood, a curl handle consists of a C pointer variable that gathers the necessary parameters to execute a request to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' As a matter of fact, the handle itself does not issue any command, but is used as a parameter inside a curl’s fetch function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' This last object is the one that actually triggers the request to the server, ranging from mail folder selection to search queries, or message fetch requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The object-oriented framework combined with the use of one curl handle per session enables mRpostman to elegantly run as a session based IMAP client, without demanding a connection reconfiguration between commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' For example, if a mail folder is selected on the current session, all requests using the same connection token will be performed on the selected folder, unless the user re-selects a different one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Software functionalities 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Configuring an IMAP connection As we demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 1, the first step for using mRpostman is to configure an IMAP connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' It consists of creating a connection token object of class ImapCon that will retain all the relevant information to issue requests toward the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' configure imap is the function used to configure and create a new IMAP connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The mandatory arguments are three character strings: url, username, and password for plain authentication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' or url, username, and xoauth2 bearer for OAuth2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='0 authentication4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The following example illustrates how to configure a connection to a Mi- crosoft Exchange IMAP 4 server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' more specifically, to an Office 365 Outlook account using plain authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' library("mRpostman") 3The curl package is a binding for the libcurl[17] C library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 4Please refer to the “IMAP OAuth2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='0 authentication in mRpostman” vignette in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 5 con <- configure_imap(url = "imaps://outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='office365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com", username = "user@agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='gov", password = rstudioapi::askForPassword()) We opted for using an Outlook Office 365 account as an example in order to highlight the difference between mRpostman and the other two CRAN packages which, although also capable of receiving e-mails, are restricted to Gmail accounts and fewer IMAP functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Although mRpostman is able to theoretically connect to any mail provider5, the Outlook Office 365 service is broadly used by universities and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' This enriches the range of data analyses applications of this package, thus justifying our choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In a hypothetical situation where the user needs to simultaneously con- nect to more than one e-mail account (in different providers or not) in the same R session, it can be easily attained by creating and configuring multiple connection tokens, such as con1, con2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Selecting a mail folder Mailboxes are structured as folders in the IMAP protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' This allows us to replicate many of the operations done in a local folder such as creating, renaming or deleting folders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' As messages are kept inside the mail folders, users need to select one of them whenever they intend to execute a search, fetch or other message-related operation, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In this sense, the select folder method is one of the key features of this package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' It selects a mail folder for the current IMAP section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The mandatory argument is a character string containing the name of the folder to be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Supposing that we want to select the "INBOX" folder and considering that we are going to use the same connection object (con) that has been previously created, the command would be: con$select_folder(name = "INBOX") Further details on other important mailbox management features are pro- vided in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Message search The IMAP protocol is designed to allow the execution of single or multi- criteria queries on the mailboxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' This package implements a vast range of 5Besides Outlook Office 365, the package has been already successfully tested with Gmail, Yahoo, Yandex, AOL, and Hotmail accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 6 IMAP search commands, which consist of a critical feature for performing data analysis on email content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' As of its version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='0, mRpostman has five types of single-criterion search methods implemented: by date;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' string;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' flag, size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' and span of time (WITHIN extension)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The custom-search, on the other hand, enables the execution of multi-criteria queries by allowing the combination of two or more types of search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' However, in this article, we will focus on the single- criterion search-by-string type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The search string method searches messages that contain a specific string or expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' One or more specific sections of a message, such as the TEXT section or the TO header field, for example, must be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In the following code snippet, we search for messages from senders whose mail domain is "@ksu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='edu".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' ids <- con$search_string(expr = "@ksu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='edu", where = "FROM") The resulting object is a vector containing the matched unique ids (UID) or the message sequence numbers7 such as presented below: [1] 60 145 147 159 332 333 336 338 341 428 Further details on the other single-search methods and the custom-search method available in this package are provided in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Message fetch After executing a search query, users may be interested in fetching the full content or some part of the messages indicated in the search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In this regard, mRpostman implements six types of fetch features: fetch body Fetches the message body (message’s full content), or an specified MIME level, which can refer to the text or the attachments if there are any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' fetch header Fetches the message header, which comprises all the com- ponents of the HEADER section of a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Besides the traditional ones (from, to, cc, subject), it may include several more fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' fetch metadata Fetches the message metadata, which consists of some message’s attributes such as the internal date, and the envelope (from, to, cc, and subject fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 6The WITHIN extension is not supported by all IMAP servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' A call to the list - server capabilities method will present all the IMAP extensions supported by the mail provider[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 7More details on the message identification methodology deployed by the IMAP pro- tocol are provided in [19, 12, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 7 fetch text Fetches the message text section, which can comprise attach- ment MIME levels if applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Each of these methods can be seamlessly integrated into a previous search operation so that the returned ids are used as input for the fetch method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Above all, these methods consist of a powerful source of information for performing data analysis on e-mail content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Here, we mimic the extraction of the TEXT portion of a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Although there is a fetch text method, the recommended approach is to use fetch body(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=', mime level = 1L) because the former may collect attachment parts along with the message text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' out <- ids %>% fetch_body(mime_level = 1L) Once the messages are fetched, the text can be cleaned and decoded with the clean msg text helper function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' A subsequent call to the writeLines base R function produces a clean printing of the fetched text: cleaned_text <- clean_msg_text(msg_list = out) writeLines(cleaned_text[[1]]) Receipt Number: XXXXXXX Customer: Vieira de Castro Quadros, Allan Kansas State University Current Date: 04/15/2020 Description Amount -------------------------------------------------------------------------------- HOUSING & DINING $30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='00 User Number: XXXXXXXXX Total $30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='00 Payments Received Amount -------------------------------------------------------------------------------- 07 CREDIT CARD PAYMENTS $30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='00 Visa XXXXXXXXXXXX8437 Authorization # XXXXXX Total $30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='00 Thank you for the payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Besides other applications, the exported function clean msg text can be used to decode hexadecimal and base 64 characters in the text and other parts of the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In some locales such as French, German or Portuguese speaking countries, message parts may contain non-ASCII characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' SMTP servers, then, encode it using the RFC 2047 specifications when sending the e-mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In these cases, clean msg text is capable of correctly decoding the non-ASCII characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Attachment extraction In its pretension to be an IMAP client for R, mRpostman provides meth- ods that enable users to list and download message payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' This feature can be particularly critical for automating the analysis of attachment data files, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Attachments can be downloaded using two different approaches in this package: extending the fetch text/body operation by adding an attach- ment extraction step at the end of the workflow with get attachments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' or directly fetching attachment parts via the fetch attachments method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In this article, we focus on the first type of attachment methods, adding a step to our previous workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The get attachments method extracts attachment files from the fetched messages and saves these files to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' In the following code excerpt, we extract attachments in a unique pipeline that gathers fetching and search steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' con$search_string(expr = "@ksu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='edu", where = "FROM") %>% con$fetch_text() %>% con$get_attachments() During the execution, the software locally saves the extracted attach- ments into sub-folders inside the user’s working directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' These sub-folders are named following the messages’ ids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The attachments are placed into their respective messages’ sub-folders as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Note that the parent levels are named after the informed username and the selected mail folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' For more information on the other attachment-related methods, the reader should refer to the documentation in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Illustrative Examples To demonstrate the capabilities of the proposed software, we explore two use cases of this package in support of data analysis tasks: a simple study of the frequency of e-mails grouped by senders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' and a sentiment analysis run on a set of e-mails received during a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The R scripts needed for reproducing these examples are provided in the appendixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Although the results cannot be exactly reproduced once it reflects the author’s mailbox contents, they can be easily adapted to the reader’s context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' (working directory) user@company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com INBOX 141 final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='zip prob plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='svg staa2072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='pdf 144 app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='R image001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='png recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='mp4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 2: Local directory tree for the extracted attachment files 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Frequency analysis of e-mail data In the first example, we run a simple analysis of the e-mail frequency with regard to senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' This can be especially useful in professional fields, such as marketing and customer service offices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' A period of analysis was defined, and a search-by-date is performed using the search period method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Then, senders’ information for the returned ids are fetched via fetch metadata, using the ENVELOPE attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' After some basic manipulation with regular expressions, the data is ready to be plotted as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 10 omitted@tbs−education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='fr omitted@lsbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='uk omitted@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com cortana@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com no−reply@researchgatemail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='net E−mail Frequency (by sender) count 0 2 4 6 8 10 12 14 ResearchGate Cortana Claudio Piga Chen, Daqing MANTOVANI Andrea Period: 01−Nov to 01−Dec−2020 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 3: An example of e-mail frequency analysis grouped by sender The same kind of analysis can be replicated for the messages’ subjects with only a few modifications in the regular expressions code chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Con- sidering that some companies/users deal with subject-standardized e-mails, this approach can be useful to analyze the frequency of e-mails with regard to different categories of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Sentiment analysis on e-mail data For the sentiment analysis example, we also define a period of analysis and run a search period query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Then, we retrieve the text part of the messages by fetching the first MIME level with fetch body(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=', mime level = 1L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The texts go trough a first cleaning step with a call to the clean msg text function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' After further cleaning procedures, we use a lexicon[20] via the syuzhet package[21] to evaluate the sentiment of each e-mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The output below is a subset of the resulting data frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The last two columns indicate, respectively, the counts of negative and positive words for each message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The other columns provide counts related to detailed emotions, which are not necessarily positive nor negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' anger anticipation disgust fear joy sadness surprise trust negative positive body91 1 5 1 1 2 2 0 9 1 13 body92 0 1 0 0 1 0 0 3 0 1 body93 0 3 0 2 0 1 2 2 1 3 body94 0 1 0 1 0 0 1 4 1 4 body95 0 5 0 0 3 0 2 8 0 13 body96 0 0 0 0 0 0 0 0 0 0 body97 4 20 4 11 13 11 4 25 16 51 11 body98 0 3 0 0 2 0 1 4 0 6 body99 3 9 1 6 1 5 2 16 14 24 body100 4 15 1 13 6 7 6 15 16 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Impact As we have demonstrated, mRpostman clearly fills an existent gap of a broad, complete, and, at the same time, easy-to-use IMAP client for the R language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The package has consolidated itself as an important tool for collecting massive e-mail content, thus contributing to data analysis tasks in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Although all sort of users have been taking advantage of this package, we are inclined to think that its use has been prevailing amid companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' We have received a considerable number of feedback from enterprise users who deploy mRpostman as an additional feature for automatically produc- ing daily reports based on attachment data files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Besides this, there are important applications for marketing and post-sales departments, for exam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' They can also deploy this package to collect e-mail data for analyzing e-mail frequency, or performing sentiment analysis, as we have demonstrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Conclusions mRpostman aims to provide an easy-to-use IMAP client for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Its design allows the efficient, elegant, and intuitive execution of several IMAP com- mands on a wide range of mail providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Consequently, users cannot only manage their mailboxes but also conduct e-mail data analysis from inside R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Finally, because IMAP is such a complex protocol, this package is in con- stant development, which means that new features are to be implemented in future versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Conflict of Interest No conflict of interest exists: We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its out- come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Acknowledgements The author would like to acknowledge the Department of Statistics at Kansas State University (K-State) for the assistantship provided for his doc- torate studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' He wants to especially thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Christopher Vahl and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 12 Michael Higgins for the academic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The author also acknowledges the academic guidance of Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' George von Borries at the University of Brasilia (UnB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' The contents of this article are the responsibility of the author and do not reflect the views of K-State or UnB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Code for example 1 library(mRpostman) con <- configure_imap( url="imaps://outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='office365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com", username="user@company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com", password=rstudioapi::askForPassword() ) con$select_folder(name = "INBOX") meta_res <- con$search_period(since_date_char = "01-Nov-2020", before_date_char = "01-Dec-2020") %>% con$fetch_metadata(attribute = "ENVELOPE") # cleaning # step 1 clean_meta <- lapply(meta_res, function(x){ regmatches(x, regexpr(pattern = "\\\\(\\\\(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='*\\"(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' )\\"\\\\)\\\\)", x, perl = TRUE)) }) # step 2 # cleaning Ccs senders1 <- lapply(clean_meta, function(x){ gsub(")) NIL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' *$|)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' *$|))$", "", x) }) # step 3 senders1 <- lapply(senders1, function(x){ gsub(’^\\\\(\\\\(|\\"+’, "", x) }) # splitting name <- c() email <- c() for (i in seq_along(senders1)){ # i = 1 out <- unlist(strsplit(senders1[[i]], " NIL ")) name <- c(name, out[1]) email <- c(email, gsub(" ", "@", out[2])) } df <- data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='frame(name, email) df$name <- decode_mime_header(string = as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='character(df$name)) df2 <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='frame(table(df$email)) colnames(df2) <- c("email", "count") df2 <- df2[order(-df2[,2]), ][1:5,] df2$name <- unique(df$name[df$email %in% df2$email]) par(mar=c(5,13,4,1)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='1) pal_cols <- c(’#3B4992FF’, ’#EE0000FF’, ’#008B45FF’, ’#631879FF’, ’#008280FF’) barplot(rev(df2$count), main = "E-mail Frequency (by sender)", xlab = "count", names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='arg = rev(df2$email), las = 1, col = pal_cols, horiz = TRUE) mysubtitle <- "Period: 01-Nov to 01-Dec-2020" legend(x = "bottomright", legend = df2$name, fill = rev(pal_cols), bty = "n", y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='intersp = 1) mtext(side=3, line=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='3, at=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='07, adj=0, cex=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='9, mysubtitle) 13 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Code for example2 library(mRpostman) con <- configure_imap(url="imaps://outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='office365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com", username="user@company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' password=rstudioapi::askForPassword(),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' timeout_ms = 20000 ) con$select_folder("INBOX") ids <- con$search_period(since_date_char = "10-Oct-2020",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' before_date_char = "20-Dec-2020") fetch_res2 <- ids %>% con$fetch_body(mime_level = 1L) cleaned_text_list <- clean_msg_text(msg_list = fetch_res2) cleaned_text_list[[4]] for (i in seq_along(cleaned_text_list)) { clean_text <- gsub("\\r\\n",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' " ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' cleaned_text_list[[i]]) clean_text <- unlist(strsplit(clean_text,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' " ")) words <- clean_text[!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='grepl("\\\\d|_|http|www|nbsp|@|(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='<=[[:lower:]])(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='=[[:upper:]])", clean_text, perl = TRUE)] words <- tolower(gsub("\\\\W+", "", words)) words <- gsub(’[^a-zA-Z|[:blank:]]’, "", words) cleaned_text_list[[i]] <- paste(words, collapse = " ") } cleaned_text_df <- do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='call(’rbind’, cleaned_text_list) library(syuzhet) email_sentiment_df <-get_nrc_sentiment(cleaned_text_df) rownames(email_sentiment_df) <- rownames(cleaned_text_df) head(email_sentiment_df,10) References 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words and phrases: Using mechanical turk to create an emotion lexicon, in: CAAGET ’10: Proceedings of the NAACL HLT 2010 Workshop on Computational Approaches to Analysis and Generation of Emotion in Text, Los Angeles, California, 2010, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' 26–34, june, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' URL http://saifmohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com/WebPages/lexicons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='html [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' Jockers, Syuzhet: Extract Sentiment and Plot Arcs from Text, r package version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='4 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content=' URL https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} +page_content='com/mjockers/syuzhet 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfrAWz/content/2301.03350v1.pdf'} diff --git a/I9FAT4oBgHgl3EQfux7Z/content/tmp_files/load_file.txt b/I9FAT4oBgHgl3EQfux7Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c37c7ce3c73025b1f81f13201e37daf618d13337 --- /dev/null +++ b/I9FAT4oBgHgl3EQfux7Z/content/tmp_files/load_file.txt @@ -0,0 +1,802 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf,len=801 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='08672v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='CT] 20 Jan 2023 ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES OLIVIA MONJON, J´ERˆOME SCHERER, AND FLORENCE STERCK Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The correspondence between the concept of conditional flatness and admissibil- ity in the sense of Galois appears in the context of localization functors in any semi-abelian category admitting a fiberwise localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' It is then natural to wonder what happens in the category of crossed modules where fiberwise localization is not always available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In this article, we establish an equivalence between conditional flatness and admissibility in the sense of Galois (for the class of regular epimorphisms) for regular-epi localization functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We use this equivalence to prove that nullification functors are admissible for the class of regular epimorphisms, even if the kernels of their localization morphisms are not acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Introduction It is a natural question to ask whether the pullback of a nice extension inherits these nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' When working with localization functors or reflections one particularly nice feature for an extension is flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We say that an extension is L-flat, for a localization functor L, if applying L to the extension yields another extension, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The question is thus to understand when the pullback of an L-flat extension is again L-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Such questions have been studied first in a homotopical context by Berrick and Farjoun, [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' For homotopical localization functors in the category of topological spaces (in the sense of Bousfield, [5], see also Farjoun’s book [13]), preservation of L-flatness (for fiber sequences) under pullbacks was shown to be equivalent for L to be a so-called nullification functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The situation is surprisingly more delicate in the category of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Farjoun and the second author proved for example that all nilpotent quotient functors have this nice property, which they called conditional flatness, see [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The standard strategy to establish conditional flatness for a localization functor consists in a few reduction steps culminating in a simpler form, which Gran identified as admissibility in the sense of Galois for the class of regular epimorphisms [17, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This shifted the study of conditional flatness in homotopy theory to that of admissibility in semi-abelian categories, see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Admissibility had been introduced by Janelidze and Kelly in [17] and has since then played a central role in the categorical study of extensions, let us mention for example Everaert, Gran, and Van der Linden’s work in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In this article we study admissibility for localization functors in the category of crossed modules (of groups), a category of interest to both topologists due to Whitehead’s work on connected 2-types, [25], and algebraists since Brown and Spencer [7] proved the equivalence between crossed modules and internal groupoids in the category of groups (a result that they credit to Verdier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This equivalence relates two interesting notions and allows one to deal with the concept of internal groupoid in an alternative way, that is useful for compu- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Moreover, crossed modules form a semi-abelian category in the sense of Janelidze, 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' 18G45, 55P60, 18E50, 55R70, 18E13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Crossed modules, Localization functors, Admissibility, Regular epimorphisms, Conditional flatness, Nullifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' 1 2 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK M´arki and Tholen, [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We adopt the algebraic point of view here and continue our work started in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Indeed, among the reduction steps we have mentioned above, the first one calls on fiberwise localization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' For group theoretical localization and homotopy localization functors, it allows one to reduce the study to extensions with local kernel (fiber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Fiberwise localization techniques are available in the category of groups thanks to work of Casacuberta and Descheemaeker, [10], but we proved in [22] that they are not at hand in general for crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Our aim in this article is thus to modify the strategy to be able to study admissibility in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We focus on localization functors such that the co-augmentation morphism ℓT: T → LT is a regular epimorphism for all crossed modules T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We call them regular-epi localization and notice that many examples of interest are provided by nullification functors, as defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Any crossed module A determines a nullification functor PA that “kills” all morphisms from A and there are other regular-epi localization functors such as abelianization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' One first important observation which makes the reduction strategy viable is that, even though fiberwise localization does not exist in general, even for nullification functors, we can use this tool for certain extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a regular-epi localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let (1) T Q N 1 1 κ α be an L-flat exact sequence of crossed modules and g : Q′ → Q a morphism of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then, we can construct the fiberwise localization of the pullback of (1) along g: N N T′ T Q′ Q 1 1 1 1 κ πT κ′ g πQ′ α This allows us to relate conditional flatness with admissibility, in the same spirit as what was done in the category of groups, [14], or in the wider context of semi-abelian categories where fiberwise localization exists, [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A localization functor L is said to be admissible for the class of regular epimorphisms if it preserves any pullback of the form LT T′ Q LQ πLT ℓQ πQ α where α is a regular epimorphism between L-local objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a regular-epi localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then the following statements are equivalent (1) L is conditionally flat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (2) L is admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' One difference between groups and crossed modules and maybe the main source of com- plication is highlighted by the behavior of kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This was already the reason why one ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 3 cannot always construct fiberwise localization and we were also surprised to find examples of nullification functors for which the kernel of the nullification morphism ℓT : T → PAT is not always PA-acyclic, see [22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' For groups and spaces, this property actually characterizes nullification functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Still we prove here that acyclic kernels implies admissibility and in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3, that if the kernels of the localization morphisms are Lf-acyclic, then Lf is a nullification functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Well behaved nullification functors are therefore admissible, but what about arbitrary nullification functors, for which fiberwise localization does not necessarily exist and for which the kernel of the nullification is not necessarily acyclic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By carefully looking at the inductive construction of PAT we show our main result, namely that all nullification functors are admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let A be any crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The nullification functor PA is admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We end this introduction with a short outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The first section consists of preliminaries that we use in the rest of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then in Section 2 we introduce L-flat exact sequences and conditionally flat localization functors in the context of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We show how to construct fiberwise localization of L-flat exact sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The third section is essential in the development of a simpler characterisation of conditional flatness: It provides an equivalence with the notion of admissibility in the specific context of regular-epi localization functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In Section 4 the link between L-acyclicity and admissibility is established and the last section is devoted to the proof that every nullification functor is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We would like to thank Marino Gran for sharing his insight about admissibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Preliminaries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The semi-abelian category of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In this subsection, following Norrie [23] and Brown-Higgins [6], we provide the basic definitions and notation concerning crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' [25] A crossed module of groups is a pair of groups T1 and T2, an action by group automorphisms of T2 on T1, denoted by T2 × T1 → T1 : (b, t) �→ bt, together with a group homomorphism ∂T : T1 → T2 such that for any b in T2 and any t, s in T1, (2) ∂T( bt) = b∂T(t)b−1, (3) ∂T(t)s = tst−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Hence we often write a crossed module as a triple (T1, T2, ∂T), or simply T for short, and we refer sometimes to ∂T as the connecting morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let N := (N1, N2, ∂N) and M := (M1, M2, ∂M) be two crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A morphism of crossed modules α: N → M is a pair of group homomorphisms α1: N1 → M1 and α2 : N2 → M2 such that the two following diagrams commute N2 N1 M1 M2 ∂N ∂M α1 α2 M2 × M1 N2 × N1 N1 M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (α2, α1) α1 4 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK where the horizontal arrows in the diagram on the right are the respective group actions of the two crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We write XMod for the category of crossed modules of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' There is an embedding of the category of groups in this category via two functors which are respectively left and right adjoint to the truncation functor Tr: XMod → Grp that sends a crossed module T := (T1, T2, ∂T) to T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The functor X: Grp → XMod which sends a group G to the crossed module XG = (1, G, 1) reduced to the group G at level 2 is the left adjoint functor and the functor R: Grp → XMod: G �→ (G, G, IdG) is the right ajoint functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This will help us to import group theoretical results into XMod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' There is an obvious notion of subcrossed module, see [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' One simply requires the sub- object to be made levelwise of subgroups, the connecting homomorphism and the action are induced by the given connecting homomorphism and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The notion of normality is less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A subcrossed module N := (N1, N2, ∂N) of T := (T1, T2, ∂T) is normal if the following three conditions hold (1) N2 is a normal subgroup of T2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (2) for any t2 ∈ T2 and n1 ∈ N1, we have t2n1 ∈ N1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (3) [N2, T1] := ⟨ n2t1t−1 1 | t1 ∈ T1, n2 ∈ N2⟩ ⊆ N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In contrast to limits, which are built component-wise, colimits are generally more delicate to construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In particular, the construction of cokernels is not straightforward, but when N is a normal subcrossed module of T the cokernel is simply the levelwise quotient by the normal subgroups N1 ⊳ T1 and N2 ⊳ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The category of crossed modules shares many nice properties with the category of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The traditional homological lemmas, [2], the Split Short Five Lemma, [3], and the Noether Isomorphism Theorems, [2], hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' One can recognize pullbacks by looking at kernels or cokernels, [2, Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5], and in fact Xmod is a semi-abelian category, as introduced by Janelidze, M´arki, and Tholen in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This is shown in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' There is one result we will use several times in this article, namely [2, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4], which we recall now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let C be a semi-abelian (or homological) category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Consider the following diagram of exact rows: T ′ Q′ N′ T Q N 1 1 1 (2) w u v κ α κ′ α′ Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (1) If u is an isomorphism then (2) is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (2) If u and w are regular epimorphisms then v is also a regular epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Localization functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In this subsection we recall the definition of localization func- tors in the category of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We also recall some important properties of such functor as well as some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 5 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A localization functor in the category of crossed modules is a coaugmented idempotent functor L: XMod → XMod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The coaugmentation ℓ: Id → L is a natural transfor- mation such that ℓLX and LℓX are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In particular we have ℓLX = LℓX, see [9, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A crossed module T is L-local if ℓT : T → LT is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A morphism f : N → M is an L-equivalence if Lf is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We recall a few basic and useful closure properties of L-equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (1) The pushout of an L-equivalence is an L-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (2) The composition of L-equivalences is an L-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (3) A κ-filtered colimit of a diagram Tβ of L-equivalences Tβ → Tβ+1 for all successor ordinals β + 1 < κ yields an L-equivalence T0 → Tκ = colimβ<κTβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (4) Let F be an I-indexed diagram of L-equivalences in the category of morphisms of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then the colimit colimIF is an L-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Sometimes a localization functor L is associated to a full reflexive subcategory L of XMod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The pair of adjoint functors U: L ⇆ XMod: F provides a localization functor L = FU, as Cassidy, H´ebert, and Kelly do in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Some other times there is a morphism f one wishes to invert so as to construct a localization functor often written Lf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let f be a morphism of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A crossed module T is Lf-local if Hom(f, T) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A morphism g in XMod is an Lf-equivalence if Hom(g, T) is an isomorphism for any Lf-local crossed module T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Such localization functors exist in XMod, see for example Bousfield’s foundational work [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Local objects and local equivalences coincide then with the notions introduced in Defini- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='8 is the analogue of Hirschhorn’s [16, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='20 and Propo- sition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' If the codomain of the morphism f is the trivial crossed module, the functor Lf is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let A be a crossed module and f be the morphism A → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The localization functor Lf is then written PA and is called a nullification functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' An f-local object is called A-null, or A-local and a crossed module T is A-acyclic if PAT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The localization morphism ℓT : T → PAT is written pT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let A and T be crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then there exists an ordinal λ depending on A such that PAT is constructed as a transfinite filtered colimit of a diagram of the form T = T0 → T1 → · · · → Tβ → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' for β < λ where all morphisms are PA-equivalences and regular epimorphims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This inductive construction has been carefully described in [22, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The rea- son why each step is a PA-equivalence and a regular epimorphism is that Tβ+1 is constructed from Tβ by taking the cokernel of all morphisms A → Tβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We recall the details and use them in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' There is a larger class of localization functors we investigate in this se- quel to [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' They share with PA the property that the localization morphism is a regular epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A localization functor L is a regular-epi localization if for any crossed module T the coaugmentation ℓT: T → LT is a regular epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' 6 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In the category of crossed modules, a morphism α = (α1, α2) is a regular epimorphism (a coequalizer of a pair of parallel arrows) if and only if both α1 and α2 are sur- jective group homomorphisms [20, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A surjective homomorphism of crossed modules is an epimorphism but there exist epimorphisms that are not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In a pointed protomodular category such as XMod, regular epimorphisms and normal epimorphisms (the cokernel of some morphism) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We present now some interesting examples of localization functors that will illustrate our results in the rest of the article, see also the end of [22, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The nullification functor PXZ with respect to the crossed module XZ is given by: PXZ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N1 N2 ∂ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = N1/[N2, N1] 1 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The abelianization functor Ab: XMod → XMod is already described in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' It is defined by: Ab \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N1 N2 ∂ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = N1/[N2, N1] N2/[N2, N2] ˜∂ Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Our third and last example of localization functor of crossed modules is I: XMod → XMod, see [22, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='15]: I \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N1 N2 ∂N \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = N2 N2 IdN2 This functor is induced by the adjunction between the truncation functor Tr: XMod → Grp, defined by Tr(T1, T2, ∂T) = T2, see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3, and its right adjoint R: Grp → XMod that sends a group T to (T, T, IdT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The functor considered in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='14 is a regular-epi localization, since all nullification functors are so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' However regular-epi localizations are not nullification functors in general as illustrated by the functor Ab in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Indeed, if Ab were a nullification PA, then A = (A1, A2, ∂A) would be a perfect crossed module, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' one such that Ab(A) = (1, 1, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In particular, the group A2 would be a perfect group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' But then PA(XS3) = XS3 since there are no non-trivial homomorphisms from a perfect group to the symmetric group S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' But we know that Ab(XS3) = XC2, where C2 is the cyclic group of order two, so abelianization is not a nullification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We finally note that a localization functor Lf is a regular-epi localization functor if f itself is a regular epimorphism, an analogous observation appears in [8] for groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' To conclude these preliminaries, let us recall the notion of fiberwise localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We introduced this for crossed modules in [22, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1], but this is not new, for spaces a good reference is [13, Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 7 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L: XMod → XMod be a localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' An exact sequence T Q N 1 1 κ α admits a fiberwise localization if there exists a commutative diagram of horizontal exact sequences T Q N E Q LN 1 1 1 1 κ j ℓN p α g where g is an L-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The following theorem is a fusion of two results from [22] namely Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' From now on, every localization functor that we consider is a regular-epi localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L: XMod → XMod be a regular-epi localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' An exact se- quence of crossed modules (4) T Q N 1 1 κ α admits a fiberwise localization if and only if we have the following inclusion (5) [κ2(ker(ℓN 2 )), T1] ⊆ κ1(ker(ℓN 1 )) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Fiberwise localization and flatness In this section, we investigate the fiberwise localization of L-flat exact sequences and their pullbacks in the context of regular-epi localization functors of crossed modules L: XMod → XMod (even if this notion is not defined only for regular-epi functor as we will see in Propo- sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This section will be essential to study the link between conditionally flatness and admissibility in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' First, let us recall the definitions of L-flat and conditionally flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a localization functor, a short exact sequence T Q N 1 1 κ α is called L-flat if the sequence LT LQ LN L(κ) L(α) is a short exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We recall that limits are computed componentwise in the category of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In the case of pullbacks in XMod they are built as follows [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let α: T → Q and g : Q′ → Q be two morphisms of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then the pullback of α along g is given by the following square T T′ Q′ Q πT g πQ′ α The object part T′ of the pullback is built component-wise as in the case of groups (T1 ×Q1 Q′ 1, T2 ×Q2 Q′ 2, ∂′), 8 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK where ∂′ and the action are induced by the universal property of the pullbacks in Grp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The projections are the natural ones, given also component-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Following the terminology introduced in [14] for groups and spaces, we define the notion of conditional flatness for localization functors in crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We say that this functor is conditionally flat if the pullback of any L-flat exact sequence is L-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In Section 3 we provide a characterization of conditional flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' To achieve this goal we will use a similar strategy to the one applied to groups and topological spaces in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The authors exploit heavily the existence of fiberwise localization in the categories of groups and spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' However, in our article [22], we observed that fiberwise localization does not always exist for a given localization functor and a given exact sequence in XMod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Fortunately, when we work with L-flat exact sequences we can show that it is always possible to construct a fiberwise localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a regular-epi localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then any L-flat exact sequence of crossed modules admits a fiberwise localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let T Q N 1 1 κ α be an L-flat exact sequence of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The L-flatness of the sequence implies in particular that Lκ is a monomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Consider the following diagram of exact sequences: 1 1 ker(ℓT) (1) ker(ℓN) N T LN LT κ Lκ ℓN ℓT We conclude from [2, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (1)] that (1) is a pullback since Lκ is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then we have that κ(ker(ℓN)) is a normal subcrossed module of T as it can be seen as the intersection of the normal subcrossed modules N and ker(ℓT) of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Therefore, we can apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='19 □ To understand conditional flatness we must study the pullback of an L-flat exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' It will thus be very handy in Section 3 to know that any such pullback admits a fiberwise localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a regular-epi localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let (6) T Q N 1 1 κ α be an L-flat exact sequence of crossed modules and g : Q′ → Q a morphism of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then, we can construct the fiberwise localization of the pullback of (6) along g N N T′ T Q′ Q 1 1 1 1 κ πT κ′ g πQ′ α ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 9 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In the rest of the article, and in particular in the following proof, we identify N with the normal subcrossed module κ(N) of T and with κ′(N), normal subcrossed module of T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We will therefore omit the us of κ and κ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' For example an element of the group N1 that we want to consider in T′ 1 will be denoted (n1, 1) instead of κ′ 1(n1) = (κ1(n1), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We need to verify that ker(ℓN) is a normal crossed module of T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Since N is a subcrossed module of T′, we just need to verify (5) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let (t1, q1) be an element in T ′ 1 and (x2, 1) be an element of ker(ℓN 2 ), then we have the following equality (x2,1)(t1, q1)(t1, q1)−1 = ( x2t1t−1 1 , q1q−1 1 ) = ( x2t1t−1 1 , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Indeed, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4 we know that the original sequence (6) admits a fiberwise localization which then implies by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='19 that [ker(ℓN 2 ), T1] ⊂ ker(ℓN 1 ) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='e for any x2 ∈ ker(ℓN 2 ) and t1 ∈ T1 we have x2t1t−1 1 ∈ ker(ℓN 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' But then, with the notation introduced in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6, this is equivalent to say that the element ( x2t1t−1 1 , 1) belongs to ker(ℓN 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ This lemma is not trivial since the fiberwise localization of an exact sequence of crossed modules does not always exist as we have proved in [22, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' If we want the strategy for groups and spaces to be also viable in the study of conditional flatness for crossed modules, we need a final ingredient, namely a commutation rule for the fiberwise localization and the pullback operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let us consider an L-flat exact sequence where L is a regular-epi localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then, the pullback of its fiberwise localization is the fiberwise localization of its pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let N N T′ T Q′ Q 1 1 1 1 κ πT κ′ g πQ′ α be the pullback of an L-flat exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then we construct the fiberwise localizations of the two sequences by quotienting out the kernel of the localization morphism ℓN as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' N 1 T′ Q′ 1 N 1 T Q 1 LN 1 T′/ker(ℓN) Q′ 1 LN 1 T/ker(ℓN) Q 1 κ′ πQ′ κ α g πT j j′ p p′ g f ′ f ℓN ℓN We complete the diagram by defining a morphism δ: T′/ker(ℓN) → T/ker(ℓN) via the universal property of the cokernel since f ◦ πT ◦ κ′|ker(ℓN) = 1, where κ′|ker(ℓN) : ker(ℓN) → T′ is 10 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK the inclusion of the kernel of ℓN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ker(ℓN) ker(ℓN) T′ T T ′/ker(ℓN) T/ker(ℓN) 1 1 1 1 κ|ker(ℓN) πT κ′|ker(ℓN) f ′ f δ N 1 T′ Q′ 1 N 1 T Q 1 LN 1 T′/ker(ℓN) Q′ 1 LN 1 T/ker(ℓN) Q 1 κ′ πQ′ κ α g πT j j′ p p′ g δ f ′ f lN lN We can check that δ makes the two front faces commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Indeed, the right and left faces commute by using the fact that ℓN and f ′ are epimorphisms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The commutativity of the above diagram and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5 implies that LN T′/ker(ℓN) Q′ 1 1 j′ p′ is the pullback of T/ker(ℓN) Q LN 1 1 j p along g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In [14], the construction of the fiberwise localization in the category of groups was functorial, therefore from the morphism T′ → T between the pullback sequence and the sequence itself we have directly a morphism between the fiberwise localization of the pullback sequence and the fiberwise localization of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In other words the map δ comes for free in contrast to the category of crossed modules where we have to build the map δ explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Conditional flatness and admissibility In this section, we develop a simpler characterisation of conditional flatness, thanks to the results of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We introduce the notion of admissibility for the class of regular epimorphisms and show that it is equivalent to conditional flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' With this equivalence, we can easily establish conditional flatness for a given localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We observe that some properties of localization functors, such as right-exactness, imply directly admissibility for the class of regular epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The first step allows us to restrict the definition of conditional flatness (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3) to fiberwise localizations of L-flat exact sequences (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' More precisely, we show that the pullback of an L-flat exact sequence is L-flat if and only if the pullback of its fiberwise localization is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 11 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a regular-epi localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then L is conditionally flat if and only if for any L-flat exact sequence T Q N 1 1 κ α with N an L-local crossed module, the pullback sequence along any morphism Q′ → Q is L-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This is clear since f ′ and ℓN are L-equivalences in this diagram: LN N T′ T′/ker(ℓN) Q′ Q′ 1 1 1 1 j′ f ′ ℓN κ′ πQ′ p′ The top row is thus L-flat if and only if so is the bottom row and we conclude by Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ The previous lemma allows us to follow the approach introduced in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' For the sake of completeness, we give an explicit proof of the following results even if the arguments are similar to the group theoretical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a regular-epi localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then L is conditionally flat if and only if the pullback of any exact sequence of L-local objects is L-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By the previous lemma it is sufficient to consider exact sequence with an L-local kernel LN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Consider thus an L-flat exact sequence T Q LN 1 1 j p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We build the following diagram where g : Q′ → Q is any morphism of crossed modules and (1) is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' LN LN LN T′ (1) (2) T LT Q′ Q LQ 1 1 1 1 1 1 j L(j) ℓT ℓQ πT j′ g πQ′ L(p) p We observe that since each row is exact, (2) is a pullback by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5, and then (1) + (2) is also a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Hence, the top row is the pullback of the bottom exact sequence of L-local objects along the map ℓQ ◦ g, which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A localization functor L is said to be admissible for the class of regular epimorphisms if it preserves any pullback of the form LT T′ Q LQ πLT ℓQ πQ α 12 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK where α is a regular epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L be a regular-epi localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then the following statements are equivalent (1) L is conditionally flat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (2) L is admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The implication (1) ⇒ (2) is trivial, so let us prove (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Consider any exact sequence of L-local objects LT LQ LN 1 1 α and any morphism g : A → LQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2 conditional flatness is established if we prove that the pullback of the exact sequence along g is L-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let us first observe that this morphism g factors through LA via the universal property of the localization: A LA LQ ℓA g ˜g Hence, we can first construct the pullback of LT LQ LN 1 1 α along ˜g and then pullback the resulting sequence along ℓA: LN LN LN T′′ T′ LT A LA LQ 1 1 1 1 1 1 g πLT ˜g ℓA πA α πLA Since the category of L-local objects is closed under pullbacks, T′ is L-local and we can apply condition (2) to conclude that the upper row is L-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This observation implies that the pullback of LT LQ LN 1 1 α along g is an L-flat sequence as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ The above theorem gives an easier characterisation of conditionally flatness in the category of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' It will be useful in rest of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Admissibility for the class of regulars epimorphisms in the context of semi- abelian categories is studied in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Similar results are proven for functors of localizations that admit a functorial fiberwise localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Note that their result does not imply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4 since localization functors of crossed modules do not admit functorial fiberwise localizations in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' However, the implication “(1) implies (2)”, in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4, holds even for not necessarily regular-epi localization functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 13 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' If L: XMod → XMod is a localization functor that is right exact in XMod, then L is admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let us consider the following pullback of an L-flat exact sequence of crossed modules along a morphism g : Q′ → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' T′ Q′ N T Q N 1 1 1 1 (1) g πT κ f κ′ πQ′ By applying L to this diagram, we obtain (since L is right exact) the following diagram LT′ LQ′ LN LT LQ LN 1 1 1 L(g) L(πT) L(κ) L(f) L(κ′) L(πQ′) Since L(κ) = L(πT)◦L(κ′) is a (normal) monomorphism, we conclude that L(κ′) is a monomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Normality follows then by right-exactness and we conclude by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ Note that this proof holds in any semi-abelian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The functor of abelianization Ab: XMod → XMod is admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The functor of abelianization Ab: XMod → XMod is right exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Since the exactness can be shown component-wise, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ Sometimes it is handy to rely on our group theoretical knowledge to construct simple examples of localization functors and how they behave on crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The proof of the following proposition is based on a counter-example coming from groups via the functor X defined in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' There are regular-epi localization functors L: XMod → XMod that are not admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We export via X: Grp → XMod the example in [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1] of a localization functor in groups that is not admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let Lφ be the localization functor induced by the projection φ: C4 → C2, where Cn denotes a cyclic group of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' It gives rise to a localization functor LXφ : XMod → XMod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In particular, if we apply X to the extension of Lφ-local groups considered in [14], we obtain an exact sequence of LXφ-local crossed modules: (1, Z) (1, C2) (1, Z) 1 1 If we pullback along the morphism of crossed modules Xφ, we obtain the following exact sequence (1, Z × C2) (1, C4) (1, Z) 1 1 14 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK We conclude from [22, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4] that this exact sequence is not LXφ-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Indeed, if it was the case we would have a contradiction with the group theoretical observation in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Admissibility and acyclicity In the categories of groups and topological spaces, the localization functor L is a nullification functor if and only if the kernels of the localization morphisms are L-acyclic (which means that Lker(ℓM) is trivial for any M ∈ XMod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This characterisation implies in particular that any nullification functor is admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' It is interesting to notice that even if nullification functors of crossed modules do not have acyclic kernels, we have a similar result in XMod: the L-acyclicity of the kernels of localization morphisms implies the admissibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let L: XMod → XMod be a regular-epi localization functor such that ker(ℓM : M → LM) is L-acyclic for any M ∈ XMod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then L is admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Consider the pullback of LT LQ LN 1 1 κ f along ℓQ: Q → LQ: LN LN T′ LT Q LQ 1 1 1 1 κ πLT κ′ ℓQ πQ f We need to prove that πLT is an L-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Since XMod is a pointed protomodular category and ℓQ is a regular epi by assumption, we know that πLT is the cokernel of ker(ℓQ) ∼= ker(πLT) → T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let Y be a local object, for any g : T′ → Y we have the following diagram: ker(ℓQ) Lker(ℓQ) = 1 T′ LT Y g′ πLT g ˜g By the universal property of the localization there exists g′: 1 → Y that makes the left square commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Hence, by the universal property of the cokernel there exists a unique ˜g: LT → Y such that the triangle commutes and we conclude that πLT is an L-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ However, localization functors of crossed modules do not behave like localization functors of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' As explained above, in the category of groups (but also of topological spaces), the kernels of the localization morphisms are L-acyclic if and only if L is a nullification functor [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In the context of crossed modules, we do not have such a characterization of nullification functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We know by [22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6] that there are nullification functors, for example PXZ defined in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='14, such that the kernels of their localization morphisms are not acyclic in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Still, in the next proposition, we prove that if the kernel of the localization morphism is L-acyclic, as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1, then the localization functor is a nullification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 15 The cardinal in the next proof is chosen exactly as in Bousfield’s [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4] for spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let f : B → C be a morphism of crossed modules and Lf : XMod → XMod be a regular-epi localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' If the kernels of the localization morphisms are Lf- acyclic, then Lf is a nullifcation functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Our strategy is to construct a crossed module A such that we can compare the functor Lf with the nullification functor PA (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='10) via a natural transformation ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We choose κ to be the first infinite ordinal greater than the number of chosen generators of B and C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=', generators of the groups B1, B2, C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We construct the crossed module A := � Aα, where Aα are all the Lf-acyclic crossed modules with less than 2κ generators, see [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The first step of this proof is to show that if a crossed module X is Lf-local then it is A-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let φ be a morphism in Hom(A, X) and construct by naturality the following commutative diagram 1 = LfA A X LfX Lfφ ∼= φ By hypothesis, we have an isomorphism between X and LfX and by construction of A, we obtain LfA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Therefore, φ factors through the zero object and hence Hom(A, X) = 1, which is equivalent to say that X is A-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Now consider the PA-equivalence pT: T → PAT and the Lf-local object LfT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By the above observation, we have that LfT is A-local and by the universal property we have the desired morphism ψT T PAT LfT pT ℓT ψT We construct next the fiberwise A-nullification of the following exact sequence T LfT ker(ℓT) 1 1 ℓT By assumption ker(ℓT) is Lf-acyclic, hence also PA-acyclic by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This implies that ker � pT: ker(ℓT) → PAker(ℓT) � is equal to ker(ℓT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Hence, the exact sequence satisfies condi- tion (5) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='19 and we obtain the following fiberwise nullification T LfT ker(ℓT) T/ker(ℓT) LfT 1 1 1 1 1 pT f ∼= ℓT 16 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK Since f is a PA-equivalence, so is ℓT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Hence, we obtain a morphism ϕT in the following commutative diagram: T LfT PAT ψT ℓT pT ϕT By universal property, we can conclude that the two compositions of ψT and ϕT are isomorphic to identities so that LfT ∼= PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' A similar argument shows the naturality of ψ and ϕ and therefore Lf is a nullification functor, namely PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Nullification functors and admissibility In the category of groups, the fact that kernels of localization morphisms are L-acyclic was fundamental to prove that nullification functors are admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This fact is not true in general for nullification functors in the category of crossed modules as shown in [22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6], it is thus natural to ask whether nullifica- tion functors are admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We provide an affirmative answer in this final section, but let us first prove that our counter-example PXZ is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The nullification functor PXZ is admissible for the class of regular epimor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1 in [15] implies that PXZ is admissible provided that the reflective category of PXZ-local objects is a Birkhoff subcategory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=', it is closed under regular quotients and subobjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Here PXZ-local objects are crossed modules of the form A → 1 where A is any abelian group and the connecting homomorphism is the trivial homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Therefore it is clearly closed under subobjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Moreover, the quotient of A → 1 by a normal subcrossed modules N → 1 is the crossed module A/N → 1 that is PXZ-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ The remaining part of the section is devoted to the proof that all nullification functors are admissible for the class of regular epmorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Consider a nullification functor PA where A = (A1, A2, ∂) is a crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' To show the admissibility, it is enough to prove that the pullback of an exact sequence of PA-local crossed modules along the coaugmentation map is PA-flat, in other words that the map f in the following commutative diagram of crossed modules is a PA-equivalence W Q PAN PAT PAQ PAN 1 1 1 1 (1) pQ f h g where (1) is a pullback and g and h are regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' To do so we follow step by step the inductive construction of PAQ = colimQβ as presented in [22, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='8], see also Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' For each successor ordinal β + 1 we obtain Qβ+1 from Qβ by killing all morphisms out of A so let us start with the construction of Q1 from Q0 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 17 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let ϕ : A → Q be a morphism of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The crossed module Q1 is the quotient of Q by the normal closure KQ in Q of the image of ev: � ϕ∈Hom(A,Q) A = M −→ Q which is defined by ϕ on the copy of A indexed by ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The idea behind the construction we perform next is that we do not need to kill all morphisms from A to the extension W in order to construct its nullification PAW, it is sufficient to take care of those factoring through Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Beware that given an extension N → T → Q with N an A-acyclic crossed module, it is not true in general that all morphisms from A to T factor through Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By definition of pQ we have the following equality for the composition pQ ◦ ϕ = 1 = h ◦ 1 as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Therefore, any morphism from A to Q induces one from A to W: (7) PAT W Q PAQ A h pQ f g 1 ϕ ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='ψ We call ψ the morphism determined by ϕ and it makes sense now to consider KW, the normal closure in W of the image of M → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' With the same notation as in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2, we have an isomorphism KW ∼= KQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Limits are computed levelwise for crossed modules, so the pullback W consists of compatible pairs (x, q) for x ∈ (PAT)i and q ∈ Qi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By construction of ψ we have ψ(a) = (1, ϕ(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Now, we compute the kernels of the cokernels of ev: M → Q and (1, ev): M → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We have the two following descriptions of the kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' KQ = � ev1(M1)Q2[ev2(M2)Q2, Q1], ev2(M2)Q2, ∂ � KW = � (1, ev1)(M1)W2[(1, ev2)(M2)W2, W1], (1, ev2)(M2)W2, ∂′� The second group of the crossed module KW is the easier one: (1, ev2)(M2)W2 = {(t2,q2)(1, ev2(m2)) | (t2, q2) ∈ W2, m2 ∈ M2} = {(1, q2ev2(m2)) | q2 ∈ Q2, m2 ∈ M2} = 1 × ev2(M2)Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' where the second equality holds since h is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Via similar computations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' we see that (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ev1)(M1)W1 = 1 × ev1(M1)Q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' so we are left with proving that [(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ev2)(M2)W2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' W1] = 1 × [ev2(M2)Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Q1] 18 OLIVIA MONJON,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' J´ER ˆOME SCHERER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' AND FLORENCE STERCK This we do via the following equalities: [(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ev2)(M2)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' W1] = [(1 × ev2(M2)Q2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' W1] = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='x2)(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' q1)(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' q1)−1 | x2 ∈ ev2(M2)Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' q1) ∈ W1} = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='x2 q1q−1 1 | x2 ∈ ev2(M2)Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' q1 ∈ Q1} = 1 × [(ev2(M2)Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Q1] So finally we can conclude that KW = 1 × KQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' in particular KW and KQ are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' For any ordinal β, we have a commutative diagram PAT Wβ Qβ PAQ W Q (2) hβ g fβ pQ β h where (2) is a pullback square, the maps fβ : W → Wβ and pQ β : Q → Qβ are PA-equivalences, and hβ is a regular epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We prove it by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Since the nullification uses possibly a transfinite construc- tion we have to initialize the induction, but the case β = 0 holds by assumption, and then check the statement for successor and limit ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The successor case Suppose that for an ordinal β the lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Then we consider the kernels KW β and KQ β of the cokernels of the evaluation maps ev : � Hom(A,Qβ) A −→ Qβ and ev : � Hom(A,Qβ) A −→ Wβ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' They fit in the following diagram of exact rows: Wβ+1 Wβ Qβ Qβ+1 KW β KQ β (2) pQ (β→β+1) f(β→β+1) hβ ∼= iW iQ ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='hβ+1 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3 applies here and gives us the isomorphism between KW β and KQ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The composition pQ (β→β+1) ◦ hβ ◦ iW : KW β → Qβ+1 is zero by commutativity, yielding by the universal property of the cokernel the morphism hβ+1: Wβ+1 → Qβ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The isomorphism between the kernels implies that (2) is a pullback (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By induction hypothesis hβ is a regular epimorphism and the composition pQ (β→β+1) ◦ hβ : Wβ → Qβ+1 is also a regular epimorphism, hence so is hβ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We show now that pQ (β→β+1) and f(β→β+1) are PA-equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' ADMISSIBILITY OF LOCALIZATIONS OF CROSSED MODULES 19 For the first one we write the cokernel Qβ+1 as the pushout along the evaluation morphism: 1 � A Qβ Qβ+1 pQ (β→β+1) 1 ϕ inc where the coproduct is taken over Hom(A, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The trivial map A → 1 is a PA-equivalence, thus so is the pushout pQ (β→β+1) : Qβ → Qβ+1 by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='8 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By composing with the PA-equivalence Q → Qβ we see that pQ β+1 : Q → Qβ+1 is a PA-equivalence as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The same argument shows that fβ+1 : W → Wβ+1 is also a PA-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By the universal property of the localization, we obtain two maps, one from Wβ+1 to PAT and the other from Qβ+1 to PAQ such that (2) commutes: PAT Wβ+1 Qβ+1 PAQ W Q (2) (1) hβ+1 g fβ+1 pQ β+1 h f pQ Since (1) and the outer rectangle are pullbacks and hβ+1 is a regular epimorphism, we can conclude by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4 in [2] that (2) is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The limit case To prove the statement for a general transfinite induction we need to prove it for a limit ordinal as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let γ be a limit ordinal and Qγ = colimα<γQα Wγ = colimα<γWα We have shown that pQ (α−1→α) : Qα−1 → Qα is a PA-equivalence for all α < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Hence the composition pQ α : Q → Qα is also a PA-equivalence and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='8 (3), implies that pQ γ : Q → Qγ is a PA-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The same reasoning holds for fγ : W → Wγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The existence of the maps f : W → PAT and pQ : Q → PAQ give us two maps Wγ → PAT and Qγ → PAQ as shown on the diagram below (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The nullification PAQ is constructed as filtered colimit of the Qα, see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Filtered colimits commutes with finite limits, in particular with kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Therefore KQ γ := ker(Q → Qγ) ∼= colimα<γker(Q → Qα) where ker(Q → Qα) will be denoted KQ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The category XMod is a variety of algebras (also called algebra category of fixed type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Hence, by [21, Proposition IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='2], we know that the forgetful functor U : XMod → Set creates filtered colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In other words we have : U(colimα<γKQ α) = colimα<γUKQ α = � α<γ UKQ α where the colimit in the first term lies in the category of crossed modules and the second colimit in the category of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This means that we know the structure of colimα<γKQ α as a 20 OLIVIA MONJON, J´ER ˆOME SCHERER, AND FLORENCE STERCK set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Now since KQ α ∼= KW α for all α < γ and KQ γ can be written as a union of KQ α (as well as KW γ ) we conclude that KQ γ ∼= KW γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We consider now the diagram: (8) PAT Wγ Qγ PAQ W Q (1) (2) hγ g fγ pQ γ h f pQ Since the kernels of fγ and pQ γ are isomorphic we deduce that (2) is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' As we have shown that every map pQ (α→α+1) : Qα → Qα+1 is a regular epimorphism, the morphism pQ α : Q → Qα is also a regular epimorphism, being a composition of regular epimorphisms in a regular category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The colimit functor being a left adjoint functor, it preserves colimits and in particular cokernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In a pointed protomodular category, any regular epimorphism is a cokernel, therefore pQ γ : Q → Qγ is a regular epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The composition pQ γ ◦ g is also a regular epimorphism, and we conclude that so is hγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' With the same argument as for the successor step, we get that (1) is a pullback, which ends the induction proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ We are ready now for the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let A be any crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The nullification functor PA is admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let W be the pullback of a regular epimorphism h: PAT → PAQ between PA-local crossed modules along the localization morphism pQ : Q → PAQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let λ be the ordinal such that Qλ ∼= PAQ (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4 we have a diagram: PAT Wλ Qλ PAQ W Q (2) ∼= hλ g h fλ pQ λ where the outer rectangle is a pullback, the morphisms fλ and pQ λ are PA-equivalences, and (2) is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Since isomorphisms are stable under pullbacks, we have an isomorphism Wλ ∼= PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We have thus proved that the map f : W → PAT is a PA-equivalence, which means that the functor PA is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ In this article we have focused on regular-epi localization functors because they appear nat- urally when studying conditional flatness and admissibility in the category of groups, crossed modules, or more general semi-abelian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We conclude this section by observing that the notion of conditional flatness can also be defined for non regular-epi localization functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' The next proposition gives an example of such a localization functor which is conditionally REFERENCES 21 flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let us stress that we will not a priori have an equivalence with admissiblity, as was the case for regular-epi localization functors by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In the proof of the following proposition we have thus to verify the more general condition for conditional flatness, as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' There exists a non regular-epi localization functor which is nevertheless conditionally flat and therefore admissible for the class of regular epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We consider the functor I defined in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='16 which sends any crossed module (N1, N2, ∂N) to (N2, N2, IdN2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' This functor is not regular-epi because if we consider a crossed module for which the connecting morphism is not surjective then the localization morphism will not be a regular epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We prove now that I is conditional flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Let T Q N 1 1 κ α be any exact sequence of crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' We see that I((N1, N2, ∂N)) = (N2, N2, IdN2) is a normal subcrossed module of (T2, T2, IdT2) = I((T1, T2, ∂T) and that I((Q1, Q2, ∂Q)) = (Q2, Q2, IdQ2) is the cokernel of κ: N → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Therefore any exact sequence of crossed modules is I-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In particular any pullback along any morphism of crossed modules of an I-flat exact sequence is I-flat, hence I is conditionally flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' □ References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Berrick and E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' King’s College, University of London, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Whitehead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' “Combinatorial homotopy II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' In: Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Soc 55 (1949), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' 453–496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content=' Mathematics, Ecole Polytechnique F´ed´erale de Lausanne, EPFL, Switzerland Email address: olivia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='monjon@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='com Mathematics, Ecole Polytechnique F´ed´erale de Lausanne, EPFL, Switzerland Email address: jerome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='scherer@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='ch Institut de Recherche en Math´ematique et Physique, Universit´e catholique de Louvain, Belgium Email address: florence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='sterck@uclouvain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} +page_content='be' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FAT4oBgHgl3EQfux7Z/content/2301.08672v1.pdf'} diff --git a/ItFJT4oBgHgl3EQfFyzw/content/tmp_files/load_file.txt b/ItFJT4oBgHgl3EQfFyzw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eece57cd297589c85d33f5ec80dced3119363300 --- /dev/null +++ b/ItFJT4oBgHgl3EQfFyzw/content/tmp_files/load_file.txt @@ -0,0 +1,1386 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf,len=1385 +page_content='3DShape2VecSet: A 3D Shape Representation for Neural Fields and Generative Diffusion Models BIAO ZHANG, KAUST, Saudi Arabia JIAPENG TANG, TU Munich, Germany MATTHIAS NIESSNER, TU Munich, Germany PETER WONKA, KAUST, Saudi Arabia Input Reconstruction Input Reconstruction Condition Generation “the tallest chair” car Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Left: Shape autoencoding results (surface reconstruction from point clouds) Right: the various down-stream applications of 3DShape2VecSet (from top to down): (a) category-conditioned generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (b) point clouds conditioned generation (shape completion from partial point clouds);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (c) image conditioned generation (shape reconstruction from single-view images);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (d) text-conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We introduce 3DShape2VecSet, a novel shape representation for neural fields designed for generative diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our shape representation can en- code 3D shapes given as surface models or point clouds, and represents them as neural fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The concept of neural fields has previously been combined with a global latent vector, a regular grid of latent vectors, or an irregu- lar grid of latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our new representation encodes neural fields on top of a set of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We draw from multiple concepts, such as the ra- dial basis function representation and the cross attention and self-attention function, to design a learnable representation that is especially suitable for processing with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our results show improved performance in 3D shape encoding and 3D shape generative modeling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We demon- strate a wide variety of generative applications: unconditioned generation, category-conditioned generation, text-conditioned generation, point-cloud completion, and image-conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Additional Key Words and Phrases: 3D Shape Generation, 3D Shape Repre- sentation, Diffusion Models, Shape Reconstruction, Generative models Authors’ addresses: Biao Zhang, KAUST, Saudi Arabia, biao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='zhang@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='sa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Jia- peng Tang, TU Munich, Germany, jiapeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='tang@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Matthias Nießner, TU Munich, Germany, niessner@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Peter Wonka, KAUST, Saudi Arabia, peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='wonka@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 1 INTRODUCTION The ability to generate realistic and diverse 3D content has many potential applications, including computer graphics, gaming, and vir- tual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' To this end, many generative models have been explored, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', generative adversarial networks, variational autoencoders, nor- malizing flows, and autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Recently, diffusion mod- els have emerged as one of the most popular method with fantastic results in the 2D image domain [Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] and have shown their superiority over other generative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For instance, it is possible to do unconditional generation [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022], text conditioned generation [Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022], and generative image inpainting [Lug- mayr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' However, the success in the 2D domain has not yet been matched in the 3D domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In this work, we will study diffusion models for 3D shape genera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' One major challenge in adapting 2D diffusion models to 3D is the design of a suitable shape representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The design of such a shape representation is the major focus of our work, and we will discuss several design choices that lead to the development of our proposed representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='11445v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='CV] 26 Jan 2023 Different from 2D images, there are several predominant ways to represent 3D data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', voxels, point clouds, meshes, and neu- ral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In general, we believe that surface-based representations are more suitable for downstream applications than point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Among the available choices, we choose to build on neural fields as they have many advantages: they are continuous, represent com- plete surfaces and not only point samples, and they enable many interesting combinations of traditional data structure design and representation learning using neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Two major approaches for 2D diffusion models are to either use a compressed latent space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', latent diffusion [Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022], or to use a sequence of diffusion models of increasing resolution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', [Ramesh 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' While both of these ap- proaches seem viable in 3D, our initial experiments indicated that it is much easier to work with a compressed latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We therefore follow the latent diffusion approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A subsequent design choice for a latent diffusion approach is to de- cide between a learned representation or a manually designed repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A manually designed representation such as wavelets [Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] is easier to design and more lightweight, but in many con- texts learned representations have shown to outperform manually designed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We therefore opt to explore learned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' This requires a two-stage training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The first stage is an autoencoder (variational autoencoder) to encode 3D shapes into a latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The second stage is training a diffusion model in the learned latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In the case of training diffusion models for 3D neural fields, it is even more necessary to generate in latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' First, diffusion models often work with data of fixed size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', images of a given fixed resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Second, a neural field is a continuous real-valued function that can be seen as an infinite-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For both reasons, we decide to find a way to encode shapes into latent space before all else (as well as a decoding method for reverting latents back to shapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Finally, we have to design a suitable learned neural field rep- resentation that provides a good trade-off between compression and reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Such a design typically requires three components: a spatial data structure to store the latent information, a spatial interpolation method, and a neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' There are multiple options proposed in the literature shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Early methods used a single global latent vector in combination with an MLP network [Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' This concept is simple and fast but generally struggles to reconstruct high-quality shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Better shape details can be achieved by using a 3D regular grid of latents [Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] together with tri-linear interpolation and an MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' However, such a representation is too large for generative models and it is only possible to use grids of very low resolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', 8×8×8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' By introducing sparsity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', [Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022], latents are arranged in an irregular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The latent size is largely reduced, but there is still a lot of room for improvement which we capitalize on in the design of 3DShape2VecSet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The design of 3DShape2VecSet combines ideas from neural fields, radial basis functions, and the network architecture of attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Similar to radial basis function representation for continuous functions, we can also re-write existing methods in a similar form (linear combination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Inspired by cross attention in the transformer network [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017], we derived the proposed latent rep- resentation which is a fixed-size set of latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' There are two main reasons that we believe contribute to the success of the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' First, the representation is well-suited for the use with transformer-based networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' As transformer-based networks tend to outperform current alternatives, we can better benefit from this network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Instead of only using MLPs to process latent information, we use a linear layer and cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Sec- ond, the representation no longer uses explicitly designed positional features, but only gives the network the option to encode positional information in any form it considers suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' This is in line with our design principle of favoring learned representations over manually designed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2 e) for the proposed latent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Using our novel shape representation, we can train diffusion mod- els in the learned 3D shape latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our results demonstrate an improved shape encoding quality and generation quality compared to the current state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' While pioneering work in 3D shape generation using diffusion models already showed unconditional 3D shape generation, we show multiple novel applications of 3D dif- fusion models: category-conditioned generation, text-conditioned shape generation, shape reconstruction from single-view image, and shape reconstruction from partial point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' To sum up, our contributions are as follows: (1) We propose a new representation for 3D shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Any shape can be represented by a fixed-length array of latents and processed with cross-attention and linear layers to yield a neural field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (2) We propose a new network architecture to process shapes in the proposed representation, including a building block to aggregate information from a large point cloud using cross- attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (3) We improve the state of the art in 3D shape autoencoding to yield a high fidelity reconstruction including local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (4) We propose a latent set diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We improve the state of the art in 3D shape generation as measured by FID, KID, FPD, and KPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (5) We show 3D shape diffusion for category-conditioned gener- ation, text-conditioned generation, point-cloud completion, and image-conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2 RELATED WORK In this section, we briefly review the literature of 3D shape learning with various data representations and 3D shape generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1 3D Shape Representations We mainly discuss the following representations for 3D shapes, including voxels, point clouds, and neural fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Voxel grids, extended from 2D pixel grids, simply repre- sent a 3D shape as a discrete volumetric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Due to their regular structure, early works take advantage of 3D transposed convolution operators for shape prediction [Brock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Choy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Girdhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016, 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A draw- back of the voxels-based decoders is that the computational and memory costs of neural networks cubicly increases with respect to 2 (x𝑖, 𝜆𝑖 ) x 𝜙 (x, x𝑖 ) (a) RBF (b) Global Latent x (x𝑖, f𝑖 ) (c) Latent Grid (x𝑖, f𝑖 ) x 𝜙 (x, x𝑖 ) (d) Irregular Latent Grid x f𝑖 𝜙 (x, f𝑖 ) = exp � q(x)⊺k(f𝑖 )/ √ 𝑑 � (e) Latent Set (Ours) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Continuous function representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Scalars are represented with spheres while vectors are cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The arrows show how spatial interpolation is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' x𝑖 and x are the coordinates of an anchor and a querying point respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 𝜆𝑖 is the SDF value of the anchor point x𝑖 in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' f𝑖 is the associate feature vector located in x𝑖 in (c)(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The queried SDF/feature of x is based on the distance function 𝜙 (x, x𝑖) in (a)(c)(d), while our proposed latent set representation (e) utilizes the similarity 𝜙 (x, f𝑖) between querying coordinate and anchored features via cross attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' # Latents Latent Position Methods OccNet [Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019] DeepSDF [Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019] Single Global IM-Net [Chen and Zhang 2019] ConvOccNet [Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] IF-Net [Chibane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] LIG [Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] DeepLS [Chabra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] SA-ConvOccNet [Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021] Multiple Regular Grid NKF [Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] LDIF [Genova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] Point2Surf [Erler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] DCC-DIF [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] 3DILG [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] Multiple Irregular Grid POCO [Boulch and Marlet 2022] Multiple Global Ours Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Neural Fields for 3D Shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We show different types how la- tents are positioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' the grid resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Thus, most voxel-based methods are limited to low-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Octree-based decoders [Häne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Meagher 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Riegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Tatarchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017, 2018] and sparse hash-based decoders [Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] take 3D space sparsity into account, alleviating the efficiency issues and supporting high-resolution outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Point Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Early works on neural-network-based point cloud processing include PointNet [Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017a,b] and DGCNN [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' These works are built upon per-point fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' More recently, transformers [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017] were pro- posed for point cloud processing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', [Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' These works are inspired by Vision Trans- formers (ViT) [Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021] in the image domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Points are firstly grouped into patches to form tokens and then fed into a transformer with self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In this work, we also introduce a network for processing point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Improving upon previous works, we compress a given point cloud to a small representation that is more suitable for generative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Neural Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A recent trend is to use neural fields as a 3d data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The key building block is a neural network which accepts a 3D coordinate as input, and outputs a scalar [Chen and Generative 3D Models Representation 3D-GAN [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016] GAN Voxels l-GAN [Achlioptas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2018] GAN★ Point Clouds IM-GAN [Chen and Zhang 2019] GAN★ Fields PointFlow [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019] NF Point Clouds GenVoxelNet [Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] EBM Voxels PointGrow [Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] AR Point Clouds PolyGen [Nash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] AR Meshes GenPointNet [Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021] EBM Point Clouds 3DShapeGen [Ibing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021] GAN★ Fields DPM [Luo and Hu 2021] DM Point Clouds PVD [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021] DM Point Clouds AutoSDF[Mittal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] AR★ Voxels CanMap [Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] AR★ Point Clouds ShapeFormer[Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] AR★ Fields 3DILG [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] AR★ Fields LION [Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] DM★ Point Clouds SDF-StyleGAN [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] GAN Fields NeuralWavelet [Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] DM★ Fields TriplaneDiffusion [Shue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022]⋄ DM★ Fields DiffusionSDF [Chou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022]⋄ DM★ Fields Ours DM★ Fields ★ Generative models in latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' ⋄ Works in submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Generative models for 3d shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zhang 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Michalkiewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019] or a vector [Mildenhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A 3D object is then implicitly defined by this neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Neural fields have gained lots of popularity as they can generate objects with arbitrary topolo- gies and infinite resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The methods are also called neural implicit representations or coordinate-based networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For neural fields for 3d shape modeling, we can categorize methods into global methods and local methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 1) The global methods encode a shape with a single global latent vector [Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Usually the capacity of these kind of methods is limited and 3 they are unable to encode shape details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2) The local methods use localized latent vectors which are defined for 3D positions defined on either a regular [Chibane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021] or irregular grid [Boulch and Marlet 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Genova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In contrast, we propose a latent representation where latent vectors do not have associated 3D positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Instead, we learn to represent a shape as a list of latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='2 Generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We have seen great success in different 2D image generative mod- els in the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Popular deep generative methods include generative adversarial networks (GANs) [Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2014], variational autoencoers (VAEs) [Kingma and Welling 2014], nor- malizing flows (NFs) [Rezende and Mohamed 2015], energy-based models [LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016], autoregressive models (ARs) [Esser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Van Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017] and more re- cently, diffusion models (DMs) [Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] which are the chosen generative model in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In 3D domain, GANs have been popular for 3D generation [Achliop- tas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Chen and Zhang 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Ibing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022], while only a few works are using NFs [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019] and VAEs [Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A lot of recent work employs ARs [Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Mittal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Nash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' DMs for 3D shapes are relatively unexplored compared to other generative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' There are several DMs dealing with point cloud data [Luo and Hu 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Due to the high freedom degree of regressed coordinates, it is always difficult to obtain clean manifold surfaces via post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' As mentioned before, we believe that neural fields are generally more suitable than point clouds for 3D shape generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The area of combining DMs and neural fields is still underexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The recent NeuralWavelet [Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] first encodes shapes (represented as signed distance fields) into the frequency domain with the wavelet transform, and then train DMs on the frequency coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' While this formulation is elegant, generative models generally work better on learned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Some concurrent works [Chou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Shue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] in submission also utilize DMs in a latent space for neural field generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The TriplaneDif- fusion [Shue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] trains an autodecoder first for each shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' DiffusionSDF [Chou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] runs a shape autoencoder based on triplane features [Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Summary of 3D generation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We list several 3d generation methods in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2, highlighting the choice of generative model (GAN, DM, EBM, NF, or AR) and the choice of data structure to represent 3D shapes (point clouds, meshes, voxels or fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 3 PRELIMINARIES An attention layer [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017] has three types of inputs: queries, keys, and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Queries Q = [q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' , q𝑁𝑞] ∈ R𝑑×𝑁𝑞 and keys K = [k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' , k𝑁𝑘 ] ∈ R𝑑×𝑁𝑘 are first compared to produce coefficients q⊺ 𝑗 k𝑖/ √ 𝑑 (they need to be normalized with the softmax function), 𝐴𝑖,𝑗 = q⊺ 𝑗 k𝑖/ √ 𝑑 �𝑁𝑘 𝑖=1 exp � q⊺ 𝑗 k𝑖/ √ 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' � (1) The coefficients are then used to (linearly) combine values V = [v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' , v𝑁𝑘 ] ∈ R𝑑𝑣×𝑁𝑘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We can write the output of an attention layer as follows, Attention(Q, K, V) = �o1 o2 · · o𝑁𝑞 � ∈ R𝑑𝑣×𝑁𝑞 = � 𝑁𝑘 ∑︁ 𝑖=1 𝐴𝑖,1v𝑖 𝑁𝑘 ∑︁ 𝑖=1 𝐴𝑖,2v𝑖 · · 𝑁𝑘 ∑︁ 𝑖=1 𝐴𝑖,𝑁𝑞v𝑖 � (2) Cross Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Given two sets A = � a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' , a𝑁𝑎 � ∈ R𝑑𝑎×𝑁𝑎 and B = � b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' , b𝑁𝑏 � ∈ R𝑑𝑏×𝑁𝑏 , the query vectors Q are con- structed with a linear function q(·) : R𝑑𝑎 → R𝑑 by taking elements of A as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Similarly, we construct K and V with k(·) : R𝑑𝑏 → R𝑑 and v(·) : R𝑑𝑏 → R𝑑, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The inputs of both k(·) and v(·) are from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Each column in the output of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (2) can be written as, o(a𝑗, B) = 𝑁𝑏 ∑︁ 𝑖=1 v(b𝑖) · 1 𝑍 (a𝑗, B) exp � q(a𝑗)⊺k(b𝑖)/ √ 𝑑 � , (3) where 𝑍 (a𝑗, B) = �𝑁𝑏 𝑖=1 exp � q(a𝑗)⊺k(b𝑖)/ √ 𝑑 � is a normalizing fac- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The cross attention operator between two sets is, CrossAttn(A, B) = �o(a1, B) o(a2, B) · · o(a𝑁𝑎, B)� ∈ R𝑑×𝑁𝑎 (4) Self Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In the case of self attention, we let the two sets be the same A = B, SelfAttn(A) = CrossAttn(A, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (5) 4 LATENT REPRESENTATION FOR NEURAL FIELDS Our representation is inspired by radial basis functions (RBFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We will therefore describe our surface representation design using RBFs as a starting point, and how we extended them using concepts from neural fields and the transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A continuous function can be represented with a set of weighted points in 3D using RBFs: ˆORBF(x) = 𝑀 ∑︁ 𝑖=1 𝜆𝑖 · 𝜙(x, x𝑖) (6) where 𝜙(x, x𝑖) is a radial basis function (RBF) and typically repre- sents the similarity (or dissimilarity) between two inputs, 𝜙(x, x𝑖) = 𝜙(∥x − x𝑖 ∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (7) Given ground-truth occupancies of x𝑖, the values of 𝜆𝑖 can be ob- tained by solving a system of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In this way, we can represent the continuous function O(·) as a set of 𝑀 points including their corresponding weights, � 𝜆𝑖 ∈ R, x𝑖 ∈ R3�𝑀 𝑖=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (8) However, in order to retain the details of a 3d shape, we often need a very large number of points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', 𝑀 = 80, 000 in [Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2001]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 4 Shape Encoding (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1) Shape Decoding (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='3) KL (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='2) latent queries Point Cloud Position Embeddings Surface Sampling Cross Attention K, V Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' latents KL Regularization Self Attention .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Self Attention .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Self Attention Query Points Position Embeddings Cross Attention K, V Q Target · · Isosurface Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Shape autoencoding pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Given a 3D ground-truth surface mesh as the input, we first sample a point cloud that is mapped to positional embeddings and encode them into a set of latent codes through a cross-attention module (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Next, we perform (optional) compression and KL- regularization in the latent space to obtain structured and compact latent shape representations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Finally, the self-attention is carried out to aggregate and exchange the information within the latent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' And a cross-attention module is designed to calculate the interpolation weights of query points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The interpolated feature vectors are fed into a fully connected layer for occupancy prediction (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' This representation does not benefit from recent advances in repre- sentation learning and cannot compete with more compact learned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We therefore want to modify the representation to change it into a neural field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' One approach to neural fields is to represent each shape as a separate neural network (making the network weights of a fixed size network the representation of a shape) and train a diffusion process as hypernetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A second approach is to have a shared encoder-decoder network for all shapes and represent each shape as a latent computed by the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We opt for the second approach, as it leads to more compact representations because it is jointly learned from all shapes in the data set and the network weights themselves do not count towards the latent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Such a neural field takes a tuple of coordinates x and 𝐶-dimensional latent f as input and outputs occupancy, ˆONN(x) = NN(x, f), (9) where NN : R3 × R𝐶 → [0, 1] is a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A first approach was to use a single global latent f, but a major limitation is the ability to encode shape details [Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Some follow-up works study coordinate-dependent latents [Chibane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] that combine traditional data structures such as regular grids with the neural field concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Latent vectors are arranged in a spatial data structure and then interpolated (trilinearly) to obtain the coordinate-dependent latent fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A recent work 3DILG [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] proposed a sparse representation for 3D shapes, using latents f𝑖 arranged in an irregular grid at point locations x𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The final coordinate-dependent latent fx is then estimated by kernel regression, fx = ˆFKN(x) = 𝑀 ∑︁ 𝑖=1 f𝑖 · 1 𝑍 � x, {x𝑖}𝑀 𝑖=1 � 𝜙(x, x𝑖), (10) where 𝑍 � x, {x𝑖}𝑀 𝑖=1 � = �𝑀 𝑖=1 𝜙(x, x𝑖) is a normalizing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Thus the representation for a 3D shape can be written as � f𝑖 ∈ R𝐶, x𝑖 ∈ R3�𝑀 𝑖=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (11) After that, an MLP : R𝐶 → [0, 1] is applied to project the approxi- mated feature ˆFKN(x) to occupancy, ˆO3DILG(x) = MLP � ˆFKN(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (12) Neural networks with latent sets (proposed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We initially explored many variations for 3D shape representation based on irregular and regular grids as well as tri-planes, frequency compositions, and other factored representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Ultimately, we could not improve on existing irregular grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' However, we were able to achieve a significant improvement with the following chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We aim to keep the structure of an irregular grid and the interpolation, but without representing the actual spatial position explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We let the net- work encode spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Both the representations (RBF in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (6) and 3DILG in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (10)) are composed by two parts, values and similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We keep the structure of the interpolation, but elmini- tate explicit point coordinates and integrate cross attention from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The result is the following learnable function approximator, ˆF (x) = 𝑀 ∑︁ 𝑖=1 v(f𝑖) · 1 𝑍 � x, {f𝑖}𝑀 𝑖=1 � 𝑒q(x)⊺k(f𝑖)/ √ 𝑑, (13) where 𝑍 � x, {f𝑖}𝑀 𝑖=1 � = �𝑀 𝑖=1 𝑒q(x)⊺k(f𝑖)/ √ 𝑑 is a normalizing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Similar to the MLP in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 12, we apply a single fully connected layer to get desired occupancy values, ˆO(x) = FC � ˆF (x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (14) Compared to 3DILG and all other coordinate-latent-based methods, we dropped the dependency of the coordinate set {x𝑖}𝑀 𝑖=1, the new 5 Cross Attention K, V Q Learnable (a) Learnable Queries Cross Attention K, V Q Subsample and Copy (b) Point Queries Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Two ways to encode a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (a) uses a learnable query set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (b) uses a downsampled version of input point embeddings as the query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' representation only contains a set of latents, � f𝑖 ∈ R𝐶�𝑀 𝑖=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (15) An alternative view of our proposed function approximator is to see it as cross attention between query points x and a set of latents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5 NETWORK ARCHITECTURE FOR SHAPE REPRESENTATION LEARNING In this section, we will discuss how we design a variational autoen- coder based on the latent representation proposed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The architecture has three components discussed in the following: a 3D shape encoder, KL regularization block, and a 3D shape decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1 Shape encoding We sample the surfaces of 3D input shapes in a 3D shape dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' This results in a point clouds of size 𝑁 for each shape, {x𝑖 ∈ R3}𝑁 𝑖=1 or in matrix form X ∈ R3×𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' While the dataset used in the paper originally represents shapes as triangle meshes, our framework is directly compatible with other surface representations, such as scanned point clouds, spline surfaces, or implicit surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In order to learn representations in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (15), the first challenge is to aggregate the information contained in a possibly large point cloud {x𝑖}𝑁 𝑖=1 into a smaller set of latent vectors {f𝑖}𝑀 𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We design a set-to-set network to this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A popular solution to this problem in previous work is to divide the large point cloud into a smaller set of patches and to learn one latent vector per patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Although this is a very well researched and standard component in many networks, we discovered a more successful way to aggregate features from a large point cloud that is better compatible with the transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We considered two options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' One way is to define a learnable query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Inspired by DETR [Car- ion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020] and Perceiver [Jaegle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021], we use the cross attention to encode X, Enclearnable(X) = CrossAttn(L, PosEmb(X)) ∈ R𝐶×𝑀, (16) where L ∈ R𝐶×𝑀 is a learnable query set where each entry is 𝐶- dimensional, and PosEmb : R3 → R𝐶 is a column-wise positional embedding function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Another way is to utilize the point cloud itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We first subsample the point cloud X to a smaller one with furthest point sampling, X0 = FPS(X) ∈ R3×𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The cross attention is applied to X0 and X, Encpoints(X) = CrossAttn(PosEmb(X0), PosEmb(X)), (17) which can also be seen as a “partial” self attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 4 for an illustration of both design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Intuitively, the number 𝑀 affects the reconstruction performance: the larger the 𝑀, the better reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' However, 𝑀 strongly affects the training time due to the transformer architecture, so it should not be too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In our final model, the number of latents 𝑀 is set as 512, and the number of channels 𝐶 is 512 to provide a trade off between reconstruction quality and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='2 KL regularization block Latent diffusion [Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] proposed to use a variational autoencoder (VAE) [Kingma and Welling 2014] to compress images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We adapt this design idea for our 3D shape representation and also regularize the latents with KL-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We should note that the KL regularization is optional and only necessary for the second-stage diffusion model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' If we just want a method for surface reconstruction from point clouds, we do not need the KL regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We first linear project latents to mean and variance by two net- work branches, respectively, FC𝜇 (f𝑖) = �𝜇𝑖,𝑗 � 𝑗 ∈[1,2,···,𝐶0] FC𝜎 (f𝑖) = � log𝜎2 𝑖,𝑗 � 𝑗 ∈[1,2,···,𝐶0] (18) where FC𝜇 : R𝐶 → R𝐶0 and FC𝜎 : R𝐶 → R𝐶0 are two linear projection layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We use a different size of output channels 𝐶0, where 𝐶0 ≪ 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' This compression enables us to train diffusion models on smaller latents of total size 𝑀 · 𝐶0 ≪ 𝑀 · 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We can write the bottleneck of the VAE formally, ∀𝑖 ∈ [1, 2, · · · , 𝑀], 𝑗 ∈ [1, 2, · · · ,𝐶0], 𝑧𝑖,𝑗 = 𝜇𝑖,𝑗 + 𝜎𝑖,𝑗 · 𝜖, (19) where 𝜖 ∼ N (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The KL regularization can be written as, Lreg � {f𝑖}𝑀 𝑖=1 � = 1 𝑀 · 𝐶0 𝑀 ∑︁ 𝑖=1 𝐶0 ∑︁ 𝑗=1 1 2 � 𝜇2 𝑖,𝑗 + 𝜎2 𝑖,𝑗 − log𝜎2 𝑖,𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (20) In practice, we set the weight for KL loss as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='001 and report the performance for different values of𝐶0 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our recommended setting is 𝐶0 = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='3 Shape decoding To increase the expressivity of the network, we add a latent learning network between the two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Because our latents are a set of vectors, it is natural to use transformer networks here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Thus, the proposed network here is a series of self attention blocks, {f𝑖}𝑀 𝑖=1 ← SelfAttn(𝑙) � {f𝑖}𝑀 𝑖=1 � , for 𝑖 = 1, · · · , 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (21) The SelfAttn(·) with a superscript (𝑙) here means 𝑙-th block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The latents {f𝑖}𝑀 𝑖=1 obtained using either Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (16) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (17) are fed into the self attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Given a query x, the corresponding latent is interpolated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (13), and the occupancy is obtained with a fully connected layer as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' KL regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Given a set of latents {f𝑖 ∈ R𝐶 }𝑀 𝑖=1 obtained from the shape encoding in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1, we employ two linear projection layers FC𝜇, FC𝜎 to predict the mean and variance of a low-dimensional latent space, where a KL regularization commonly used in VAE training is applied to constrain the feature diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Then, we obtain smaller latents {z𝑖 ∈ R𝐶0 } of size 𝑀 · 𝐶0 ≪ 𝑀 · 𝐶 via reparametrization sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Finally, the compressed latents are mapped back to the original space by FCup to obtain a higher dimensionality for the shape decoding in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Forward Diffusion Process Reverse Diffusion Process Add Noise Add Noise Add Noise Denoise Denoise Denoise Condition Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Latent set diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The diffusion model operates on compressed 3D shapes in the form of a regularized set of latent vectors {z𝑖 }𝑀 𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Self Attention Self Attention · · (a) Unconditional Denoising Network Self Attention Cross Attention K V Q · · Condition (b) Conditional Denoising Network Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Denoising network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our denoising network is composed of several denoising layers (a box in the figure denotes a layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The denoising layer for unconditional generation contains two sequential self attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The denoising layer for conditional generation contains a self attention and a cross attention block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The cross attention is for injecting condition information such as categories, images or partial point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We optimize the binary cross entropy loss between our approximated function and the ground-truth indicator function as in prior works [Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Lrecon � {f𝑖}𝑀 𝑖=1, O � = Ex∈R3 � BCE � ˆO(x), O(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (22) Surface reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We sample query points in a grid of res- olution 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The final surface is reconstructed with Marching Cubes [Lorensen and Cline 1987].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 6 SHAPE GENERATION Our proposed diffusion model combines design decisions from latent diffusion (the idea of the compressed latent space), EDM [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] (most of the training details), and our shape representation design (the architecture is based on attention and self-attention instead of convolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We train diffusion models in the latent space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', the bottleneck in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Following the diffusion formulation in EDM [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022], our denoising objective is En𝑖∼N(0,𝜎2I) 1 𝑀 𝑀 ∑︁ 𝑖=1 ���Denoiser � {z𝑖 + n𝑖}𝑀 𝑖=1, 𝜎, C � 𝑖 − z𝑖 ��� 2 2 , (23) where Denoiser(·, ·, ·) is our denoising neural network, 𝜎 is the noise level, and C is the optional conditional information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', categories, images, partial point clouds and texts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We denote the corresponding output of z𝑖 +n𝑖 with the subscript 𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Denoiser(·, ·, ·)𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We should minimize the loss for every noise level 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The sampling is done by solving ordinary/stochastic differential equations (ODE/SDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 6 for an illustration and EDM [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] for a detailed description for both the forward (training) and reverse (sampling) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The function Denoiser(·, ·, ·) is a set denoising network (set-to-set function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The network can be easily modeled by a self-attention transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Each layer consists of two attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The first one is a self attention for attentive learning of the latent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The second one is for injecting the condition information C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 7 (b)) as in prior works [Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For simple information like categories, C is a learnable embedding vector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=', 55 different embedding vectors for 55 categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For a single-view image , we use ResNet-18 [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016] as the context encoder to extract a global feature vector as condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For text conditioning, we use BERT [Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2018] to learn a global feature vector as C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For partial point clouds, we use the shape encoder introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1 to obtain a set of latent embeddings as C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In the case of unconditional generation, the cross attention degrades to self attention (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 7 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 7 EXPERIMENTAL SETUP We use the dataset of ShapeNet-v2 [Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2015] as a bench- mark, containing 55 categories of man-made objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We use the training/val splits in [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We preprocess shapes as in [Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Each shape is first converted to a water- tight mesh, and then normalized to its bounding box, from which we further sample a dense surface point cloud of size 50,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' To learn the neural fields, we randomly sample 50,000 points with occupan- cies in the 3D space, and 50,000 points with occupancies in the near surface region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For the single-view object reconstruction, we use the 2D rendering dataset provided by 3D-R2N2 [Choy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2016], where each shape is rendered into RGB images of size of 224 × 224 from 24 random viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For text-driven shape generation, we use the text prompts of ShapeGlot [Achlioptas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For data preprocess of shape completion training, we create partial point clouds by sampling point cloud patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1 Baselines For shape auto-encoding, we conduct experiments against state- of-the-art methods for implicit surface reconstruction from point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We use OccNet [Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019], ConvOccNet [Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020], IF-Net [Chibane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2020], and 3DILG [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The OccNet is the first work of learning neural fields from a single global latent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' ConvOccNet and IF-Net 7 OccNet ConvOccNet IF-Net 3DILG Ours Learned Queries Point Queries table 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='823 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='975 mean (selected) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='979 F-Score ↑ mean (all) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='970 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Shape autoencoding (surface reconstruction from point clouds) on ShapeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We show averaged metrics on all 55 categories and individual metrics for the 7 largest categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 𝑀 = 512 𝑀 = 256 𝑀 = 128 𝑀 = 64 IoU ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='940 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='916 Chamfer ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='049 F-Score ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='929 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Results for different number of latents 𝑀 for shape autoencoding 𝐶0 = 1 𝐶0 = 2 𝐶0 = 4 𝐶0 = 8 𝐶0 = 16 𝐶0 = 32 𝐶0 = 64 IoU ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='964 Chamfer ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='038 F-Score ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='970 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Ablation study of compression via the number of channels𝐶0 for shape (variational) autoencoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Grid-83 3DILG Ours 𝐶0 = 8 𝐶0 = 16 𝐶0 = 32 𝐶0 = 64 Surface-FPD ↓ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='97 Surface-KPD (×103) ↓ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='11 Rendering-FID ↓ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='78 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='83 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='25 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='26 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='08 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='24 Rendering-KID (×103) ↓ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='51 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='60 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='76 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Unconditional generation on full ShapeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' PVD Ours Surface-FPD ↓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='63 Surface-KPD (×103) ↓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='53 Rendering-FID ↓ 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='64 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='08 Rendering-KID (×103) ↓ 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='75 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Unconditional generation on full ShapeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' learn local neural fields based on latent vectors arranged in a regular grid, while 3DILG uses latent vectors on an irregular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For 3D shape generation, we compare against recent state-of-the- art generative models, including PVD [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021], 3DILG [Zhang 8 Input GT OccNet ConvONet IF-Net 3DILG Proposed Learnable Queries Point Queries Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Visualization of shape autoencoding results (surface reconstruction from point clouds from ShapeNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022], and NeuralWavelet [Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' PVD is a diffusion model for 3D point cloud generation, and 3DILG utilizes autore- gressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' NeuralWavelet utilized diffusion models in the frequency domain of shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 9 Ours 3DILG Grid-83 PVD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Unconditional generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' All models are trained on full ShapeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' airplane chair table car sofa 3DILG NW Ours 3DILG NW Ours 3DILG NW Ours 3DILG NW Ours 3DILG NW Ours Surface-FID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='77 Surface-KID (×103) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='87 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='70 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Category conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' NW is short for NeuralWavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The dash sign “-” means the method NeuralWavelet does not release models trained on these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='2 Evaluation metrics To evaluate the reconstruction accuracy of shape auto-encoding from point clouds, we adopt Chamfer distance, volumetric Intersection- over-Union (IoU), and F-score as primary evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' IoU is computed based on the occupancy predictions of 50𝑘 querying points sampled in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Chamfer distance and F-score are cal- culated between two sampled point clouds with the size of 50𝑘 respectively from reconstructed and ground-truth surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For IoU and F-score, higher is better, while for Chamfer, lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' To measure the mesh quality of unconditional and conditional shape generation, we follow [Ibing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Shue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] to adapt the Fréchet Inception Distance (FID) and Kernel Inception Distance (KID) commonly used to assess the image gener- ative models to rendered images of 3d shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' To calculate FID and KID of rendered images, we render each shape from 10 viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The metrics are named as Rendering-FID and Rendering-KID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The Rendering-FID is defined as, Rendering-FID = ∥𝜇g − 𝜇r∥ +𝑇𝑟 � Σ𝑔 + Σ𝑟 − 2(Σ𝑔Σ𝑟)1/2� (24) where 𝑔 and 𝑟 denotes the generated and training datasets respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 𝜇 and Σ are the statistical mean and covariance matrix of the feature distribution extracted by the Inception network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The Rendering-KID is defined as, Rendering-KID = MMD � 1 |R| ∑︁ x∈R max y∈G 𝐷(x, y) �2 (25) where 𝐷(x, y) is a polynomial kernel function to evaluate the simi- larity of two samples, G and R are feature distributions of generated set and reference set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The function MMD(·) is Maxi- mum Mean Discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' However, the rendering-based FID and KID are essentially designed to understand 3D shapes from 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Thus, they have the inherent issue of not accurately understanding shape compositions in the 3D world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' To compensate their draw- backs, we also adapt the FID and KID to 3D shapes directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For each generated or groud-truth shape, we sample 4096 points (with nor- mals) from the surface mesh and then feed them into a pre-trained PointNet++ [Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2017b] to extract a global latent vector, repre- senting the global structure of the 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The PointNet++ is first pretrained on shape classification on ShapeNet-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' As we use point clouds, we call the FID and KID for 3D shapes as Fréchet PointNet++ Distance (FPD) and Kernel PointNet++ Distance (KPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The two metrics are defined similarly as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (24) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' (25), except that the features are extracted from a PointNet++ network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 10 Ours NW 3DILG Grid-83 Ours NW 3DILG Grid-83 Ours NW 3DILG Grid-83 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Category-conditional generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' From top to bottom, we show category (airplane, chair, table) conditioned generation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='3 Implementation For the shape auto-encoder, we use the point cloud of size 2048 as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' At each iteration, we individually sample 1024 query points from the bounding volume ([−1, 1]3) and the other 1024 points from near surface region for the occupancy values prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The shape auto-encoder is trained on 8 A100, with batch size of 512 for 𝑇 = 1, 600 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The learning rate is linearly increased to 𝑙𝑟max = 5𝑒 − 5 in the first 𝑡0 = 80 epochs, and then gradually decreased using the cosine decay schedule 𝑙𝑟max ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='51+𝑐𝑜𝑠 ( 𝑡−𝑡0 𝑇 −𝑡0 ) until reaching the minimum value of 1𝑒 − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The diffusion models are trained on 4 A100 with batch size of 256 for 𝑇 = 8, 000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The learning rate is linearly increased to 𝑙𝑟𝑚𝑎𝑥 = 1𝑒 − 4 in the first 𝑡0 = 800 epochs, and then gradually decreased using the above mentioned decay schedule until reaching 1𝑒 − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We use the default settings for the hyperparameters of EDM [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' During sampling, we obtain the final latent set via only 18 denoising steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8 RESULTS We present our results for multiple applications: 1) shape auto- encoding, 2) unconditional generation, 3) category-conditioned generation, 4) text-conditioned generation, 5) shape completion, 11 6) image-conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Finally, we perform a shape nov- elty analysis to validate that we are not overfitting to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1 Shape Auto-Encoding We show the quantitative results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 3 for a deterministic au- toencoder without the KL block described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In particular, we show results for the largest 7 categories as well as averaged re- sults over the categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The two design choices of shape encoding described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='1 are also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The case of using the subsampled point cloud as queries is better than learnable queries in all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Thus we use subsampled point clouds in our later ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The visualization of reconstruction results can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We visualize some extremely difficult shapes from the datasets (test split).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' These shapes often contain some thin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' However, our method still performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Ablation study of the number of latents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The number 𝑀 is the number of latent vectors used in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Intuitively, a larger 𝑀 leads to a better reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We show results of 𝑀 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Thus, in all of our experiments, 𝑀 is set to 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We are limited by computation time to work with larger 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Ablation study of the KL block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We described the KL block in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='2 that leads to additional compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In addition, this block changes the deterministic shape encoding into a variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The introduced hyperparameter is 𝐶0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' A smaller 𝐶0 leads to a higher compression rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The choice of𝐶0 is ablated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Clearly, larger 𝐶0 gives better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The reconstruction results of𝐶0 = 8, 16, 32, 64 are very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' However, they differ significantly in the second stage, because a larger latent size could make the training of diffusion models more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' This result is very encouraging for our model, because it indicates that aggressively increasing the compression in the KL block does not decrease reconstruction performance too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We can also see that compressing with the KL block by de- creasing 𝐶0 is much better than compressing using fewer latent vectors 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='2 Unconditional Shape Generation Comparison with surface generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We evaluate the task of un- conditional shape generation with the proposed metrics in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We also compared our method with a baseline method proposed in [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The method is called Grid-83 because the latent grid size is 83, which is exactly the same as in AutoSDF [Mittal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The table also shows the results of different 𝐶0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our results are best when 𝐶0 = 32 in all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' When 𝐶0 = 64 the results become worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' This also aligns with our conjecture that a larger latent size makes the training more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Comparison with point cloud generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Additionally, we compare our method with PVD [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021] which is a point cloud diffusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We re-train PVD using the official released code on our preprocessed dataset and splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We use the same evaluation protocol as before but with one major difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Since PVD can only generate point clouds without normals, we use another pretrained PointNet++ (without normals) as the feature extractor to calculate Surface-FPD and Surface-KPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 7 shows we can beat PVD by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Additionally, we also show the metrics calculated AutoSDF Ours “horizontal slats on top of back” “one big hole between back and seat” “this chair has wheels” “vertical back ribs” Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Text conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For each text prompt, we generate 3 shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our results (Right) are compared with AutoSDF (Left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' on rendered images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Visualization of generated results can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='3 Category-conditioned generation We train a category-conditioned generation model using our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We evaluate our models in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We should note that the competitor method NeuralWavelet [Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] trains models for categories separately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' thus, NeuralWavelet is not a true category-conditioned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We also visualize some results (airplane, chair, and table) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our training is more challenging, as we train on a dataset that is an order of magnitude larger and we train for all classes jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' While NeuralWavelet already has good results, the joint training is necessary / beneficial for many subsequent applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='4 Text-conditioned generation The results of our text-conditioned generation model can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Since the model is a probabilistic model, we can sample shapes given a text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The results are very encouraging and they constitute the first demonstration of text-conditioned 3D shape generation using diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' To the best of our knowledge, there are no published competing methods at the point of submitting this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='5 Probabilistic shape completion We also extend our diffusion model for probablistic shape comple- tion by using a partial point cloud as conditioning input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The compar- ison against ShapeFormer [Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022] is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' As seen, our latent set diffusion can predict more accurate completion, and we also have the ability to achieve more diverse generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 12 GT Condition ShapeFormer Ours Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Point cloud conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We show three generated results given a partial cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The ground-truth point cloud and the partial point cloud used as condition are shown in Left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We compare our results (Right) with ShapeFormer (Middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Condition IM-Net OccNet Ours Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Image conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In the left column we show the condition image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In the middle we show results obtained by the method IM-Net and OccNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our generated results are shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='6 Image-conditioned shape generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' We also provide comparisons on the task of single-view 3D object reconstruction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Compared to other deterministic methods including OccNet [Mescheder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2019] and IM-Net [Chen and Zhang 2019], our latent set diffusion can not only reconstruct more accurate surface details, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' long rods and tiny holes in the back), but also support multi-modal prediction, which is a desired property to deal with severe occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Ref Gen Ref Gen Ref Gen Ref Gen Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Shape generation novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For a generated shape, we retrieve the top-1 similar shape in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The similarity is measured using Chamfer distance of sampled surface point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In each pair, we show the retrieved shape (left) and the generated shape (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The generated shapes are from our category-conditioned generation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='7 Shape novelty analysis We use shape retrieval to demonstrate that we are not simply over- fitting to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Given a generated shape, we measure the Chamfer distance between it and training shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' The visualization of retrieved shapes can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Clearly, the model can synthesize new shapes with realistic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content='8 Limitations While our method shows convincing results on a variety of tasks, our design choices also have drawbacks that we would like to dis- cuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' For instance, we require a two stage training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' While this leads to improved performance in terms of generation quality, training the first stage is more time consuming than relying on manually-designed features such as wavelets [Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In addition, the first stage might require retraining if the shape data in consideration changes, and for the second stage – the core of our diffusion architecture – training time is also relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Overall, we believe that there is significant potential for future research av- enues to speed up training, in particular, in the context of diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 9 CONCLUSION We have introduced 3DShape2VecSet, a novel shape representation for neural fields that is tailored to generative diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' To this end, we combine ideas from radial basis functions, previous neural field architectures, variational autoencoding, as well as cross attention and self-attention to design a learnable representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Our shape representation can take a variety of inputs including triangle meshes and point clouds and encode 3D shapes as neu- ral fields on top of a set of latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' As a result, our method demonstrates improved performance in 3D shape encoding and 3D shape generative modeling tasks, including unconditioned genera- tion, category-conditioned generation, text-conditioned generation, point-cloud completion, and image-conditioned generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In future work, we see many exciting possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Most impor- tantly, we believe that our model further advances the state of the art in point cloud and shape processing on a large variety of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In particular, we would like to employ the network architecture of 3DShape2VecSet to tackle the problem of surface reconstruction from scanned point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In addition, we can see many applica- tions for content-creation tasks, for example 3D shape generation of textured models along with their material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Finally, we 13 would like to explore editing and manipulation tasks leveraging pretrained diffusion models for 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' Linqi Zhou, Yilun Du, and Jiajun Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 3d shape generation and completion through point-voxel diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 5826–5835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItFJT4oBgHgl3EQfFyzw/content/2301.11445v1.pdf'} diff --git a/JdAyT4oBgHgl3EQffviy/vector_store/index.pkl b/JdAyT4oBgHgl3EQffviy/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..23bddf8f258de7538439089572213c6b712a79f2 --- /dev/null +++ b/JdAyT4oBgHgl3EQffviy/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1b9ac8305deaec0f6c7246bd2c401c1e5a8c78e60d5c9c05865af58b838daf7b +size 78949 diff --git a/LNAyT4oBgHgl3EQfgPhD/content/tmp_files/2301.00354v1.pdf.txt b/LNAyT4oBgHgl3EQfgPhD/content/tmp_files/2301.00354v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6a5c3f20ed57baaefcb748ab1c196f8c4efb303 --- /dev/null +++ b/LNAyT4oBgHgl3EQfgPhD/content/tmp_files/2301.00354v1.pdf.txt @@ -0,0 +1,1714 @@ +RiskProp: Account Risk Rating on Ethereum via De-anonymous +Score and Network Propagation +Dan Lin +School of Software Engineering, +Sun Yat-sen University +Zhuhai, China +lind8@mail2.sysu.edu.cn +Jiajing Wu∗ +School of Computer Science and +Engineering, Sun Yat-sen University +Guangzhou, China +wujiajing@mail.sysu.edu.cn +Qishuang Fu +School of Computer Science and +Engineering, Sun Yat-sen University +Guangzhou, China +fuqsh6@mail2.sysu.edu.cn +Zibin Zheng +School of Software Engineering, +Sun Yat-sen University +Zhuhai, China +zhzibin@mail.sysu.edu.cn +Ting Chen +University of Electronic Science and +Technology of China +Guangzhou, China +brokendragon@uestc.edu.cn +ABSTRACT +As one of the most popular blockchain platforms supporting smart +contracts, Ethereum has caught the interest of both investors and +criminals. Differently from traditional financial scenarios, executing +Know Your Customer verification on Ethereum is rather difficult +due to the pseudonymous nature of the blockchain. Fortunately, +as the transaction records stored in the Ethereum blockchain are +publicly accessible, we can understand the behavior of accounts or +detect illicit activities via transaction mining. Existing risk control +techniques have primarily been developed from the perspectives of +de-anonymizing address clustering and illicit account classification. +However, these techniques cannot be used to ascertain the potential +risks for all accounts and are limited by specific heuristic strate- +gies or insufficient label information. These constraints motivate +us to seek an effective rating method for quantifying the spread +of risk in a transaction network. To the best of our knowledge, +we are the first to address the problem of account risk rating on +Ethereum by proposing a novel model called RiskProp, which in- +cludes a de-anonymous score to measure transaction anonymity +and a network propagation mechanism to formulate the relation- +ships between accounts and transactions. We demonstrate the ef- +fectiveness of RiskProp in overcoming the limitations of existing +models by conducting experiments on real-world datasets from +Ethereum. Through case studies on the detected high-risk accounts, +we demonstrate that the risk assessment by RiskProp can be used +to provide warnings for investors and protect them from possible +financial losses, and the superior performance of risk score-based +account classification experiments further verifies the effectiveness +of our rating method. +KEYWORDS +Abnormal detection, network propagation, Ethereum, risk control, +de-anonymization +1 +INTRODUCTION +Ethereum [30] has the second-largest market cap in the blockchain +ecosystem. The account model is adopted on Ethereum, and the +native cryptocurrency on Ethereum is named Ether (abbreviated as +∗Corresponding author. +“ETH”), which is widely accepted as payments and transferred from +one account to another. It is known that Ethereum accounts are +indexed according to pseudonyms, and the creation of accounts is +almost cost-free. This anonymous nature and the lack of regulation +result in the bad reputation of Ethereum and other blockchain sys- +tems for breeding malicious behaviors and enabling fraud, thereby +resulting in large property losses for investors. As reported in a +Chainalysis Crime Report, the illicit share of all cryptocurrency +activities was valued at nearly USD 2.7 billion in 2020. These losses +illustrate that Know-Your-Customer (KYC) and risk control of ac- +counts are critical and necessary. Risk control [23] not only helps +wallet customers identify risky accounts and avoid losses but also +plays a vital role in the anti-money laundering of virtual asset +service providers, such as cryptocurrency exchanges. +Therefore, a wealth of efforts have been expended in risk con- +trol on Ethereum in recent years. In September 2020, the Financial +Action Task Force (FATF) published a recommendation report on +virtual assets and released information on Red Flag Indicators [11] +related to transactions, anonymity, senders or recipients, the source +of funds, and geographical risks. In addition, researchers in the aca- +demic community have proposed various techniques from the per- +spectives of address clustering and illicit account classification. Ad- +dress clustering techniques perform entity identification of anony- +mous accounts. For example, Victor [27] proposes several clustering +heuristics for Ethereum accounts and clusters 17.9% of all active ex- +ternally owned accounts. Illicit account detection techniques focus +on training classifiers based on well-designed features extracted +from transactions [8, 10, 32]. Moreover, some researchers have de- +veloped methods for automatic feature extraction incorporating +structural information [19, 21, 29, 33]. +However, there are still some limitations (L) associated with +these techniques. L1: Account clustering techniques can only be +applied to part of accounts and therefore have limited applicability, +and most accounts beyond heuristic rules cannot thus be identified. +L2: In the existing methods for illicit account detection, binary +classifiers are usually trained via supervised learning. However, as +only a very small percentage of risky nodes have clear labels, which +are required for these methods, the vast majority of accounts that +may be involved in malicious events are unlabeled. In particular, +arXiv:2301.00354v1 [cs.SI] 1 Jan 2023 + +WWW ’23, April 30–May 4, 2023, Austin, TX, US +Anonymous author(s) +TxnHash: 0x0bf742... +From: 0x3da2b... +To: 0x41b53... +Timestamp: 1525153486 +Value: 0.1 Ether +TxnFee: 0.000861 Ether +Customer +... +... +Scammer +Scammer +Exchange +Txns +Txns +Txns +... +Account risk rating +Public transactions +Ethereum blockchain +0x3da2b... +0x41b53... +Figure 1: The procedure of ETH transfer in Ethereum. +“From” denotes the sender, “To” denotes the receiver, and +“Txn” denotes “Transaction”. +risky accounts with few transactions or unseen patterns are likely +to be misidentified in practical use. +To address the limitations presented above, we explore risk con- +trol on Ethereum from a new perspective: Account risk rating. In +traditional financial scenarios, credit scoring is usually conducted +by authorized financial institutions, which perform audits on their +customers to fully understand their identity, background, and fi- +nancial credit standing. Similarly to credit scoring, risk rating on +Ethereum can help us quantify the latent risk of a transaction or +account with a quantitative score, thereby combating money laun- +dering and identifying potential scams before new victims emerge. +In terms of the abovementioned L1, in contrast to the traditional +account clustering method, which can only de-anonymize a small +number of accounts, the account risk method proposed in this paper +can obtain quantitative risk indicators for all accounts. Regarding +L2, the proposed risk rating method can achieve decent perfor- +mance in an unsupervised manner without feeding labels. The +output of the proposed method is risk values, which are provided +continuously and allow evaluation of the severity of risk. +Compared with traditional financial scenarios, several unique +challenges (C) are encountered in the task of account risk rating on +Ethereum. C1: Nature of anonymity. Transactions on Ethereum +do not require real-name verification. Even worse, perpetrators of +some malicious activities deliberately enhance their anonymity to +counter the impact of de-anonymizing clustering techniques [29]. +C2: Complex transaction relationship. Compared with tradi- +tional financial scenarios, a user or entity on Ethereum may control +a large number of accounts at almost no cost, and the transaction +relationship between accounts is also more complex. How to quan- +tify the impact of trading behavior between accounts on account +risk is a challenging core problem. +To overcome the challenges mentioned above, we propose a +novel approach called Risk Propagation (RiskProp) for Ethereum ac- +count rating. It comprises two core designs, namely de-anonymous +score and a network propagation mechanism. To resolve C1, de- +anonymous score measures the degree to which transactions remain +anonymous. For example, both the payer and the payee of an illicit +transaction prefer to have a small number of transactions to en- +sure anonymity-preserving protection. In contrast, both sides of a +licit transaction may participate in numerous interactions without +evading the impact of the de-anonymized clustering algorithm. Af- +terward, to resolve C2, we model the massive transaction records +as a directed bipartite graph and introduce a network propagation +mechanism with three interdependent metrics, namely Confidence +of the de-anonymous score, Trustiness of the payee, and Reliability +of the payer. Intuitively, payees with higher trustiness receive trans- +actions with higher de-anonymous scores, and payers with higher +reliability will send transactions with higher confidence. Clearly, +reliability, trustiness, and confidence are related to each other, so +we define five items of prior knowledge that these metrics should +satisfy and propose three mutually recursive equations to estimate +the values of these metrics. To verify the effectiveness of the pro- +posed risk rating method and further illustrate the significance of +rating for risk control on Ethereum, we evaluate the effect of the +risk rating system via experiments from two aspects, i.e., analysis of +risk rating results and rating score-based illicit/licit classification. +Overall, our contributions are summarized as follows: +• A new perspective for Ethereum risk control. This paper is +the first to propose tackling the problem of Ethereum risk control +via the perspective of account risk rating. +• A novel risk metric for transactions. We creatively develop a +metric called de-anonymous score for transactions, which measures +the degree of de-anonymization to quantify the risk of a transaction. +• An effective method and interesting insights. We implement +a novel risk rating method called RiskProp and demonstrate its su- +perior effectiveness and efficiency via experiments on a real-world +Ethereum transaction dataset together with theoretical analysis. By +analyzing the rating results and case studies on high-risk accounts, +we obtain interesting insights into the Ethereum ecosystem and +further show how our method could prevent financial losses ahead +of blacklisting malicious accounts. +2 +PRELIMINARY +2.1 +Ethereum Financial Background +Ether is the native “currency” on Ethereum and plays a fundamen- +tal part in the Ethereum payment system. Ether can be paid or +received in financial activities, just like currency in real life. In +conventional financial scenarios, a Know Your Customer (KYC) +check is the mandatory process to identify and verify a customer’s +identity when opening an account and to periodically understand +the legitimacy of the involved funds over time. However, unlike +traditional transaction systems, where customers’ identity informa- +tion is required and obtained in KYC checks, Ethereum accounts +are designed as pseudonymous addresses identified by 20 bytes of +public key information generated by cryptographic algorithms, for +example, “0x99f154f6a393b088a7041f1f5d0a7cbfa795d301”. +Figure 1 depicts the risky scenario of Ether transfer in aspects of +data acquisition. It includes three layers: 1) Ethereum blockchain. +The Ethereum historical data are irreversible and publicly trace- +able on the chain. 2) Public transactions. The transaction denotes +a signed data package from an account to another account, in- +cluding the sending address, receiver address, transferred Ether +amount, etc. 3) Account risk rating. Usually, the identities who con- +trol the accounts are not labeled. Customers may become involved +in suspicious financial crimes or be vulnerable to frauds and scams. +Furthermore, the illicit funds can be laundered and cashed out via +exchanges. In this procedure, our proposed RiskProp is implemented +to measure the risk of unlabeled accounts that may have ill inten- +tions and alert customers when engaging in suspicious, potentially +illegal transactions. + +RiskProp +WWW ’23, April 30–May 4, 2023, Austin, TX, US +2.2 +The Nature of Blockchain: Anonymity +It is known that the Ethereum account is identified as a pseudony- +mous address. However, if customers repeatedly use the same ad- +dress as on-chain identification, the relationship between addresses +becomes linkable via public transaction records. Accounts that +participate in more transactions and connect with more accounts +experience degrading anonymity [29]. To reduce the likelihood +of exposure, criminals naturally tend to initiate transactions with +fewer accounts. Here is an example on Ethereum: The two accounts +of transaction 0x9a9d have only three transactions and became in- +active thereafter. These two accounts are considered suspicious and +reported as relevant accounts of Upbit exchange hack. On the con- +trary, entities who do not deliberately take anonymity-preserving +measures are likely to be normal [29]. Thus, the transaction is +scored based on the fact of whether the accounts are trying to hide +or not, which is the de-anonymous score. +Definition 1 (De-anonymous score, abbreviated as “score”). +The de-anonymous score of a transaction from account𝑢 to 𝑣 where +there is no intention to hide is defined as +𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) =1 +2 ( 2 log |𝑂𝑢𝑡𝑇𝑥𝑛(𝑢)| − log𝑚𝑎𝑥𝑂𝑢𝑡 +log𝑚𝑎𝑥𝑂𝑢𝑡 ++ 2 log |𝐼𝑛𝑇𝑥𝑛(𝑣)| − log𝑚𝑎𝑥𝐼𝑛 +log𝑚𝑎𝑥𝐼𝑛 +), +(1) +where 𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) represents the outgoing transactions (payments) +of payer 𝑢, 𝐼𝑛𝑇𝑥𝑛(𝑣) represents the incoming transactions (recep- +tions) of payee 𝑣, and | × | denotes the size of a set. The minimum +value of |𝑂𝑢𝑡𝑇𝑥𝑛(𝑢)| and |𝐼𝑛𝑇𝑥𝑛(𝑣)| is 1. Let 𝑚𝑎𝑥𝑂𝑢𝑡 and 𝑚𝑎𝑥𝐼𝑛 +be the largest number of payments and receptions, respectively. The +de-anonymous scores of a transaction (𝑢, 𝑣) range from −1 (very +high anonymity, abnormal) to 1 (very low anonymity, normal). +Intuitively, the score of (𝑢, 𝑣) increases as the transaction num- +bers of either payer or payee grow. Note that tricky criminals may +camouflage themselves by deliberately conducting low-anonymity +transactions [20]. +2.3 +Transaction Network Construction +First, each transaction on Ethereum has one payer (i.e., sender) and +one payee (i.e., receiver). Any account can be the role of payer or +payee, just as a person in real life has different roles. The payee is a +passive role and, therefore, we consider the incoming transactions +to indicate the trustiness of an account. For instance, exchange +accounts that receive more transactions are considered to be more +trustworthy. In contrast, the payer is an active role and, thus, the +outgoing transactions embody the intention of an account. For ex- +ample, a scam account subjectively wants to transfer stolen money +to its partners. +Next, the transaction records are modeled as a directed bipartite +graph 𝐺 = (𝑈,𝑉,𝑆), where 𝑈 , 𝑉 , and 𝑆 represent the set of all +payers, payees, and scores, respectively. A weighted edge (𝑢, 𝑣) +denotes the transfer of Ethers from account 𝑢 ∈ 𝑈 to account 𝑣 ∈ 𝑉 +with 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) ∈ 𝑆. The graph construction procedure is shown +in Figure 2. +Then, the ego network of a payer 𝑢 is introduced. It is formed by +its outgoing scores and corresponding payee neighbors, formulated +Money transfer +Accounts +Tnx Score +Accounts +Payer +Payee +Tnx Score +Accounts +(A) Ethereum +transaction records +(B) De-anonymous +score calculation +(C) Payer-payee graph +Figure 2: The transformation from the raw transaction +records to the directed bipartite graph. “Txn” denotes +“Transaction”. +Payee set +V +Payer A +Payer B +Payee X +Payee Y +Score(A, X) +Score(B, Y) +Payer set +U +Score(B, X) +In( ) = { Score(A, X), Score(B, X) } +Payee X +Out( ) = { Score(B, X), Score(B, Y) } +Payer B +Figure 3: A toy example of the directed bipartite graph estab- +lished from transactions and the illustration of functions 𝐼𝑛 +and 𝑂𝑢𝑡. +as 𝑂𝑢𝑡(𝑢) ∪ {𝑣|(𝑢, 𝑣) ∈ 𝑂𝑢𝑡(𝑢)}, where 𝑂𝑢𝑡(𝑢) is the set of scores +connected with 𝑢. It is similar for the ego network of a payee, +formulated as 𝐼𝑛(𝑣)∪{𝑢|(𝑢, 𝑣) ∈ 𝐼𝑛(𝑣)}. Figure 3 shows an example +in which there are two payers, two payees, and three transactions. +3 +MODEL +In this section, we describe the prior knowledge that establishes the +relationships among accounts and transactions and then propose +risk propagation formulations that satisfy the prior knowledge. It +is worth noticing that the proposed algorithm does not require +handcraft feature engineering. +3.1 +Problem Definition and Model Overview +Given raw transaction records of Ethereum, we model the transac- +tion relationships between accounts as a directed bipartite graph +𝐺 = (𝑈,𝑉,𝑆) with payers and payees as nodes and prepossessed +de-anonymous scores as weights of edges. We believe that accounts +have intrinsic metrics to quantify their reliability and trustworthi- +ness and transactions have intrinsic metrics to measure the con- +fidence of their calculated de-anonymous scores. Naturally, those +metrics are interdependent and interplay with each other via the +risk propagation mechanism: +• Payers vary in terms of their Reliability, which indicates how +motivated they are. A licit payer without malicious intent usually +does not hide himself or disguise its intentions during transactions. +Specifically, a reliable payer has harmless intentions regardless of +whether it is transferring money to an exchange or to a scammer +account (being gypped). In contrast, a perpetrator (e.g., a scammer) +hopes to cover up its traces [27]. The reliability metric 𝑅(𝑢) of a +payer 𝑢 lies in [0, 1], ∀𝑢 ∈ 𝑈 . A value of 1 denotes a 100% reliable +payer and 0 denotes a 0% reliable payer. + +WWW ’23, April 30–May 4, 2023, Austin, TX, US +Anonymous author(s) +Table 1: An example of propagation. 𝑅0 is initial value, 𝑅𝑓 𝑖𝑛𝑎𝑙 +and Risk𝑓 𝑖𝑛𝑎𝑙 are the results after convergence. +Account +Label +𝑅0 +𝑅𝑓 𝑖𝑛𝑎𝑙 +Risk𝑓 𝑖𝑛𝑎𝑙 +0xa768 +Contract-deployer +0.7 +0.8575 +1.425 +0x8271 +Exchange +0.7 +0.9526 +0.474 +0xebdc +Phish-hack +0.7 +0.1195 +8.805 +0xfe34 +Phish-hack +0.7 +0.2330 +7.670 +• Payees vary in their trustworthiness level, measured by a metric +called Trustiness, which indicates how trustworthy they are. Intu- +itively, a cryptocurrency service provider with a better reputation +will receive more licit transactions (with higher scores) from well- +motivated payers. Trustiness of a payee 𝑇 (𝑣) ranges from 0 (very +untrustworthy) to 1 (very trustworthy) ∀𝑣 ∈ 𝑉 . +• De-anonymous scores vary in terms of Confidence, which re- +flects the confidence in the estimated risk probability of a trans- +action. The confidence metric 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) ranges from 0 (lack of +confidence) to 1 (very confident). +The connection between the reliability and risk of accounts: We +define Reliability to characterize the risk rating of accounts because +an account’s intention can be inferred by its (active) sending be- +havior, rather than by its (passive) receiving behavior. A scammer +transferring stolen money to its gang is a better reflection of its +evil intention than the receipt of stolen money from victims. In the +later section, we calculate the risk rating of accounts based on the +Reliability of payer roles. +3.2 +Network Propagation Mechanism +Given a cryptocurrency payer–payee graph, all intrinsic metrics +are unknown but are interdependent. Here, we introduce five items +of prior knowledge that establish the relationships and how the net- +work propagation mechanism is specially designed for our problem. +The first two items of prior knowledge reflect the interdependency +between a payee and the de-anonymous scores that they receive. +[Prior knowledge 1] Payees with higher trustiness receive trans- +actions with higher de-anonymous scores. Intuitively, a payee +receiving transactions with high de-anonymous scores is more +likely to be trustworthy. Formally, if two payees 𝑣1 and 𝑣2 have +a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2) and 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > +𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) > 𝑇 (𝑣2). +[Prior knowledge 2] Payees with higher trustiness receive trans- +actions with more positive confident scores. For two payees 𝑣1 and +𝑣2 with identical de-anonymous score networks, if the confidence +of the in-transactions of payee 𝑣1 is higher than that of payee 𝑣2, +the trustiness of payee 𝑣1 should be higher. Formally, if two pay- +ees 𝑣1 and 𝑣2 have a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2) +and 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) > 𝐶𝑜𝑛𝑓 (ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) > +𝑇 (𝑣2). +According to the above prior knowledge, we develop the Trusti- +ness formulation for ∀𝑣 ∈ 𝑉 of our RiskProp algorithm: +𝑇 (𝑣) = +� +(𝑢,𝑣) ∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) +|𝐼𝑛(𝑣)| +. +(2) +The next item of prior knowledge defines the relationship be- +tween the score of a transaction and the connected payer–payee +pair using the anonymous nature of cryptocurrency. +Algorithm 1 RiskProp Algorithm +1: Input: Directed Bipartite Graph 𝐺 = (𝑈,𝑉,𝑆) +2: Output: Risk of accounts +3: Initialize 𝑇 0 = 0.5, 𝑅0 = 0.7,𝐶𝑜𝑛𝑓 0 = 0.5,𝑡 = 0, Δ = 1 +4: while Δ ≥ 0.01 do +5: +𝑡 = 𝑡 + 1 +6: +Update 𝑡𝑟𝑢𝑠𝑡𝑖𝑛𝑒𝑠𝑠 of payees using Equation 2 +7: +Update 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑖𝑡𝑦 of payers using Equation 4 +8: +Update 𝑐𝑜𝑛𝑓 𝑖𝑑𝑒𝑛𝑐𝑒 of transactions using Equation 3 +9: +Δ𝑇 = � +𝑣∈𝑉 |𝑇 𝑡 (𝑣) −𝑇 𝑡−1(𝑣) | +10: +Δ𝑅 = � +𝑢∈𝑈 |𝑅𝑡 (𝑢) − 𝑅𝑡−1(𝑢) | +11: +Δ𝐶 = � +(𝑢,𝑣)∈𝑆 |𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) | +12: +Δ = max{Δ𝑇 , Δ𝑅, Δ𝐶 } +13: end while +14: 𝑅𝑖𝑠𝑘 (𝑢) = (1 − 𝑅(𝑢)) × 10, ∀𝑢 ∈ 𝑈 +15: return +[Prior knowledge 3] Confident de-anonymous scores of transac- +tions are closely linked with the connected payee’s trustiness. For- +mally, if two scores (𝑢1, 𝑣1) and (𝑢2, 𝑣2) are such that𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) = +𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2),𝑅(𝑢1) = 𝑅(𝑢2), and |𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) −𝑇 (𝑣1)| ⩽ |𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2) +−𝑇 (𝑣2)|, then 𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) ⩾ 𝐶𝑜𝑛𝑓 (𝑢2, 𝑣2). +We imply that different transactions sent by the same payers +can have different intentions and anonymity. Even scammers on +Ethereum can have transactions that seem normal. +[Prior knowledge 4] Transactions with higher confidence de- +anonymous scores are sent by more reliable payers. Formally, if two +scores (𝑢1, 𝑣1) and (𝑢2, 𝑣2) are such that𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) = 𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2), +𝑇 (𝑣1) = 𝑇 (𝑣2), and𝑅(𝑢1) ⩾ 𝑅(𝑢2), then𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) ⩾ 𝐶𝑜𝑛𝑓 (𝑢2, 𝑣2). +This prior knowledge incorporates the payer’s intention in mea- +suring the confidence of transaction scores. In this way, payees +may have different confidence in receiving transactions with the +same anonymity. For instance, exchanges on Ethereum receive +funds from payers with different motivations—some are ordinary +investors and some are suspicious accounts. +Below, we propose the Confidence formulation that satisfies the +above items of prior knowledge: +𝐶𝑜𝑛𝑓 (𝑢, 𝑣) = 𝑅(𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) −𝑇 (𝑣)|) +2 +. +(3) +Then, we describe how to quantify the Reliability metric of a +payer by the transactions it sends. +[Prior knowledge 5] Payers with higher reliability send transac- +tions with higher confidence. For two payers 𝑢1 and 𝑢2 with equal +scores, if payer 𝑢1 has higher confidence for all out transaction +scores than𝑢2, then payer𝑢1 has a higher reliability. Formally, if two +payers 𝑢1 and 𝑢2 have ℎ : 𝑂𝑢𝑡(𝑢1) → 𝑂𝑢𝑡(𝑢2) and 𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) > +𝐶𝑜𝑛𝑓 (𝑢2,ℎ(𝑣)) ∀(𝑢1, 𝑣) ∈ 𝑂𝑢𝑡(𝑢1), then 𝑅(𝑢1) > 𝑅(𝑢2). The corre- +sponding formulation of Reliability metric for ∀𝑢 ∈ 𝑈 is defined +as +𝑅(𝑢) = +� +(𝑢,𝑣) ∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) +|𝑂𝑢𝑡(𝑢)| +. +(4) +Finally, the risk rating of an account is calculated by 𝑅𝑖𝑠𝑘(𝑢) = +(1 − 𝑅(𝑢)) × 10. The pseudo-code of RiskProp network propagation +is described in Algorithm 1. Let 𝑇 0, 𝐶𝑜𝑛𝑓 0, 𝑅0 be initial values +and 𝑡 be the number of interactions. In the beginning, we have +initial reliability 𝑅0 ∀𝑢 ∈ 𝑈 , initial trustiness 𝑇 0 = 0.5 ∀𝑣 ∈ 𝑉 , and + +RiskProp +WWW ’23, April 30–May 4, 2023, Austin, TX, US +Account Risk Rating +Trustiness +Confidence +Risk +Reliability +Update +Propagation Mechanism + Risk Rating +Results Analysis +Ablation +Study +Risk +Threshold +Guarantees +for Practice +Comparative +Evaluation + Further + + Analysis +Results Analysis +Data Acquisition +Labeled data +Etherscan +Ethereum +Transactions +Data Pre-processing +Directed Bipartite Graph +Construction +De-anonymous Score +Calculation +Figure 4: The workflow of account risk rating on Ethereum. +initial confidence 𝐶𝑜𝑛𝑓 0 = 0.5 for all transactions. Then, we keep +updating metrics using Equations 2–4 until Δ is less than 0.01. +RiskProp+: A Semi-supervised Version. Sometimes, we have +partial information about the labels of fraudulent accounts (verified, +phishing scams, etc.) and licit accounts. We can take advantage of +such prior information and incorporate them into our approach +in a semi-supervised manner. In the semi-supervised RiskProp+, +we initialize the Reliability metrics only for the training accounts. +According to the risk levels of services reported by Chainalysis [26], +we set 𝑅0 = 0.9 for ICO wallet, Converter, and Mining, 𝑅0 = 0.7 for +Exchange, 𝑅0 = 0.4 for Gambling, 𝑅0 = 0 for Phish/Hack, and set +𝑅0 = 0.7 for testing accounts. The reliability values of labeled illicit +accounts are unchanged during the training procedure. +Example. Here, we use a small real-world dataset on Ethereum +to intuitively show the results of RiskProp+ after interactions. We +collect transactions of 10 accounts (6 for training and 4 for testing), +including 28,598 accounts and 52,733 transactions in total. Table 1 +shows how the reliability of the 4 testing accounts varies over +interactions (we omit trustiness and confidence for brevity). These +testing accounts have the same reliability values at the beginning +(𝑅0 = 0.7). After convergence, accounts labeled as “phish/hack” +get a lower value of reliability, and other licit accounts get higher +reliability. Confirming our intuition, RiskProp learns that accounts +0xebdc and 0xfe34 are high-risk accounts that investors need to be +aware of. +Workflow for Account Risk Rating. Figure 4 shows the work- +flow of account risk rating on Ethereum, which contains four mod- +ules: (i) Data acquisition collects accounts, transactions, and la- +bels from Ethereum and Etherscan. Only a few labels are provided, +and these labels are not available in the unsupervised setting. (ii) +Data pre-processing of raw transaction data described in Fig- +ure 1 is conducted in two steps: de-anonymous score calculation +and directed bipartite graph construction (i.e., payer–payee net- +work). (iii) Account risk rating recursively calculates the Relia- +bility, Trustiness of accounts, and Confidence of transaction scores +until convergence, updated by the propagation mechanism. (iv) Re- +sults analysis contains risk rating results analysis, comparative +evaluation, and further analysis. +4 +EXPERIMENTS +To investigate the effectiveness of RiskProp, we conduct experi- +ments on a real-world Ethereum transaction dataset. As risk rating +is an issue without any ground truth, we verify the effectiveness +and significance of the risk rating results of RiskProp via three tasks: +1) risk rating analysis, which includes distribution of risk rating +results and case studies of transaction pattern; 2) comparative +evaluation, which reports on the classification performance of +labeled accounts compared with various baselines; and 3) further +analysis, which contains ablation study, impact of risk threshold, +and guarantees for practical use. RiskProp is open source and repro- +ducible, and the code and dataset are publicly available after the +paper is accepted. +4.1 +Data Collection +We first obtain 803 ground truth account labels from an official +Ethereum explorer and then include all the accounts and transac- +tions that are within the one-hop and two-hop neighborhood of +each labeled account. Next, we filter out the zero-ETH transactions +and construct the records into a graph, retaining the largest weakly +connected component for experiments. As a result, there are 1.19 +million accounts and 4.13 million transactions in the network. In the +dataset, 0.02 percent (243) are labeled illicit (e.g., phishing scam), +whereas 0.05 percent (560) are labeled licit (e.g., exchanges). The +remaining unknown accounts are not labeled with regards to licit +versus illicit. +4.2 +Effectiveness of De-anonymous Score +We use one-way analysis of variance (ANOVA) to assess whether +there is a significant difference between illicit and licit transactions +in the proposed de-anonymous score in Equation (1). We consider +a transaction as illicit (versus licit) if its payer is marked as illicit +(versus licit). Table 2 shows that compared with the random score, +our proposed score achieves a larger mean square (MS) between +groups and smaller MS within groups; in addition, our proposed +score has a higher F value, and the 𝑝-value equals 0. These results +suggest that the de-anonymous score is a useful metric for assessing +the quality of transactions. +Table 2: ANOVA of random scores and de-anonymous +scores. +Random scores +De-anonymous score +Src of var. +MS +F +𝑝-value +MS +F +𝑝-value +Between groups +8.8 × 10−1 +2.6 × 101 +1.0 × 10−1 +7.8 × 102 +7.7 × 103 +0 +Within groups +3.3 × 10−1 +- +- +1.0 × 10−1 +- +- +4.3 +Analysis of Risk Rating Results +The principal task of RiskProp is to rate Ethereum accounts based on +how ill-disposed they are. Given the account risk rating obtained by +RiskProp, we first review the results and investigate the capability +of RiskProp in discovering new risky accounts. Then, we dig deeper +into the predicted high-risk accounts and obtain some insights. +4.3.1 +Distribution of risk rating results. The risk value of an account +ranges from 0 (low risk) to 10 (high risk). The distribution of the +predicted risk scores is as follows: 33.58% are located at (0,2], 63.45% + +WWW ’23, April 30–May 4, 2023, Austin, TX, US +Anonymous author(s) +(b) +(a) +(c) +(d) +Exchange +(e) +Phish_contract +Victims +Victims +Exchange +Scammer +Scammers +(f) +16 ETH +16 ETH +37 ETH +37 ETH +8 ETH +8 ETH +8 ETH +Create +0.29 ETH +0.29 ETH +Figure 5: Visualization showing some typical transaction +patterns of risky accounts (in red circles). +are located at (2,4], 2.03% are located at (4,6], 0.78% are located +at (6, 8], and 0.19% are located at (8, 10]. This is consistent with +expectations: The risk value of the Ethereum transaction network +meets the power distribution law, indicating that the overwhelming +majority of accounts act normally, and only very few accounts have +abnormal behaviors. We are interested in whether the high-risk +accounts predicted by RiskProp are actually questionable. Thus, we +first manually check the top 150 accounts with the highest risk +(with both in-coming and out-going transactions). The finding is +that 119 out of 150 (approximately 80%) accounts have abnormal +behaviors. Among these 119 illicit accounts, 43 accounts are already +labeled as “phish/hack” by Etherscan, whereas the remaining 76 +are newly discovered suspicious accounts that are not marked in +the existing label library. This result indicates the capabilities of +RiskProp in predicting undiscovered risky accounts and reducing +financial losses. +4.3.2 +Case studies of transaction pattern. We then manually veri- +fied the predicted risky accounts by investigating their abnormal +behaviors and find that there are many suspicious transaction pat- +terns in the network. In order to save space, we show 6 typical +patterns in Figure 5. These patterns are summarized from the real- +world Ethereum transaction data and guided by current research +and recommendation reports. +(a) Hacking scammers are a list of addresses related to phish- +ing and hacks. Figure 5(a) shows a pattern of phishing accounts +reported by users who suffered financial loss. A typical phishing +scam on Ethereum is the “Bee Token ICO Scam” attack, in which +the phishers sent fake emails to the investors of an ICO with a fake +Ethereum address to deposit their contributions into. For example, +account 0xe336 has been confirmed to be part of this “Bee Token” +scam, and 243 ETH has been sent to this address by 165 victims. +(b) Fund source of hacking scammers are the upstream ac- +counts of the known illicit accounts, which are collusion scam ac- +counts to attract victims or provide money for hacking. As shown +in Figure 5(b), the behaviors of collusion scam accounts may look +similar to victims. Nevertheless, we find that the upstream collusion +accounts appear to participate in fewer transactions with shorter +time intervals, and there are attempts to transfer the entire ETH +balance of the scammers according to the Red Flag Indicators of +FATF [11]. +(c) Money laundering of scammers are the downstream ac- +counts of the known illicit accounts, which are collusion scam +accounts to accept and transfer the stolen money, obfuscating the +true sources. As shown in Figure 5(c), account 0x78f1 received +stolen funds from several known hacking scammers, appearing +to be the account used in the “placement” stage of money laun- +dering. Another example is 0xcfdd, which receives stolen funds +from the Fake Starbase Crowdsale Contribution account 0x122c. (d) +Zero-out middle accounts are the middle accounts that serve as +a bridge defined by Li et al. [20]. As shown in Figure 5(d), most of +the received funds will be transferred out in short succession (such +as within 24 hours). See 0x126e for an example. +(e) Round transfers among exchanges denote a pattern that +an account withdraws ETH without additional activity to a pri- +vate wallet and then deposits back to the exchange, as shown in +Figure 5(e). Account 0x886e withdraws 0.4 ETH from Cryptopia +exchange and then deposits the same amount of ETH back to Cryp- +topia, which is an unnecessary step and incurs transaction fees [11]. +Such a phenomenon indicates that the exchange is misused as a +money-laundering mixer or is conducting wash trading [28]. +(f) Creators of illicit contracts are often the manipulators +behind the scenes. The Origin Protocol phishing scam contact ac- +count 0x9819 was created by account 0xff1a. After victims deposited +money into the phishing contract, the creator transfers the stolen +funds back to himself via internal transactions, which deliberately +enhances anonymity. +We observe that many illicit accounts are outside the label li- +brary and are still considered risk-free. Based on the results, we +infer that our RiskProp is able to expose unlabeled illicit accounts. +This is crucial on Ethereum, which lacks authorized and effective +regulation. In addition, the newly identified illicit accounts can +complete the current label collection for additional analysis. +4.4 +Comparative Evaluation Settings +To further evaluate the performance of our method and show the +potential application, we employ the rating scores to conduct clas- +sification experiments that divide Ethereum accounts into illicit +and licit accounts, and we compare the results with the existing +baseline methods for further verification. We wish to investigate if +RiskProp can give a higher risk rating for the known illicit accounts +and a lower rating for known licit accounts. +4.4.1 +Compared Methods. As mentioned earlier, RiskProp is the +first algorithm that explores the risk rating of blockchain accounts. +We chose a variety of methods (unsupervised and supervised) as +baselines, which are similar to the problem we want to solve. We +compare unsupervised RiskProp with (i) web page ranking, such +as PageRank [6], and (ii) bipartite graph-based fraud detection, +such as FraudEagle [2], BIRDNEST [12], and REV2 [16], which are +also unsupervised methods. +The (semi-)supervised approaches are as follows. (i) Machine +learning methods, e.g., logistic regression (LR), naïve Bayes (NB), +decision tree (DT), support vector machine (SVM), random for- +est (RF), extreme gradient boosting (XGBoost), and LightGBM. +These methods are used by [1, 4, 8, 19] for detection of abnor- +mal Ethereum accounts. (ii) Traditional graph neural network, +including DeepWalk, Node2Vec, and graph convolutional network + +RiskProp +WWW ’23, April 30–May 4, 2023, Austin, TX, US +50 100 150 200 250 300 350 400 +Top k +0 +20 +40 +60 +80 +100 +Precision@k (%) +(a) +50 100 150 200 250 300 350 400 +Top k +0 +20 +40 +60 +80 +100 +Recall@k (%) +(b) +RiskProp +PageRank +Birdnest +FraudEagle +REV2 +Figure 6: The 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 of illicit account pre- +diction with different rating methods. +(GCN) were conducted by Chen et al. [7] for detection of Ethereum +phishing scams. (iii) Graph neural network for graphs with +heterophily, such as CPGNN [34]. The application of this type of +algorithms is a recent research advancement in the task of Ethereum +account classification [14]. +4.4.2 +Evaluation Metrics. To evaluate the performance of the mod- +els, we calculate the following metrics: 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛,𝑅𝑒𝑐𝑎𝑙𝑙, 𝐹1,𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦, +and 𝐴𝑈𝐶. As we know, there are only 6 out of 10,000 (0.067 percent) +accounts labeled in the entire dataset. To measure the order of the +risk rating, we employ 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 to evaluate the +ranking order of the algorithm (@𝑘 means the top 𝑘 accounts). All +baseline methods are tested using the original codes published by +the authors. We repeat experiments 10 times and report the average +results. +4.4.3 +Implementation Details. We evaluate the methods with bi- +nary labeled accounts (illicit verse licit) and, thus, we assume ac- +counts in the top 1% to be the illicit accounts (corresponding thresh- +old: 6 for RiskProp). The reason for this threshold and percentage +setting is discussed in Section 4.7. The split of the dataset in the +(semi-)supervised setting is 𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔 : 𝑡𝑒𝑠𝑡 = 8 : 2. +4.5 +Comparative Evaluation Results +We report the 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 curves of the compared +algorithms, as shown in Figure 6. We observe that RiskProp obtains +superior precision and recall than that of baseline with different +𝑘. Up to 𝑘 = 100, the precision of RiskProp is almost 1 for illicit +account prediction, which is surprising for an unsupervised setting. +The 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 curve of RiskProp is significantly higher than the +compared methods, and also increases steadily with 𝑘. Table 3 +shows the performance of unsupervised and supervised methods +separately. We observe that RiskProp remarkably outperforms the +unsupervised graph rating baselines in terms of accuracy and AUC, +improving by 38.90% and 34.16%, respectively. Meanwhile, for the +licit account prediction, we observe that RiskProp beats the best +baseline (FraudEagle) with a 10.48% improvement in its F1-score. +These demonstrate the effectiveness of our account risk rating +method without labeling information. +Next, we turn our attention to the results of the semi-supervised +RiskProp+ compared with the existing (semi-)supervised classifica- +tion in Table 3, from which we derive the following conclusions: +1) RiskProp+ outperforms all baseline methods by 12.32% in terms +of F1-score, 13.62% in terms of AUC, and 10.56% in terms of accu- +racy. 2) The precision of licit accounts prediction is improved from +Table 3: The classification results (%) of unsupervised and +(semi-)supervised methods. +Illicit account +Licit account +Total +Methods +P +R +F1 +P +R +F1 +Acc. +AUC +PageRank +29.13 +67.49 +40.69 +67.08 +28.75 +40.25 +40.47 +48.12 +FraudEagle +12.28 +2.88 +4.670 +68.36 +91.07 +78.10 +64.38 +46.98 +BIRDNEST +22.24 +47.32 +30.26 +55.24 +28.21 +37.35 +34.00 +37.77 +REV2 +14.10 +4.527 +6.854 +68.00 +88.04 +76.73 +62.76 +46.28 +RiskProp +71.48 +71.48 +76.15 +91.44 +85.89 +88.58 +84.56 +83.69 +Illicit account +Licit account +Total +Methods +P +R +F1 +P +R +F1 +Acc. +AUC +LR +65.67 +74.58 +69.84 +83.87 +77.23 +80.41 +76.25 +75.90 +NB +59.79 +98.31 +74.36 +98.41 +61.39 +75.61 +75.00 +79.85 +DT +62.66 +54.07 +58.04 +75.79 +81.19 +78.40 +71.75 +68.39 +SVM +90.00 +45.76 +60.67 +75.38 +97.03 +84.85 +78.12 +71.40 +RF +71.52 +53.39 +61.14 +75.55 +86.93 +80.84 +74.00 +69.40 +XGBoost +67.35 +55.95 +61.11 +76.58 +84.16 +80.19 +70.05 +73.75 +LightGBM +75.77 +65.19 +69.93 +84.23 +92.57 +88.21 +81.86 +77.75 +DeepWalk +66.85 +66.30 +66.54 +86.48 +86.75 +86.61 +83.13 +81.03 +Node2Vec +62.36 +63.26 +62.76 +85.10 +84.56 +84.82 +78.13 +72.78 +GCN +20.83 +27.78 +23.81 +79.46 +68.99 +73.86 +60.63 +47.40 +CPGNN +52.17 +61.54 +56.47 +86.84 +81.82 +84.26 +76.88 +71.68 +RiskProp+ +70.91 +84.78 +77.23 +93.33 +85.96 +89.49 +85.63 +85.37 +Table 4: Illicit account prediction of ablation studies. +Methods +Precision +Recall +F1-score +RiskProp+ +0.7091 +0.8478 +0.7723 +RiskProp+ (w/o label) +0.7148 +0.8148 +0.7615 +RiskProp+ (w/o NP) +0.3811 +0.9959 +0.5513 +RiskProp+ (w/o DS) +0.4737 +0.1957 +0.2769 +82.52% (i.e., the average precision in baselines) to 93.33%, which +means more licit accounts can be correctly identified. 3) The supe- +rior performance of RiskProp is more significant in the prediction of +illicit accounts. The recall of illicit accounts prediction is improved +from 60.56% (i.e., the average recall of illicit accounts prediction in +baselines) to 84.78%. This shows the effectiveness of our framework +in the prediction of both illicit and licit accounts. +4.6 +Ablation Study +To further validate the contribution of each component of the pro- +posed RiskProp+, we conduct an ablation study as follows. +• RiskProp+ (Full model): All components of the model and label +data are included. +• w/o label: Labels are unavailable in the learning procedure, and +the model is trained in an unsupervised manner. +• w/o network propagation (NP): Remove the NP procedure +and calculate the average de-anonymous scores (𝐴𝐷𝑆) for each ac- +counts’ outgoing transactions (payer role). An account is predicted +as abnormal if its 𝐴𝐷𝑆 ⩽ 0. +• w/o de-anonymous score (DS): Replace DS with random scores, +ranging from −1 to 1. +We derive the following findings from Table 4: 1) Without the +labels, the F1-score drops only slightly, indicating that our RiskProp +does not rely on label data and can obtain good results in an un- +supervised manner. To our surprise, the full model outperforms +the RiskProp (w/o label), with a 3.3% increase in recall and 0.51% + +WWW ’23, April 30–May 4, 2023, Austin, TX, US +Anonymous author(s) +decrease in precision. This may be possibly explained by the re- +liability values of labeled illicit accounts remaining unchanged +during training in the supervised setting. 2) RiskProp (w/o NP) +has a only lower precision but a greatly improved recall, revealing +that most of the illicit accounts are correctly predicted as illicit +but that some licit accounts are misjudged to be illicit. This result +demonstrates that de-anonymous score is an effective indicator of +illicit transactions but their confidence varies among transactions. +This result also confirms why we need to consider the confidence +of the score in the propagation mechanism. 3) RiskProp (w/o DS) +yields low precision (47.37%) and severely low recall (19.57%). This +result demonstrates that, even if the network propagation model is +retained, the wrong scores of transactions will be spread through- +out the entire Ethereum transaction network, resulting in poor +prediction results. +4.7 +Risk Threshold of RiskProp +Given accounts in the reversed order of their risk ratings, a natural +question is how to classify licit or illicit accounts according to their +risk ratings for the classification task? One possible option is to +determine the percentage of known illicit labels in the dataset +and set the risk value of this percentage as a demarcation line +for account classification. However, the percentage is imprecise +because some of the illicit accounts remain unrevealed according +to the experimental results in Section 4.3.2. Therefore, we try to +establish a suitable risk threshold (𝑅𝑇𝐻) of RiskProp by conducting +classification experiments. Figure 7(a) demonstrates the results of +illicit account prediction with different risk thresholds, ranging +from 1 to 10. As expected, the precision increases while the recall +decreases with increasing 𝑅𝑇𝐻. In addition, F1, accuracy, and AUC +first increase and then decrease with the increase in 𝑅𝑇𝐻. The best +performance for F1 and AUC is when 𝑅𝑇𝐻 = 6. Thus, we set the +risk threshold 𝑅𝑇𝐻 = 6 for RiskProp. +4.8 +Guarantees for Practical Use +Here, we present guarantees for RiskProp in practical use regard- +ing the following aspects: 1) guarantee of convergence; 2) time +complexity; and 3) linear scalability. +Convergence and Uniqueness. We present the theoretical prop- +erties of RiskProp, including the proofs of prior knowledge, con- +vergence, and uniqueness of the proposed metrics, i.e., reliability, +trustiness, and confidence. Proofs are shown in the Appendix due to +lack of space. +Time complexity. In each interaction, the RiskProp updates the +reliability, Trustiness metrics of accounts and Confidence metric of +transactions. Therefore, the complexity of each iteration is O(|𝑈 | + +|𝑆|) = O(|𝑆|), |𝑆| is the total edges in the payer–payee network. +Thus, for 𝑘 iterations, the total running time is O(𝑘|𝑆|). +Linear scalability. We have shown that RiskProp is linear in run- +ning time in the number of nodes. To show this experimentally as +well, we create random networks of an increasing number of nodes +and edges and compute the running time of the algorithm until +convergence. Figure 7(b) shows that the running time increases lin- +early with the number of nodes in the network. Therefore, we can +conclude that RiskProp is a scalable rating method that is suitable +for applications on large-scale transaction networks. +1 2 3 4 5 6 7 8 9 10 +RTH +0 +20 +40 +60 +80 +100 +Performance(%) +Precision +Recall +F1 +Accuracy +AUC +(a) Impact of different RTH +103 +104 +105 +106 +107 +Number of nodes +10 +1 +100 +101 +102 +103 +104 +Run time (seconds) +(b) Scalability of RiskProp +Figure 7: Further analysis of RiskProp. +Analysis of incorrect predictions. Furthermore, the results of the +RiskProp+ experiment showed that 39 out of 46 (85%) phishing ac- +counts were correctly predicted as illicit accounts. To understand +why the remaining accounts failed to be detected as illicit by our +model, we manually checked their transactions and neighbors and +obtained the following results: (i) For one of the accounts, we have +a risk score of 5.72, and in practice, the system will also warn about +such accounts that are close to the risk threshold (𝑅𝑇𝐻 = 6). (ii) +One account is set as the default risk value because the phishing +account has no outgoing transactions for the time being, and in +practice, we can make correct predictions as soon as the phish- +ing account starts laundering money. (iii) Among the remaining +five accounts, one account has a high transaction volume of 154. +The remaining four accounts have a high volume of transactions +along with withdrawals of ETH from exchanges, which directly +contributed to the high de-anonymity score of transactions. How- +ever, there are two sides to the story: regulation and fraud are a +game of confrontation. For hackers, reusing accounts reduces the +probability of being identified as high risk and, at the same time, +reusing accounts and withdrawing money from exchanges increase +the risk of exposure and fund freezing. +5 +RELATED WORK +Risk control studies in cryptocurrency. In recent years, there +has been growing interest in account clustering and detecting il- +licit activities (e.g., financial scams, money laundering) in cryp- +tocurrency transaction networks [31]. Victor [27] is the first to +propose clustering heuristics for the Ethereum’s account model, +including deposit address reuse, airdrop multi-participation, and +self-authorization. A recent review of the literature on cryptocur- +rency scams [5] showed that the existing methods (e.g., [9], [10], +and [15]) are mainly based on supervised classifiers fed with hand- +crafted features. Many attempts have been made [25, 29, 32] to +incorporate structural information by learning the latent repre- +sentations of accounts. Some researchers have investigated and +modeled the money flow from a network perspective [18, 21] to +better identify illicit activities. After all, there is still a black area +regarding the estimation of the risk value of Ethereum accounts, +which is the key task in alerting about suspicious accounts and +transactions on the chain. +Rating and ranking on graph data. The aim of ratings and +rankings on graph data is to provide a score or an order for each +node in a graph. Currently, the main solutions are based on link +analysis technique [24], Bayesian model [12], and iterative learn- +ing [13], etc. Similarly to the proposed RiskProp algorithm, [16] + +RiskProp +WWW ’23, April 30–May 4, 2023, Austin, TX, US +proposed axioms and iterative formulations to establish the rela- +tionship between ratings. In [22], the authors measured the bias +and prestige of nodes in networks based on trust scores. In [17], the +authors highlighted that graph-based approaches provide unique +solution opportunities for financial crime and fraud detection. A +review on this topic [3] described the problems in current studies: +lack of ground truths, imbalanced class, and large-scale network. +These challenges also exist in our risk rating problem on Ethereum +transaction networks. +6 +CONCLUSIONS AND FUTURE WORK +In this paper, we present the first systematic study to assess the +account risk via a rating system named RiskProp. In RiskProp, we +modeled transaction records of Ethereum as a bipartite graph, pro- +posed a novel metric called de-anonymous score to quantify the +transaction risk, and designed a network propagation mechanism +based on transaction semantics. By analyzing the rating results and +manually checking the accounts with high risk, we evaluated the +performance of RiskProp and obtained new insights about transac- +tion risks on Ethereum. In addition, we employed the obtained risk +scores to conduct illicit/licit account classification experiments on +labeled data, and the superiority of this method over baseline meth- +ods further verified the effectiveness of RiskProp in risk estimation. +For future work, we plan to integrate the transaction amounts and +temporal information in our model, develop a web page or online +tool for querying risk values of accounts, and share the details of +risky cases with the Ethereum community. +REFERENCES +[1] Rachit Agarwal, Shikhar Barve, and Sandeep Kumar Shukla. 2021. Detecting ma- +licious accounts in permissionless blockchains using temporal graph properties. +Applied Network Science 6, 1 (Dec. 2021), 9. +[2] Leman Akoglu, Rishi Chandy, and Christos Faloutsos. 2013. Opinion fraud +detection in online reviews by network effects. In Proceedings of the International +AAAI Conference on Web and Social Media, Vol. 7. 2–11. +[3] Leman Akoglu, Hanghang Tong, and Danai Koutra. 2015. 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ACM, 23– +32. +[29] Mark Weber, Giacomo Domeniconi, Jie Chen, Daniel Karl I. Weidele, Claudio +Bellei, Tom Robinson, and Charles E. Leiserson. 2019. Anti-money laundering in +Bitcoin: Experimenting with graph convolutional networks for financial forensics. +arXiv preprint arXiv:1908.02591 (2019). +[30] Gavin Wood. 2014. Ethereum: A secure decentralised generalised transaction +ledger. Ethereum Project Yellow Paper 151 (2014), 1–32. +[31] Jiajing Wu, Jieli Liu, Yijing Zhao, and Zibin Zheng. 2021. Analysis of cryptocur- +rency transactions from a network perspective: An overview. Journal of Network +and Computer Applications 190 (2021), 103139. +[32] Jiajing Wu, Qi Yuan, Dan Lin, Wei You, Weili Chen, Chuan Chen, and Zibin +Zheng. 2022. Who are the phishers? Phishing scam detection on Ethereum via +network embedding. IEEE Transactions on Systems, Man, and Cybernetics: Systems +52, 2 (2022), 1156–1166. +[33] Zihao Yuan, Qi Yuan, and Jiajing Wu. 2020. 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Formally, if two pay- +ees 𝑣1 and 𝑣2 have a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2) +and 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) > +𝑇 (𝑣2). +The formulation to be used to show that the prior knowledge is +satisfied is Equations 2, 3, and 4 in the main paper. +𝑇 (𝑣) = +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) +|𝐼𝑛(𝑣) | +𝑅(𝑢) = +� +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) +|𝑂𝑢𝑡 (𝑢) | +𝐶𝑜𝑛𝑓 (𝑢, 𝑣) = 𝑅(𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 (𝑣) |) +2 +Proof. To prove the Prior Knowledge 1, let us take two payees 𝑣1 +and 𝑣2 that have identically ego networks and a one-to-one mapping +ℎ, such that |𝐼𝑛(𝑣1)| = |𝐼𝑛(𝑣2)|, 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) = 𝐶𝑜𝑛𝑓 (ℎ(𝑢), 𝑣2), and +𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1). +According to Equation 2, we have +𝑇 (𝑣1) −𝑇 (𝑣2) = +� +(𝑢,𝑣1)∈𝐼𝑛(𝑣1) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣1) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) +|𝐼𝑛(𝑣1) | +− +� +(𝑢,𝑣2)∈𝐼𝑛(𝑣2) 𝑆𝑐𝑜𝑟𝑒 (ℎ(𝑢), 𝑣2) × 𝐶𝑜𝑛𝑓 (ℎ(𝑢), 𝑣2) +|𝐼𝑛(𝑣2) | += +� +(𝑢,𝑣1)∈𝐼𝑛(𝑣1) (𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣1) − 𝑆𝑐𝑜𝑟𝑒 (ℎ(𝑢), 𝑣2)) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) +|𝐼𝑛(𝑣1) | +As 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2), so +𝑇 (𝑣1) −𝑇 (𝑣2) > +� +(𝑢,𝑣1)∈𝐼𝑛(𝑣1) 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) +|𝐼𝑛(𝑣1) | +As 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) ≥ 0 because Confidence are non-negative, +𝑇 (𝑣1) −𝑇 (𝑣2) > 0 ⇒ 𝑇 (𝑣1) > 𝑇 (𝑣2) +The other items of prior knowledge have very similar and straight- +forward proof. +□ +A.2 +Proof for Convergence +Before the proof of convergence, we first discuss the boundary of +proposed metrics. At the end of iteration 𝑡 of Algorithm 1, and by +equation 2, 3, and 4, we get, +𝑇 𝑡 (𝑣) = +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) +|𝐼𝑛(𝑣) | +𝑅𝑡 (𝑢) = +� +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) +|𝑂𝑢𝑡 (𝑢) | +𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) = 𝑅𝑡 (𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 𝑡 (𝑣) | +2 +) +𝑇 ∞(𝑣), 𝑅(∞) (𝑢),𝐶𝑜𝑛𝑓 (∞) (𝑢, 𝑣) are their final values after con- +vergence. +Lemma A.1. (Boundary discussion) Set the maximum score in the +transaction network as 𝑀, namely: +𝑀 = max +(𝑢,𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) +then |𝑀| < 1. +The difference between a payee𝑣’s final Trustiness and its Trustiness +after the first iteration is +|𝑇 ∞(𝑢) −𝑇 1(𝑢) | ≤ |𝑀 | +Similarly, +|𝑅∞(𝑢) − 𝑅1(𝑢) | ≤ 1 +|𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1(𝑢, 𝑣) | ≤ 1 + |𝑀 | +2 += 𝛼 (𝛼 ≤ 1) +Proof. First, we state that |𝑀| is strictly less than 1 in prac- +tice. According to the formulation of 𝑆𝑐𝑜𝑟𝑒 of the main paper, we +can see that 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) = 1 when 𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) = 𝑚𝑎𝑥𝑂𝑢𝑡 and +𝐼𝑛𝑇𝑥𝑛(𝑣) = 𝑚𝑎𝑥𝐼𝑛, it is an extreme situation where the largest +number of payments and receptions of the entire network appears +in one transaction. The other case is 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) = −1 when that +𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) = 1 and 𝐼𝑛𝑇𝑥𝑛(𝑣) = 1 at the same time. This situation +presents to be some isolated transactions, however, they do not +propagate risk and thus do not influence convergence. These situa- +tions are out of our consideration. So we get |𝑀| is strictly smaller +than 1. +Then, we prove that 𝑇 (𝑣) is bounded during the iterations: +|𝑇 ∞(𝑣) −𝑇 1(𝑣) | = | +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) +|𝐼𝑛(𝑣) | +− +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 0(𝑢, 𝑣) +|𝐼𝑛(𝑣) | +| +Since |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|, we get, +|𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × (𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢, 𝑣)) | +|𝐼𝑛(𝑣) | +Since |𝑥 × 𝑦| = |𝑥| × |𝑦|, we have, +|𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × (𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢, 𝑣)) | +|𝐼𝑛(𝑣) | +(5) +Since |𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣)| ≤ |𝑀| ≤ 1, and |(𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣)−𝐶𝑜𝑛𝑓 0(𝑢, 𝑣))| ≤ +1, we get, +|𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤ |𝑀 | × |𝐼𝑛(𝑣) | +|𝐼𝑛(𝑣) | = |𝑀 | +Next, we conduct the proof on 𝑅(𝑢): +|𝑅∞ (𝑢) − 𝑅1 (𝑢) | = +| � +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − � +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 0 (𝑢, 𝑣) | +|𝑂𝑢𝑡 (𝑢) | +Again, since |𝑥 × 𝑦| = |𝑥| × |𝑦|, we get, +|𝑅∞ (𝑢) − 𝑅1 (𝑢) | ≤ +� +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢, 𝑣) | +|𝑂𝑢𝑡 (𝑢) | +(6) +Similarly, since |(𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 0(𝑢, 𝑣))| ≤ 1, we have, +|𝑅∞ (𝑢) − 𝑅1 (𝑢) | ≤ |𝑂𝑢𝑡 (𝑢) | +|𝑂𝑢𝑡 (𝑢) | = 1 + +RiskProp +WWW ’23, April 30–May 4, 2023, Austin, TX, US +Finally, we calculate the bound of 𝐶𝑜𝑛𝑓 (𝑢, 𝑣): +|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢, 𝑣) | = +|𝑅∞ (𝑢) − 𝑅1 (𝑢) + |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 1 (𝑣) | − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 ∞ (𝑣) || +2 +Since |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|, we have +|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢, 𝑣) | ≤ +|𝑅∞ (𝑢) − 𝑅1 (𝑢) | + ( ||𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 1 (𝑣) | − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 ∞ (𝑣) ||) +2 +Since ||𝑥| − |𝑦|| ≤ |𝑥 − 𝑦|, it follows that, +|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢, 𝑣) | ≤ |𝑅∞ (𝑢) − 𝑅1 (𝑢) | + |𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | +2 +(7) +Since |𝑅∞(𝑢) − 𝑅1(𝑢)| ≤ 1, and|𝑇 ∞(𝑣) −𝑇 1(𝑣)| ≤ |𝑀|, we get, +|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢, 𝑣) | ≤ 1 + |𝑀 | +2 +For convenience, we let 1+|𝑀 | +2 += 𝛼. Since |𝑀| < 1, then 𝛼 < 1. +□ +Theorem A.2. Convergence of Propagation: The difference during +iterations is bounded as as |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣)| ≤ 𝛼𝑡 (𝛼 = +1+|𝑀 | +2 +< 1), ∀(𝑢, 𝑣) ∈ 𝑆. As 𝑡 increases, the difference decreases and +𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) converges to |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣). Similarly, |𝑇 ∞(𝑣) −𝑇𝑡 (𝑣)| ≤ +𝛼𝑡−1, ∀𝑣 ∈ 𝑉 , |𝑅∞(𝑢) − 𝑅𝑡 (𝑢)| ≤ 𝛼𝑡−1, ∀𝑢 ∈ 𝑈 . +Proof. Similar to Equations 5, 6, and 7, we have, +|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) | ≤ |𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | + |𝑇 ∞ (𝑣) −𝑇𝑡 (𝑣) | +2 +(8) +|𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | ≤ +� +(𝑢,𝑣,)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) | +|𝑂𝑢𝑡 (𝑢) | +(9) +|𝑇 ∞ (𝑣) −𝑇𝑡 (𝑣) | +≤ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) | +|𝐼𝑛(𝑣) | +(10) +First, we will prove the convergence of Confidence using mathe- +matical induction. +Base case of induction. +When 𝑡 = 1, as we proved in Lemma A.1, we get: +|𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1(𝑢, 𝑣) | ≤ 𝛼1 +Induction step. +We assume by hypothesis that +|𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) | ≤ 𝛼𝑡−1, +which is consistent with the base case already. +Then, by substituting Equations 9 and 10 into Equation 8, for the +case in the next iteration where time is 𝑡, we have, +|𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) | +≤ +� +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) | +2 × |𝑂𝑢𝑡 (𝑢) | +) ++ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) | +2 × |𝐼𝑛(𝑣) | +≤ 1 +2 × +� +( 1 + |𝑀 | +2 +)𝑡−1 + +|𝑀 | × � +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) | +|𝐼𝑛(𝑣) | +� +≤ 1 +2 × +� +( 1 + |𝑀 | +2 +)𝑡−1 + |𝑀 | × ( 1 + |𝑀 | +2 +)𝑡−1 +� +≤ ( 1 + |𝑀 | +2 +)𝑡 = 𝛼𝑡 +Therefore, |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣)| ≤ 𝛼𝑡. +|𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | ≤ +� +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) | +|𝑂𝑢𝑡 (𝑢) | +≤ +� +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) ( 1+|𝑀| +2 +)𝑡−1 +|𝑂𝑢𝑡 (𝑢) | +≤ ( 1 + |𝑀 | +2 +)𝑡−1 = 𝛼𝑡−1 +|𝑇 ∞ (𝑣) −𝑇 (𝑣)𝑡 | +≤ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) | +|𝐼𝑛(𝑣) | +≤ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) | +|𝐼𝑛(𝑣) | +≤ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) ( 1+|𝑀| +2 +)𝑡−1 +|𝐼𝑛(𝑣) | +≤ ( 1 + |𝑀 | +2 +)𝑡−1 = 𝛼𝑡−1 +As discussed in the Lemma A.1. we know that |𝑀| is strictly +smaller than 1, then we have 𝛼 < 1. As 𝑡 increases, 𝛼𝑡−1 → 0 +and 𝛼𝑡 → 0, so after t iterations, 𝐶𝑜𝑛𝑓 (𝑢, 𝑣)𝑡 → 𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣), +𝑅(𝑢)𝑡 → 𝑅∞(𝑢), and 𝑇 (𝑣)𝑡 → 𝑇 ∞(𝑣), the algorithm converges. +□ +A.3 +Proof for Uniqueness +In this part, we provides proofs that Reliability, Trustiness, and +Confidence are unique. +Theorem A.3. Confidence, Reliability, and Trustiness converge to +the unique value. +Proof. First, we consider the uniqueness of Confidence using +mathematical contradiction. +Let the 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) converges to different values. So, let (𝑢, 𝑣) +be the transaction with maximum Confidence difference, 𝐷 (with +𝐷 ≥ 0), between its two possible 𝐶𝑜𝑛𝑓1(𝑢, 𝑣) and 𝐶𝑜𝑛𝑓2(𝑢, 𝑣). +According to Equation 8, we get, +𝐷 = |𝐶𝑜𝑛𝑓 ∞ +1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞ +2 (𝑢, 𝑣) | +≤ |𝑅∞ +1 (𝑢) − 𝑅∞ +2 (𝑢) | + |𝑇 ∞ +1 (𝑣) −𝑇 ∞ +2 (𝑣) | +2 +(11) +Then, according to Equation 9 and 10, we have, +|𝑅∞ +1 (𝑢) − 𝑅∞ +2 (𝑢) | ≤ +� +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ +1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞ +2 (𝑢, 𝑣) | +|𝑂𝑢𝑡 (𝑢) | +≤ 𝐷 +(12) + +WWW ’23, April 30–May 4, 2023, Austin, TX, US +Anonymous author(s) +|𝑇 ∞ +1 (𝑣) −𝑇 ∞ +2 (𝑣) | ≤ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |𝐶𝑜𝑛𝑓 ∞ +1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞ +2 (𝑢, 𝑣) | +|𝐼𝑛(𝑣) | +≤ |𝑀 | × 𝐷 +(13) +We substitute Equation 12 and 13 into Equation (11), and get, +𝐷 = |𝐶𝑜𝑛𝑓 ∞ +1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞ +2 (𝑢, 𝑣) | +≤ 1 +2 × ( +� +(𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ +1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞ +2 (𝑢, 𝑣) | +|𝑂𝑢𝑡 (𝑢) | ++ +� +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |𝐶𝑜𝑛𝑓 ∞ +1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞ +2 (𝑢, 𝑣) | +|𝐼𝑛(𝑣) | +≤ 1 +2 × +� +𝐷 + +|𝑀 | × � +(𝑢,𝑣)∈𝐼𝑛(𝑣) |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) | +|𝐼𝑛(𝑣) | +� +≤ 1 +2 × (𝐷 + |𝑀 | × 𝐷) +≤ ( 1 + |𝑀 | +2 +) × 𝐷 += 𝛼 × 𝐷 +Thus, by solving 𝐷 ≤ 𝛼 × 𝐷(𝛼 ≠ 0) and with the condition that +𝐷 ≥ 0, we obtain 𝐷 = 0. Then, |𝐶𝑜𝑛𝑓 ∞ +1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞ +2 (𝑢, 𝑣)| = 0 +and converge value of Confidence is unique. The uniqueness of +Trustiness and Reliability have similar proof. +□ + diff --git a/LNAyT4oBgHgl3EQfgPhD/content/tmp_files/load_file.txt b/LNAyT4oBgHgl3EQfgPhD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..02095c9ab88b9b4c28c6b9c28918d10856c7482d --- /dev/null +++ b/LNAyT4oBgHgl3EQfgPhD/content/tmp_files/load_file.txt @@ -0,0 +1,1015 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf,len=1014 +page_content='RiskProp: Account Risk Rating on Ethereum via De-anonymous Score and Network Propagation Dan Lin School of Software Engineering, Sun Yat-sen University Zhuhai, China lind8@mail2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='cn Jiajing Wu∗ School of Computer Science and Engineering, Sun Yat-sen University Guangzhou, China wujiajing@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='cn Qishuang Fu School of Computer Science and Engineering, Sun Yat-sen University Guangzhou, China fuqsh6@mail2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='cn Zibin Zheng School of Software Engineering, Sun Yat-sen University Zhuhai, China zhzibin@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='cn Ting Chen University of Electronic Science and Technology of China Guangzhou, China brokendragon@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='cn ABSTRACT As one of the most popular blockchain platforms supporting smart contracts, Ethereum has caught the interest of both investors and criminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Differently from traditional financial scenarios, executing Know Your Customer verification on Ethereum is rather difficult due to the pseudonymous nature of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Fortunately, as the transaction records stored in the Ethereum blockchain are publicly accessible, we can understand the behavior of accounts or detect illicit activities via transaction mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Existing risk control techniques have primarily been developed from the perspectives of de-anonymizing address clustering and illicit account classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' However, these techniques cannot be used to ascertain the potential risks for all accounts and are limited by specific heuristic strate- gies or insufficient label information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These constraints motivate us to seek an effective rating method for quantifying the spread of risk in a transaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To the best of our knowledge, we are the first to address the problem of account risk rating on Ethereum by proposing a novel model called RiskProp, which in- cludes a de-anonymous score to measure transaction anonymity and a network propagation mechanism to formulate the relation- ships between accounts and transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We demonstrate the ef- fectiveness of RiskProp in overcoming the limitations of existing models by conducting experiments on real-world datasets from Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Through case studies on the detected high-risk accounts, we demonstrate that the risk assessment by RiskProp can be used to provide warnings for investors and protect them from possible financial losses, and the superior performance of risk score-based account classification experiments further verifies the effectiveness of our rating method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' KEYWORDS Abnormal detection, network propagation, Ethereum, risk control, de-anonymization 1 INTRODUCTION Ethereum [30] has the second-largest market cap in the blockchain ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The account model is adopted on Ethereum, and the native cryptocurrency on Ethereum is named Ether (abbreviated as ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' “ETH”), which is widely accepted as payments and transferred from one account to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' It is known that Ethereum accounts are indexed according to pseudonyms, and the creation of accounts is almost cost-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This anonymous nature and the lack of regulation result in the bad reputation of Ethereum and other blockchain sys- tems for breeding malicious behaviors and enabling fraud, thereby resulting in large property losses for investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As reported in a Chainalysis Crime Report, the illicit share of all cryptocurrency activities was valued at nearly USD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 billion in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These losses illustrate that Know-Your-Customer (KYC) and risk control of ac- counts are critical and necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Risk control [23] not only helps wallet customers identify risky accounts and avoid losses but also plays a vital role in the anti-money laundering of virtual asset service providers, such as cryptocurrency exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Therefore, a wealth of efforts have been expended in risk con- trol on Ethereum in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In September 2020, the Financial Action Task Force (FATF) published a recommendation report on virtual assets and released information on Red Flag Indicators [11] related to transactions, anonymity, senders or recipients, the source of funds, and geographical risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In addition, researchers in the aca- demic community have proposed various techniques from the per- spectives of address clustering and illicit account classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Ad- dress clustering techniques perform entity identification of anony- mous accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For example, Victor [27] proposes several clustering heuristics for Ethereum accounts and clusters 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='9% of all active ex- ternally owned accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Illicit account detection techniques focus on training classifiers based on well-designed features extracted from transactions [8, 10, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Moreover, some researchers have de- veloped methods for automatic feature extraction incorporating structural information [19, 21, 29, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' However, there are still some limitations (L) associated with these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' L1: Account clustering techniques can only be applied to part of accounts and therefore have limited applicability, and most accounts beyond heuristic rules cannot thus be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' L2: In the existing methods for illicit account detection, binary classifiers are usually trained via supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' However, as only a very small percentage of risky nodes have clear labels, which are required for these methods, the vast majority of accounts that may be involved in malicious events are unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In particular, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='00354v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='SI] 1 Jan 2023 WWW ’23, April 30–May 4, 2023, Austin, TX, US Anonymous author(s) TxnHash: 0x0bf742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' From: 0x3da2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To: 0x41b53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Timestamp: 1525153486 Value: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1 Ether TxnFee: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='000861 Ether Customer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Scammer Scammer Exchange Txns Txns Txns .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Account risk rating Public transactions Ethereum blockchain 0x3da2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 0x41b53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Figure 1: The procedure of ETH transfer in Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' “From” denotes the sender, “To” denotes the receiver, and “Txn” denotes “Transaction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' risky accounts with few transactions or unseen patterns are likely to be misidentified in practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To address the limitations presented above, we explore risk con- trol on Ethereum from a new perspective: Account risk rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In traditional financial scenarios, credit scoring is usually conducted by authorized financial institutions, which perform audits on their customers to fully understand their identity, background, and fi- nancial credit standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Similarly to credit scoring, risk rating on Ethereum can help us quantify the latent risk of a transaction or account with a quantitative score, thereby combating money laun- dering and identifying potential scams before new victims emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In terms of the abovementioned L1, in contrast to the traditional account clustering method, which can only de-anonymize a small number of accounts, the account risk method proposed in this paper can obtain quantitative risk indicators for all accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Regarding L2, the proposed risk rating method can achieve decent perfor- mance in an unsupervised manner without feeding labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The output of the proposed method is risk values, which are provided continuously and allow evaluation of the severity of risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Compared with traditional financial scenarios, several unique challenges (C) are encountered in the task of account risk rating on Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' C1: Nature of anonymity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Transactions on Ethereum do not require real-name verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Even worse, perpetrators of some malicious activities deliberately enhance their anonymity to counter the impact of de-anonymizing clustering techniques [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' C2: Complex transaction relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Compared with tradi- tional financial scenarios, a user or entity on Ethereum may control a large number of accounts at almost no cost, and the transaction relationship between accounts is also more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' How to quan- tify the impact of trading behavior between accounts on account risk is a challenging core problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To overcome the challenges mentioned above, we propose a novel approach called Risk Propagation (RiskProp) for Ethereum ac- count rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' It comprises two core designs, namely de-anonymous score and a network propagation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To resolve C1, de- anonymous score measures the degree to which transactions remain anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For example, both the payer and the payee of an illicit transaction prefer to have a small number of transactions to en- sure anonymity-preserving protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In contrast, both sides of a licit transaction may participate in numerous interactions without evading the impact of the de-anonymized clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Af- terward, to resolve C2, we model the massive transaction records as a directed bipartite graph and introduce a network propagation mechanism with three interdependent metrics, namely Confidence of the de-anonymous score, Trustiness of the payee, and Reliability of the payer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Intuitively, payees with higher trustiness receive trans- actions with higher de-anonymous scores, and payers with higher reliability will send transactions with higher confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Clearly, reliability, trustiness, and confidence are related to each other, so we define five items of prior knowledge that these metrics should satisfy and propose three mutually recursive equations to estimate the values of these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To verify the effectiveness of the pro- posed risk rating method and further illustrate the significance of rating for risk control on Ethereum, we evaluate the effect of the risk rating system via experiments from two aspects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', analysis of risk rating results and rating score-based illicit/licit classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Overall, our contributions are summarized as follows: A new perspective for Ethereum risk control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This paper is the first to propose tackling the problem of Ethereum risk control via the perspective of account risk rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' A novel risk metric for transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We creatively develop a metric called de-anonymous score for transactions, which measures the degree of de-anonymization to quantify the risk of a transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' An effective method and interesting insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We implement a novel risk rating method called RiskProp and demonstrate its su- perior effectiveness and efficiency via experiments on a real-world Ethereum transaction dataset together with theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' By analyzing the rating results and case studies on high-risk accounts, we obtain interesting insights into the Ethereum ecosystem and further show how our method could prevent financial losses ahead of blacklisting malicious accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2 PRELIMINARY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1 Ethereum Financial Background Ether is the native “currency” on Ethereum and plays a fundamen- tal part in the Ethereum payment system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Ether can be paid or received in financial activities, just like currency in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In conventional financial scenarios, a Know Your Customer (KYC) check is the mandatory process to identify and verify a customer’s identity when opening an account and to periodically understand the legitimacy of the involved funds over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' However, unlike traditional transaction systems, where customers’ identity informa- tion is required and obtained in KYC checks, Ethereum accounts are designed as pseudonymous addresses identified by 20 bytes of public key information generated by cryptographic algorithms, for example, “0x99f154f6a393b088a7041f1f5d0a7cbfa795d301”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Figure 1 depicts the risky scenario of Ether transfer in aspects of data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' It includes three layers: 1) Ethereum blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The Ethereum historical data are irreversible and publicly trace- able on the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2) Public transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The transaction denotes a signed data package from an account to another account, in- cluding the sending address, receiver address, transferred Ether amount, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 3) Account risk rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Usually, the identities who con- trol the accounts are not labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Customers may become involved in suspicious financial crimes or be vulnerable to frauds and scams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Furthermore, the illicit funds can be laundered and cashed out via exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In this procedure, our proposed RiskProp is implemented to measure the risk of unlabeled accounts that may have ill inten- tions and alert customers when engaging in suspicious, potentially illegal transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' RiskProp WWW ’23, April 30–May 4, 2023, Austin, TX, US 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2 The Nature of Blockchain: Anonymity It is known that the Ethereum account is identified as a pseudony- mous address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' However, if customers repeatedly use the same ad- dress as on-chain identification, the relationship between addresses becomes linkable via public transaction records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Accounts that participate in more transactions and connect with more accounts experience degrading anonymity [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To reduce the likelihood of exposure, criminals naturally tend to initiate transactions with fewer accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Here is an example on Ethereum: The two accounts of transaction 0x9a9d have only three transactions and became in- active thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These two accounts are considered suspicious and reported as relevant accounts of Upbit exchange hack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' On the con- trary, entities who do not deliberately take anonymity-preserving measures are likely to be normal [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Thus, the transaction is scored based on the fact of whether the accounts are trying to hide or not, which is the de-anonymous score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Definition 1 (De-anonymous score, abbreviated as “score”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The de-anonymous score of a transaction from account𝑢 to 𝑣 where there is no intention to hide is defined as 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) =1 2 ( 2 log |𝑂𝑢𝑡𝑇𝑥𝑛(𝑢)| − log𝑚𝑎𝑥𝑂𝑢𝑡 log𝑚𝑎𝑥𝑂𝑢𝑡 + 2 log |𝐼𝑛𝑇𝑥𝑛(𝑣)| − log𝑚𝑎𝑥𝐼𝑛 log𝑚𝑎𝑥𝐼𝑛 ), (1) where 𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) represents the outgoing transactions (payments) of payer 𝑢, 𝐼𝑛𝑇𝑥𝑛(𝑣) represents the incoming transactions (recep- tions) of payee 𝑣, and | × | denotes the size of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The minimum value of |𝑂𝑢𝑡𝑇𝑥𝑛(𝑢)| and |𝐼𝑛𝑇𝑥𝑛(𝑣)| is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Let 𝑚𝑎𝑥𝑂𝑢𝑡 and 𝑚𝑎𝑥𝐼𝑛 be the largest number of payments and receptions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The de-anonymous scores of a transaction (𝑢, 𝑣) range from −1 (very high anonymity, abnormal) to 1 (very low anonymity, normal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Intuitively, the score of (𝑢, 𝑣) increases as the transaction num- bers of either payer or payee grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Note that tricky criminals may camouflage themselves by deliberately conducting low-anonymity transactions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3 Transaction Network Construction First, each transaction on Ethereum has one payer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', sender) and one payee (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', receiver).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Any account can be the role of payer or payee, just as a person in real life has different roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The payee is a passive role and, therefore, we consider the incoming transactions to indicate the trustiness of an account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For instance, exchange accounts that receive more transactions are considered to be more trustworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In contrast, the payer is an active role and, thus, the outgoing transactions embody the intention of an account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For ex- ample, a scam account subjectively wants to transfer stolen money to its partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Next, the transaction records are modeled as a directed bipartite graph 𝐺 = (𝑈,𝑉,𝑆), where 𝑈 , 𝑉 , and 𝑆 represent the set of all payers, payees, and scores, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' A weighted edge (𝑢, 𝑣) denotes the transfer of Ethers from account 𝑢 ∈ 𝑈 to account 𝑣 ∈ 𝑉 with 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) ∈ 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The graph construction procedure is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Then, the ego network of a payer 𝑢 is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' It is formed by its outgoing scores and corresponding payee neighbors, formulated Money transfer Accounts Tnx Score Accounts Payer Payee Tnx Score Accounts (A) Ethereum transaction records (B) De-anonymous score calculation (C) Payer-payee graph Figure 2: The transformation from the raw transaction records to the directed bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' “Txn” denotes “Transaction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Payee set V Payer A Payer B Payee X Payee Y Score(A, X) Score(B, Y) Payer set U Score(B, X) In( ) = { Score(A, X), Score(B, X) } Payee X Out( ) = { Score(B, X), Score(B, Y) } Payer B Figure 3: A toy example of the directed bipartite graph estab- lished from transactions and the illustration of functions 𝐼𝑛 and 𝑂𝑢𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' as 𝑂𝑢𝑡(𝑢) ∪ {𝑣|(𝑢, 𝑣) ∈ 𝑂𝑢𝑡(𝑢)}, where 𝑂𝑢𝑡(𝑢) is the set of scores connected with 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' It is similar for the ego network of a payee, formulated as 𝐼𝑛(𝑣)∪{𝑢|(𝑢, 𝑣) ∈ 𝐼𝑛(𝑣)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Figure 3 shows an example in which there are two payers, two payees, and three transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 3 MODEL In this section, we describe the prior knowledge that establishes the relationships among accounts and transactions and then propose risk propagation formulations that satisfy the prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' It is worth noticing that the proposed algorithm does not require handcraft feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1 Problem Definition and Model Overview Given raw transaction records of Ethereum, we model the transac- tion relationships between accounts as a directed bipartite graph 𝐺 = (𝑈,𝑉,𝑆) with payers and payees as nodes and prepossessed de-anonymous scores as weights of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We believe that accounts have intrinsic metrics to quantify their reliability and trustworthi- ness and transactions have intrinsic metrics to measure the con- fidence of their calculated de-anonymous scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Naturally, those metrics are interdependent and interplay with each other via the risk propagation mechanism: Payers vary in terms of their Reliability, which indicates how motivated they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' A licit payer without malicious intent usually does not hide himself or disguise its intentions during transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Specifically, a reliable payer has harmless intentions regardless of whether it is transferring money to an exchange or to a scammer account (being gypped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In contrast, a perpetrator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', a scammer) hopes to cover up its traces [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The reliability metric 𝑅(𝑢) of a payer 𝑢 lies in [0, 1], ∀𝑢 ∈ 𝑈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' A value of 1 denotes a 100% reliable payer and 0 denotes a 0% reliable payer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' WWW ’23, April 30–May 4, 2023, Austin, TX, US Anonymous author(s) Table 1: An example of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑅0 is initial value, 𝑅𝑓 𝑖𝑛𝑎𝑙 and Risk𝑓 𝑖𝑛𝑎𝑙 are the results after convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Account Label 𝑅0 𝑅𝑓 𝑖𝑛𝑎𝑙 Risk𝑓 𝑖𝑛𝑎𝑙 0xa768 Contract-deployer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='8575 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='425 0x8271 Exchange 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='9526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='474 0xebdc Phish-hack 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1195 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='805 0xfe34 Phish-hack 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2330 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='670 Payees vary in their trustworthiness level, measured by a metric called Trustiness, which indicates how trustworthy they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Intu- itively, a cryptocurrency service provider with a better reputation will receive more licit transactions (with higher scores) from well- motivated payers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Trustiness of a payee 𝑇 (𝑣) ranges from 0 (very untrustworthy) to 1 (very trustworthy) ∀𝑣 ∈ 𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' De-anonymous scores vary in terms of Confidence, which re- flects the confidence in the estimated risk probability of a trans- action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The confidence metric 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) ranges from 0 (lack of confidence) to 1 (very confident).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The connection between the reliability and risk of accounts: We define Reliability to characterize the risk rating of accounts because an account’s intention can be inferred by its (active) sending be- havior, rather than by its (passive) receiving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' A scammer transferring stolen money to its gang is a better reflection of its evil intention than the receipt of stolen money from victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In the later section, we calculate the risk rating of accounts based on the Reliability of payer roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2 Network Propagation Mechanism Given a cryptocurrency payer–payee graph, all intrinsic metrics are unknown but are interdependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Here, we introduce five items of prior knowledge that establish the relationships and how the net- work propagation mechanism is specially designed for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The first two items of prior knowledge reflect the interdependency between a payee and the de-anonymous scores that they receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' [Prior knowledge 1] Payees with higher trustiness receive trans- actions with higher de-anonymous scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Intuitively, a payee receiving transactions with high de-anonymous scores is more likely to be trustworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Formally, if two payees 𝑣1 and 𝑣2 have a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2) and 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) > 𝑇 (𝑣2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' [Prior knowledge 2] Payees with higher trustiness receive trans- actions with more positive confident scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For two payees 𝑣1 and 𝑣2 with identical de-anonymous score networks, if the confidence of the in-transactions of payee 𝑣1 is higher than that of payee 𝑣2, the trustiness of payee 𝑣1 should be higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Formally, if two pay- ees 𝑣1 and 𝑣2 have a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2) and 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) > 𝐶𝑜𝑛𝑓 (ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) > 𝑇 (𝑣2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' According to the above prior knowledge, we develop the Trusti- ness formulation for ∀𝑣 ∈ 𝑉 of our RiskProp algorithm: 𝑇 (𝑣) = � (𝑢,𝑣) ∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) |𝐼𝑛(𝑣)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (2) The next item of prior knowledge defines the relationship be- tween the score of a transaction and the connected payer–payee pair using the anonymous nature of cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Algorithm 1 RiskProp Algorithm 1: Input: Directed Bipartite Graph 𝐺 = (𝑈,𝑉,𝑆) 2: Output: Risk of accounts 3: Initialize 𝑇 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='5, 𝑅0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7,𝐶𝑜𝑛𝑓 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='5,𝑡 = 0, Δ = 1 4: while Δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='01 do 5: 𝑡 = 𝑡 + 1 6: Update 𝑡𝑟𝑢𝑠𝑡𝑖𝑛𝑒𝑠𝑠 of payees using Equation 2 7: Update 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑖𝑡𝑦 of payers using Equation 4 8: Update 𝑐𝑜𝑛𝑓 𝑖𝑑𝑒𝑛𝑐𝑒 of transactions using Equation 3 9: Δ𝑇 = � 𝑣∈𝑉 |𝑇 𝑡 (𝑣) −𝑇 𝑡−1(𝑣) | 10: Δ𝑅 = � 𝑢∈𝑈 |𝑅𝑡 (𝑢) − 𝑅𝑡−1(𝑢) | 11: Δ𝐶 = � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝑆 |𝐶𝑜𝑛𝑓 𝑡 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | 12: Δ = max{Δ𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Δ𝑅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Δ𝐶 } 13: end while 14: 𝑅𝑖𝑠𝑘 (𝑢) = (1 − 𝑅(𝑢)) × 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' ∀𝑢 ∈ 𝑈 15: return [Prior knowledge 3] Confident de-anonymous scores of transac- tions are closely linked with the connected payee’s trustiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For- mally, if two scores (𝑢1, 𝑣1) and (𝑢2, 𝑣2) are such that𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) = 𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2),𝑅(𝑢1) = 𝑅(𝑢2), and |𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) −𝑇 (𝑣1)| ⩽ |𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2) −𝑇 (𝑣2)|, then 𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) ⩾ 𝐶𝑜𝑛𝑓 (𝑢2, 𝑣2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We imply that different transactions sent by the same payers can have different intentions and anonymity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Even scammers on Ethereum can have transactions that seem normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' [Prior knowledge 4] Transactions with higher confidence de- anonymous scores are sent by more reliable payers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Formally, if two scores (𝑢1, 𝑣1) and (𝑢2, 𝑣2) are such that𝑆𝑐𝑜𝑟𝑒(𝑢1, 𝑣1) = 𝑆𝑐𝑜𝑟𝑒(𝑢2, 𝑣2), 𝑇 (𝑣1) = 𝑇 (𝑣2), and𝑅(𝑢1) ⩾ 𝑅(𝑢2), then𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) ⩾ 𝐶𝑜𝑛𝑓 (𝑢2, 𝑣2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This prior knowledge incorporates the payer’s intention in mea- suring the confidence of transaction scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In this way, payees may have different confidence in receiving transactions with the same anonymity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For instance, exchanges on Ethereum receive funds from payers with different motivations—some are ordinary investors and some are suspicious accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Below, we propose the Confidence formulation that satisfies the above items of prior knowledge: 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) = 𝑅(𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) −𝑇 (𝑣)|) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (3) Then, we describe how to quantify the Reliability metric of a payer by the transactions it sends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' [Prior knowledge 5] Payers with higher reliability send transac- tions with higher confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For two payers 𝑢1 and 𝑢2 with equal scores, if payer 𝑢1 has higher confidence for all out transaction scores than𝑢2, then payer𝑢1 has a higher reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Formally, if two payers 𝑢1 and 𝑢2 have ℎ : 𝑂𝑢𝑡(𝑢1) → 𝑂𝑢𝑡(𝑢2) and 𝐶𝑜𝑛𝑓 (𝑢1, 𝑣1) > 𝐶𝑜𝑛𝑓 (𝑢2,ℎ(𝑣)) ∀(𝑢1, 𝑣) ∈ 𝑂𝑢𝑡(𝑢1), then 𝑅(𝑢1) > 𝑅(𝑢2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The corre- sponding formulation of Reliability metric for ∀𝑢 ∈ 𝑈 is defined as 𝑅(𝑢) = � (𝑢,𝑣) ∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) |𝑂𝑢𝑡(𝑢)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (4) Finally, the risk rating of an account is calculated by 𝑅𝑖𝑠𝑘(𝑢) = (1 − 𝑅(𝑢)) × 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The pseudo-code of RiskProp network propagation is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Let 𝑇 0, 𝐶𝑜𝑛𝑓 0, 𝑅0 be initial values and 𝑡 be the number of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In the beginning, we have initial reliability 𝑅0 ∀𝑢 ∈ 𝑈 , initial trustiness 𝑇 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='5 ∀𝑣 ∈ 𝑉 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' and RiskProp WWW ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' April 30–May 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' US Account Risk Rating Trustiness Confidence Risk Reliability Update Propagation Mechanism Risk Rating Results Analysis Ablation Study Risk Threshold Guarantees for Practice Comparative Evaluation Further Analysis Results Analysis Data Acquisition Labeled data Etherscan Ethereum Transactions Data Pre-processing Directed Bipartite Graph Construction De-anonymous Score Calculation Figure 4: The workflow of account risk rating on Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' initial confidence 𝐶𝑜𝑛𝑓 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='5 for all transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Then, we keep updating metrics using Equations 2–4 until Δ is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' RiskProp+: A Semi-supervised Version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Sometimes, we have partial information about the labels of fraudulent accounts (verified, phishing scams, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=') and licit accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We can take advantage of such prior information and incorporate them into our approach in a semi-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In the semi-supervised RiskProp+, we initialize the Reliability metrics only for the training accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' According to the risk levels of services reported by Chainalysis [26], we set 𝑅0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='9 for ICO wallet, Converter, and Mining, 𝑅0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 for Exchange, 𝑅0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='4 for Gambling, 𝑅0 = 0 for Phish/Hack, and set 𝑅0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 for testing accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The reliability values of labeled illicit accounts are unchanged during the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Here, we use a small real-world dataset on Ethereum to intuitively show the results of RiskProp+ after interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We collect transactions of 10 accounts (6 for training and 4 for testing), including 28,598 accounts and 52,733 transactions in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Table 1 shows how the reliability of the 4 testing accounts varies over interactions (we omit trustiness and confidence for brevity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These testing accounts have the same reliability values at the beginning (𝑅0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' After convergence, accounts labeled as “phish/hack” get a lower value of reliability, and other licit accounts get higher reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Confirming our intuition, RiskProp learns that accounts 0xebdc and 0xfe34 are high-risk accounts that investors need to be aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Workflow for Account Risk Rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Figure 4 shows the work- flow of account risk rating on Ethereum, which contains four mod- ules: (i) Data acquisition collects accounts, transactions, and la- bels from Ethereum and Etherscan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Only a few labels are provided, and these labels are not available in the unsupervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (ii) Data pre-processing of raw transaction data described in Fig- ure 1 is conducted in two steps: de-anonymous score calculation and directed bipartite graph construction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', payer–payee net- work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (iii) Account risk rating recursively calculates the Relia- bility, Trustiness of accounts, and Confidence of transaction scores until convergence, updated by the propagation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (iv) Re- sults analysis contains risk rating results analysis, comparative evaluation, and further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4 EXPERIMENTS To investigate the effectiveness of RiskProp, we conduct experi- ments on a real-world Ethereum transaction dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As risk rating is an issue without any ground truth, we verify the effectiveness and significance of the risk rating results of RiskProp via three tasks: 1) risk rating analysis, which includes distribution of risk rating results and case studies of transaction pattern;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2) comparative evaluation, which reports on the classification performance of labeled accounts compared with various baselines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' and 3) further analysis, which contains ablation study, impact of risk threshold, and guarantees for practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' RiskProp is open source and repro- ducible, and the code and dataset are publicly available after the paper is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1 Data Collection We first obtain 803 ground truth account labels from an official Ethereum explorer and then include all the accounts and transac- tions that are within the one-hop and two-hop neighborhood of each labeled account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Next, we filter out the zero-ETH transactions and construct the records into a graph, retaining the largest weakly connected component for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As a result, there are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='19 million accounts and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='13 million transactions in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In the dataset, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='02 percent (243) are labeled illicit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', phishing scam), whereas 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='05 percent (560) are labeled licit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', exchanges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The remaining unknown accounts are not labeled with regards to licit versus illicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2 Effectiveness of De-anonymous Score We use one-way analysis of variance (ANOVA) to assess whether there is a significant difference between illicit and licit transactions in the proposed de-anonymous score in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We consider a transaction as illicit (versus licit) if its payer is marked as illicit (versus licit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Table 2 shows that compared with the random score, our proposed score achieves a larger mean square (MS) between groups and smaller MS within groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' in addition, our proposed score has a higher F value, and the 𝑝-value equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These results suggest that the de-anonymous score is a useful metric for assessing the quality of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Table 2: ANOVA of random scores and de-anonymous scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Random scores De-anonymous score Src of var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' MS F 𝑝-value MS F 𝑝-value Between groups 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='8 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='6 × 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='0 × 10−1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='8 × 102 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 × 103 0 Within groups 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='0 × 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3 Analysis of Risk Rating Results The principal task of RiskProp is to rate Ethereum accounts based on how ill-disposed they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Given the account risk rating obtained by RiskProp, we first review the results and investigate the capability of RiskProp in discovering new risky accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Then, we dig deeper into the predicted high-risk accounts and obtain some insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1 Distribution of risk rating results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The risk value of an account ranges from 0 (low risk) to 10 (high risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The distribution of the predicted risk scores is as follows: 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='58% are located at (0,2], 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='45% WWW ’23, April 30–May 4, 2023, Austin, TX, US Anonymous author(s) (b) (a) (c) (d) Exchange (e) Phish_contract Victims Victims Exchange Scammer Scammers (f) 16 ETH 16 ETH 37 ETH 37 ETH 8 ETH 8 ETH 8 ETH Create 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='29 ETH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='29 ETH Figure 5: Visualization showing some typical transaction patterns of risky accounts (in red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' are located at (2,4], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='03% are located at (4,6], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='78% are located at (6, 8], and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='19% are located at (8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This is consistent with expectations: The risk value of the Ethereum transaction network meets the power distribution law, indicating that the overwhelming majority of accounts act normally, and only very few accounts have abnormal behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We are interested in whether the high-risk accounts predicted by RiskProp are actually questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Thus, we first manually check the top 150 accounts with the highest risk (with both in-coming and out-going transactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The finding is that 119 out of 150 (approximately 80%) accounts have abnormal behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Among these 119 illicit accounts, 43 accounts are already labeled as “phish/hack” by Etherscan, whereas the remaining 76 are newly discovered suspicious accounts that are not marked in the existing label library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This result indicates the capabilities of RiskProp in predicting undiscovered risky accounts and reducing financial losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2 Case studies of transaction pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We then manually veri- fied the predicted risky accounts by investigating their abnormal behaviors and find that there are many suspicious transaction pat- terns in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In order to save space, we show 6 typical patterns in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These patterns are summarized from the real- world Ethereum transaction data and guided by current research and recommendation reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (a) Hacking scammers are a list of addresses related to phish- ing and hacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Figure 5(a) shows a pattern of phishing accounts reported by users who suffered financial loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' A typical phishing scam on Ethereum is the “Bee Token ICO Scam” attack, in which the phishers sent fake emails to the investors of an ICO with a fake Ethereum address to deposit their contributions into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For example, account 0xe336 has been confirmed to be part of this “Bee Token” scam, and 243 ETH has been sent to this address by 165 victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (b) Fund source of hacking scammers are the upstream ac- counts of the known illicit accounts, which are collusion scam ac- counts to attract victims or provide money for hacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As shown in Figure 5(b), the behaviors of collusion scam accounts may look similar to victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Nevertheless, we find that the upstream collusion accounts appear to participate in fewer transactions with shorter time intervals, and there are attempts to transfer the entire ETH balance of the scammers according to the Red Flag Indicators of FATF [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (c) Money laundering of scammers are the downstream ac- counts of the known illicit accounts, which are collusion scam accounts to accept and transfer the stolen money, obfuscating the true sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As shown in Figure 5(c), account 0x78f1 received stolen funds from several known hacking scammers, appearing to be the account used in the “placement” stage of money laun- dering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Another example is 0xcfdd, which receives stolen funds from the Fake Starbase Crowdsale Contribution account 0x122c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (d) Zero-out middle accounts are the middle accounts that serve as a bridge defined by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As shown in Figure 5(d), most of the received funds will be transferred out in short succession (such as within 24 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' See 0x126e for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (e) Round transfers among exchanges denote a pattern that an account withdraws ETH without additional activity to a pri- vate wallet and then deposits back to the exchange, as shown in Figure 5(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Account 0x886e withdraws 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='4 ETH from Cryptopia exchange and then deposits the same amount of ETH back to Cryp- topia, which is an unnecessary step and incurs transaction fees [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Such a phenomenon indicates that the exchange is misused as a money-laundering mixer or is conducting wash trading [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (f) Creators of illicit contracts are often the manipulators behind the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The Origin Protocol phishing scam contact ac- count 0x9819 was created by account 0xff1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' After victims deposited money into the phishing contract, the creator transfers the stolen funds back to himself via internal transactions, which deliberately enhances anonymity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We observe that many illicit accounts are outside the label li- brary and are still considered risk-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Based on the results, we infer that our RiskProp is able to expose unlabeled illicit accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This is crucial on Ethereum, which lacks authorized and effective regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In addition, the newly identified illicit accounts can complete the current label collection for additional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='4 Comparative Evaluation Settings To further evaluate the performance of our method and show the potential application, we employ the rating scores to conduct clas- sification experiments that divide Ethereum accounts into illicit and licit accounts, and we compare the results with the existing baseline methods for further verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We wish to investigate if RiskProp can give a higher risk rating for the known illicit accounts and a lower rating for known licit accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1 Compared Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As mentioned earlier, RiskProp is the first algorithm that explores the risk rating of blockchain accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We chose a variety of methods (unsupervised and supervised) as baselines, which are similar to the problem we want to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We compare unsupervised RiskProp with (i) web page ranking, such as PageRank [6], and (ii) bipartite graph-based fraud detection, such as FraudEagle [2], BIRDNEST [12], and REV2 [16], which are also unsupervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The (semi-)supervised approaches are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (i) Machine learning methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', logistic regression (LR), naïve Bayes (NB), decision tree (DT), support vector machine (SVM), random for- est (RF), extreme gradient boosting (XGBoost), and LightGBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These methods are used by [1, 4, 8, 19] for detection of abnor- mal Ethereum accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (ii) Traditional graph neural network, including DeepWalk, Node2Vec, and graph convolutional network RiskProp WWW ’23, April 30–May 4, 2023, Austin, TX, US 50 100 150 200 250 300 350 400 Top k 0 20 40 60 80 100 Precision@k (%) (a) 50 100 150 200 250 300 350 400 Top k 0 20 40 60 80 100 Recall@k (%) (b) RiskProp PageRank Birdnest FraudEagle REV2 Figure 6: The 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 of illicit account pre- diction with different rating methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (GCN) were conducted by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' [7] for detection of Ethereum phishing scams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (iii) Graph neural network for graphs with heterophily, such as CPGNN [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The application of this type of algorithms is a recent research advancement in the task of Ethereum account classification [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2 Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To evaluate the performance of the mod- els, we calculate the following metrics: 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛,𝑅𝑒𝑐𝑎𝑙𝑙, 𝐹1,𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦, and 𝐴𝑈𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As we know, there are only 6 out of 10,000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='067 percent) accounts labeled in the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To measure the order of the risk rating, we employ 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 to evaluate the ranking order of the algorithm (@𝑘 means the top 𝑘 accounts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' All baseline methods are tested using the original codes published by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We repeat experiments 10 times and report the average results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3 Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We evaluate the methods with bi- nary labeled accounts (illicit verse licit) and, thus, we assume ac- counts in the top 1% to be the illicit accounts (corresponding thresh- old: 6 for RiskProp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The reason for this threshold and percentage setting is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The split of the dataset in the (semi-)supervised setting is 𝑡𝑟𝑎𝑖𝑛𝑖𝑛𝑔 : 𝑡𝑒𝑠𝑡 = 8 : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='5 Comparative Evaluation Results We report the 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛@𝑘 and 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 curves of the compared algorithms, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We observe that RiskProp obtains superior precision and recall than that of baseline with different 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Up to 𝑘 = 100, the precision of RiskProp is almost 1 for illicit account prediction, which is surprising for an unsupervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The 𝑅𝑒𝑐𝑎𝑙𝑙@𝑘 curve of RiskProp is significantly higher than the compared methods, and also increases steadily with 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Table 3 shows the performance of unsupervised and supervised methods separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We observe that RiskProp remarkably outperforms the unsupervised graph rating baselines in terms of accuracy and AUC, improving by 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='90% and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='16%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Meanwhile, for the licit account prediction, we observe that RiskProp beats the best baseline (FraudEagle) with a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='48% improvement in its F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These demonstrate the effectiveness of our account risk rating method without labeling information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Next, we turn our attention to the results of the semi-supervised RiskProp+ compared with the existing (semi-)supervised classifica- tion in Table 3, from which we derive the following conclusions: 1) RiskProp+ outperforms all baseline methods by 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='32% in terms of F1-score, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='62% in terms of AUC, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='56% in terms of accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2) The precision of licit accounts prediction is improved from Table 3: The classification results (%) of unsupervised and (semi-)supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Illicit account Licit account Total Methods P R F1 P R F1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' AUC PageRank 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='13 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='49 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='69 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='08 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='75 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='25 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='47 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='12 FraudEagle 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='88 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='670 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='36 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='07 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='10 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='38 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='98 BIRDNEST 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='24 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='32 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='26 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='24 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='21 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='35 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='00 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='77 REV2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='527 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='854 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='00 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='04 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='73 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='76 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='28 RiskProp 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='48 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='48 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='15 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='44 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='89 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='58 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='56 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='69 Illicit account Licit account Total Methods P R F1 P R F1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' AUC LR 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='67 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='58 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='84 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='87 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='23 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='41 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='25 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='90 NB 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='79 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='31 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='36 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='41 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='39 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='61 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='00 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='85 DT 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='66 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='07 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='04 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='79 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='19 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='40 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='75 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='39 SVM 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='00 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='76 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='67 75.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='99 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='86 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='63 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='40 CPGNN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='17 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='54 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='47 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='84 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='82 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='26 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='88 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='68 RiskProp+ 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='91 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='78 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='23 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='33 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='96 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='49 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='63 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='37 Table 4: Illicit account prediction of ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Methods Precision Recall F1-score RiskProp+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='8478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7723 RiskProp+ (w/o label) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='8148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7615 RiskProp+ (w/o NP) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='9959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='5513 RiskProp+ (w/o DS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='4737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2769 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='52% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', the average precision in baselines) to 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='33%, which means more licit accounts can be correctly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 3) The supe- rior performance of RiskProp is more significant in the prediction of illicit accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The recall of illicit accounts prediction is improved from 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='56% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', the average recall of illicit accounts prediction in baselines) to 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='78%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This shows the effectiveness of our framework in the prediction of both illicit and licit accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='6 Ablation Study To further validate the contribution of each component of the pro- posed RiskProp+, we conduct an ablation study as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' RiskProp+ (Full model): All components of the model and label data are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' w/o label: Labels are unavailable in the learning procedure, and the model is trained in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' w/o network propagation (NP): Remove the NP procedure and calculate the average de-anonymous scores (𝐴𝐷𝑆) for each ac- counts’ outgoing transactions (payer role).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' An account is predicted as abnormal if its 𝐴𝐷𝑆 ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' w/o de-anonymous score (DS): Replace DS with random scores, ranging from −1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We derive the following findings from Table 4: 1) Without the labels, the F1-score drops only slightly, indicating that our RiskProp does not rely on label data and can obtain good results in an un- supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To our surprise, the full model outperforms the RiskProp (w/o label), with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3% increase in recall and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='51% WWW ’23, April 30–May 4, 2023, Austin, TX, US Anonymous author(s) decrease in precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This may be possibly explained by the re- liability values of labeled illicit accounts remaining unchanged during training in the supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2) RiskProp (w/o NP) has a only lower precision but a greatly improved recall, revealing that most of the illicit accounts are correctly predicted as illicit but that some licit accounts are misjudged to be illicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This result demonstrates that de-anonymous score is an effective indicator of illicit transactions but their confidence varies among transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This result also confirms why we need to consider the confidence of the score in the propagation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 3) RiskProp (w/o DS) yields low precision (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='37%) and severely low recall (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='57%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This result demonstrates that, even if the network propagation model is retained, the wrong scores of transactions will be spread through- out the entire Ethereum transaction network, resulting in poor prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='7 Risk Threshold of RiskProp Given accounts in the reversed order of their risk ratings, a natural question is how to classify licit or illicit accounts according to their risk ratings for the classification task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' One possible option is to determine the percentage of known illicit labels in the dataset and set the risk value of this percentage as a demarcation line for account classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' However, the percentage is imprecise because some of the illicit accounts remain unrevealed according to the experimental results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Therefore, we try to establish a suitable risk threshold (𝑅𝑇𝐻) of RiskProp by conducting classification experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Figure 7(a) demonstrates the results of illicit account prediction with different risk thresholds, ranging from 1 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As expected, the precision increases while the recall decreases with increasing 𝑅𝑇𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In addition, F1, accuracy, and AUC first increase and then decrease with the increase in 𝑅𝑇𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The best performance for F1 and AUC is when 𝑅𝑇𝐻 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Thus, we set the risk threshold 𝑅𝑇𝐻 = 6 for RiskProp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='8 Guarantees for Practical Use Here, we present guarantees for RiskProp in practical use regard- ing the following aspects: 1) guarantee of convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2) time complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' and 3) linear scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Convergence and Uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We present the theoretical prop- erties of RiskProp, including the proofs of prior knowledge, con- vergence, and uniqueness of the proposed metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', reliability, trustiness, and confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Proofs are shown in the Appendix due to lack of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In each interaction, the RiskProp updates the reliability, Trustiness metrics of accounts and Confidence metric of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Therefore, the complexity of each iteration is O(|𝑈 | + |𝑆|) = O(|𝑆|), |𝑆| is the total edges in the payer–payee network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Thus, for 𝑘 iterations, the total running time is O(𝑘|𝑆|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Linear scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We have shown that RiskProp is linear in run- ning time in the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To show this experimentally as well, we create random networks of an increasing number of nodes and edges and compute the running time of the algorithm until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Figure 7(b) shows that the running time increases lin- early with the number of nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Therefore, we can conclude that RiskProp is a scalable rating method that is suitable for applications on large-scale transaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 RTH 0 20 40 60 80 100 Performance(%) Precision Recall F1 Accuracy AUC (a) Impact of different RTH 103 104 105 106 107 Number of nodes 10 1 100 101 102 103 104 Run time (seconds) (b) Scalability of RiskProp Figure 7: Further analysis of RiskProp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Analysis of incorrect predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Furthermore, the results of the RiskProp+ experiment showed that 39 out of 46 (85%) phishing ac- counts were correctly predicted as illicit accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To understand why the remaining accounts failed to be detected as illicit by our model, we manually checked their transactions and neighbors and obtained the following results: (i) For one of the accounts, we have a risk score of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='72, and in practice, the system will also warn about such accounts that are close to the risk threshold (𝑅𝑇𝐻 = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (ii) One account is set as the default risk value because the phishing account has no outgoing transactions for the time being, and in practice, we can make correct predictions as soon as the phish- ing account starts laundering money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (iii) Among the remaining five accounts, one account has a high transaction volume of 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The remaining four accounts have a high volume of transactions along with withdrawals of ETH from exchanges, which directly contributed to the high de-anonymity score of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' How- ever, there are two sides to the story: regulation and fraud are a game of confrontation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For hackers, reusing accounts reduces the probability of being identified as high risk and, at the same time, reusing accounts and withdrawing money from exchanges increase the risk of exposure and fund freezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 5 RELATED WORK Risk control studies in cryptocurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In recent years, there has been growing interest in account clustering and detecting il- licit activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', financial scams, money laundering) in cryp- tocurrency transaction networks [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Victor [27] is the first to propose clustering heuristics for the Ethereum’s account model, including deposit address reuse, airdrop multi-participation, and self-authorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' A recent review of the literature on cryptocur- rency scams [5] showed that the existing methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', [9], [10], and [15]) are mainly based on supervised classifiers fed with hand- crafted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Many attempts have been made [25, 29, 32] to incorporate structural information by learning the latent repre- sentations of accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Some researchers have investigated and modeled the money flow from a network perspective [18, 21] to better identify illicit activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' After all, there is still a black area regarding the estimation of the risk value of Ethereum accounts, which is the key task in alerting about suspicious accounts and transactions on the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Rating and ranking on graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The aim of ratings and rankings on graph data is to provide a score or an order for each node in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Currently, the main solutions are based on link analysis technique [24], Bayesian model [12], and iterative learn- ing [13], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Similarly to the proposed RiskProp algorithm, [16] RiskProp WWW ’23, April 30–May 4, 2023, Austin, TX, US proposed axioms and iterative formulations to establish the rela- tionship between ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In [22], the authors measured the bias and prestige of nodes in networks based on trust scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In [17], the authors highlighted that graph-based approaches provide unique solution opportunities for financial crime and fraud detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' A review on this topic [3] described the problems in current studies: lack of ground truths, imbalanced class, and large-scale network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These challenges also exist in our risk rating problem on Ethereum transaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 6 CONCLUSIONS AND FUTURE WORK In this paper, we present the first systematic study to assess the account risk via a rating system named RiskProp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In RiskProp, we modeled transaction records of Ethereum as a bipartite graph, pro- posed a novel metric called de-anonymous score to quantify the transaction risk, and designed a network propagation mechanism based on transaction semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' By analyzing the rating results and manually checking the accounts with high risk, we evaluated the performance of RiskProp and obtained new insights about transac- tion risks on Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In addition, we employed the obtained risk scores to conduct illicit/licit account classification experiments on labeled data, and the superiority of this method over baseline meth- ods further verified the effectiveness of RiskProp in risk estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' For future work, we plan to integrate the transaction amounts and temporal information in our model, develop a web page or online tool for querying risk values of accounts, and share the details of risky cases with the Ethereum community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' REFERENCES [1] Rachit Agarwal, Shikhar Barve, and Sandeep Kumar Shukla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2021.' metadata={'source': 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transaction ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Ethereum Project Yellow Paper 151 (2014), 1–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' [31] Jiajing Wu, Jieli Liu, Yijing Zhao, and Zibin Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Analysis of cryptocur- rency transactions from a network perspective: An overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Journal of Network and Computer Applications 190 (2021), 103139.' metadata={'source': 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Nedim Lipka, Nesreen K Ahmed, and Danai Koutra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Graph neural networks with heterophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 11168–11176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' WWW ’23, April 30–May 4, 2023, Austin, TX, US Anonymous author(s) A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1 Proof for Prior Knowledge In this part, we provide proofs that the proposed metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=', Re- liability, Trustiness, and Confidence satisfy Prior knowledge 1 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Prior knowledge 1 in the main paper is the following: [Prior knowledge 1] Payees with higher trustiness receive trans- actions with higher de-anonymous scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Formally, if two pay- ees 𝑣1 and 𝑣2 have a one-to-one mapping, ℎ : 𝐼𝑛(𝑣1) → 𝐼𝑛(𝑣2) and 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1), then 𝑇 (𝑣1) > 𝑇 (𝑣2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The formulation to be used to show that the prior knowledge is satisfied is Equations 2, 3, and 4 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑇 (𝑣) = � (𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) |𝐼𝑛(𝑣) | 𝑅(𝑢) = � (𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) |𝑂𝑢𝑡 (𝑢) | 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) = 𝑅(𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 (𝑣) |) 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' To prove the Prior Knowledge 1, let us take two payees 𝑣1 and 𝑣2 that have identically ego networks and a one-to-one mapping ℎ, such that |𝐼𝑛(𝑣1)| = |𝐼𝑛(𝑣2)|, 𝐶𝑜𝑛𝑓 (𝑢, 𝑣1) = 𝐶𝑜𝑛𝑓 (ℎ(𝑢), 𝑣2), and 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢), 𝑣2) ∀(𝑢, 𝑣1) ∈ 𝐼𝑛(𝑣1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' According to Equation 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we have 𝑇 (𝑣1) −𝑇 (𝑣2) = � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣1)∈𝐼𝑛(𝑣1) 𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣1) × 𝐶𝑜𝑛𝑓 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣1) |𝐼𝑛(𝑣1) | − � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣2)∈𝐼𝑛(𝑣2) 𝑆𝑐𝑜𝑟𝑒 (ℎ(𝑢),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣2) × 𝐶𝑜𝑛𝑓 (ℎ(𝑢),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣2) |𝐼𝑛(𝑣2) | = � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣1)∈𝐼𝑛(𝑣1) (𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣1) − 𝑆𝑐𝑜𝑟𝑒 (ℎ(𝑢),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣2)) × 𝐶𝑜𝑛𝑓 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣1) |𝐼𝑛(𝑣1) | As 𝑆𝑐𝑜𝑟𝑒(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣1) > 𝑆𝑐𝑜𝑟𝑒(ℎ(𝑢),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' so 𝑇 (𝑣1) −𝑇 (𝑣2) > � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣1)∈𝐼𝑛(𝑣1) 𝐶𝑜𝑛𝑓 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣1) |𝐼𝑛(𝑣1) | As 𝐶𝑜𝑛𝑓 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣1) ≥ 0 because Confidence are non-negative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑇 (𝑣1) −𝑇 (𝑣2) > 0 ⇒ 𝑇 (𝑣1) > 𝑇 (𝑣2) The other items of prior knowledge have very similar and straight- forward proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2 Proof for Convergence Before the proof of convergence, we first discuss the boundary of proposed metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' At the end of iteration 𝑡 of Algorithm 1, and by equation 2, 3, and 4, we get, 𝑇 𝑡 (𝑣) = � (𝑢,𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) × 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) |𝐼𝑛(𝑣) | 𝑅𝑡 (𝑢) = � (𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) |𝑂𝑢𝑡 (𝑢) | 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) = 𝑅𝑡 (𝑢) + (1 − |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) −𝑇 𝑡 (𝑣) | 2 ) 𝑇 ∞(𝑣), 𝑅(∞) (𝑢),𝐶𝑜𝑛𝑓 (∞) (𝑢, 𝑣) are their final values after con- vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' (Boundary discussion) Set the maximum score in the transaction network as 𝑀, namely: 𝑀 = max (𝑢,𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) then |𝑀| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The difference between a payee𝑣’s final Trustiness and its Trustiness after the first iteration is |𝑇 ∞(𝑢) −𝑇 1(𝑢) | ≤ |𝑀 | Similarly, |𝑅∞(𝑢) − 𝑅1(𝑢) | ≤ 1 |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1(𝑢, 𝑣) | ≤ 1 + |𝑀 | 2 = 𝛼 (𝛼 ≤ 1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' First, we state that |𝑀| is strictly less than 1 in prac- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' According to the formulation of 𝑆𝑐𝑜𝑟𝑒 of the main paper, we can see that 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) = 1 when 𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) = 𝑚𝑎𝑥𝑂𝑢𝑡 and 𝐼𝑛𝑇𝑥𝑛(𝑣) = 𝑚𝑎𝑥𝐼𝑛, it is an extreme situation where the largest number of payments and receptions of the entire network appears in one transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The other case is 𝑆𝑐𝑜𝑟𝑒(𝑢, 𝑣) = −1 when that 𝑂𝑢𝑡𝑇𝑥𝑛(𝑢) = 1 and 𝐼𝑛𝑇𝑥𝑛(𝑣) = 1 at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' This situation presents to be some isolated transactions, however, they do not propagate risk and thus do not influence convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' These situa- tions are out of our consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' So we get |𝑀| is strictly smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we prove that 𝑇 (𝑣) is bounded during the iterations: |𝑇 ∞(𝑣) −𝑇 1(𝑣) | = | � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) × 𝐶𝑜𝑛𝑓 ∞(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) |𝐼𝑛(𝑣) | − � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) 𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) × 𝐶𝑜𝑛𝑓 0(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) |𝐼𝑛(𝑣) | | Since |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤ � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) × (𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣)) | |𝐼𝑛(𝑣) | Since |𝑥 × 𝑦| = |𝑥| × |𝑦|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤ � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | × (𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣)) | |𝐼𝑛(𝑣) | (5) Since |𝑆𝑐𝑜𝑟𝑒(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣)| ≤ |𝑀| ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' and |(𝐶𝑜𝑛𝑓 ∞(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣)−𝐶𝑜𝑛𝑓 0(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣))| ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | ≤ |𝑀 | × |𝐼𝑛(𝑣) | |𝐼𝑛(𝑣) | = |𝑀 | Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we conduct the proof on 𝑅(𝑢): |𝑅∞ (𝑢) − 𝑅1 (𝑢) | = | � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝑂𝑢𝑡 (𝑢) 𝐶𝑜𝑛𝑓 0 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | |𝑂𝑢𝑡 (𝑢) | Again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' since |𝑥 × 𝑦| = |𝑥| × |𝑦|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝑅∞ (𝑢) − 𝑅1 (𝑢) | ≤ � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 0 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | |𝑂𝑢𝑡 (𝑢) | (6) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' since |(𝐶𝑜𝑛𝑓 ∞(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 0(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣))| ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝑅∞ (𝑢) − 𝑅1 (𝑢) | ≤ |𝑂𝑢𝑡 (𝑢) | |𝑂𝑢𝑡 (𝑢) | = 1 RiskProp WWW ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' April 30–May 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' US Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we calculate the bound of 𝐶𝑜𝑛𝑓 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣): |𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | = |𝑅∞ (𝑢) − 𝑅1 (𝑢) + |𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) −𝑇 1 (𝑣) | − |𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) −𝑇 ∞ (𝑣) || 2 Since |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we have |𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | ≤ |𝑅∞ (𝑢) − 𝑅1 (𝑢) | + ( ||𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) −𝑇 1 (𝑣) | − |𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) −𝑇 ∞ (𝑣) ||) 2 Since ||𝑥| − |𝑦|| ≤ |𝑥 − 𝑦|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' it follows that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | ≤ |𝑅∞ (𝑢) − 𝑅1 (𝑢) | + |𝑇 ∞ (𝑣) −𝑇 1 (𝑣) | 2 (7) Since |𝑅∞(𝑢) − 𝑅1(𝑢)| ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' and|𝑇 ∞(𝑣) −𝑇 1(𝑣)| ≤ |𝑀|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | ≤ 1 + |𝑀 | 2 For convenience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we let 1+|𝑀 | 2 = 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Since |𝑀| < 1, then 𝛼 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' □ Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Convergence of Propagation: The difference during iterations is bounded as as |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣)| ≤ 𝛼𝑡 (𝛼 = 1+|𝑀 | 2 < 1), ∀(𝑢, 𝑣) ∈ 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As 𝑡 increases, the difference decreases and 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) converges to |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Similarly, |𝑇 ∞(𝑣) −𝑇𝑡 (𝑣)| ≤ 𝛼𝑡−1, ∀𝑣 ∈ 𝑉 , |𝑅∞(𝑢) − 𝑅𝑡 (𝑢)| ≤ 𝛼𝑡−1, ∀𝑢 ∈ 𝑈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Similar to Equations 5, 6, and 7, we have, |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢, 𝑣) | ≤ |𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | + |𝑇 ∞ (𝑣) −𝑇𝑡 (𝑣) | 2 (8) |𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | ≤ � (𝑢,𝑣,)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) | |𝑂𝑢𝑡 (𝑢) | (9) |𝑇 ∞ (𝑣) −𝑇𝑡 (𝑣) | ≤ � (𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) | |𝐼𝑛(𝑣) | (10) First, we will prove the convergence of Confidence using mathe- matical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Base case of induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' When 𝑡 = 1, as we proved in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1, we get: |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 1(𝑢, 𝑣) | ≤ 𝛼1 Induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' We assume by hypothesis that |𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1(𝑢, 𝑣) | ≤ 𝛼𝑡−1, which is consistent with the base case already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' by substituting Equations 9 and 10 into Equation 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' for the case in the next iteration where time is 𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | ≤ � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | 2 × |𝑂𝑢𝑡 (𝑢) | ) + � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | 2 × |𝐼𝑛(𝑣) | ≤ 1 2 × � ( 1 + |𝑀 | 2 )𝑡−1 + |𝑀 | × � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) |𝐶𝑜𝑛𝑓 ∞ (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | |𝐼𝑛(𝑣) | � ≤ 1 2 × � ( 1 + |𝑀 | 2 )𝑡−1 + |𝑀 | × ( 1 + |𝑀 | 2 )𝑡−1 � ≤ ( 1 + |𝑀 | 2 )𝑡 = 𝛼𝑡 Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝐶𝑜𝑛𝑓 ∞(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 𝑡 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣)| ≤ 𝛼𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝑅∞ (𝑢) − 𝑅𝑡 (𝑢) | ≤ � (𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣) | |𝑂𝑢𝑡 (𝑢) | ≤ � (𝑢,𝑣)∈𝑂𝑢𝑡 (𝑢) ( 1+|𝑀| 2 )𝑡−1 |𝑂𝑢𝑡 (𝑢) | ≤ ( 1 + |𝑀 | 2 )𝑡−1 = 𝛼𝑡−1 |𝑇 ∞ (𝑣) −𝑇 (𝑣)𝑡 | ≤ � (𝑢,𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢, 𝑣) | × |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) | |𝐼𝑛(𝑣) | ≤ � (𝑢,𝑣)∈𝐼𝑛(𝑣) |(𝐶𝑜𝑛𝑓 ∞ (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 𝑡−1 (𝑢, 𝑣)) | |𝐼𝑛(𝑣) | ≤ � (𝑢,𝑣)∈𝐼𝑛(𝑣) ( 1+|𝑀| 2 )𝑡−1 |𝐼𝑛(𝑣) | ≤ ( 1 + |𝑀 | 2 )𝑡−1 = 𝛼𝑡−1 As discussed in the Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we know that |𝑀| is strictly smaller than 1, then we have 𝛼 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' As 𝑡 increases, 𝛼𝑡−1 → 0 and 𝛼𝑡 → 0, so after t iterations, 𝐶𝑜𝑛𝑓 (𝑢, 𝑣)𝑡 → 𝐶𝑜𝑛𝑓 ∞(𝑢, 𝑣), 𝑅(𝑢)𝑡 → 𝑅∞(𝑢), and 𝑇 (𝑣)𝑡 → 𝑇 ∞(𝑣), the algorithm converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3 Proof for Uniqueness In this part, we provides proofs that Reliability, Trustiness, and Confidence are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Confidence, Reliability, and Trustiness converge to the unique value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' First, we consider the uniqueness of Confidence using mathematical contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Let the 𝐶𝑜𝑛𝑓 (𝑢, 𝑣) converges to different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' So, let (𝑢, 𝑣) be the transaction with maximum Confidence difference, 𝐷 (with 𝐷 ≥ 0), between its two possible 𝐶𝑜𝑛𝑓1(𝑢, 𝑣) and 𝐶𝑜𝑛𝑓2(𝑢, 𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' According to Equation 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝐷 = |𝐶𝑜𝑛𝑓 ∞ 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 ∞ 2 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | ≤ |𝑅∞ 1 (𝑢) − 𝑅∞ 2 (𝑢) | + |𝑇 ∞ 1 (𝑣) −𝑇 ∞ 2 (𝑣) | 2 (11) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' according to Equation 9 and 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' |𝑅∞ 1 (𝑢) − 𝑅∞ 2 (𝑢) | ≤ � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 ∞ 2 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | |𝑂𝑢𝑡 (𝑢) | ≤ 𝐷 (12) WWW ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' April 30–May 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' US Anonymous author(s) |𝑇 ∞ 1 (𝑣) −𝑇 ∞ 2 (𝑣) | ≤ � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | × |𝐶𝑜𝑛𝑓 ∞ 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 ∞ 2 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | |𝐼𝑛(𝑣) | ≤ |𝑀 | × 𝐷 (13) We substitute Equation 12 and 13 into Equation (11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' and get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝐷 = |𝐶𝑜𝑛𝑓 ∞ 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 ∞ 2 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | ≤ 1 2 × ( � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝑂𝑢𝑡 (𝑢) |𝐶𝑜𝑛𝑓 ∞ 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 ∞ 2 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | |𝑂𝑢𝑡 (𝑢) | + � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) |𝑆𝑐𝑜𝑟𝑒 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | × |𝐶𝑜𝑛𝑓 ∞ 1 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 ∞ 2 (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | |𝐼𝑛(𝑣) | ≤ 1 2 × � 𝐷 + |𝑀 | × � (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content='𝑣)∈𝐼𝑛(𝑣) |𝐶𝑜𝑛𝑓 ∞(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) − 𝐶𝑜𝑛𝑓 ∞(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' 𝑣) | |𝐼𝑛(𝑣) | � ≤ 1 2 × (𝐷 + |𝑀 | × 𝐷) ≤ ( 1 + |𝑀 | 2 ) × 𝐷 = 𝛼 × 𝐷 Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' by solving 𝐷 ≤ 𝛼 × 𝐷(𝛼 ≠ 0) and with the condition that 𝐷 ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' we obtain 𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' Then, |𝐶𝑜𝑛𝑓 ∞ 1 (𝑢, 𝑣) − 𝐶𝑜𝑛𝑓 ∞ 2 (𝑢, 𝑣)| = 0 and converge value of Confidence is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' The uniqueness of Trustiness and Reliability have similar proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} +page_content=' □' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfgPhD/content/2301.00354v1.pdf'} diff --git a/PNE3T4oBgHgl3EQfxguY/content/tmp_files/load_file.txt b/PNE3T4oBgHgl3EQfxguY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d899b9a0379b62527833826a2439c8a90032ba4c --- /dev/null +++ b/PNE3T4oBgHgl3EQfxguY/content/tmp_files/load_file.txt @@ -0,0 +1,1696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf,len=1695 +page_content='Generated using the official AMS LATEX template v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='1 two-column layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This work has been submitted for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Copyright in this work may be transferred without further notice, and this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' An analytic formula for entraining CAPE in mid-latitude storm environments John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Petersa , Daniel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Chavasb , Chun-Yian Sua , Hugh Morrisonc , and Brice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Cofferd a Department of Meteorology and Atmospheric Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' PA b Department of Earth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Atmospheric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' and Planetary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' IN c National Center for Atmospheric Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' CO d Department of Marine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Earth and Atmospheric Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' North Carolina State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Raleigh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' NC ABSTRACT: This article introduces an analytic formula for entraining convective available potential energy (ECAPE) with an entrainment rate that is determined directly from the storm environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Extending previous formulas derived in Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2020a), entrainment is connected to the background environment via an analytic manipulation of the equations of motion that yields a direct correspondence between the storm relative flow and the updraft radius, and an inverse scaling between the updraft radius squared and entrainment rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' These concepts, combined with the assumption of adiabatic conservation of moist static energy, yield an explicit analytic equation for ECAPE that depends entirely on state variables in an atmospheric profile and a few constant parameters with values that are established in past literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Using a simplified Bernoulli-like equation, a second formula is derived that accounts for updraft enhancement via kinetic energy extracted from the cloud’s background environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' CAPE and ECAPE can be viewed as predictors of the maximum vertical velocity 𝑤𝑚𝑎𝑥 in an updraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, these formulas are evaluated using 𝑤𝑚𝑎𝑥 from past numerical modeling studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Both of the new formulas improve predictions of 𝑤𝑚𝑎𝑥 substantially over undiluted CAPE, ECAPE with a prescribed entrainment rate, and the ECAPE formula from Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The formula that incorporates environmental kinetic energy contribution to the updraft correctly predicts instances of exceedance of √ 2CAPE by 𝑤𝑚𝑎𝑥 in simulations, and provides a conceptual explanation for why such exceedance is rare among past simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' These formulas are potentially useful in nowcasting and forecasting thunderstorms and as thunderstorm proxies in climate change studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' SIGNIFICANCE STATEMENT: Substantial mixing occurs between the upward moving air currents in thun- derstorms (updrafts) and the surrounding comparatively dry environmental air, through a process called entrain- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Entrainment controls thunderstorm intensity via its diluting effect on the buoyancy of air within updrafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' A challenge to representing entrainment in forecasting and predictions of the intensity of updrafts in future climates is to determine how much entrainment will occur in a given thunderstorm environment without a computationally ex- pensive high resolution simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To address this gap, this article derives a new formula that computes entrain- ment from the properties of an updraft’s background envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This formula is shown to predict updraft vertical velocity more accurately than past diagnostics, and can be used in forecasting and climate prediction to improve predictions of thunderstorm behavior and impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Introduction Middle-to-upper1 tropospheric vertical velocities in deep convective updrafts influence a variety of storm- related societal impacts, including precipitation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Jo Corresponding author: John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters, John.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='Peters@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='edu 1We contrast middle-to-upper tropospheric vertical velocities, which are primarily buoyantly driven, with lower tropospheric vertical veloc- ities which are often dynamically driven in squall lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Bryan and Rotunno 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Jeevanjee and Romps 2015) and supercells (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Weisman and Rotunno 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' and Lasher-Trapp 2022), hail (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Danielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Lin and Kumjian 2022), electrification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Romps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2015), downdraft and cold pool inten- sity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Marion and Trapp 2019), tropospheric convec- tive mass flux (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020b), and the flux of mass, aerosols, and water vapor across the tropopause (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Mullendore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The magnitude of vertical veloc- ities in the upper reaches of deep convective updrafts are strongly influenced by updraft buoyancy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Morrison and Peters 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Jeevanjee 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' It is well known that entrainment-driven dilution of deep con- vective updrafts substantially influences updraft buoyancy and vertical velocity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Zipser 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Romps and Kuang 2010a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, weakly sheared deep convective updrafts with large fractional entrainment rates are sub- stantially diluted and often only realize a small fraction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', 20-30 %) of their convective available potential en- ergy (CAPE) as updraft kinetic energy KE (Romps and Kuang 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In contrast, more organized modes of deep convection such as squall lines and supercells with smaller fractional entrainment rates and less dilution can realize much larger fractions of their CAPE as KE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', 80-100 % Lebo and Morrison 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Mulhol- land et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, storm-to-storm variations in entrainment substantially alter how much CAPE a storm is able to process, and consequently its updraft kinetic en- ergy and vertical velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' These storm-to-storm variations in entrainment also generally supersede the influences of variations in other updraft processes and environment fac- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='04712v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='ao-ph] 11 Jan 2023 2 tors on vertical velocity that receive substantial attention in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Lebo 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Grabowski and Morrison 2021), such as aerosol effects, pressure perturbations, and precipitation behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, the atmospheric sciences community would benefit from an accurate representation of entrainment in research and forecasting diagnostic pa- rameters, such as CAPE, so that the parameters can more accurately characterize the intensity of convective updrafts that might form in a given environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' CAPE calculations that include entrainment effects are referred to as entraining CAPE, or ECAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Whereas CAPE is often viewed as the theoretical maximum kinetic energy that can be extracted by an isolated parcel from its envi- ronment, ECAPE makes additional assumptions about up- draft steadiness and mixing to estimate how the efficiency of this kinetic energy extraction is affected by entrain- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Various ECAPE-like calculations have been used for the better part of the last century, primarily in the cli- mate, tropical meteorology, and cumulus parameterization communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, simple plume models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Squires and Turner 1962) for moist convective updrafts predict profiles of buoyancy that include entrainment ef- fects, which can be vertically integrated to obtain ECAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The “cloud work function”, which is an essential element of many cumulus parameterizations (Arakawa and Schu- bert 1974), uses the buoyancy of a diluted parcel within its calculation, and yields a quantity that is analogous to ECAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' ECAPE is used as diagnostic tool in the research of tropical environments to explain the sensitivity of deep convection initiation to free tropospheric moisture (Brown and Zhang 1997), and in the closure formulation of cumu- lus parameterizations (Zhang 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The zero-buoyancy plume model, in which buoyancy is assumed to be exactly extinguished by entrainment, yields analytic solutions for the mean state thermal structure of the tropical atmosphere (Singh and O’Gorman 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The range of fractional en- trainment rates in the tropics is typically smaller than that of the mid latitudes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, using an ECAPE calculated with an empirically obtained constant fractional entrainment rate provides reasonably accurate predictions of deep convective updraft character- istics in the tropics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Gregory 2001) There are also a few scattered applications of ECAPE in the weather forecasting community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, the spatial distribution of ECAPE has been shown to better identify the tornadic regions of tropical (Sueki and Niino 2016) and extratropical cyclones (Tochimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2019) than undiluted CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' ECAPE has also been used to pre- dict vertical velocities in supercells more accurately than standard CAPE calculations (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' There is substantially larger variability in fractional entrainment in the continental mid-latitudes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Lasher-Trapp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021) than in the tropics, meaning that ECAPE computed with a sin- gle fractional entrainment rate cannot accurately describe all midliatude convective environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This makes using ECAPE in midlatudes more dif- ficult than in the tropics, because it is not always clear what entrainment rate should be used in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To address the issue over what choice of fractional en- trainment rate to use in the midlatitudes, Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2020a) (hereafter P20) developed an analytic formula for maximum updraft vertical velocity (which is equal to √ 2𝐸𝐶𝐴𝑃𝐸) that calculated entrainment from attributes of a storm’s background environment, rather than requiring that the user specify an entrainment rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The connection between entrainment and the background environment in this formula was based on the previously-established nega- tive correspondence between vertical wind shear and frac- tional entrainment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2019, 2020c, 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' That is, mature deep convective updrafts tend to be wider in environments with strong vertical wind shear and have accordingly smaller fractional entrainment rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This for- mula more accurately predicted maximum updraft vertical velocities than standard ECAPE computed with constant pre-specified fractional entrainment rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' There are several shortcomings of the P20 study that warrant a revisit of the concepts contained therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' First, the expression derived in the paper uses a hodgepodge of formulas from previous studies, such as Morrison (2017) and Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2019) as a starting point2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The assump- tions underlying these formulas from previous studies are not explicitly discussed in P20, nor are they even thor- oughly scrutinized in their source articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Because of this rooting in past studies, a few of the terms that end up in the P20 equation are complicated and lack obvious physical underpinning, which is challenging for end users of this formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Second, the end formula for maximum updraft vertical velocity is a third-order polynomial equation that must either be solved explicitly with the complicated quartic equation, or with a numerical root finding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' End users of the formula found this quartic solution difficult to efficiently incorporate into software routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This 3rd order polynomial equation results from the assumption that fractional entrainment 𝜀 scales with the inverse of updraft radius 𝑅−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, there is now evidence that 𝜀 ∼ 𝑅−2 is a more realistic scaling (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Mulholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Re-formulating the P20 equation with 𝜀 ∼ 𝑅−2 yields a 2nd-order polynomial equation that is much easier to solve, as will be shown in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Third, the title of that paper, which is “A formula for the maximum vertical velocity in supercell updrafts”, ob- scures the take-home messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The title does not contain 2Note a litany of constants are carried over into P20 from these past formulas, and some of the symbols used (such as 𝐻𝑣 for the latent heat of vaporization) are inconsistent with the symbols used in some of our more recent articles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', 𝐿𝑣 for the latent heat of vaporization Peters and Chavas 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2022c,a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3 the terms entrainment or CAPE, so it is not obvious that the parameter derived in the paper essentially modifies CAPE to account for the effects of entrainment (which is by definition ECAPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The concepts contained within the paper apply to any isolated deep convective updraft exist- ing within moderate to strong vertical wind shear – they are not limited to supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' There is no assumption about up- draft rotation within the mathematical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, the inclusion of the term supercell in the title made the application of the formula sound unnecessarily restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Our goal in this article is to revisit the concepts of P20 to derive ECAPE formulas (Sections 2-3) that improve upon the concepts in the P20 study in the following ways: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The buoyancy formula in the present study is de- rived directly from the assumed conservation of moist static energy, which differs from the P20 formula which used the supersaturation tendency equation from Politovich and Cooper (1988) as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This methodological alteration requires less severe assumptions and results in formulas with greater ac- curacy in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The new formula uses the 𝜀 ∼ 𝑅−2 scaling, with fur- ther improves accuracy over the P20 formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We also account for additional processes that were not considered by P20, such as the contribution to updraft kinetic energy from the kinetic energy an up- draft extracts from its inflow via pressure gradient accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The new ECAPE formulas are evaluated with output from four past numerical modeling studies that included 141 simulations (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The formulas and their con- stituent terms, along with recommended parameter values, are summarized in the discussion and conclusions (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Derivation of analytic ECAPE formula The derivation relies on three underlying concepts: a scaling between entrainment and updraft radius (section 2a), an analytic relationship between ECAPE and entrain- ment (section 2b), and an analytic relationship between updraft radius and state variables within an atmospheric sounding (sections 2c-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Combining these components allows us to eliminate entrainment and updraft radius to express ECAPE as a function of the state variables within a sounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We will need to make numerous approximations through the course of the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To evaluate the accuracy of these approximations, we will first establish a benchmark calculation of both buoyancy and ECAPE computed with as few approximations as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This benchmark cal- culation uses the adiabatic unsaturated and saturated lapse rate equations derived in Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2022c), eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 19 and 24 from that article respectively, with a mixed-phase layer in the parcel temperature range of 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='5 K to 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='15 K (see that study for details on the mixed-phase calculation), and the bulk plume entrainment approximation for the mix- ing of individual state variables with that of a horizontally invariant background environment (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 36-38 in that study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The formulas are evaluated using the severe weather proximity sounding dataset of Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This dataset includes 1028 atmospheric profiles taken near severe weather events that ranged from disorganized deep convection to tornadic supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In each profile, the par- cel with the largest undiluted CAPE in lowest 5 km of the atmosphere is lifted to calculate buoyancy, CAPE, and ECAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Connecting fractional entrainment to updraft radius Our first step is to establish a relationship between up- draft radius and the fractional entrainment rate 𝜀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We ac- complish this by deriving an expression for passive tracer dilution in the cloud core assuming that entrained air has a tracer value of zero, and assuming that detrained air has a tracer value equal to that locally in the cloud core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Here 𝜀 is the fractional entrainment rate needed to produce a vertical profile of cloud core passive tracer consistent with the dilution it undergoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The derivation closely follows that of Morrison (2017) (hereafter M17), section 2a therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We first consider a passive tracer 𝐶, whose mixing ratio (in kg kg-1) is 1 in a cloud’s effective inflow layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', the layer of nonzero CAPE Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Nowotarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020), and 0 above this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Conceptually, the passive tracer value represents the degree to which a parcel has been diluted via entrainment, with 𝐶 ≈ 1 indicating undiluted air, and 𝐶 << 1 indicating highly diluted air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The anelastic Lagrangian tendency equation for 𝐶 may be written in cylindrical coordinates as: 𝑑𝐶 𝑑𝑡 = 𝜕𝐶 𝜕𝑡 + 1 𝑟 𝜕𝑟𝑢𝐶 𝜕𝑟 + 1 𝑟 𝜕𝑣𝐶 𝜕𝜙 + 1 𝜌0 𝜕𝜌0𝑤𝐶 𝜕𝑧 = 0, (1) where 𝑟, 𝜙, and 𝑧 are the radial, azimuthal, and vertical coordinates, 𝑢, 𝑣, and 𝑤 are the corresponding radial, az- imuthal, and vertical velocities, and 𝜌0(𝑧) is a reference density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Azimuthally averaging this equation, and then Reynolds averaging, yields: 𝑑𝐶 𝑑𝑡 = −1 𝑟 𝜕𝑟𝑢′𝐶′ 𝜕𝑟 − 1 𝜌0 𝜕𝜌0𝑤′𝐶′ 𝜕𝑧 (2) where overbar denotes a spatial average with a filter scale similar to that of the updraft width (tyically on the order of 1-2 km), primes denote deviations smaller than the filter scale, and 𝑑𝐶 𝑑𝑡 = 𝜕𝐶 𝜕𝑡 + 𝑢 𝜕𝐶 𝜕𝑟 + 𝑣 𝜕𝐶 𝜕𝜙 + 𝑤 𝜕𝐶 𝜕𝑧 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Physically, the overbar terms correspond to updraft-scale flow patterns, 4 whereas the ′ terms correspond to turbulent fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We neglect the vertical turbulent flux term since recent large eddy simulations have supported a dominant role of lateral mixing in entrainment (Böing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' All quantities are valid at the updraft horizontal center unless explicitly stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Following M17 and De Rooy and Siebesma (2010), we assume that 𝑢′𝐶′ varies linearly over a turbulent mixing length scale 𝐿𝑚𝑖𝑥 and vanishes at the updraft center, such that 𝑢′𝐶′(𝑟) = 𝑢′𝐶′ ��� 𝐿𝑚𝑖𝑥 � 𝑟 𝐿𝑚𝑖𝑥 � , where the 𝑢′𝐶′ ��� 𝐿𝑚𝑖𝑥 de- notes the value of 𝑢′𝐶′ at distance 𝐿𝑚𝑖𝑥 from the updraft center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Finally, we use the chain rule to write 𝑑 𝑑𝑡 = 𝑤 𝑑 𝑑𝑧 , where 𝑑 𝑑𝑧 is the rate of change of a quantity as the parcel changes height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Making these approximations allows us to write eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2 as: 𝑑𝐶 𝑑𝑧 = −2 𝑢′𝐶′ ��� 𝐿𝑚𝑖𝑥 𝑤𝐿𝑚𝑖𝑥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (3) In the eddy diffusivity approximation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Kuo 1962), we assume that turbulent fluxes act to diffuse a quantity down- gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Using this approach, we may write 𝑢′𝐶′ ��� 𝐿𝑚𝑖𝑥 ≈ − 𝑘2𝐿2 𝑚𝑖𝑥 𝑃𝑟 �� 𝜕𝑤 𝜕𝑟 �� 𝜕𝐶 𝜕𝑟 (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5-6 in M17) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3 as: 𝑑𝐶 𝑑𝑧 = 2 𝑘2𝐿𝑚𝑖𝑥 𝑤𝑃𝑟 ���� 𝜕𝑤 𝜕𝑟 ���� 𝜕𝐶 𝜕𝑟 , (4) where 𝑘2 is the von Karman constant and 𝑃𝑟 is the turbulent Prandtl number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Finally, we use linear approximations to the lateral gradients in 𝐶 and 𝑤, such that 𝜕𝐶 𝜕𝑟 = 𝐶0−𝐶 𝑅 and �� 𝜕𝑤 𝜕𝑟 �� = |𝑤0−𝑤 | 𝑅 , and assume that 𝑤0 = 0 and 𝐶0 = 0 to write: 𝑑𝐶 𝑑𝑧 = −𝜀𝐶, (5) where 𝜀 = 2𝑘2𝐿𝑚𝑖𝑥 𝑃𝑟 𝑅2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (6) Equation 5 takes the form of a classical steady-state plume equation (Squires and Turner 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Betts 1975), where 𝜀 is the fractional entrainment inverse length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This term represents the rate at which 𝐶 is diluted with height by entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' There is some debate in past literature over how 𝐿𝑚𝑖𝑥 should be interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, in Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2020), P20, and Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2020b), we simply set 𝐿𝑚𝑖𝑥 ∼ 𝑅, which from Equation 6 results in a 𝜀 ∼ 𝑅−1 scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, analysis of large eddy simulations (LES) in our more recent work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Mulholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2022) indicates that 𝜀 ∼ 𝑅−2, suggesting from Equation 6 that 𝐿𝑚𝑖𝑥 should be viewed as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, we set 𝐿𝑚𝑖𝑥 to a fixed value following Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The eddy diffusivity approximation for lateral mixing implicitly neglects the entrainment of air occurring within organized updraft-scale flow, which is known as dynamic entrainment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', De Rooy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, our past work has shown that dynamic entrainment primarily affects updraft properties below the height of maximum 𝑤 where flow is laterally convergent into the updraft (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Morrison 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, it is reasonable to neglect dynamic entrainment in our present objective of deriving an expression for ECAPE, which pertains to the maximum kinetic energy achieved by the updraft that coincides with the position of maximum 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Derivation of analytic expressions for the buoyancy and 𝐸𝐶𝐴𝑃𝐸 of an entraining parcel Our next step is to express ECAPE as an analytic func- tion of 𝜀, wherein 𝜀 is not contained within integrals or differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We begin with the first law of thermody- namics for a rising parcel, which may be written as (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Emanuel 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Romps 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2022c): 𝑐 𝑝𝑚 𝑑𝑇 𝑑𝑧 − 1 𝜌 𝑑𝑝 𝑑𝑧 + 𝐿𝑣 𝑑𝑞𝑣 𝑑𝑧 − 𝐿𝑖 𝑑𝑞𝑖 𝑑𝑧 = 𝑄 (7) where 𝑐 𝑝𝑚 is the moist heat capacity that depends on water vapor and condensates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝑇 is temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝜌 is density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝑝 is pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝐿𝑣 is the temperature dependent latent heat of vaporization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝑞𝑣 is the water vapor mass fraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝐿𝑖 is the temperature dependent latent heat of freezing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝑞𝑖 is the ice mass fraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝑄 represents all diabatic effects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' and 𝑑 𝑑𝑧 represents the rate at which a quantity changes as a parcel changes its vertical position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We simplify this equation by making a series of approx- imations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' First, we replace the moist heat capacity 𝑐 𝑝𝑚 with the constant dry-air heat capacity 𝑐 𝑝𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Second, we use the hydrostatic equation to write 1 𝜌 𝑑𝑝 𝑑𝑧 = −𝑔, where 𝑔 is the acceleration of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Third, we neglect ice (𝑞𝑖 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Fourth, we replace the temperature-dependent latent heat of vaporization with its reference value at the triple point temperature 𝐿𝑣,𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Fifth, we assume that the only diabatic effect is the mixing of a parcel with its far-field environmen- tal profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Using these approximations, we may re-write eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 7 as: 𝑑ℎ 𝑑𝑧 = −𝜀 (ℎ − ℎ0) , (8) where ℎ is the moist static energy, defined as ℎ = 𝑐 𝑝𝑑𝑇 + 𝐿𝑣,𝑟𝑞 +𝑔𝑧, (9) ℎ0 is the moist static energy of the background environ- ment, defined as: ℎ0 = 𝑐 𝑝𝑑𝑇0 + 𝐿𝑣,𝑟𝑞0 +𝑔𝑧, (10) 5 the subscripts 0 denote the height-dependent background environmental profile, and we have dropped the 𝑣 subscript on 𝑞 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The −𝜀 (ℎ − ℎ0) term represents dilu- tion of ℎ with height due to entrainment, and is expressed in a manner consistent with a classical plume updraft model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Betts 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Note that for an adiabatic parcel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', 𝜀 → 0), ℎ is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, ℎ is analogous to equivalent potential temperature (𝜃𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' It will also be useful later to define the saturated moist static energy of the environment ℎ∗ 0 as: ℎ∗ 0 = 𝑐 𝑝𝑑𝑇0 + 𝐿𝑣,𝑟𝑞∗ 0 +𝑔𝑧, (11) where 𝑞∗ is the saturation mass fraction defined via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 10 in Bolton (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Finally, we define the buoyancy 𝐵 of an updraft air parcel as: 𝐵 = 𝑔𝑇 −𝑇0 𝑇0 , (12) which neglects the effects of water vapor and condensate loading on buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To evaluate the accuracy of these approximate equa- tions, we integrate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8 upward using a forward Euler integration scheme with a vertical grid spacing of 100 m, and solve for 𝑇 at each height using a numerical nonlinear equation solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We use 𝑑𝑞 𝑑𝑧 = −𝜀 (𝑞 − 𝑞0) during the un- saturated part of parcel ascent, and set 𝑞 = 𝑞∗ during the saturated part of parcel ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Quantities such as buoy- ancy and ECAPE computed with 8 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 12 are referred to as “approximate”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The vertical distributions of ℎ0 and ℎ∗ 0 in a typical deep convective environment are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Much like the typical vertical distribution of 𝜃𝑒, ℎ has a local maximum in the lower troposphere when nonzero CAPE is present, a local minimum in the middle troposphere, and becomes large again in the lower stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' An undiluted parcel lifted from the surface has larger ℎ than its surroundings until it reaches the lower stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In an entraining parcel, ℎ gradually relaxes to that of the background environment as the parcel ascends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Profiles of approximate buoyancy are compared to bench- mark buoyancy, calculated from equations in Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2022c) as described earlier in this section, for undiluted and diluted parcels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Despite the assumptions made thus far, the approximate and benchmark buoyancy profiles are comparable, having similar profile shapes and magnitudes at all heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Combining eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 9, 10, and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 11 yields: 𝐵 = 𝑔 𝑐 𝑝𝑑𝑇0 �ℎ − ℎ∗ 0 � − 𝑔𝐿𝑣,𝑟 𝑐 𝑝𝑑𝑇0 �𝑞∗ − 𝑞∗ 0 � , (13) where we have assumed that the updraft parcel is saturated, such that 𝑞 = 𝑞∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The second term on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 13 is often small relative to the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 13 suggests that 𝐵 > 0 when ℎ > ℎ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This agrees with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1a-b, which shows approximate coincidence between the vertical extent of ℎ > ℎ∗ 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1a) and the vertical extent of 𝐵 > 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' An entrainment term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', 𝜀) does not show up explicitly in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 13, but is included implicitly via the moist static energy of the updraft parcel ℎ, which is affected by entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To make 𝜀 show up explicitly, we find the particular solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8 with ℎ = ℎ0 at 𝑧 = 0, which may be written as: ℎ = 𝑒−𝜀𝑧 � ℎ𝑢𝑑 + ∫ 𝜉=𝑧 𝜉=0 𝜀𝑒𝜀 𝜉 ℎ0𝑑𝜉 � , (14) where ℎ𝑢𝑑 is the moist static energy of an undiluted parcel (or equivalently the moist static energy of the entraining parcel at its origin height since we assume ℎ is conserved for undilute ascent), 𝜉 is a dummy variable of integration, and we defined the parcel starting height as 𝑧 = 0 for sim- plicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Combining eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 14 with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 13 yields the following: 𝐵 = 𝑔 𝑐 𝑝𝑑𝑇0 � 𝑒−𝜀𝑧 � ℎ𝑢𝑑 + ∫ 𝜉=𝑧 𝜉=0 𝜀𝑒𝜀 𝜉 ℎ0𝑑𝜉 � − ℎ∗ 0 � − 𝑔𝐿𝑣,𝑟 𝑐 𝑝𝑑𝑇0 �𝑞∗ − 𝑞∗ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (15) The term 𝜀 now shows up explicitly in the equation, but is contained within integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We will need to make some additional approximations to bring this term out of the integrals to obtain our desired analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 15 can be re-arranged to express 𝐵 as a modification to the undiluted buoyancy 𝐵𝑢𝑑 using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 13 evaluated with ℎ = ℎ𝑢𝑑 and 𝑞 = 𝑞𝑢𝑑: 𝐵 = 𝐵𝑢𝑑𝑒−𝜀𝑧 + 𝑔 𝑐 𝑝𝑑𝑇0 � 𝑒−𝜀𝑧 ∫ 𝜉=𝑧 𝜉=0 𝜀𝑒𝜀 𝜉 ℎ0𝑑𝜉 − (1− 𝑒−𝜀𝑧) ℎ∗ 0 � − 𝑔𝐿𝑣,𝑟 𝑐 𝑝𝑑𝑇0 �𝑞∗ − 𝑞∗ 0 � + 𝑒−𝜀𝑧 𝑔𝐿𝑣,𝑟 𝑐 𝑝𝑑𝑇0 �𝑞∗ 𝑢𝑑 − 𝑞∗ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (16) This re-arrangement provides us with the opportunity to use the the undiluted buoyancy computed with the bench- mark parcel to calculate 𝐵𝑢𝑑 rather than the approximate 𝐵𝑢𝑑 when evaluating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 16 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', the black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1 in- stead of the red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This substitution generally improves the accuracy of the formula, and is used in all subsequent calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We note that the two terms on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 16 will cancel each other in the limit of 𝜀 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In the opposite limit of 𝜀 → ∞, each of these terms individual vanish because 𝑞∗ → 𝑞∗ 0 and 𝑒−𝜀𝑧 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We assume these terms are small in the intermediary range of 𝜀, and consequently neglect them to simplify the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Using integration by parts and neglecting the aforementioned terms, we may re-write eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 16 as: 𝐵 = 𝐵𝑢𝑑𝑒−𝜀𝑧 + 𝑔 𝑐 𝑝𝑑𝑇0 � 𝜀𝑧 �ℎ0 + 𝑒−𝜀𝑧𝜀2 ∫ 𝜉=𝑧 𝜉=0 �ℎ0𝜉𝑒𝜀 𝜉 𝑑𝜉 − (1− 𝑒−𝜀𝑧) ℎ∗ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (17) 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panel a: profiles of environmental ℎ0, ℎ∗ 0, and ℎ of an undiluted parcel, and the ℎ of a diluted parcel with 𝜀 = 1×10−4 m-1 (“h dil.”), computed using the tornadic supercell composite profile from Parker (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Moist static energies have been divided by 𝑐𝑝𝑑 to yield “energy temperature" with units of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panel b: buoyancy of the diluted (dashed lines) and undiluted (solid lines) parcels, computed using the benchmark parcel (black, described in the beginning of this section) and from the approximate formula for ℎ calculated by numerically integrating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='8 as described in the text (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' where �ℎ0(𝜉) ≡ 1 𝜉 ∫ 𝜉 ∗=𝜉 𝜉 ∗=0 ℎ0𝑑𝜉∗ is the average of ℎ0 below height 𝜉 and �ℎ0 in the first term in the parentheses on the RHS is evaluated at 𝜉 = 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' If we assume that �ℎ0 is approxi- mately constant with height3 in the integral term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 17, the equation simplifies dramatically to the following: 𝐵 = 𝐵𝑢𝑑𝑒−𝜀𝑧 + 𝑔 𝑐 𝑝𝑑𝑇0 (1− 𝑒−𝜀𝑧) � �ℎ0 − ℎ∗ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (18) This equation is an analytic function of 𝐵𝑢𝑑, 𝜀, and the state variables within a sounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The first term on the RHS represents the direct dilution of the updraft’s temper- ature perturbation via entrained air with no temperature perturbation, whereas the second term encapsulates the reduced condensation rate resulting from the entrainment of unsaturated air by the updraft, relative to an undiluted parcel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Before moving on to an analytic formula for ECAPE, we evaluate the accuracy of this analytic buoyancy formula by comparing the average buoyancy 𝐵 between the level of free convection (LFC) and the level of neutral buoyancy (LNB) to that of the benchmark buoyancy profile and the formula from P20 (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4-5 therein4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Here, the LFC is the highest instance of zero buoyancy below the height of maximum buoyancy, and the LNB is the highest instance of zero buoyancy in the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We define three metrics for evaluation: Pearson correlation coefficient 𝐶𝐶 among soundings of 𝐵 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 18 with 𝐵 from the more accurate benchmark lapse rate formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' the fractional reduction in undiluted 𝐵 by entrainment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' and normalized root-mean- 3This assumption is reasonable, given that vertical variations in � ℎ0 are on the order of 1×104 J kg-1, whereas the typical magnitude of this quantity is on the order of 1×106 J kg-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4We also use the 𝐵𝑢𝑑 computed with the benchmark parcel in the P20 formula to maximize this formula’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' square-error (NRMSE), defined as the the average over all soundings of the squared difference between 𝐵 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 18 and 𝐵 from the benchmark lapse rate formula, divided by the magnitude of 𝐵 from the benchmark formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' These metrics are plotted as a function of 𝜀 and updraft radius 𝑅 on the 𝑥 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We relate 𝑅 to 𝜀 using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6, with 𝑘2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='18, 𝑃𝑟 = 1 3, and 𝐿𝑚𝑖𝑥 = 120 m following Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The 𝐶𝐶 of the new formula with the benchmark calcu- lation is very close to 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2a) for all 𝑅 > 750 m and for fractional reductions in CAPE of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='9 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', updrafts that realize 10 % or more of their CAPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2c), which is the range of fractional reductions expected in midlatitude deep convection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Lasher-Trapp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For 𝑅 less than 750 m and when fractional re- ductions approach 1, 𝐶𝐶 begins to drop, suggesting that the formula is less accurate for strongly entraining weak con- vection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The story is similar for NRMSE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2e), which is relatively small in magnitude (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='1) for 𝑅 > 750 m, but increases when 𝑅 falls below 750 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Compared to the P20 formula, the new formula derived here has smaller NRMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2e) and larger 𝐶𝐶 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2a), indicating that we have made an improvement in accuracy in the present derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This improvement over the P20 formula is pri- marily due to an over-estimation of the fractional reduction in buoyancy via entrainment in the P20 formula that does not occur in the one derived here (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This difference is particularly noticeable when we restrict our analysis to soundings with less than 1000 J kg−1 of undiluted CAPE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2b,d,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In this low CAPE regime, the NRMSE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2f) and 𝐶𝐶 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2b) of the new formula are comparable to the errors for the whole sounding data set, whereas the P20 formula performs considerably worse with respect to both 𝐶𝐶 and errors in the low CAPE regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Comparison of vertically-averaged buoyancy 𝐵 calculated using the formula from the present study (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 18, red), the P20 buoyancy formula (gray), and the benchmark parcel (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panels a,b show 𝐶𝐶, c,d the fractional reduction in 𝐵, and e,f the normalized error NRMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝐶𝐶 and NRMSE are calculated relative to the benchmark parcel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Left panels show results from all Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2003) soundings, and right panels show results from only soundings with < 1000 J kg-1 undiluted CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Our next task is to use eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 18 to obtain an expression for ECAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We formally define ECAPE as: ECAPE = ∫ 𝑧=𝐿𝑁 𝐵 𝑧=𝐿𝐹𝐶 𝐵𝑑𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (19) Vertically integrating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 18 from the LFC to the LNB and combining with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 19 yields: ECAPE = ∫ 𝑧=𝐿𝑁 𝐵 𝑧=𝐿𝐹𝐶 𝐵𝑢𝑑𝑒−𝜀𝑧𝑑𝑧+ ∫ 𝑧=𝐿𝑁 𝐵 𝑧=𝐿𝐹𝐶 𝑔 𝑐 𝑝𝑑𝑇0 (1− 𝑒−𝜀𝑧) � �ℎ0 − ℎ∗ 0 � 𝑑𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (20) It will become advantageous later to have the integral bounds on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 20 extend to the equilibrium level for an undiluted parcel5 𝐻, rather than to the LNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We note that the integral of the first term from the LNB to the 𝐻 will always be positive, since 𝐵𝑢𝑑 is positive below the 𝐻 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' On the other hand, the integral of the second term over this range is typically negative (as will be discussed shortly), and at least partially cancels the contribution of the integral of the first term over this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, we extend the upper bounds of these integrals to the 𝐻, assuming that the partial cancellation between the terms mitigates the resulting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5The equilibrium level is typically denoted with the acronym EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We instead use the symbol 𝐻 for compactness in equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To pull 𝜀 out of the integrals in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 20, we use integration by parts and these integral definitions to write the first term on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 20 as: ∫ 𝑧=𝐻 𝑧=𝐿𝐹𝐶 𝐵𝑢𝑑𝑒−𝜀𝑧𝑑𝑧 = 𝑒−𝜀𝐻CAPE+𝜀 ∫ 𝑧=𝐻 𝑧=𝐿𝐹𝐶 𝑒−𝜀𝑧𝐵𝑢𝑑𝑑𝑧 (21) where CAPE = ∫ 𝑧=𝐻 𝑧=𝐿𝐹𝐶 𝐵𝑢𝑑𝑑𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (22) We make the approximation that 𝐵𝑢𝑑 is linear with height on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 21: 𝐵𝑢𝑑 ≈ � 𝐵𝑢𝑑 (𝑧 − 𝐿𝐹𝐶) , (23) where � 𝐵𝑢𝑑 is the average undilute 𝐵 between the 𝐿𝐹𝐶 and 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We then vertically integrate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 21, assume that 𝐿𝐹𝐶 << 𝐻 and hence 𝐻 − 𝐿𝐹𝐶 ≈ 𝐻, and neglect entrain- ment below the LFC such that 𝑒−𝜀𝐿𝐹𝐶 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We apply analogous assumptions to the 2nd term on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Modifying eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 20 with these assumptions yields: ECAPE = �1− 𝑒−𝜀𝐻 𝜀𝐻 � CAPE− � 1− 1− 𝑒−𝜀𝐻 𝜀𝐻 � NCAPE (24) 8 where NCAPE = − ∫ 𝑧=𝐻 𝑧=𝐿𝐹𝐶 𝑔 𝑐 𝑝𝑑𝑇0 � �ℎ0 − ℎ∗ 0 � 𝑑𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (25) NCAPE represents the buoyancy dilution potential of the free troposphere: the potential buoyancy loss that could be induced by entrainment mixing due principally to the saturation deficit of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' It is a purely environ- mental quantity that does not depend on parcel properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' As defined here with �ℎ0, it specifically measures the energy difference between the saturation MSE at a given level and the mean MSE of the free troposphere below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The latter captures the environment through which a parcel would have to rise, and potentially mix with, prior to reaching a particular level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Because ℎ∗ 0 is comparable to or larger than �ℎ0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3a), NCAPE is typically (but not always) positive (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The difference term in the integral �ℎ0 − ℎ∗ 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3a) and hence the magnitude of NCAPE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3b) will be larger when the free troposphere is dry and �ℎ0 is far smaller than ℎ∗ 0, compared to when the free troposphere is moist and �ℎ0 is closer in magnitude to ℎ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' A warm free tro- posphere at a given RH generally increases the difference between ℎ∗ 0 and �ℎ0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3c) compared to a situation when the free troposphere is cool at the same RH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For a fixed RH, this makes NCAPE larger when the free troposphere is warm, relative to when it is cool (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, NCAPE generally encapsulates the effects of tropospheric dryness and temperature on buoyancy via entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 24 achieves the stated purpose of this derivation, since 𝜀 is now outside of the integral terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' It will become advantageous in the next sub-section to further simplify the exponential terms in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' One may consider making first order Taylor series approximations for the exponential terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance 1−𝑒−𝜀𝐻 𝜀𝐻 ≈ 1 − 𝜀𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, the ex- ponential functions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 24 are strongly nonlinear with respect to 𝜀𝐻 in the range of 0 < 𝜀𝐻 < 10, which is the typical range we would encounter in our analysis, mak- ing the first order Taylor series approximation inaccurate (compare the blue and black lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Instead, we invert the exponential term 1−𝑒−𝜀𝐻 𝜀𝐻 , approximate its inverse with a first order Taylor series, and then invert the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance: 𝜀𝐻 1− 𝑒𝜀𝐻 ≈ 1+ 𝜀𝐻 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (26) and consequently: 1− 𝑒𝜀𝐻 𝜀𝐻 ≈ 1 1+ 𝜀𝐻 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (27) This approximation is far more accurate (compare the red and black lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Substituting these approxima- tions into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 24 and re-arranging yields: ECAPE = CAPE− 𝜀𝐻 2 NCAPE 1+ 𝜀𝐻 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (28) As a sanity check, examine the behavior of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 28 under limiting scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, in the limit of no entrain- ment where 𝜀 → 0, ECAPE → CAPE, which makes sense given that ECAPE for an undiluted parcel intuitively con- verges to the CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In the converse limit of 𝜀 → ∞, we may use L’Hôpital’s rule to deduce that ECAPE → NCAPE, which is inconsistent with the definition of CAPE as a quantity greater than or equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, this situa- tion is easily remedied by simply setting ECAPE to 0 if eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Finally, the case NCAPE = 0 yields ECAPE = 𝐶 𝐴𝑃𝐸 1+ 𝜀𝐻 2 , indicating that ECAPE is still smaller than CAPE when 𝜀 ≠ 0 and hence dilution still reduces buoyancy in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Indeed, for a saturated parcel to be positively buoyant in the first place requires ℎ > ℎ∗ 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 13), and since ℎ∗ 0 ≥ ℎ0 by definition, then ℎ > ℎ0 and entrainment will dilute ℎ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' and by extension, 𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' One spe- cific example of this situation is an adiabatic atmosphere (dry or saturated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' constant ℎ0), in which a parcel must be warmed in order to become positively buoyant and have non-zero CAPE, but in doing so the parcel will also have higher energy than the environment at all levels through which it rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The analytic formula for ECAPE in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 28 loses a bit of accuracy relative to the numerically integrated analytic buoyancy equation at larger values of 𝜀 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', smaller up- draft radii;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4b-d), but remains more accurate than the formula for maximum updraft vertical velocity 𝑤𝑚𝑎𝑥 from P20 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 18 therein), which is converted to ECAPE via 𝑤2 𝑚𝑎𝑥 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' These errors stem from a slight underestimation of the fractional reduction in undiluted CAPE at large 𝜀 val- ues (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4c) that results from our changing of the integral bounds in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 20 from the LNB to 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Despite these errors, this formula is quite accurate over the range of 𝑅 and 𝜀 that typify deep moist convection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', fractional reductions of no greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='8, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Relating fractional entrainment to environmental vari- ables It will be convenient later in the derivation to manipulate a nondimensional form of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We define the nondimen- sional ECAPE as �𝐸 ≡ ECAPE CAPE , the nondimensional NCAPE as �𝑁 ≡ NCAPE CAPE , and the nondimensional fractional entrain- ment rate �𝜀 ≡ 𝜀𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Using these definitions, we re-write eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 28 as: �𝐸 = 1− �𝜀 2 �𝑁 1+ �𝜀 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (29) Our next task is to eliminate �𝜀 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 28 by expressing this term as function of other updraft and environmental 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Demonstrations of the sensitivities of NCAPE to relative humidity (RH) and free tropospheric temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panel a: profiles of ℎ∗ 0 (red, divided by 𝑐𝑝𝑑 to yield units of K), and � ℎ0 (blue, K) for the baseline sounding (solid), RH increased by 20 % (dashed blue), and RH decreased by 20 % (dotted blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panel b: profiles of NCAPE (J kg-1) corresponding to panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panels c-d: analogous to panels a-b, but showing differences in ℎ∗ 0 and � ℎ0 resulting from an increase in 𝑇 by 2 K with RH held constant (dashed), and a decrease in 𝑇 of 2 K with RH held constant (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We proceed by defining �𝑅 ≡ 𝑅 𝐻 and use eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6 to write: �𝜀 = 𝜖 �𝑅−2, (30) where 𝜖 = 2𝑘2𝐿𝑚𝑖𝑥 𝐻𝑃𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (31) Combining eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 30 with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 29 yields: �𝐸 = 1− 𝜖 2 �𝑅2 �𝑁 1+ 𝜖 2 �𝑅2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (32) Following P20 and Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2022a), we may express �𝑅 as a function of updraft and environmental attributes by making the following assumptions about updraft geometry and inflow: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Updrafts are cylindrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Updraft radius 𝑅 is constant with height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Numerous previous studies show this to be approximately valid (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Sherwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hernandez-Deckers and Sherwood 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We assume that all environmental storm-relative wind V𝑆𝑅 that encounters the cross-sectional area of the updraft on the upstream side becomes inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Past studies also show this assumption to be reasonable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2019, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The updraft maximum vertical velocity 𝑤𝑚𝑎𝑥 is pro- portional to the horizontally averaged vertical velocity < 𝑤 > at the same height, such that < 𝑤 >= 𝛼𝑤𝑚𝑎𝑥, where 0 < 𝛼 < 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Morrison 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Morrison and Peters 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The updraft maximum vertical velocity is primarily determined by updraft buoyancy, such that 𝑤𝑚𝑎𝑥 = √ 2ECAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This assumption is supported by (Mor- rison and Peters 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Jeevanjee 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2019, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The maximum vertical velocity occurs at height 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panel a: comparison of the scale factor in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 24 (solid black) with its first order Taylor series approximation (blue dashed), and the first order Taylor series approximation of its inverse (dashed red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panels b-d: analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2a,b,c, but evaluating ECAPE from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 28 (red, the present article), ECAPE from P20 (gray), and ECAPE from numerically integrating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 18 (black), all relative to the benchmark calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' With these assumptions at hand, we start by writing the anelastic continuity equation in cylindrical coordinates as: 𝜌0 𝜕𝑟𝑢 𝜕𝑟 + 𝜌0 𝜕𝑣 𝜕𝜙 +𝑟 𝜕𝜌0𝑤 𝜕𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (33) Azimuthally integrating from 𝜙 = 0 to 𝜙 = 2𝜋, radially integrating from 𝑟 = 0 to the updraft radius at 𝑟 = 𝑅, and vertically integrating from the surface to 𝐻 (assuming 𝑤 = 0 at 𝑧 = 0) and dividing by 2𝜋 yields: 𝐻 �𝜌0�𝑢𝑅 + 𝑅 𝜌0,𝐻 < 𝑤𝐻 > 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (34) where �𝑢𝑅 = 1 2𝜋 ∫ 𝑧=𝐻 𝑧=0 𝜌0 ∫ 𝜙=2𝜋 𝜙=0 𝑢𝑑𝜙𝑑𝑧 ∫ 𝑧=𝐻 𝑧=0 𝜌0𝑑𝑧 (35) is the density-weighted vertical average of 𝑢 at radius 𝑅, and between the surface and height 𝐻, and represents the average inflow speed, < 𝑤 >= 1 𝜋𝑅2 ∫ 𝑟=𝑅 𝑟=0 ∫ 𝜙=2𝜋 𝜙=0 𝑟𝑤𝑑𝜙𝑑𝑟 (36) is the area average of 𝑤 within radius 𝑅, �𝜌0 is the vertical average of 𝜌0 between the surface and height 𝐻, and 𝜌0,𝐻 is 𝜌0 valid at height 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Making use of < 𝑤 >= 𝛼𝑤𝑚𝑎𝑥 (as- sumption 4) at height H and 𝑤2 𝑚𝑎𝑥 2 = 𝐸𝐶𝐴𝑃𝐸 (assumption 5), and re-arranging eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 34 yields: �𝑅 = −2𝜎 𝛼 �𝑢𝑅 √ 2ECAPE , (37) where 𝜎 = � 𝜌0 𝜌0,𝐻 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We may relate �𝑢𝑅 to the horizontal storm-relative wind speed 𝑉𝑆𝑅 = |V𝑆𝑅|, where V𝑆𝑅 is the storm-relative wind vector, by first defining the upstream flank of the updraft as the range from 𝜙 = − 𝜋 2 to 𝜙 = 𝜋 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We next assume that all inflow is accomplished by the cloud- relative wind entering the upstream updraft flank, and the radial component of the environmental cloud-relative wind at the updraft edge is 𝑢 = −𝑉𝑆𝑅 cos𝜙 and 𝑢 = 0 m s-1 on the downstream edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' These assumptions allow us to re-write eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 35 as: �𝑢𝑅 = − 1 2𝜋 ∫ 𝑧=𝐻 𝑧=0 ∫ 𝜙= 𝜋 2 𝜙=− 𝜋 2 𝜌0𝑉𝑆𝑅 cos𝜙𝑑𝜙𝑑𝑧 ∫ 𝑧=𝐻 𝑧=0 𝜌0𝑑𝑧 = � 𝑉𝑆𝑅 𝜋 , (38) where � 𝑉𝑆𝑅 is the density weighted vertical average of 𝑉𝑆𝑅 below height 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In defining �𝑣 ≡ � 𝑉𝑆𝑅 √ 2CAPE, combining eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 37 and 38 and the definition of 𝜖, and squaring and inverting the result, we obtain �𝑅−2 = 𝛼2𝜋2 4𝜎2 �𝐸 �𝑣2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (39) 11 combining eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 39 with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 32 to eliminate 𝑅 yields: �𝐸2 𝜓 �𝑣2 + �𝐸 � 1+ 𝜓 �𝑣2 �𝑁 � −1 = 0, (40) where 𝜓 = 𝑘2𝛼2𝜋2𝐿𝑚𝑖𝑥 4𝑃𝑟𝜎2𝐻 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (41) Solving for �𝐸 using the quadratic formula gives: �𝐸 = −1− 𝜓 �𝑣2 �𝑁 + √︂� 1+ 𝜓 �𝑣2 �𝑁 �2 +4 𝜓 �𝑣2 2 𝜓 �𝑣2 , (42) where we have neglected the negative quadratic root that yields an imaginary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Solutions for �𝐸, which rep- resent the fractional reduction of undiluted CAPE by en- trainment, are contoured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5a as a function of �𝑣 (non-dimensional storm-relative flow speed) and �𝑁 (non- dimensional NCAPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In general, �𝐸 increases from left-to- right in the figure as �𝑣 becomes large, indicating stronger storm-relative inflow, wider updrafts, and hence smaller fractional entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' From bottom-to-top on the figure, �𝐸 decreases as �𝑁 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This trend occurs because larger �𝑁 implies a drier and/or warmer mean free tropo- sphere, both of which amplify entrainment-driven dilution relative to situations with a cooler and/or moister free tro- posphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In dimensional form, eq 42 is: ECAPE = −1− 2𝜓 𝑉 2 𝑆𝑅 NCAPE+ √︂� 1+ 2𝜓 𝑉 2 𝑆𝑅 NCAPE �2 + 8𝜓 𝑉 2 𝑆𝑅 CAPE 4 𝜓 𝑉 2 𝑆𝑅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (43) Solutions for ECAPE from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 43 as a function of 𝑉𝑆𝑅 and CAPE are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5b,c,d for NCAPE=500 J kg-1, 1000 J kg-1, and 5000 J kg-1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In general, curves of ECAPE take on hyperbolic shapes with respect to the 𝑥 and 𝑦 axes, with contours of ECAPE parallelling the 𝑥 axis for large 𝑉𝑆𝑅, and the 𝑦 axis for small 𝑉𝑆𝑅 and large CAPE, and with the largest values coinciding with the largest 𝑉𝑆𝑅 and undiluted CAPE in the upper-right corners of the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This pattern means that different combi- nations of 𝑉𝑆𝑅 and undiluted CAPE may result in similar ECAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, an environment with 1000 J kg-1 of undiluted CAPE, a 𝑉𝑆𝑅 of 30 m s-1, and an NCAPE of 5000 J kg-1, has an ECAPE of roughly 1000 J kg-1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Mature isolated deep convective updrafts in this en- vironment will be sufficiently wide, due to their large 𝑉𝑆𝑅, such that their cores are approximately undiluted and they realize nearly all of their undiluted CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' A contrasting environment with 6000 J kg-1 of undiluted CAPE and an NCAPE of -5000 J kg-1, but with a 𝑉𝑆𝑅 of only 5 m s-1 will have a similar ECAPE of 1000 J kg-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Despite the large undiluted CAPE in the second environment, updrafts are narrow and substantially diluted by entertainment because of small 𝑉𝑆𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Consistent with the dependence of �𝐸 on �𝑁 seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5a, the fractional reduction in undiluted CAPE by ECAPE increases as NCAPE increases, particularly for smaller val- ues of undiluted CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This is most evident as a movement to the right of the contours of �𝐸 (black) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5b-d as NCAPE increases, indicating that an updraft with a given combination of undiluted CAPE and 𝑉𝑆𝑅 will realize less of its CAPE when NCAPE is large, compared to when NCAPE is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Accounting for kinetic energy the storm derives from its environment While it is somewhat infrequent, past studies have doc- umented instances in supercells where the maximum up- draft 𝑤 exceeds √ 2CAPE for extended periods of time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Fiedler 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, there are factors, such as vertical pressure gradient accelerations, that can explain why updrafts are sometimes more intense than buoyancy alone would suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This section introduces a simple ad- justment factor to the ECAPE formula to represent of how such pressure effects redirect environmental kinetic energy into the updraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To derive this adjustment factor, we must make the following assumptions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The Lagrangian evolution of kinetic energy following an air parcel is well described by the Boussinesq ap- proximation, meaning that 𝜌0 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Past studies have shown that errors related to an over-estimation of 𝜌0 aloft in deep convective environments have a small effect on analytic solutions for vertical velocity, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Morrison 2016a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Perturbation pressure accelerations in the middle-to- upper troposphere are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Pressure pertur- bations aloft may be large, but they typically oc- cur within the toroidal circulations of moist thermals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Romps and Charn 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Morrison and Peters 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters and Chavas 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' As parcels ascend through these thermals, they experience an upward acceleration below the minimum in 𝑝′, and then a commensurate downward acceleration above the min- imum in 𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, any temporary 𝐾𝐸 gained by the interaction of a parcel with these pressure perturba- tions is quickly lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We therefore neglect pressure perturbations at the height of maximum 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Direct dilution of 𝐾𝐸 via entrainment is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This assumption is also supported by past studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Sherwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Note that entrainment will still indirectly affect KE via the entrainment- driven dilution of updraft buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panel a: � 𝐸 (shading) as a function of �𝑣 (𝑥 axis) and � 𝑁 (𝑦 axis), with 𝐻 set to 12,000 m, 𝐿 = 120 m, 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='8, 𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='131, 𝑘2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='18, and 𝑃𝑟 = 1 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panels b-d: ECAPE (shading, J kg-1) as a function of 𝑉𝑆𝑅 (𝑥 axis, m s-1) and undiluted CAPE (𝑦 axis, J kg-1), and � 𝐸 (black contours), with NCAPE = 500 J kg-1 (panel a), NCAPE = 1000 J kg-1 (panel b), and NCAPE = 5000 J kg-1 (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In panels b-d, 𝐻 is determined via 𝐻 = 5808+96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='12 √ 2𝐶 𝐴𝑃𝐸, based on a linear regression between these variables among the soundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' All other parameters are the same as in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Updrafts are approximately steady, such that 𝜕 𝜕𝑡 of quantities are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The magnitude of convective inhibition (CIN) is neg- ligable relative to the magnitude of ECAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Horizontal storm-relative flow vanishes at the height of 𝑤𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We may use the first assumption to write eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 15 in Peters and Chavas (2021), which describes the Lagrangian tendency for 𝐾𝐸, as as: 𝑑𝐾𝐸 𝑑𝑡 = V· ∇ � 𝑝′ 𝜌0 � + 𝑤𝐵 (44) where 𝑝′ is a pressure perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We define 𝐾𝐸 here in an updraft relative sense, such that 𝐾𝐸 = 𝑢2 𝐶𝑅+𝑣2 𝐶𝑅+𝑤2 2 , where 𝑢𝐶𝑅 and 𝑣𝐶𝑅 are the 𝑢 and 𝑣 cloud-relative wind components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Because of the steady state assumption, we may substitute 𝑑 𝑑𝑡 � 𝑝′ 𝜌0 � = V · ∇ � 𝑝′ 𝜌0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We further use the chain rule to write 𝑑 𝑑𝑡 = 𝑤 𝑑 𝑑𝑧 , where 𝑑 𝑑𝑧 is the rate of change of a quantity as a parcel changes height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Making these assumptions and substitutions, and integrating from a parcel starting position (defined as 𝑧 = 0) to an ending position at the height of 𝑤𝑚𝑎𝑥 yields the following form of the classical Bernoulli equation: 𝐾𝐸𝐿𝑁 𝐵 − 𝐾𝐸0 = 𝑝′ 𝐿𝑁 𝐵 𝜌 − 𝑝′ 0 𝜌 + ∫ 𝑧=𝐿𝑁 𝐵 𝑧=0 𝐵𝑑𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (45) If a parcel originates within an updraft’s unmodified background environmental flow then 𝑝′ = 0, 𝑤 = 0, and 𝐾𝐸0 = 𝑉 2 𝑆𝑅 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We may also neglect 𝑝′ 𝐿𝑁 𝐵 𝜌 because of as- sumption (2) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Finally, we note that ∫ 𝑧=𝐿𝑁 𝐵 𝑧=0 𝐵𝑑𝑧 = ECAPE + ECIN, where ECIN is the convective inhibi- tion for an entraining parcel (ECAPE here is defined via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Combining all these assumptions and substi- tutions, neglecting ECIN, and assuming that horizontal storm-relative flow vanishes at the height of 𝑤𝑚𝑎𝑥 gives: ECAPE𝐴 = 𝑤2 𝑚𝑎𝑥 2 = 𝑉2 𝑆𝑅 2 +ECAPE (46) where the subscript 𝐴 indicates “adjusted”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' According to this equation, the role of low-level pressure perturbations is to preserve the incoming cloud-relative horizontal kinetic energy, deflecting it into the vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Further, the maxi- mum updraft kinetic energy at the height of 𝑤𝑚𝑎𝑥 consists 13 of the sum of the kinetic energy gained from the release of ECAPE and the kinetic energy of the redirected inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Nondimensionalizing by the undiluted CAPE yields: �𝐸𝐴 = �𝑣2 + �𝐸, (47) where �𝐸𝐴 is the nondimensional analogy to ECAPE𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Recall that in the derivation in the previous sub-section, we neglected pressure effects and assumed that ECAPE = 𝑤2 𝑚𝑎𝑥 2 when deriving the expression for 𝑅−2 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Now we must account for the influence of the added contribution to 𝑤𝑚𝑎𝑥 from velocity from environmental kinetic energy on updraft radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, we set ECAPE𝐴 = 𝑤2 𝑚𝑎𝑥 2 , and adjust eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 39 using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 47 to: �𝑅−2 = 𝛼2𝜋2 4𝜎2 𝑤2 𝑚𝑎𝑥 𝑉2 𝑆𝑅 = 𝛼2𝜋2 4𝜎2 � �𝐸 �𝑣2 +1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (48) Combining eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 47-48 with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 32 yields: �𝐸2 𝜓 �𝑣2 + �𝐸 � 1+𝜓 + 𝜓 �𝑣2 �𝑁 � −1+𝜓 �𝑁 = 0, (49) Solving �𝐸 using the quadratic formula and then plugging the result into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 47 to solve for �𝐸𝐴 gives: �𝐸𝐴 =�𝑣2+ −1−𝜓 − 𝜓 �𝑣2 �𝑁 + √︂� 1+𝜓 + 𝜓 �𝑣2 �𝑁 �2 +4 𝜓 �𝑣2 � 1−𝜓 �𝑁 � 2 𝜓 �𝑣2 , (50) which may be written dimensionally as: ECAPE𝐴 = 𝑉2 𝑆𝑅 2 + −1−𝜓 − 2𝜓 𝑉 2 𝑆𝑅 NCAPE 4 𝜓 𝑉 2 𝑆𝑅 + √︄� 1+𝜓 + 2𝜓 𝑉 2 𝑆𝑅 𝑁𝐶𝐴𝑃𝐸 �2 +8 𝜓 𝑉 2 𝑆𝑅 (CAPE−𝜓NCAPE) 4 𝜓 𝑉 2 𝑆𝑅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (51) The solution for �𝐸𝐴 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 51 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6a) is similar to that of �𝐸 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 42 at small values of �𝑣, but diverges notably from �𝐸 at large �𝑣, exceeding 1 (indicating that ECAPEA surpasses CAPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Similar behavior is evident in the solutions for ECAPEA as a function of 𝑉𝑆𝑅 and CAPE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Notably, ECAPEA is similar to ECAPE at smaller values of 𝑉𝑆𝑅, but larger than ECAPEA at large values of 𝑉𝑆𝑅, which is evident as a persistent downward slant of ECAPEA as one moves from left-to-right on the fig- ure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Again, we see that drastically different combinations of 𝑉𝑆𝑅 and CAPE can yield the same value of ECAPEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, an environment with NCAPE of 500 J kg-1, 1000 J kg-1 of CAPE, and a 𝑉𝑆𝑅 of 45 m s-1 will have an ECAPEA of 2000 J kg-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' A starkly contrasting environ- ment with NCAPE of 5000 J kg-1, 7000 J kg-1 of CAPE, and a 𝑉𝑆𝑅 of 7 m s-1 will also have an ECAPEA of 2000 J kg-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To illustrate the circumstances under which pressure ac- celerations (as they have been formulated here) have the greatest enhancement effect on updrafts, we examine the quantity 𝐹 = √︃ ECAPE𝐴 ECAPE − 1, which is equal to the ratio of the fractional enhancement in 𝑤𝑚𝑎𝑥 due to pressure ac- celerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Fractional enhancement is quite small (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='1) for most combinations of 𝑉𝑆𝑅 and CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' It only becomes larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='1 for smaller values of CAPE and/or larger values of 𝑉𝑆𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Physically, when CAPE is large and/or 𝑉𝑆𝑅 is small, the kinetic energy generation from buoyancy dom- inates the updraft kinetic energy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Whereas, when CAPE is small and/or 𝑉𝑆𝑅 is large, the kinetic energy input from the environmental wind becomes comparable to the kinetic energy generation from buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Given this dis- tribution of 𝐹, a potential explanation for why many past studies have found that 𝑤𝑚𝑎𝑥 is primarily determined by buoyancy is that the CAPE and 𝑉𝑆𝑅 in these simulations fell within the region of the parameter space where 𝐹 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In other words, the kinetic energy input into the updraft via the background environmental flow is insignif- icant compared to the kinetic energy generation via the release of CAPE in most storm environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Evaluation of the formulas a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Comparison of predicted 𝑤𝑚𝑎𝑥 with the output from past simulations We will compare the formula’s predictions to the vertical velocities from simulations to evaluate the ECAPE and ECAPE𝐴 formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The simulations, which featured a mix of supercells and multicellular clusters, originate from four past studies: Coffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2022) (C23, 9 simulations), Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2023) (P23, 32 simulations), Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2020d) (P20, 48 simulations), and Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2019) (54 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' All simulations used Cloud Model 1 (CM1 Bryan and Fritsch 2002) and were initialized with soundings that featured a variety of different wind and thermodynamic profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Horizontal grid spacing was 100 m in P23 and C23, and 250 m in P20, and P19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Vertical grid spacing was 100 m or less in the troposphere in all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Additional details of the model configurations are omitted here to save room, but are available in the studies referenced in this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We computed all subsequent quantities with the initial model thermodynamic and wind profiles and storm mo- tions in past simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Predictions of 𝑤𝑚𝑎𝑥 were derived by taking the square root of half of the predicted CAPE and ECAPE values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We compared the predicted values of 𝑤𝑚𝑎𝑥 to the median 𝑤𝑚𝑎𝑥 during the 1-3 hour time range in the simulations, excluding tornadic periods in the P23 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6, but showing � 𝐸𝐴 (panel a), and ECAPE𝐴 (panels b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 𝐹 (shading, nondimensional) as a function of 𝑉𝑆𝑅 (𝑥 axis, m s-1) and CAPE (𝑦 axis, J kg-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Colored dots indicate the 𝑉𝑆𝑅 and CAPE from the simulated storms analyzed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' and C23 simulations (see those studies for definitions of “tornadic periods").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The parameter 𝑉𝑆𝑅 was computed by subtracting the tracked motion vector of simulated updrafts from the initial model profile, and averaging the resulting storm-relative wind profile in the 0-1 km layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Other layer averages, including 0-500 m, 0-2 km, 0-3 km, and the density weighted average from the surface to the EL gave nearly identical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We will first see how well √ 2CAPE, which is the tradi- tional “thermodynamic speed limit”, predicts 𝑤𝑚𝑎𝑥 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This parameter loosely captures the differences in 𝑤𝑚𝑎𝑥 among groups of simulations, but does not cap- ture any of the variability in 𝑤𝑚𝑎𝑥 among simulations that 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' All panels: predicted 𝑤𝑚𝑎𝑥 (𝑥 axis, m s-1) versus simulated 𝑤𝑚𝑎𝑥 (𝑦 axis, m s-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Predictors are: the traditional “thermodynamic speed limit" √ 2CAPE (panel a), ECAPE with the fixed 𝜀 that minimized the RMSE (panel b), a multi-linear regression with 𝑉𝑆𝑅 and √ 2CAPE as predictors (panel c), ECAPE from P20 (panel d), ECAPE from the present study (panel e), and ECAPEA from the present study (panel f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Bias, RMSE and 𝑅2 values are shown in the title of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Colors correspond to the study where the simulations originated (see the legend in panel e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' shared the same CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Most 𝑤𝑚𝑎𝑥 were less than the tra- ditional thermodynamic speed limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', below the 1-to-1 line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, the bulk of the P23 simulations and a few of the P19 simulations exceeded this threshold, by up to 15 m s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The 𝑉𝑆𝑅 and CAPE of these simulations puts them in the portion of the parameter space where our theoretical representation of pressure effects predicts that their 𝑤𝑚𝑎𝑥 should exceed √ 2CAPE (see the gray and red dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The coefficient of determination (𝑅2) of √ 2CAPE with simulated 𝑤𝑚𝑎𝑥 was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='38, with a root-mean-square-error (RMSE) of roughly 15 m s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To see if we can do a better job of predicting 𝑤𝑚𝑎𝑥 with ECAPE that uses a fixed entrainment rate, we found the 𝜀 that yielded the smallest RMSE between predictions by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 24 and simulated 𝑤𝑚𝑎𝑥 (this value was 𝜀 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='25×10−5 m-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This prediction reduces the RMSE to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='2 m s-1, but does not improve the 𝑅2 much (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, with no knowledge of how the variations in environmental wind profiles affect entrainment, ECAPE with a fixed entrain- ment rate only slightly improves predictions of the mean 𝑤𝑚𝑎𝑥 among groups of simulations, but does not capture any of the variance in 𝑤𝑚𝑎𝑥 within a particular group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We can do a better job of predicting 𝑤𝑚𝑎𝑥 by forming a mult-linear regression with √ 2CAPE and 𝑉𝑆𝑅 as predic- tors, and 𝑤𝑚𝑎𝑥 as a predictand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This regression equation takes the form 𝑤𝑚𝑎𝑥,𝑝𝑟𝑒𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='7823 √ 2CAPE+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='503𝑉𝑆𝑅 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='3437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The predictions by this formula reduce RMSE to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='95 m s-1 and increase the 𝑅2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='7 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This formula also produces an improved subjective correspon- dence between predicted and simulated 𝑤𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The ECAPE formula from P20, computed using all the procedures and parameter values described in that study, also better captures the variability in 𝑤𝑚𝑎𝑥 among simu- lations with the same CAPE value than the √ 2CAPE and ECAPE with a fixed entrainment rate, with a 𝑅2 with 𝑤𝑚𝑎𝑥 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The RMSE of 13 m s-1, however, is inferior to that of the linear regression and comparable to that of √ 2CAPE and ECAPE with a fixed entrainment rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This large error stems from a low bias in predictions from this formula, rel- ative to the values in simulations, which is demonstrated by the dots mostly falling to the left of the one-to-one line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Recall that P20 used a 𝜀 ∼ 𝑅−1 scaling, and the buoyancy formula from that study consequently over- estimated the fractional reduction in undiluted buoyancy by 16 entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Both of these factors may have contributed to the formula’s bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To evaluate the ECAPE and ECAPEA derived in the present study, we set 𝐿𝑚𝑖𝑥 = 120 m when evaluating the ECAPE formulas derived in the present study against the P23 and C23 simulations, and 𝐿𝑚𝑖𝑥 = 250 m when evalu- ating against the P20, N20, and P19 simulations to account for their coarser grid spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' All other parameter values were the same as those used to generate Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The new ECAPE formula improves correspondence (𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='79), reduces the low bias in prediction, and substantially de- creases RMSE (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='2 m s-1) relative to the formula from P20 and the linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8c fall close to the 1-1 line, suggesting that the 𝜀 ∼ 𝑅−2 scaling better reflects the trends in entrainment-driven dilution in the simulations than 𝜀 ∼ 𝑅−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The ECAPEA formula further improves correspondence between predicted and simulated 𝑤𝑚𝑎𝑥 (𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='82), de- creases RMSE to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='4 m s-1, and brings points closer to the 1-to-1 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The most notable difference between ECAPEA and ECAPE occurs with the P23 simulations, whose 𝑤𝑚𝑎𝑥 substantially exceeded √ 2CAPE (red dots above the 1-to-1 line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8a) and was under-predicted by the ECAPE formulas from both P20 (red dots above the 1-to-1 line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8b) and the present study (red dots above the 1-to-1 line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The ECAPEA brings the red dots much closer to the 1-to-1 line, correctly reflecting that 𝑤𝑚𝑎𝑥 in many of these simulations exceeded √ 2CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The take home message is that the two formulas derived in the present study are superior predictors of 𝑤𝑚𝑎𝑥 when compared to CAPE and ECAPE with a fixed entrainment rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' They also perform better than a simple linear re- gression that includes CAPE and 𝑉𝑆𝑅, suggesting that the additional information contained in our formula about the environmental thermodynamic profile via the NCAPE pa- rameter is critical to accurately representing the effects of entrainment on 𝑤𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Finally, the new ECAPE formulas correct a low bias in the older P20 formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Properties of ECAPE in severe weather proximity soundings Our final analysis examines the distribution of ECAPE𝐴 within the Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2003) sounding dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Once again, we use the 0-1 km mean 𝑉𝑆𝑅 computed with the ob- served storm motion in our formulas, though we evaluate other definitions of 𝑉𝑆𝑅 later in this sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The dis- tribution of ECAPE𝐴 for all nonsupercell severe weather events is plotted against undiluted CAPE in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Con- tours of �𝐸𝐴 (the fraction of CAPE “realized") are also shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' There is substantial variability �𝐸𝐴, with ECAPE𝐴 ≈ CAPE (�𝐸𝐴 ≈ 1) in some events, and ECAPE𝐴 << CAPE (�𝐸𝐴 << 1) in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Furthermore, case-to-case variations in ECAPE𝐴 and CAPE only loosely corresponded with one another, with 𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='46 based on a linear fit of these two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In most events, particu- larly those with significant CAPE (> 1000 𝐽/𝑘𝑔), ECAPE𝐴 was less than CAPE suggesting that most nonsupercell storms only realize a fraction of their available CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In contrast with nonsupercell events, there is a much closer correspondence between ECAPE𝐴 and CAPE in su- percell events, with 𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='90 between these two variables (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Furthermore, �𝐸𝐴 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='5 for nearly every supercell sounding, and this quantity was close to 1 in many cases, and exceeded 1 in a handful of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This corroborates the idea, proposed by Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2019), that supercells re- alize a larger percentage of their environmental CAPE than nonsupercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The primary reason for this difference is the larger vertical wind shear, and consequently storm-relative flow, in supercell environments relative to nonsupercell environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, CAPE may be a better predictor of storm-to-storm variations in updraft intensity in supercells than it is in nonosupercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, there is still substan- tial variability in the correspondence between ECAPE and CAPE, particular for larger CAPE values, which suggests that ECAPE provides added value over CAPE in supercell environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' To evaluate the sensitivity of ECAPE to how 𝑉𝑆𝑅 is cal- culated, we re-computed ECAPE𝐴 with the 0-3 km mean 𝑉𝑆𝑅 with the observed storm motion, the density weighted average of 𝑉𝑆𝑅 below the LFC with the observed storm motion, the 0-1 km mean 𝑉𝑆𝑅 computed using the storm motion estimate of Bunkers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2000) which includes components of storm motion driven by advection and prop- agation, and the advective storm motion only, estimated as half the 0-6 km bulk wind difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Results with the 𝑉𝑆𝑅 measures that use the observed storm motion yield nearly identical results to one another in both nonsupercells (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 9c) and supercells (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 9d), with 𝑅2 ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='96 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In the case of supercells, the ECAPE𝐴 computed with the observed storm motion corresponded well with the ECAPE𝐴 computed using the Bunkers storm motion esti- mate and half the bulk wind difference (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 9d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, this correspondence was degraded slightly in nonsupercell events, with the 𝑅2 ranging form 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='71 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='75 between ECAPE𝐴 computed with the observed storm-motion, with that computed using the bunkers estimate and bulk wind difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This likely reflects the fact that the motion of nonsupercell storms is more often influenced by extraneous factors like outflow and airmass boundaries, than in super- cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, sounding-based estimates for storm motion do not correspond with actual storm motions as well in nonsupercell events as they do in supercell events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In many contexts where this formula would be used, such as in forecasting, the storm motion is unknown and must be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This analysis suggests that estimating storm motion with the method of Bunkers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2000) or half the 0-6 km BWD are both viable choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Top panels: scatter plots of ECAPE𝐴 (𝑥 axis, J kg-1) versus CAPE (𝑦 axis, J kg-1), computed with the Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2003) soundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panel a: 351 nonsupercell events, and panel b: 834 supercell events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Contours of � 𝐸𝐴 are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Panels c-d: 𝑅2 between solutions for ECAPE𝐴 computed using different definitions of 𝑉𝑆𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' A given cell shows the correlation coefficient between ECAPE𝐴 computed with the 𝑉𝑆𝑅 definition on the 𝑥 axis, with that on the corresponding 𝑦 axis, with colors corresponding to the relative magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Summary, conclusions, and discussion In summary, we have derived a formula for ECAPE that depends entirely on state variables available within an atmospheric sounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This formula relies on three concepts: a scaling between fractional entrainment and updraft radius of 𝜀 ∼ 𝑅−2, the adiabatic conservation of moist static energy, and a direct correspondence between the cloud relative flow and the updraft radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Finally, we have accounted for the potential enhancement of updraft kinetic energy via pressure accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We recommend using the following steps to compute this quantity in a software routine: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Set the following constant values: 𝑐 𝑝 = 1005 J kg-1 K-1, 𝐿𝑣,𝑟 = 2,501,000 J kg-1, 𝑔 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='81 m s-1, 𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='6, 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='8, 𝑘2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='18, 𝑃𝑟 = 1 3, and 𝐿𝑚𝑖𝑥 = 120 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Compute CAPE, the 𝐿𝐹𝐶, and the 𝐸𝐿 for an undi- luted parcel from an atmospheric profile using an ex- isting software routine (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', SHARPy, Metpy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Compute the following parameter: 𝜓 = 𝑘2𝛼2𝜋2𝐿𝑚𝑖𝑥 𝑃𝑟𝜎2𝐻 , (52) where 𝐻 is the equilibrium level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Compute 𝑉𝑆𝑅 from an atmospheric profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' We rec- ommend averaging 𝑉𝑆𝑅 in the 0-1 km layer, using the method for estimating storm motion described by Bunkers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Evaluate the following formula, using a numerical integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' �ℎ0(𝑧) = 1 𝑧 ∫ 𝑧∗=𝑧 𝑧∗=0 �𝑐 𝑝𝑑𝑇0 + 𝐿𝑣,𝑟𝑞0 +𝑔𝑧∗� 𝑑𝑧∗, (53) This procedure only needs to be done once in a given profile, and yields < ℎ0 > as a function of height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Compute NCAPE, using the following formula: NCAPE = − ∫ 𝑧=𝐸𝐿 𝑧=𝐿𝐹𝐶 𝑔 𝑐 𝑝𝑑𝑇0 � �ℎ0 − ℎ∗ 0 � 𝑑𝑧, (54) 18 NCAPE is positive in most contexts though it may become negative in environments with large free tro- pospheric relative humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Compute ECAPE𝐴, using the following formula: ECAPE𝐴 = 𝑉2 𝑆𝑅 2 + −1−𝜓 − 2𝜓 𝑉 2 𝑆𝑅 NCAPE 4 𝜓 𝑉 2 𝑆𝑅 + √︄� 1+𝜓 + 2𝜓 𝑉 2 𝑆𝑅 𝑁𝐶𝐴𝑃𝐸 �2 +8 𝜓 𝑉 2 𝑆𝑅 (CAPE−𝜓NCAPE) 4 𝜓 𝑉 2 𝑆𝑅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (55) In the case of a negative solution to this equation, set the ECAPE𝐴 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Our results show that ECAPE provides a more accu- rate prediction of updraft intensity than standard CAPE when forecasting severe weather hazards that depend on middle-to-upper tropospheric vertical velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Exam- ples of these situations include forecasting heavy precipi- tation, large hail, and intense cold pools and downdrafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, it would benefit the forecasting community to dis- play this quantity alongside standard CAPE on websites that provide numerical weather prediction model output graphics, such as the storm-prediction center Mesoanaly- sis site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In addition, �𝐸𝐴, which is the fraction of CAPE realized, is a powerful discriminator of supercellular from nonsupercellular storm mode, with a True Skill Statistic (TSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', section 2 in Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020d) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='76 in this prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This is on par with the TSS for 0-1 km 𝑉𝑆𝑅, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='79 (these values are not statistically different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The physical reason behind this discriminatory skill re- lates to the conclusions of Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' (2019), who showed that supercells realize larger fractions of their CAPE than nonsupercells (and hence have larger �𝐸𝐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' A variety of research applications would also benefit from the consideration of ECAPE, in addition to standard CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, studies in past literature often contrast storm dynamics in high-shear low-CAPE severe weather events with events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Schneider and Dean 2008) occur- ring in environments with higher CAPE (and sometimes weaker shear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The premise behind this distinction is, be- cause of the small updraft buoyancy in low-CAPE events, the updrafts accelerations in these storms are dominated by dynamic pressure accelerations rather than buoyancy (Wade and Parker 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, it is possible that because of the extreme shear in many low-CAPE severe weather outbreaks, updrafts in these scenarios realize a higher percentage of their CAPE than their counterparts in high CAPE environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, ECAPE may more ac- curately distinguish between storms with large and small buoyancy than standard CAPE, and a reconsideration of the analyses in these past studies with distinctions drawn between high ECAPE and low ECAPE events may yield additional insights into storm dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' ECAPE may also yield novel insight into the influence of climate change on thunderstorms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, a subset of studies that investigate the influence of climate change on severe storm behavior use proxy analyses in global cli- mate model (GCM) simulations, assessing the impacts of global warming on parameters like CAPE and CIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Future changes to free tropospheric relative humidity, tempera- ture, and vertical wind shear are also likely to influence thunderstorms via the connection between these environ- mental attributes and entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Investigating changes to the climatology of ECAPE in future climates is a con- cise way of encapsulating these yet-to-be explored climate change influences on storm entrainment, and consequently storm intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Efforts to quantify the effects of climate change among the authors of the present study are currently underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Some of the intermediary formulas that express buoy- ancy and ECAPE as an analytic function of fractional entrainment may be useful in cumulus parameterization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' For instance, multi-plume schemes like the scheme of Arakawa and Schubert (1974), the Relaxed Arakawa-Schubert scheme Moorthi and Suarez (1992), the EDMF𝑁 scheme Neggers (2015), and the MAP scheme (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020b) require the computation of diluted buoyancy and ECAPE for each plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' In the traditional approach for computing ECAPE, these schemes would ex- ecute two numerical vertical integrations for each plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This procedure, however, is dramatically simplified by us- ing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 24 in the present study, where only 3 vertical integrations per grid cell are needed to obtain CAPE and NCAPE, and then the ECAPE associated with each plume is computed analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The MAP scheme from (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020b) was also formulated to use the formula from P20 as part of its closure for convective mass flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The formula presented here is a more accurate alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' A potential caveat to using this parameter operationally is that ECAPE𝐴 vanishes in the absence of 𝑉𝑆𝑅, whereas we know that deep convection is possible in the absence of substantial 𝑉𝑆𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This discrepancy is likely a consequence of the primary controls on updraft width shifting away from vertical wind shear to other environmental factors when shear is weak, such as the planetary boundary layer (PBL) depth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Mulholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021a) or the width scale of terrain features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Kirshbaum 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' A potential way to circumvent this issue is to revert to a standard ECAPE calculation (with a user-prescribed 𝜀) in these weakly sheared environments, setting the updraft radius to scale with the PBL depth or to a constant value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', 1500 m, as was done in Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Some may debate the semantics over whether the formu- las derived are more appropriately described as predictive equations for the maximum updraft vertical velocity, rather than a modified CAPE that accounts for entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Some 19 view CAPE as pertaining only to an isolated ascending parcel with no explicit assumptions about updraft structure and behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Hence, our inclusions of updraft dynamics in our ECAPE calculation makes this calculation concep- tually distinct from that of CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' However, we argue that there are a variety of conceptual definitions of CAPE in past literature, and that this quantity is often used in the fore- casting community to predict how a given thermodynamic environment may affect updraft vertical velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Because of the familiarity of forecasters with CAPE, ECAPE (with units of J kg-1) is a more relatable quantity to forecasters than 𝑤𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' This is the primary reason why we have adver- tised the quantity derived here as an ECAPE, rather than a predictor of 𝑤𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' 20 Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Peters’s efforts were supported by National Science Foundation (NSF) grants AGS- 1928666, AGS-1841674, and the Department of Energy Atmospheric System Research (DOE ASR) grants DE- SC0000246356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Chavas was supported by National Science Foundation (NSF) grants 1648681 and 2209052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Morrison was supported by DOE ASR grant DE- SC0020104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' The National Center for Atmospheric Re- search is sponsored by NSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Data availability statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Matlab code to compute ECAPE using an atmospheric sounding as input is available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='6084/m9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='21859818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' References Arakawa, A.' metadata={'source': 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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Morrison, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Nelson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Marquis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Mul- holland, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Nowotarski, 2022a: The influence of shear on deep convection initiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' part i: Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' Journal of the At- mospheric Sciences, 79 (6), 1669 – 1690, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} 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On “Hot Towers” after 50 Years of Tropical Field Programs and Two Years of TRMM Data, 49–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content=' American Meteorological Society, Boston, MA, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='1007/978-1-878220-63-9_5, URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} +page_content='1007/978-1-878220-63-9_5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfxguY/content/2301.04712v1.pdf'} diff --git a/QNE0T4oBgHgl3EQfkQEN/content/tmp_files/2301.02469v1.pdf.txt b/QNE0T4oBgHgl3EQfkQEN/content/tmp_files/2301.02469v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..459e20234394732f3e8e34b493852e3e2a75e223 --- /dev/null +++ b/QNE0T4oBgHgl3EQfkQEN/content/tmp_files/2301.02469v1.pdf.txt @@ -0,0 +1,795 @@ +1 +Cox Point Processes for Multi-Altitude LEO +Satellite Networks +Chang-Sik Choi and Franc¸ois Baccelli +Abstract—We propose a simple analytical approach to describe +the locations of low earth orbit (LEO) satellites based on a +Cox point process. We develop a variable-altitude Poisson orbit +process by accounting for the fact that satellites are always +located on circular orbits and these orbits may have different +altitudes. Then, the satellites on these orbits are modeled as +the Poisson point processes conditionally on the orbit process. +For this model, we derive the distribution of the distance to +the nearest visible satellite, the outage probability, the Laplace +functional of the proposed satellite Cox point process, and the +Laplace transform of the interference under a general fading. The +derived statistics allow one to evaluate the performance of such +LEO satellite communication systems as functions of network +parameters. +Index Terms—LEO satellite communications, Stochastic geom- +etry, Cox point process, Nearest distance, Total interference +I. INTRODUCTION +A. Motivation and Background +LEO satellites provide global connectivity to millions of +devices on earth [1]–[5]. The applications of LEO satellite net- +works are numerous [1]: they provide Internet connections to +devices where ground infrastructure is unavailable [2]; local- +ization and emergency communications of aerial and ground +devices can be enabled by LEO satellites [3]; LEO satellite +networks provide cheaper Internet connections to developing +countries [4]. LEO satellite networks can even be integrated +with terrestrial networks to enable reliable connections to +devices in a small area [5]. To support these applications, LEO +satellite networks will have a very large number of satellites. +The viability and performance of LEO satellite communi- +cations are significantly determined by the way satellites are +distributed in space. Various evaluation methodologies have +been proposed to obtain the performance of LEO satellite +communication networks. For satellite layout, some studies +used probabilistic approaches including a binomial point pro- +cess [6]–[9]. In contrast to the simulation-based approach, +the benefits of employing such analytical models lie in the +fact that they presents large-scale behaviors as functions of +network key parameters such as the mean number of satel- +lites, their altitudes, etc. Nevertheless, the binomial satellite +point processes in [6]–[9] were not able to incorporate the +fact that the satellites are located on approximately circular +trajectories around the earth, namely their orbits. In this paper, +we provide a tractable model that incorporates this fact in the +multi-altitude LEO satellite case, by generalizing the work in +Chang-Sik Choi is with Hongik University, South Korea. Franc¸ois +Baccelli is with Inria Paris and Telecom Paris, France. (email: chang- +sik.choi@hongik.ac.kr, francois.baccelli@inria.fr) +[10] where all orbits are at the same altitude. Specifically, +we present an analytical framework leveraging a Cox point +process so that orbits are created first according to a Poisson +point process on a cuboid and then satellites are distributed +as Poisson point processes conditionally on these orbits. We +derive key statistical properties of the proposed network model +that are critical to obtain the performance of such satellite +networks as functions of the altitude distribution, of the mean +number of orbits, of the number of satellites, and of the +Laplace transform of the random variable representing fading. +B. Contributions +Modeling of variable orbit LEO satellite constellations: +This paper accounts for the geometric properties of practical +LEO satellite systems that (i) satellites are always on orbits +around the earth and (ii) such orbits are possibly at different +altitudes. By developing a nonhomogeneous Poisson point +process of mean λ in a cuboid, we creates a Poisson orbit +process of orbits in the Euclidean space. Then, conditionally +on the orbit process, satellites are distributed as linear Poisson +point processes of mean µ on these orbits. Our motivation is +to represent a general LEO satellite network where satellites +are located at different altitude bands. +Statistical properties of the proposed Cox point pro- +cess: The proposed satellite Cox point process is built to be +invariant by all rotations of the reference plane. This makes +the statistical properties of the network to be the same for all +perspectives seen from all points on earth. Leveraging this, we +obtain the probability distribution function of the distance from +the typical user to its nearest visible satellite and then derive +the outage probability of the proposed network model. Using +it, we derive the Laplace functional of the proposed satellite +Cox point process and then give an integral expression for the +Laplace transform of the total interference. These formulas +are directly used to assess the network performance metrics +such as the Signal-to-interference-plus-noise ratio (SINR) of +the typical user. +II. COX-MODELED SATELLITES +A. Satellite Distribution +The center of the earth is O = (0, 0, 0) and it is of radius +re. The xy-plane is the reference plane and the x-axis is +longitude reference direction. In this paper, we only focus on +the snapshot of the network geometry and the movement of +satellites is out of the scope. +Consider a cuboid C = [ra, rb] × [0, π) × [0, π) where ra ≤ +rb the minimum and maximum altitudes and a Poisson point +arXiv:2301.02469v1 [eess.SP] 6 Jan 2023 + +2 +Reference: xy-plane +x-axis +A +θ +l(ρ,θ,φ) +φ +X: satellite +ω +O +~ +y-axis +ρ +z-axis +Fig. 1. +The orbital plane meets the reference plane at two points and the +point with angle less than π is A. The angle θ is measured from the x-axis +to the segment OA. The inclination ˜ϕ is measured from the reference plane +to the orbital plane and the azimuth ϕ is given by π/2 − ˜ϕ. The angle ω for +satellite X is measured from OA to OX over the orbital plane. +process Ξ of intensity measure λν(dρ)/π2 in the cuboid C. +We have +� rb +ra ν(dρ) = 1. Then, we build an orbit process by +mapping each point of Ξ, say (ρ, θ, ϕ) into an orbit l(ρ, θ, ϕ) +in the Euclidean space. Specifically, the first coordinate ρ is the +orbit’s radius, θ is the orbit’s longitude, and ϕ is the orbit’s +azimuth. See Fig. 1. For the Poisson point process on the +cuboid, we write Ξ = � +i Zi, where Zi is the point of Ξ. +Since there are on average λ points of Ξ, there are on average +λ orbits. The orbit process O in R3 is given by +O = +� +Zi∈Ξ +l(ρi, θi, ϕi). +(1) +Conditionally on Ξ, the locations of satellites on each orbit +l(ρi, θi, ϕi) are modeled as a homogeneous Poisson point +process ψi of intensity µ/(2πρi) on this orbit. Equivalently, +the orbital angles of satellites on each orbit are modeled as +a 1-dim homogeneous Poisson point process φi on segment +[0, 2π) of intensity µ/(2π). Since the satellites are distributed +conditionally on Ξ, the satellite point process Ψ is a Cox point +process. The satellite Cox point process is +Ψ = +� +i +ψi. +(2) +Figs. 2 – 4 depict the proposed satellite Cox point process with +λ, µ, ra and rb. In the figures, we use ν(dρ) = +dρ +rb−ra , i.e., +the radii of orbits are uniformly distributed on the interval +[ra, rb]. The proposed model can be used to represent e.g., +multiple operators of LEO satellite networks where orbits are +at different altitudes. The case of all satellites are located at +the same altitude in [10] is a special case of the proposed +model by taking ν(dρ) = δra(dρ), where ra is the radius of +orbits. +B. User Distribution +Users are located on the surface of the earth {(x, y, z)|x2 + +y2 +z2 = r2 +e} and the locations of network users are assumed +to be independent of the locations of the LEO satellites. +III. STATISTICAL RESULTS +In this section, we derive/prove (i) the mean number of +LEO satellites, (ii) the isotropy of Ψ, (iii) the distances from +Fig. 2. The proposed Cox satellite model with ra = 7000 km, rb = 7100 +km. We use λ = 60, µ = 40, and ν(dρ) = dρ/(rb − ra). +Fig. 3. The Cox-modeled satellite with ra = 7000 km and rb = 7500 km. +We use λ = 30, µ = 60, and ν(dρ) = dρ/(rb − ra). +Fig. 4. The Cox-modeled satellite with ra = 7000 km and rb = 8500 km. +We use λ = 70, µ = 30, and ν(dρ) = dρ/(rb − ra). + +3 +the LEO satellites to an arbitrarily located user, (iv) the +distribution of the distance to the nearest visible satellite, (v) +the outage probability, (vi) the Laplace functional of Ψ, and +(vii) the Laplace transform of the total interference under +general fading. These statistical properties directly determine +the performance of downlink LEO satellite communications +in this context. +Lemma 1. The average number of the proposed Cox satellite +point process is λµ. +Proof: The average number of satellites is given by +E [Ψ(S)] = E +� +� � +Zi∈Ξ +E +� +� � +Xj∈ψi +1 +������ +Ξ +� +� +� +� += E +� � +Zi∈Ξ +� 2π +0 +µ +2π dx +����� Ξ +� += µ +� +C +λ +π2 ν(dρ) dθ dϕ = λµ, +where we use Campbell’s mean value theorem [11]. +Below we show that O is invariant w.r.t. rotations. This +allows one to evaluate the performance of network seen by a +typical user at the north pole. +Lemma 2. The distribution of O and Ψ are invariant by all +rotations of the reference space (O, x, y, z). +Proof: The intensity measure of the proposed orbit pro- +cess Ξ has the product form: ν(dρ) × +� +λ/π2� +dθ dϕ. This +shows that the angles (θ, ϕ) form a homogeneous Poisson +point process on the rectangle [0, π) × [0, π). [10] proved +that the orbit process mapped by the very intensity measure +� +λ/π2� +dθ dϕ is invariant by all rotations of the reference +space. Hence, the law of O is also invariant by all rotations of +the reference space (O, x, y, z). In the same vein, the law of +Ψ is invariant by all rotations of the reference space as well. +Lemma 3. Consider a satellite X of orbital angle ωj on the +orbit l(ρi, θi, ϕi). The distance from (0, 0, re) to the satellite +X(ρi, θi, ϕi, ωj) is given by +� +ρ2 +i − 2ρire sin(ωj) cos(ϕi) + r2e. +(3) +Proof: The coordinates (x, y, z) ∈ R3 of the satellite that +has the orbital angle ωj on the orbit l(ρi, θi, ϕi) are given by +x = +� +ρ2 +i cos2(ωj) + ρ2 +i sin2(ωj) cos2( ˜ϕi) cos +� +˜θ + θi +� +, (4) +y = +� +ρ2 +i cos2(ωj) + ρ2 +i sin2(ωj) cos2( ˜ϕi) sin +� +˜θ + θi +� +, (5) +z = ρi sin(ωj) sin( ˜ϕi), +(6) +˜θ = arctan (tan(ωj) cos( ˜ϕi)) , +(7) +where ˜ϕ is the inclination: ˜ϕ = π/2 − ϕ. +As a result, the distance from (0, 0, re) to the satellite is +∥(x, y, z) − (0, 0, re)∥ = +� +ρ2 +i − 2ρire sin(ωj) cos(ϕi) + r2e. +Note the distance is independent of the variable θ. +U=(0,0,re) +O +A +C +E +B +F +A` +D +ρ +re +Bottom of spherical cap +Fig. 5. The arc of orbit l(ρ, θ, ϕ) in spherical cap C(ρ, d). +A. The Lengths of Orbits’ Arcs +Since (i) users are independent of Ψ and (ii) Ψ is invariant +by rotations (Lemma 2 ), one can consider a typical user at +(0, 0, re) and study the network performance it experiences, +which will be typical. +Let C(d) be the subset of S such that the distances from +the typical observer u to the satellites on C(d) are less than +a distance d. For any ra ≤ ρ ≤ rb, we define +C(d) = +� +ra≤ρ≤rb +C(ρ, d) += +� +ra≤ρ≤rb +� +(x, y, z) ∈ R3 |z ≥ re, x2 + y2 + z2 = ρ2, +x2 + y2 + (z − re)2 ≤ d2� +, +where z ≥ re, since satellites with z-coordinates less than re +are invisible to the user at (0, 0, re). C(ρ, d) is a spherical cap +associated with the orbit of radius ρ. See Fig. 5. +Lemma 4. The length of the arc given by the intersection of +the spherical cap C(ρ, d) and the orbit l(ρ, θ, ϕ) is +2ρ arcsin +� +� +� +1 − +�ρ2 + r2e − d2 +2ρre cos(ϕ) +�2 +� +� , +(8) +for ρ − re ≤ d ≤ +� +ρ2 − r2e. +Proof: Consider C(ρ, d). Let ξ be the angle ∠AOU in +Fig. 5. Then, we use the law of Cosine to obtain cos(ξ) = +(ρ2 + r2 +e − d2)/(2ρre). +For the triangle △BCD, we have CD = ρ cos(ξ) tan(ϕ). +Since the angle ∠BDC is π/2, we obtain +BD = +� +ρ2 sin2(ξ) − ρ2 cos2(ξ) tan2(ϕ). +For △BOD, OB = ρ and let κ′ = ∠BOD. Then we have +sin(κ′) = BD/ρ = +� +sin2(ξ) − cos2(ξ) tan2(ϕ). +Finally, the length of the arc > +BF is given by +ν(> +BF) = 2ρ arcsin( +� +1 − cos2(ξ) sec2(ϕ)). +where cos(ξ) = (ρ2 + r2 +e − d2)/(2ρre). +In downlink LEO satellite communication networks, net- +work users are meant to receive signals from their closest or + +4 +P(D > d) = exp +� +−2λ +π +� rb +ra +� ξ +0 +� +1 − e +−µπ−1 arcsin +�√ +1−cos2(ξ) sec2(ϕ) +�� +dϕν(dρ) +� +, +(9) +P(D = ∞) = exp +� +−2λ +π +� rb +ra +� arccos(re/ρ) +0 +� +1 − e +−µπ−1 arcsin +�√ +1−r2e sec2(ϕ)/ρ2 +�� +dϕν(dρ) +� +, +(10) +L(f) = exp +� +− λ +π2 +� +C +� +1 − e− µ +2π +� 2π +0 +1−exp (− ¯ +f(ρ,θ,ϕ,ω)) dω� +ν(dρ) dθ dϕ +� +, +(11) +LΨ(f)f=sH∥X−U∥−α = exp +� +− λ +π2 +� +¯C +� +1 − e− µ +2π +� +¯ +ω 1−LH(s(ρ2−2ρre sin(ω) cos(ϕ)+r2 +e)− α +2 ) dω� +ν(dρ) dθ dϕ +� +. +(12) +nearest satellites [9]. The distance D from a network user to +its closest LEO satellite is a random variable. When there is +no visible satellite, D +def += ∞. +Lemma 5. The cumulative distribution function of D is given +by Eq. (9) where cos(ξ) = (ρ2 + r2 +e − d2)/(2ρre). +Proof: For ra − re ≤ d ≤ +� +r2 +b − r2e, we have +P(D > d) +(a) += P(∥X − u∥ > d, ∀X ∈ Ψ) +(b) += P(∥Xj − u∥ > d, ∀Xj ∈ ψi, ∀Zi ∈ Ξ) += P +� +� � +Zi∈Ξ +P +� +� � +Xj∈ψi +∥Xj − u∥ > d +������ +Ξ +� +� +� +� . +To get (a), we use the fact that for R > r, all satellites should +be at distances greater than r. We have (b) by using that the +Cox satellite point process is comprised of the Poisson point +processes conditionally on orbits. We have +P +� +� � +Xj∈ψi +∥Xj − u∥ > r +������ +Ξ +� +� += exp +� +−µπ−1 arcsin +�� +1 − cos2(ξ)sec2(ϕi) +�� +, +where cos(ξ) = (ρ2 +i + r2 +e − d2)/(2ρire), as a function of the +orbits’ radius. We use the facts that (i) in order to have no point +at distance less than r, the arc created by the orbit l(ρi, ϕi, θi) +and the set C(ρi, d) has to be empty of satellite points and (ii) +the void probability of the Poisson point process of intensity +µ on the arc is given by the negative exponential of µ times +the arc length. Leveraging the facts that only the orbits with +azimuth angles ϕ < ξ1, π − ξ1 < ϕ < π meet the spherical +cap C(d), we have +P(D > d) += P +�ϕi<ξ1,π−ξ1<ϕi<π +� +Zi∈Ξ +e +−µπ−1 arcsin +�√ +1−cos2(ξ) sec2(ϕi) +�� += exp +� +−2λ +π +� rb +ra +� ξ +0 +� +1 − e +−µπ−1 arcsin +�√ +1−cos2(ξ) sec2(ϕ) +�� +dϕν(dρ) +� +, +where cos(ξ) = (ρ2 + r2 +e − d2)/(2ρre). Above, we use the +probability generating functional of the Poisson point process +Ξ of intensity measure λν(dρ)/π2 in C . +Definition 1. Outage occurs if the typical network user has +no visible satellite. Equivalently, outage occurs if D = ∞. +Lemma 6. The outage probability is given by Eq. (10). +Proof: When there is no visible satellite, D = ∞. By +using Lemma 5, the outage probability is given by +P(D = ∞) += P(∥Xj − u∥ > +� +ρ2 +i − r2e, ∀Xj ∈ ψi, ∀Zi ∈ Ξ) += P +� +� � +Zi∈Ξ +P +� +� � +Xj∈ψi +∥Xj − u∥ > +� +ρ2 +i − r2e +������ +Ξ +� +� +� +� , +where we have +P +� +� � +Xj∈ψi +∥Xj − u∥ > +� +ρ2 +i − r2e +������ +Ξ +� +� += exp +� +−µπ−1 arcsin +�� +1 − r2e sec2(ϕi)/ρ2 +i +�� +. +We use that when d = +� +ρ2 +i − r2e, cos(ξ) = re/ρi. In other +words, for a given ρ, the orbits with azimuth angles less than +arccos(re/ρ) meet the spherical cap C(ρ, +� +ρ2 − r2e). The +outage probability is then given by +P +� � +Zi∈Ξ +e−µπ−1 arcsin(√ +1−r2 +e sec2(ϕi)/ρ2 +i ) +� += exp +� +−2λ +π +� rb +ra +� arccos(re/ρ) +0 +� +1 − e +−µπ−1 arcsin +�√ +1−r2e sec2(ϕ)/ρ2 +�� +dϕν(dρ) +� +, +where we use the probability generating functional of Ξ of +intensity measure λν(dρ)/π2. +Fig. 6 shows the outage probability obtained by Lemma 6. +Lemma 7. Consider a function f(X) : R3 → R. The Laplace +functional of the Cox point process is defined by LΨ(f) = +EΨ +� +exp +� +− � +Xi∈Ψ f(Xi) +�� +. The Laplace functional is given +by Eq. (11) where C = [ra, rb] × [0, π) × [0, π). + +5 +Fig. 6. The outage probability with ra = 7000 km and rb = 7500 km. We +use λ = 72, µ = 22, and ν(dρ) = dρ/(rb − ra). +Proof: The Laplace functional of the satellite Cox point +process is given by +LΨ(f) += E +� +e− � +X∈Ψ f(X)� += EΞ +� +Eψ +� +e +− � +Zi∈Ξ +� +Xj ∈ψi f(X)��� Ξ +�� += EΞ +� � +Zi∈Ξ +exp +� +− µ +2π +� 2π +0 +1 − e− ¯ +f(ρi,θi,ϕi,ω) dω +�� += exp +� +− λ +π2 +� rb +ra +� π +0 +� π +0 +� +1 − e− µ +2π +� 2π +0 +1−exp (− ¯ +f(ρ,θ,ϕ,ω)) dω� +dϕ dθν(dρ) +� +, +where we use the function ¯f(ρ, θ, ϕ, ω)=f(X) for any satellite +X on the orbit l(ρ, θ, ϕ) with its orbital angle ω. Then, we +use the probability generating functional of the Poisson point +process of intensity measure λν(dρ)/π2 to get the result. +Consider a random variable H modeling general fading. A +received signal power of a user at u is then given by f(X) = +H∥X −u∥−α where X is the location of the satellite and α is +the path loss exponent. The total interference S is then given +by the sum of the received signal powers from all satellites. +S = +� +X∈ ¯Ψ +H∥X − u∥−α, ¯Ψ = Ψ +� +� +� +ra<ρ≤rb +C(ρ, +� +ρ2 − r2e) +� +� . +Corollary 1. The Laplace transform of the total interference +is given by Eq. (12) where ¯C = {(ρ, θ, ϕ) ∈ |l(ρ, θ, ϕ) ∩ +C( +� +r2 +b − r2a) ̸= ∅} and ¯ω = {ω ∈ [0, 2π]|X(ρ, θ, ϕ, ω) ∈ +C( +� +r2 +b − r2a), ∀(ρ, θ, ϕ) ∈ ¯C}. +Proof: The Laplace transform in question is +LΨ(f)f=sH∥X−U∥−α += EΞ +� +Eψ +� +e +− � +Zi∈Ξ +� +Xj ∈ψ sH∥Xj−u∥−α��� Ξ +�� += EΞ +� +� � +Zi∈Ξ +Eψ +� +� � +Xj∈ψi +LH(s∥Xj − u∥−α) +� +� +� +� , +where LH(κ) is the Laplace transform of the random variable +H. Using a technique similar to Lemmas 3 and 7, we obtain +the final result. +IV. CONCLUSION +This paper presents a stochastic geometry framework to +model the locations of LEO satellites with multiple altitudes +using a Cox point process. It provides analytical expressions +for essential statistical properties such as the distribution of the +distance from a typical user to the nearest satellite, the Laplace +functional of the Cox point process, and the Laplace transform +of the total interference, experienced by a typical user. These +results can directly be used to determine the performance of +multi-altitude LEO satellite communication networks. Future +work will include (i) the analysis of the coverage probability of +the typical user, (ii) the evaluation of the satellite coverage area +underneath the Cox-modeled satellites, and (iii) an extension +to a fixed-inclination orbit process. +ACKNOWLEDGMENT +The work of Chang-Sik Choi was supported by the NRF- +2021R1F1A1059666. The work of Franc¸ois Baccelli was +supported by the ERC NEMO grant 788851 to INRIA. +REFERENCES +[1] Y. Su, Y. Liu, Y. Zhou, J. Yuan, H. Cao, and J. Shi, “Broadband LEO +satellite communications: Architectures and key technologies,” IEEE +Wireless Commn., vol. 26, no. 2, pp. 55–61, 2019. +[2] Z. Qu, G. Zhang, H. Cao, and J. Xie, “LEO satellite constellation for +Internet of Things,” IEEE Access, vol. 5, pp. 18 391–18 401, 2017. +[3] J. Khalife, M. Neinavaie, and Z. M. Kassas, “The first carrier phase +tracking and positioning results with starlink LEO satellite signals,” +IEEE Trans. Aerospace and Electronic Syst., vol. 58, no. 2, pp. 1487– +1491, 2022. +[4] A. Guidotti, A. Vanelli-Coralli, M. Conti, S. Andrenacci, S. Chatzinotas, +N. Maturo, B. Evans, A. Awoseyila, A. Ugolini, T. Foggi, L. Gaudio, +N. Alagha, and S. Cioni, “Architectures and key technical challenges for +5G systems incorporating satellites,” IEEE Trans. Veh. Technol., vol. 68, +no. 3, pp. 2624–2639, 2019. +[5] 3GPP TR 38.821, “Solutions for NR to support non-terrestrial networks +(NTN),” 3GPP TR 38.821. +[6] N. Okati, T. Riihonen, D. Korpi, I. Angervuori, and R. Wichman, +“Downlink coverage and rate analysis of low earth orbit satellite con- +stellations using stochastic geometry,” IEEE Trans. Commun., vol. 68, +no. 8, pp. 5120–5134, 2020. +[7] A. Talgat, M. A. Kishk, and M.-S. Alouini, “Stochastic geometry-based +analysis of LEO satellite communication systems,” IEEE Commun. Lett., +vol. 25, no. 8, pp. 2458–2462, 2021. +[8] D.-H. Na, K.-H. Park, Y.-C. Ko, and M.-S. Alouini, “Performance +analysis of satellite communication systems with randomly located +ground users,” IEEE Trans. Wireless Commun., vol. 21, no. 1, pp. 621– +634, 2022. +[9] D.-H. Jung, J.-G. Ryu, W.-J. Byun, and J. Choi, “Performance analysis +of satellite communication system under the shadowed-rician fading: A +stochastic geometry approach,” IEEE Trans. Commun., vol. 70, no. 4, +pp. 2707–2721, 2022. +[10] C.-S. Choi and F. Baccelli, “An analytical framework for downlink LEO +satellite communications based on Cox point processes,” arXiv preprint +arXiv:2212.03549, 2022. +[11] F. Baccelli and B. Błaszczyszyn, “Stochastic geometry and wireless +networks: volume I theory,” Foundations and Trends in Networking, +vol. 3, no. 3–4, pp. 249–449, 2010. + +10-2 +outage probability +10-4 +10-6 +10 +20 +10 +20 +30 +30 +40 +50 +μ +入 \ No newline at end of file diff --git a/QNE0T4oBgHgl3EQfkQEN/content/tmp_files/load_file.txt b/QNE0T4oBgHgl3EQfkQEN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe8e1a767852c6f98b5580bac90fc73b2540ddaf --- /dev/null +++ b/QNE0T4oBgHgl3EQfkQEN/content/tmp_files/load_file.txt @@ -0,0 +1,307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf,len=306 +page_content='1 Cox Point Processes for Multi-Altitude LEO Satellite Networks Chang-Sik Choi and Franc¸ois Baccelli Abstract—We propose a simple analytical approach to describe the locations of low earth orbit (LEO) satellites based on a Cox point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We develop a variable-altitude Poisson orbit process by accounting for the fact that satellites are always located on circular orbits and these orbits may have different altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Then, the satellites on these orbits are modeled as the Poisson point processes conditionally on the orbit process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' For this model, we derive the distribution of the distance to the nearest visible satellite, the outage probability, the Laplace functional of the proposed satellite Cox point process, and the Laplace transform of the interference under a general fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The derived statistics allow one to evaluate the performance of such LEO satellite communication systems as functions of network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Index Terms—LEO satellite communications, Stochastic geom- etry, Cox point process, Nearest distance, Total interference I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' INTRODUCTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Motivation and Background LEO satellites provide global connectivity to millions of devices on earth [1]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The applications of LEO satellite net- works are numerous [1]: they provide Internet connections to devices where ground infrastructure is unavailable [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' local- ization and emergency communications of aerial and ground devices can be enabled by LEO satellites [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' LEO satellite networks provide cheaper Internet connections to developing countries [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' LEO satellite networks can even be integrated with terrestrial networks to enable reliable connections to devices in a small area [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' To support these applications, LEO satellite networks will have a very large number of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The viability and performance of LEO satellite communi- cations are significantly determined by the way satellites are distributed in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Various evaluation methodologies have been proposed to obtain the performance of LEO satellite communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' For satellite layout, some studies used probabilistic approaches including a binomial point pro- cess [6]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' In contrast to the simulation-based approach, the benefits of employing such analytical models lie in the fact that they presents large-scale behaviors as functions of network key parameters such as the mean number of satel- lites, their altitudes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Nevertheless, the binomial satellite point processes in [6]–[9] were not able to incorporate the fact that the satellites are located on approximately circular trajectories around the earth, namely their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' In this paper, we provide a tractable model that incorporates this fact in the multi-altitude LEO satellite case, by generalizing the work in Chang-Sik Choi is with Hongik University, South Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Franc¸ois Baccelli is with Inria Paris and Telecom Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (email: chang- sik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='choi@hongik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='kr, francois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='baccelli@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='fr) [10] where all orbits are at the same altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Specifically, we present an analytical framework leveraging a Cox point process so that orbits are created first according to a Poisson point process on a cuboid and then satellites are distributed as Poisson point processes conditionally on these orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We derive key statistical properties of the proposed network model that are critical to obtain the performance of such satellite networks as functions of the altitude distribution, of the mean number of orbits, of the number of satellites, and of the Laplace transform of the random variable representing fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Contributions Modeling of variable orbit LEO satellite constellations: This paper accounts for the geometric properties of practical LEO satellite systems that (i) satellites are always on orbits around the earth and (ii) such orbits are possibly at different altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' By developing a nonhomogeneous Poisson point process of mean λ in a cuboid, we creates a Poisson orbit process of orbits in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Then, conditionally on the orbit process, satellites are distributed as linear Poisson point processes of mean µ on these orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Our motivation is to represent a general LEO satellite network where satellites are located at different altitude bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Statistical properties of the proposed Cox point pro- cess: The proposed satellite Cox point process is built to be invariant by all rotations of the reference plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' This makes the statistical properties of the network to be the same for all perspectives seen from all points on earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Leveraging this, we obtain the probability distribution function of the distance from the typical user to its nearest visible satellite and then derive the outage probability of the proposed network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Using it, we derive the Laplace functional of the proposed satellite Cox point process and then give an integral expression for the Laplace transform of the total interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' These formulas are directly used to assess the network performance metrics such as the Signal-to-interference-plus-noise ratio (SINR) of the typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' COX-MODELED SATELLITES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Satellite Distribution The center of the earth is O = (0, 0, 0) and it is of radius re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The xy-plane is the reference plane and the x-axis is longitude reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' In this paper, we only focus on the snapshot of the network geometry and the movement of satellites is out of the scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Consider a cuboid C = [ra, rb] × [0, π) × [0, π) where ra ≤ rb the minimum and maximum altitudes and a Poisson point arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='02469v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='SP] 6 Jan 2023 2 Reference: xy-plane x-axis A θ l(ρ,θ,φ) φ X: satellite ω O ~ y-axis ρ z-axis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The orbital plane meets the reference plane at two points and the point with angle less than π is A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The angle θ is measured from the x-axis to the segment OA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The inclination ˜ϕ is measured from the reference plane to the orbital plane and the azimuth ϕ is given by π/2 − ˜ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The angle ω for satellite X is measured from OA to OX over the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' process Ξ of intensity measure λν(dρ)/π2 in the cuboid C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We have � rb ra ν(dρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Then, we build an orbit process by mapping each point of Ξ, say (ρ, θ, ϕ) into an orbit l(ρ, θ, ϕ) in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Specifically, the first coordinate ρ is the orbit’s radius, θ is the orbit’s longitude, and ϕ is the orbit’s azimuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' For the Poisson point process on the cuboid, we write Ξ = � i Zi, where Zi is the point of Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Since there are on average λ points of Ξ, there are on average λ orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The orbit process O in R3 is given by O = � Zi∈Ξ l(ρi, θi, ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (1) Conditionally on Ξ, the locations of satellites on each orbit l(ρi, θi, ϕi) are modeled as a homogeneous Poisson point process ψi of intensity µ/(2πρi) on this orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Equivalently, the orbital angles of satellites on each orbit are modeled as a 1-dim homogeneous Poisson point process φi on segment [0, 2π) of intensity µ/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Since the satellites are distributed conditionally on Ξ, the satellite point process Ψ is a Cox point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The satellite Cox point process is Ψ = � i ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (2) Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 2 – 4 depict the proposed satellite Cox point process with λ, µ, ra and rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' In the figures, we use ν(dρ) = dρ rb−ra , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=', the radii of orbits are uniformly distributed on the interval [ra, rb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The proposed model can be used to represent e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=', multiple operators of LEO satellite networks where orbits are at different altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The case of all satellites are located at the same altitude in [10] is a special case of the proposed model by taking ν(dρ) = δra(dρ), where ra is the radius of orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' User Distribution Users are located on the surface of the earth {(x, y, z)|x2 + y2 +z2 = r2 e} and the locations of network users are assumed to be independent of the locations of the LEO satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' STATISTICAL RESULTS In this section, we derive/prove (i) the mean number of LEO satellites, (ii) the isotropy of Ψ, (iii) the distances from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The proposed Cox satellite model with ra = 7000 km, rb = 7100 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We use λ = 60, µ = 40, and ν(dρ) = dρ/(rb − ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The Cox-modeled satellite with ra = 7000 km and rb = 7500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We use λ = 30, µ = 60, and ν(dρ) = dρ/(rb − ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The Cox-modeled satellite with ra = 7000 km and rb = 8500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We use λ = 70, µ = 30, and ν(dρ) = dρ/(rb − ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 3 the LEO satellites to an arbitrarily located user, (iv) the distribution of the distance to the nearest visible satellite, (v) the outage probability, (vi) the Laplace functional of Ψ, and (vii) the Laplace transform of the total interference under general fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' These statistical properties directly determine the performance of downlink LEO satellite communications in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The average number of the proposed Cox satellite point process is λµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Proof: The average number of satellites is given by E [Ψ(S)] = E � � � Zi∈Ξ E � � � Xj∈ψi 1 ������ Ξ � � � � = E � � Zi∈Ξ � 2π 0 µ 2π dx ����� Ξ � = µ � C λ π2 ν(dρ) dθ dϕ = λµ, where we use Campbell’s mean value theorem [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Below we show that O is invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' This allows one to evaluate the performance of network seen by a typical user at the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The distribution of O and Ψ are invariant by all rotations of the reference space (O, x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Proof: The intensity measure of the proposed orbit pro- cess Ξ has the product form: ν(dρ) × � λ/π2� dθ dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' This shows that the angles (θ, ϕ) form a homogeneous Poisson point process on the rectangle [0, π) × [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' [10] proved that the orbit process mapped by the very intensity measure � λ/π2� dθ dϕ is invariant by all rotations of the reference space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Hence, the law of O is also invariant by all rotations of the reference space (O, x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' In the same vein, the law of Ψ is invariant by all rotations of the reference space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Consider a satellite X of orbital angle ωj on the orbit l(ρi, θi, ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The distance from (0, 0, re) to the satellite X(ρi, θi, ϕi, ωj) is given by � ρ2 i − 2ρire sin(ωj) cos(ϕi) + r2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (3) Proof: The coordinates (x, y, z) ∈ R3 of the satellite that has the orbital angle ωj on the orbit l(ρi, θi, ϕi) are given by x = � ρ2 i cos2(ωj) + ρ2 i sin2(ωj) cos2( ˜ϕi) cos � ˜θ + θi � , (4) y = � ρ2 i cos2(ωj) + ρ2 i sin2(ωj) cos2( ˜ϕi) sin � ˜θ + θi � , (5) z = ρi sin(ωj) sin( ˜ϕi), (6) ˜θ = arctan (tan(ωj) cos( ˜ϕi)) , (7) where ˜ϕ is the inclination: ˜ϕ = π/2 − ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' As a result, the distance from (0, 0, re) to the satellite is ∥(x, y, z) − (0, 0, re)∥ = � ρ2 i − 2ρire sin(ωj) cos(ϕi) + r2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Note the distance is independent of the variable θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' U=(0,0,re) O A C E B F A` D ρ re Bottom of spherical cap Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The arc of orbit l(ρ, θ, ϕ) in spherical cap C(ρ, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The Lengths of Orbits’ Arcs Since (i) users are independent of Ψ and (ii) Ψ is invariant by rotations (Lemma 2 ), one can consider a typical user at (0, 0, re) and study the network performance it experiences, which will be typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Let C(d) be the subset of S such that the distances from the typical observer u to the satellites on C(d) are less than a distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' For any ra ≤ ρ ≤ rb, we define C(d) = � ra≤ρ≤rb C(ρ, d) = � ra≤ρ≤rb � (x, y, z) ∈ R3 |z ≥ re, x2 + y2 + z2 = ρ2, x2 + y2 + (z − re)2 ≤ d2� , where z ≥ re, since satellites with z-coordinates less than re are invisible to the user at (0, 0, re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' C(ρ, d) is a spherical cap associated with the orbit of radius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The length of the arc given by the intersection of the spherical cap C(ρ, d) and the orbit l(ρ, θ, ϕ) is 2ρ arcsin � � � 1 − �ρ2 + r2e − d2 2ρre cos(ϕ) �2 � � , (8) for ρ − re ≤ d ≤ � ρ2 − r2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Proof: Consider C(ρ, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Let ξ be the angle ∠AOU in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Then, we use the law of Cosine to obtain cos(ξ) = (ρ2 + r2 e − d2)/(2ρre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' For the triangle △BCD, we have CD = ρ cos(ξ) tan(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Since the angle ∠BDC is π/2, we obtain BD = � ρ2 sin2(ξ) − ρ2 cos2(ξ) tan2(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' For △BOD, OB = ρ and let κ′ = ∠BOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Then we have sin(κ′) = BD/ρ = � sin2(ξ) − cos2(ξ) tan2(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Finally, the length of the arc > BF is given by ν(> BF) = 2ρ arcsin( � 1 − cos2(ξ) sec2(ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' where cos(ξ) = (ρ2 + r2 e − d2)/(2ρre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' In downlink LEO satellite communication networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' net- work users are meant to receive signals from their closest or 4 P(D > d) = exp � −2λ π � rb ra � ξ 0 � 1 − e −µπ−1 arcsin �√ 1−cos2(ξ) sec2(ϕ) �� dϕν(dρ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (9) P(D = ∞) = exp � −2λ π � rb ra � arccos(re/ρ) 0 � 1 − e −µπ−1 arcsin �√ 1−r2e sec2(ϕ)/ρ2 �� dϕν(dρ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (10) L(f) = exp � − λ π2 � C � 1 − e− µ 2π � 2π 0 1−exp (− ¯ f(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content='ω)) dω� ν(dρ) dθ dϕ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (11) LΨ(f)f=sH∥X−U∥−α = exp � − λ π2 � ¯C � 1 − e− µ 2π � ¯ ω 1−LH(s(ρ2−2ρre sin(ω) cos(ϕ)+r2 e)− α 2 ) dω� ν(dρ) dθ dϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (12) nearest satellites [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The distance D from a network user to its closest LEO satellite is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' When there is no visible satellite, D def = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The cumulative distribution function of D is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (9) where cos(ξ) = (ρ2 + r2 e − d2)/(2ρre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Proof: For ra − re ≤ d ≤ � r2 b − r2e, we have P(D > d) (a) = P(∥X − u∥ > d, ∀X ∈ Ψ) (b) = P(∥Xj − u∥ > d, ∀Xj ∈ ψi, ∀Zi ∈ Ξ) = P � � � Zi∈Ξ P � � � Xj∈ψi ∥Xj − u∥ > d ������ Ξ � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' To get (a), we use the fact that for R > r, all satellites should be at distances greater than r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We have (b) by using that the Cox satellite point process is comprised of the Poisson point processes conditionally on orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We have P � � � Xj∈ψi ∥Xj − u∥ > r ������ Ξ � � = exp � −µπ−1 arcsin �� 1 − cos2(ξ)sec2(ϕi) �� , where cos(ξ) = (ρ2 i + r2 e − d2)/(2ρire), as a function of the orbits’ radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We use the facts that (i) in order to have no point at distance less than r, the arc created by the orbit l(ρi, ϕi, θi) and the set C(ρi, d) has to be empty of satellite points and (ii) the void probability of the Poisson point process of intensity µ on the arc is given by the negative exponential of µ times the arc length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Leveraging the facts that only the orbits with azimuth angles ϕ < ξ1, π − ξ1 < ϕ < π meet the spherical cap C(d), we have P(D > d) = P �ϕi<ξ1,π−ξ1<ϕi<π � Zi∈Ξ e −µπ−1 arcsin �√ 1−cos2(ξ) sec2(ϕi) �� = exp � −2λ π � rb ra � ξ 0 � 1 − e −µπ−1 arcsin �√ 1−cos2(ξ) sec2(ϕ) �� dϕν(dρ) � , where cos(ξ) = (ρ2 + r2 e − d2)/(2ρre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Above, we use the probability generating functional of the Poisson point process Ξ of intensity measure λν(dρ)/π2 in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Outage occurs if the typical network user has no visible satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Equivalently, outage occurs if D = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The outage probability is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Proof: When there is no visible satellite, D = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' By using Lemma 5, the outage probability is given by P(D = ∞) = P(∥Xj − u∥ > � ρ2 i − r2e, ∀Xj ∈ ψi, ∀Zi ∈ Ξ) = P � � � Zi∈Ξ P � � � Xj∈ψi ∥Xj − u∥ > � ρ2 i − r2e ������ Ξ � � � � , where we have P � � � Xj∈ψi ∥Xj − u∥ > � ρ2 i − r2e ������ Ξ � � = exp � −µπ−1 arcsin �� 1 − r2e sec2(ϕi)/ρ2 i �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We use that when d = � ρ2 i − r2e, cos(ξ) = re/ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' In other words, for a given ρ, the orbits with azimuth angles less than arccos(re/ρ) meet the spherical cap C(ρ, � ρ2 − r2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The outage probability is then given by P � � Zi∈Ξ e−µπ−1 arcsin(√ 1−r2 e sec2(ϕi)/ρ2 i ) � = exp � −2λ π � rb ra � arccos(re/ρ) 0 � 1 − e −µπ−1 arcsin �√ 1−r2e sec2(ϕ)/ρ2 �� dϕν(dρ) � , where we use the probability generating functional of Ξ of intensity measure λν(dρ)/π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 6 shows the outage probability obtained by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Consider a function f(X) : R3 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The Laplace functional of the Cox point process is defined by LΨ(f) = EΨ � exp � − � Xi∈Ψ f(Xi) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The Laplace functional is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (11) where C = [ra, rb] × [0, π) × [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The outage probability with ra = 7000 km and rb = 7500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' We use λ = 72, µ = 22, and ν(dρ) = dρ/(rb − ra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Proof: The Laplace functional of the satellite Cox point process is given by LΨ(f) = E � e− � X∈Ψ f(X)� = EΞ � Eψ � e − � Zi∈Ξ � Xj ∈ψi f(X)��� Ξ �� = EΞ � � Zi∈Ξ exp � − µ 2π � 2π 0 1 − e− ¯ f(ρi,θi,ϕi,ω) dω �� = exp � − λ π2 � rb ra � π 0 � π 0 � 1 − e− µ 2π � 2π 0 1−exp (− ¯ f(ρ,θ,ϕ,ω)) dω� dϕ dθν(dρ) � , where we use the function ¯f(ρ, θ, ϕ, ω)=f(X) for any satellite X on the orbit l(ρ, θ, ϕ) with its orbital angle ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Then, we use the probability generating functional of the Poisson point process of intensity measure λν(dρ)/π2 to get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Consider a random variable H modeling general fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' A received signal power of a user at u is then given by f(X) = H∥X −u∥−α where X is the location of the satellite and α is the path loss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The total interference S is then given by the sum of the received signal powers from all satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' S = � X∈ ¯Ψ H∥X − u∥−α, ¯Ψ = Ψ � � � ra<ρ≤rb C(ρ, � ρ2 − r2e) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The Laplace transform of the total interference is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' (12) where ¯C = {(ρ, θ, ϕ) ∈ |l(ρ, θ, ϕ) ∩ C( � r2 b − r2a) ̸= ∅} and ¯ω = {ω ∈ [0, 2π]|X(ρ, θ, ϕ, ω) ∈ C( � r2 b − r2a), ∀(ρ, θ, ϕ) ∈ ¯C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Proof: The Laplace transform in question is LΨ(f)f=sH∥X−U∥−α = EΞ � Eψ � e − � Zi∈Ξ � Xj ∈ψ sH∥Xj−u∥−α��� Ξ �� = EΞ � � � Zi∈Ξ Eψ � � � Xj∈ψi LH(s∥Xj − u∥−α) � � � � , where LH(κ) is the Laplace transform of the random variable H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Using a technique similar to Lemmas 3 and 7, we obtain the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' CONCLUSION This paper presents a stochastic geometry framework to model the locations of LEO satellites with multiple altitudes using a Cox point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' It provides analytical expressions for essential statistical properties such as the distribution of the distance from a typical user to the nearest satellite, the Laplace functional of the Cox point process, and the Laplace transform of the total interference, experienced by a typical user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' These results can directly be used to determine the performance of multi-altitude LEO satellite communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Future work will include (i) the analysis of the coverage probability of the typical user, (ii) the evaluation of the satellite coverage area underneath the Cox-modeled satellites, and (iii) an extension to a fixed-inclination orbit process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' ACKNOWLEDGMENT The work of Chang-Sik Choi was supported by the NRF- 2021R1F1A1059666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' The work of Franc¸ois Baccelli was supported by the ERC NEMO grant 788851 to INRIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQfkQEN/content/2301.02469v1.pdf'} +page_content=' Liu, Y.' metadata={'source': 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Ahmed Chowdhury,1, a) Tara Peña,2 Sobhit Singh,1 Stephen M. Wu,2, b) and Hesam +Askari1 +1) Department of Mechanical Engineering, University of Rochester, New York +2) Department of Electrical and Computer Engineering, University of Rochester, Rochester, +New York +(*Electronic mail: adey2@ur.rochester.edu) +Twisted bilayer graphene exhibits electronic properties that are highly correlated with the size and arrangement of +moiré patterns. While rigid rotation of two layers creates the topology of moiré patterns, local rearrangements of the +atoms due to interlayer van der Waals interactions result in atomic reconstruction within the moiré cells. The ability to +manipulate these patterns by controlling twist angle and/or externally applied strain provides a promising route to tune +their properties. While this phenomenon has been extensively studied for angles close to or smaller than the magic angle +(θm=1.1°), its extent for higher angles and how it evolves with strain is unknown and is believed to be mostly absent at +high angles. We use theoretical and numerical analyses to resolve reconstruction in angles above θm using interpretive +and fundamental physical measures. In addition, we propose a method to identify local regions within moiré cells and +track their evolution with strain for a range of representative high twist angles. Our results show that reconstruction is +actively present beyond the magic angle and its contribution to the evolution of the moiré cells is major. Our theoretical +method to correlate local and global phonon behavior provides further validation on the role of reconstruction at higher +angles. Our findings provide a better understanding of moiré reconstruction in large twist angles and the evolution of +moiré cells in the presence of strain, that might be very crucial for twistronics-based applications. + +I. +Introduction +Engineering two-dimensional (2D) materials by control- +ling the stacking orientation of atomic layers have emerged +as a powerful technique to manipulate their mechanical and +opto-electronic properties. Bilayer graphene (BLG) is one of +the simplest van der Waals (vdW) structures that display di- +verse physical properties such as contrasting electronic struc- +tures depending on the stacking arrangement1–4. Introduc- +ing a relative rotation between the layers forms the Twisted +Bilayer Graphene (TBG) in which the atoms create a peri- +odic hexagonal superlattice called ‘moiré pattern’ (MP)5–7. +Emergence of this pattern is due to the atoms occupying dif- +ferent relative interlayer positions compared to BLG with a +global size that is inversely correlated with the twist angle +(θ ) as Lm = a/(2 sin(θ /2)) where a is the lattice constant of +graphene. Application of other mechanical stimuli such as in- +equivalent strain to the individual layers of TBG can further +manipulate the shape of the pattern. Thus, the combination of +hetero-straining process and twist provides a promising out- +look for creating unique shapes and geometries of MPs for +exciting opto-electronic applications8–10. +The atomic arrangements within MPs are influenced by +the interlayer vdW forces between the atoms that consider- +ably influence the atomic arrangement landscape. To manifest +this influence, we can consider a hypothetical intermediate +configuration where atoms are rigidly twisted in their plane +and consequently, the well-defined BLG stacking configura- +tions of AA, AB and SP types with their spatial variations +will emerge11,12. Upon allowing atomic reconfiguration, an + + +a)These authors contributed equally to this work +b)Department of Physics and Astronomy, University of Rochester, Rochester, +New York +atomic-scale reconstruction occurs and local stacked regions +evolve to their true minimum local energy configuration. This +process is known as moiré reconstruction13,14. Previous stud- +ies have reported this phenomenon for low angle TBGs, es- +pecially in the vicinity of or below the ’magic angle’ (θm = +1.1°)15,16. As the size of MP shrinks with an increase in θ and +leaves less space for reconfiguration of atoms, experimental +observation of moiré reconstruction becomes a challenge and +is generally assumed to be absent for θ > 2°15,17,18. Since the +large angle TBGs contain the same atomic registry but only +over a smaller region compared to the small twist angles, it is +unreasonable to expect moiré reconstruction should suddenly +become absent. The interplay between the in-plane elastic en- +ergy and interlayer vdW energy is still expected to contribute +to reconstruction at higher angles due to the same fundamen- +tal physics. Nevertheless, its extent remains unknown due to +the current limitations of experimental methods. +Recent experimental studies have demonstrated the abil- +ity to control TBGs with and without strain and characterize +moiré reconstruction for smaller θ systems8,14,19–23. Imag- +ing techniques such as STM and TEM become challenging +when feature size becomes comparable to their resolution. As +the size of MP decreases with increasing twist, imaging for +θ > 2° systems become unfeasible11,24. Therefore, the cur- +rent understanding of reconstruction through experimental vi- +sualization is limited to low angle twists and largely based +on image analysis techniques rather than physical measurable +quantities. Optical procedures such as Raman spectroscopy +offer an expedient method to characterize TBGs irrespective +of their size and twist angle25–28 but such methods predomi- +nantly extract the collective behavior of TBGs spanning nu- +merous MPs. Therefore the global vibrational behavior ob- +tained by Raman cannot be readily used to infer stacking and +the extent of reconstruction without an interrelation of phonon + +− +i +An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +2 + +FIG. 1: Atomistic model. Relaxed atomistic structures illustrate how periodic moiré superlattice is formed and how its shape +evolves with strain (a & b). Unlike BLG where a single interlayer distancing is expected, twist results in spatial variations of +interlayer distancing with as shown for (c) unstrained and (d) strained TBGs. Data presented for the twist angle of θ = 6° and +uniaxial strain of 1% . (e) Real space geometric analysis demonstrating the distortion of MPs with applied uniaxial tension to +the top layer. + + +behavior between local sub-domains and the bulk of TBG. +Atomistic analyses offer an alternative tool to study atomic ar- +rangements locally with a fine resolution and allow for track- +ing of atomistic evolution with varying twist angle11,20,29–31. +Current works are heavily concentrated on or below the magic +angle and do not explain the correlation of local and global be- +havior of TBGs and moreover, have not studied the evolution +of MPs with strain. As a result, there remains an outstanding +question about the viability and the role of reconstruction at +higher angles and how local and global vibrational properties +are correlated. +In this work, we utilized a combination of first-principles +and molecular statics atomistic simulations to examine the lo- +cal domains in TBGs and how global vibrational behavior is +tied to changes in local atomic registries. Based on physi- +cal parameters that include interlayer spacing and interlayer +energy, our method associates each atom to known stacking +types of the constituent bi-layer graphene and calculates their +resultant area fraction and traces the evolution of local sub- +domains, and demonstrates evidence of moiré reconstruction +for larger θ TBG systems. This paper presents an effective set +of criteria for the identification of local stacking and recon- +struction phenomena in TBGs that are valid with or without +the application of strain. In addition, we demonstrate the cor- +relation between local and global vibrational characteristics of +TBGs and how it validates our results on reconstructed struc- +tures, especially at higher angles. The methods presented in +this paper are devised for graphene but further adaptations are + +possible for other 2D materials. + +II. Methods +A.Atomistic modeling. +All the TBG structures are constructed by rotating the top +layer of Bernal stacked bilayer graphene with respect to its +bottom layer. The moiré lattice is created by identifying a +common periodic lattice for the two layers. Using the TBG +commensurability conditions, we have modeled their real and +reciprocal space lattice parameters32,33. The ⃗q vector or re- +ciprocal lattice parameter of TBG moirlattice is given as ⃗q = +⃗b′ ⃗b, where ⃗b and ⃗b′ denote the reciprocal lattice vectors +of the bottom layer and rotated top layer respectively. When +heterostrain is applied, the strained ⃗q vector is expressed as +q⃗ε = b⃗ε −⃗b, where b⃗ε denotes the strained top layer. The +mathematical expressions of b⃗ε are deduced in Supplementary +section II. All the atomistic models are relaxed using density +functional theory (DFT) simulations, except for θ = 1.08° sys- +tem. Because of a large moiré lattice for this structure (11164 +atoms), DFT becomes forbiddingly inefficient and thus, we +use force-field potentials for relaxing this structure. +B.DFT calculations. +The real space lattices of TBG systems were constructed us- +ing ATOMISTIX TOOLKIT (QuantumATK) package34. All +the first-principles simulations were conducted with gener- +alized gradient approximation (GGA) assimilated in Quan- +tum Espresso open source package35,36. The Perdew-Burke- + +3.55 +3.5 +鞋 +3.45 +鞋 +3.4 +3.35± +± +An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +3 + +FIG. 2: Local stacking identification method. (a) Path PQ along the center of one moiré pattern to the other (θ = 6°) (b) +Illustration of interlayer energy (ILE) which is the energy contribution of vdW interactions. (c) ILE contour plot for unstrained +θ = 6° system. (d) Variation of interlayer spacing (ILS) with respect to moiré twist angles; Horizontal dotted line (magenta) +shows the minima of maximum ILS (dmax) obtained throughout a span of low and high angles TBGs. (e) Variation of ILE +difference for five representative θ (the dotted line shows the energy difference at soliton width boundary) (f) Contour plot +demonstrating individual stacking type locally, obtained after implementing classification method. + + +Ernzerhof (PBE) form along with GGA has been used as the +exchange-correlation functional37. Ion-electron interactions +for carbon atoms in TBGs have been described by ultrasoft +pseudopotentials38. All technical details about DFT parame- +ters are given in Supplementary information-Section I. + +C.MS simulations. +Molecular statics simulations were done using LAMMPS +open source software39,40. The unstrained, DFT relaxed TBG +moiré lattice was transformed into an orthogonal cell for per- +forming MS simulations. The simulation box is considered +with free surface boundary conditions allowing us to account +for the aperiodic crystal geometry (or moiré lattice mismatch) +due to strain applied to one of the layers. The uniaxial strain +was incremented by 0.1% up to the final strain magnitude of +1%. The snapshots of the structure at different strain mag- +nitudes were taken in Ovito open visualization tool. Further +computational details are mentioned in Supplementary section +I. + +III. Results and Discussions +A. Global structural analysis of pristine and strained TBGs. +We have studied a number of TBG systems between θ = +1.08° and 13.2° to perform our analysis on MPs close to θm +as well as outside the limit of small angles. For simplic- +ity, most of the presented data include three representative +TBG systems θ = 1.08°, 6° and 13.2°. The MP geometries + +are modeled using the well-defined commensurability con- +ditions of TBG systems and relaxed using first-principles or +force field optimization techniques (see Methods) (Fig. 1(a)). +Since the local domains in TBG evolve through high symme- +try BLG stacking, we can observe topographical variation in +the structure41,42 represented by interlayer spacing (ILS) con- +tour plot (Fig. 1(c)). The centers of hexagonal MPs have re- +gions of atoms where AA stacking exists13,43. These central +regions are surrounded by two domains, AB and BA stack- +ing, which are energetically degenerate but topologically in- +equivalent. Since both of these stacking represent the Bernal +graphene, they can be categorized as one44,45. The boundaries +of these AB/BA regions are separated by segments referred +as strain solitons. The shear strain which generates due to +two inequivalent stacking domains facing each other is con- +fined within those segments with characteristic width referred +to as the soliton width43. The atomic structure in the soliton +regions corresponds to SP stacking which is an intermediate +configuration between AB (or BA) and AA. A TBG system +displays an out-of-plane corrugation in its structure caused by +local ILS variation with AA regions having the highest spac- +ing followed by SP and AB regions11,12,30. +On employing heterostrain, we observed a similar topo- +graphical feature with distorted MPs due to the inequiva- +lence of strain in each layer that resulted in an oblique moiré +arrangement8 (Fig. 1(b), (d) for tension and Fig. S1 for com- + +(LE (meV/atom) +28 +17 +14 +3.6 +dmax (AA stacking) +12 +10 +3.5 +dmin (AB stacking) +8 +6 +1.10 +3.4 +3.48° +4 +4.410 +d max +60 +d min +7.34° +3.3 +4 +8 +12 +16 +20 +24 +28 +0 +0.2 +0.4 +0.6 +0.8 +Twist angle (00) +Normalized distance| +| +| +| +| +| ̸ +| +| +An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +4 + +pression). A geometric analysis is represented to explicate +the angular change due to distortion and rigid rotation (Fig. +1(e)) by deducing the expressions of their reciprocal lattice +(⃗q) vectors (see Supplementary). The change in ⃗q vector with +uniaxial strain triggers the distortion in MPs27,46. As shown +in Fig. 1(e), the boundaries of MPs resemble a hexagon. On +connecting the centers of adjacent MPs, we can draw a tri- +angle (∆ABC) with A⃗B and B⃗C as the moiré lattice vectors +and α being the angle between them. In unstrained condition, +the magnitude of vectors A⃗B = B⃗C = Lm (Lm = Length of +MP) and the angles are α = 60°, φ = 120°. As the ⃗q vector +changes with uniaxial heterostrain, ∆ABC transforms to ∆A′B +′C′ such that A⃗′B′ = B⃗′C′ . The deformed moiré lattice can +be quantified with a change in α with the applied strain (Fig. +S2). The expressions of moiré reciprocal lattice vectors, show +the geometrical changes enforced upon hetero-straining these +systems (Supplementary section II). + +B. Classification method to identify local domains. +The deformation of MP with strain gives rise to changes in +their local sub-domains and it is important to examine them +for quantifying their contribution to global physical behavior. +Traversing along a diagonal of MP (path PQ in Fig. 2(a)), +i.e., from the center of one moiré pattern to the center of its +second nearest neighbor, we expect to cross all the locally +stacked regions: AA, AB, SP, BA, and AA11,43,45. Since we +aim to develop a criteria to classify each atom into one of these +stacking, we first examined the atoms along the path PQ. To +perform the stacking identification, we initially used the ILS +parameter d because the local domains in TBGs vary in in- +terlayer distancing. Since pristine BLG stacking follows an +increasing ILS trend from AB to SP and finally the AA re- +gion, dmax (maximum ILS) and dmin (minimum ILS) in TBGs +can be respectively understood as the ILS of AA and AB re- +gions. By examining the range of ILS (dmax and dmin) over +different possible twist angles (Fig. 2(d)) we identify the min- +imum value of dmax (3.475Å) and classify atoms above this +ILS threshold as AA. It should be noted that this does not mis- +classify AB and SP because this threshold is quite above the +ILS of pristine AB (3.33Å) and SP (3.38Å). Due to the small +ILS difference between AB and SP, the same ILS parameter +cannot be used to identify the rest of the stackings. +We introduced another parameter called ′interlayer energy′ +(ILE) to distinguish between AB and SP according to their +energy, rather than ILS. The ILE is a physical measure of +vdW interaction between atoms in two different layers, as il- +lustrated by the schematic in Fig. 2(b). It is obtained by com- +puting the vdW part of the total potential energy between C- +atoms in different layers Fig. 2(c). Since these local domains +have indistinguishable and strong in-plane covalent bonds, +their total potential energy is predominantly sourced from the +in-plane interactions, which show little variance risen from +their interlayer configuration. Moreover, with applied strain, +the changes in total potential energy due to stretching and +compressing of the in-plane bonds are orders of magnitude +higher than their interlayer vdw counterparts. This motivates +the use of vdW interaction energy and its variations for iden- +tification purposes. However, being a per-atom quantity there + +are a lot of fluctuations in ILE magnitudes, most prominently +observed in AB regions (2(c)). Moreover, if the average ILE +magnitude is used with respect to their bonded neighbors, it +will result in an insignificant difference between AB and SP +sub-domains. Hence to account for this, we calculated the av- +erage ILE difference (∆EILE ) of each atom with its bonded +neighbors. Although it can be difficult to separate AA and +SP regions since they have minimal fluctuations in ILE, this +parameter easily allows to classify AB stacked atoms as they +have the highest variations in energy with neighbors. Based +on the ∆EILE analysis for five representative TBGs (Fig. 2e), +we have identified the ∆EILE threshold at the soliton bound- +ary (SP width) and classified atoms above that threshold as +AB. The infinitesimal difference in these thresholds allowed +us to define a θ -independent ∆EILE value for identifying the +two stackings (see Supplementary for details). It is important +to note that the same approach can be used for classification +in the presence of strain because the physical parameters used +do not depend on strain. Although the magnitude of inter- +layer energy can be expected to vary, we observed a negligible +change in ∆EILE threshold with strain (see Supplementary). +Thus using these criterion based on ILS and ILE, we could +classify atoms into their local stacking as shown in Fig. 2(f), +which applies to TBGs with any twist angle and strain (Fig. +3(a)). + + +On implementing the classification method, we obtained +area fractions (AF) of each sub-domains present in a TBG +structure. Using this measure to monitor the evolution of lo- +cal domains in the presence of strain, we observed that the +sub-domains’ AF remain almost unchanged (Fig. 3(b), Fig. +S4 for tension and Fig S5 for compression). It demonstrates +a characteristic tendency of these local regions to retain their +registry with an external strain applied globally. The varia- +tion of AF as a function of twist angle (Fig. 3(c)) shows that +area fraction of AB (AFAB) and SP (AFSP) increases whereas +that of AA (AFAA) decreases with decreasing θ . This can +be attributed to the potential energy of soliton (SP) regions +contributing to in-plane forces, that displace atoms to max- +imize the area of AB/BA (most stable BLG-stacking) local +domains30. Such observations are well-interpreted in exper- +iments, particularly for systems close to θm (1.08°). Hence +we compared our theoretically estimated AF for θ = 1.08° +(and additional θ = 1.21°, 1.37°) systems with experimen- +tally interpreted area fractions from graphical analysis of STM +images19, as marked in Fig. 3(c). The close similitude be- +tween these sets of area fraction values provides a valida- +tion of our stacking classification method. We believe our +approach interprets the physical behavior of sub-domains at +atomic-level and with high accuracy. Besides, as our method +is based on physical parameters such as energy, it directly en- +capsulates the underlying physics while in contrast, the previ- +ously reported data rely on a graphical interpretation of gradi- +ent in image intensity and contrast from experiments. Hence, +our methodology is more accurate and able to resolve atom- +istic insights even at a higher twist angle where the moiré cell +size shrinks drastically. + +An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +5 + +FIG. 3: Evolution of local regions with twist angle and strain. (a) Contour plot demonstrating local stacking type for +heterostrained θ = 6° system (1% tension). Area fractions of individual stacking domain with respect to (b) strain (tension) and +(c) twist angle. The red markings in (c) are extracted from reported work by Kazmierczak et. al.19 to compare our results with +data obtained by analyzing experimental measurements . + + +C. Detecting moiré reconstruction in high twist angle TBGs. +We further utilized this method to study the extent of atomic +reconstruction in TBG systems. Moiré reconstruction can be +studied by examining local regions in rigidly twisted (R-TBG) +structure and comparing with their relaxed geometry13–16. +The rigidly twisted TBG refers to its unrelaxed geometry, con- +sidered in a conceptual intermediate configuration, in which +the layers of BLG are twisted by a certain angle but the +atoms are not allowed to reconfigure to form their true equi- +librium structure. During reconstruction, local sites in the +structure prefer to diverge from energetically unfavorable AA +stacking by atomic displacements. This is achieved by rear- +rangement of the atoms to minimize vdW energy and obtain- +ing the nearly commensurate Bernal-stacked (AB/BA) BLG +structure partitioned by the SP segments after reconstruction. +The emergence of soliton (SP) domains is one of the pre- +dominant features of reconstruction phenomena in 2D mate- +rials. Previous studies have attributed the minor atomic dis- +placements of large θ relaxed TBGs to insignificant change +in atomic registry of local domains indicating the absence +of reconstruction15–17,47. However, examining TBG systems +with an atomistic insight and employing our sub-domain iden- + +tification method, we show considerable changes in the local +registries for larger θ TBGs. We utilized the area fraction +measure to capture the structural changes in local domains of +relaxed and unrelaxed geometries. The stacking identification +assessment of R-TBG is conducted similarly to the relaxed +TBG (see Supplementary). For θ = 6° structure (4(a)-(c)), the +AA regions shrink upon relaxation and conversely, the AB/BA +regions expand to approximate triangular domains. Undoubt- +edly, this structural change was expected and prominently ob- +served for θ = 1.08° system (4(d)-(f)). But we encountered +a similar observation for a large θ structure. Hence, contrary +to the general idea that reconstruction diminishes at higher +angles, we show clear evidence demonstrating moiré recon- +struction in higher θ (>2°) TBG systems. This observation +indicates that irrespective of how small the atomic displace- +ments are, the change in AF of local domains for higher θ +TBGs show pronounced variation in atomic registries upon +relaxation. + +0.5 +AA +SP +0.4 +AB +fraction +0.3 +0.2 +0.1 +0%strain +0.5%strain +1%strain +0.5 +0.4 +Area fraction +0.3 +0.2 +-→AB +-CAA +--SP +0.1 +AB(Kazmierczaket.al.) +AA(Kazmierczaket.al.) +SP(Kazmierczaket.al.) +0 +0 +2 +4 +6 +8 +10 +12 +Twist Angle (0)AFrigid +An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +6 + + +FIG. 4: Demonstration of moiré reconstruction. Stacking contour plot for rigid (a) θ = 6°, (d) θ = 1.08° and relaxed (b) θ = +6°, (e) θ = 1.08° TBG systems. (c), (f) Comparison of area fractions for each stacking , showing the change in local atomic +registries before and after relaxation that signifies the extent of reconstruction. + + +D. Analyzing extent of reconstruction in strained and +unstrained TBGs. +Using this approach, we have also studied the extent of +moiré reconstruction in high angle TBGs in the presence +of heterostrain. Lattice deformation due to heterostrain in- +duces distortion in MPs, which is minimized by sustaining the +formed domain-wall-like boundary lines (SP regions) due to +superlattice reconstruction15,23,48. Similar to the unstrained +case, we have compared the local AF of rigid and relaxed +systems under heterostrain (Fig. 5). The rigid system for +strained TBGs refers to its unrelaxed structure obtained af- +ter employing strain to the relaxed geometry of pristine TBG +structure (see Supplementary). We observed that our assess- +ment could capture the variations in local atomic registry of +strained TBGs (Fig. 5(a)-(c)). The substantial change in AF +of AA and AB regions and perpetual of SP domains, signifies +the tendency of preserving the SP boundaries with change in +local atomic registry of AA and AB domains, thus indicating +the presence of atomic reconstruction in large θ strained TBG +systems. To assess the extent of change in local registries, we +have calculated the percentage change in local AF upon re- +laxing the structures, i.e., ∆AF(%) = ( +AFrelaxed−AFrigid ) × 100. +On examining the variation of ∆AF over unstrained (Fig 5(c)) +and strained (tensile Fig. 5(e) and compressive Fig. 5(f)) +TBGs spanning a wide range of twist angles, it is observed +that ∆AF for all local stackings monotonically decreases with +increasing θ . Although this implies that, as expected, the ef- + +fect of reconstruction reduces with increasing twist angle, AFs +data shows that it can not be disregarded. It is noticed that +for both pristine and strained cases, the AB stacked domains +show ample variation in rigid and relaxed configurations even +for higher angles. This variation rapidly decreases for AA and +SP regions, especially at very high twist angles. Nonetheless, +this analysis reveals the existence of local atomic reconstruc- +tion for both unstrained and strained large θ TBG systems. +It has been previously argued that for a large twist angle, +the gaining vdW energy cannot compensate for the losing in- +tralayer elastic energy15,17,23. This results in no change of +vdW stacking energy between rigid and relaxed structures, ul- +timately indicating the absence of reconstruction. However, +our analysis of ILE over different θ (Fig. S6) clearly shows +a small but relatively significant difference between the rigid +and relaxed structures of higher θ TBGs. Although we ob- +served a quick increase and gradual decrease in energies of re- +laxed and R-TBG respectively with increasing θ , the relaxed +(or reconstructed) system has the lower energy throughout. +Thus, even for large twist angles the reconstructed structure +formed as a consequence of local atomic changes is their en- +ergetically favorable configuration, which directly establishes +the presence of reconstruction. It is not surprising that such +minor changes in atomic registries for large twist angles are +challenging to capture in experiments given the length scale +limitations. But based on our results, reconstruction should +not be neglected for higher angles and motivate the study of +the implications of reconstruction for large θ TBGs. + +0.5 +RigidTBLG +RelaxedTBLG +Area Fraction +0.4 +0.3 +0.2 +0.1 +AA +AB +SP +Stackings +RigidTBLG +0.5 +RelaxedTBLG +0.4 +0.1 +AA +AB +SP +Stackings− +An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +7 + +FIG. 5: Moiré reconstruction in hetero-strained TBGs. Stacking contour plot of (a) rigid and (b) relaxed θ = 6° structure in +the presence of 0.5% uniaxial tension. (c) Change in local stacking area fractions before and after relaxation for the strained +structure. Percentage change in local AF of rigid and relaxed θ = 6° structures (∆AF) with respect to twist angle for (d) pristine +(unstrained), (e) 1% strained (uniaxial tension) and (f) -1% strained (uniaxial compression) TBGs. Positive and negative values +of ∆AF (%) respectively indicates increase and decrease in respective local AFs + +E. Mapping local and global physical property (phonon +behavior) to changes in local atomic registry. +Further validation on the presence of reconstruction at high +angles lies within an interrelation of local stacking domains +and global vibrational properties. To accomplish this, we +have studied their phonon behavior that can be directly trans- +lated to Raman scattering frequencies, which is an efficient +experimental technique for examining these systems, espe- +cially under strain49–52. We have examined the phonon dis- +persion spectra of TBGs and their local domains with ab-inito +simulations. Initially, we obtained the phonon spectra of un- +strained TBG systems using DFT (See Methods and Supple- +mentary). As compared to phonon spectrum of BLG, the +difference in phonon modes for TBG is quite small due to +weaker interlayer interaction (Fig. S7). Although we noticed +some differences in low-frequency acoustic phonons, the ef- +fect is substantially feeble for optical modes that correspond +to the experimentally observed Raman peaks53,54. Pertaining +to our goal of probing Raman spectra of TBGs, we analyzed +the high frequency optical (Longitudinal (LO) and Transverse +(TO)) branches of its phonon spectra55. We have indepen- +dently computed the phonon behavior of each sub-domain for +comparing them to the global optical vibrational behavior (see +Supplementary) as shown in Fig. 6(a). To analyze the minute +difference between phonon frequencies of all the structures, +we have plotted the optical phonon frequency difference (∆ω) +of each stacking with respect to the whole TBG structure, +∆ω = ωTBG ωstacking (Fig. 6(b) shows ∆ω for LO). We ob- +served that the phonon frequency magnitude of AA and AB +regions are smaller than TBG, whereas larger for SP region. +A similar trend is observed while comparing the TO phonon +modes (Fig. S8). The optical phonon behavior of AB stacking +is the closest to that of TBG which indicates that AB-stacked +domains predominantly control the overall phonon behavior +in TBGs. This is because unfolded phonon branches of TBG +exhibit an infinitesimal difference when compared to that of +Bernal stacked BLG49,54. The correlation of AF measure with +local and global phonon behavior is discussed in the following +sub-sections. + +1. Correlating local area fraction measure and phonon +behavior using Bond-Order-Length-Strength theory +To further establish a connection between the optical +phonons modes of TBG and phonon frequencies of its sub- +domains with individual stacking AF, we utilized the Bond +Order Length Strength (BOLS) theory56. BOLS can correlate +Raman peaks and their shifts in terms of constitutive struc- +tural parameters such as bond length and bond energy56–58. +It explains that the intrinsic association of bonds with their +physical properties can describe the extrinsic process of opti- +cal electron scattering captured by their phonon spectra. This +theory provides an independent method of calculating phonon + +0.5 +Rigid +0.4 +Relaxed +0.1 +AE +AA +AB +SP +100 +100 +100 +&XX +0% +&XX +-0.5% +50 +50 +50 +AAF (%) +AAF (%) +△AF (%) +0 +0 +-50 +.·AA +-50 +AA +-50 +-AA +.--AB +.--AB +.--AB +.-SP ++SP +-100 +1 +-1005 +-100 +0 +2 +4 +6 +8 +10 +12 +14 +0 +2 +4 +6 +8 +10 +12 +14 +0 +2 +4 +6 +8 +10 +12 +14 +Twist angle (00) +Twist angle (00) +Twist angle (00)dz +ε=0 +ε= 1% +2g +√ +− +An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +8 +frequencies of TBG based on the AFs of each sub-domain. +TABLE I: List of β (eV 1/2Å−1) pre-factor values +Therefore, the comparison of the results from BOLS theory + + +and ab-initio phonon frequencies of TBG can further validate +the accuracy of our sub-domain categorizations. The details of +BOLS formulation and the parameters involved are explained +in Supplementary section I. To obtain the vibrational prop- +erties of various structures using BOLS correlation, we can +deduce the phonon frequency shift based on bond length (dz), +bond energy (Ez), reduced mass (µ) and atomic coordination +number (z) using the following relation: +∆ω ∝ z + +Ez +(1) +analyzed both the βTBG values using a times improvement ba- +sis (mi). Using this we compared the weighted βTBG, first by +dz +µ +taking our calculated local reconstructed AF as the weights +and second by randomly assigning equal AF (33.33% weight +for three regions) to each individual stacking. We calculated +∆ω = k + z +Ez + +(2) +∆ω = ωstructure −ωbulk = k (β ) +(3) +where, k is the proportionality constant in Eq. 1 (µ is con- +stant because we have only carbon-based systems). ∆ω is the +difference of the optical phonon frequency of a system and a +reference material considered in bulk form (see Supplemen- +tary). Hence ∆ω = kβ , where β is the pre-factor containing +the variable parameters, such that β = Ez(z/dz). The mag- +nitude of this pre-factor directly relates to the optical phonon +frequency of a structure ωstructure, and thus can help in calcu- +lating its phonon behavior in terms of the associated physical +parameters (i.e., z, dz and Ez). Hence, we have utilized this +BOLS theory based pre-factor β to study the phonon behav- +ior of TBGs and their local domains, including their strained +configurations. +The calculated β magnitudes for global TBG structure +(βTBG) and its sub-domains are listed in Table I and values +of all the parameters such as, d, z and E are listed in Table +SI. Although the β magnitudes are numerically close, they +follow a trend as βSP > βTBG > βAB > βAA, on careful in- +spection. This trend also aligns with the observation made +while comparing the optical frequencies of these structures +(6(b)). Interestingly, this shows how effectively the BOLS +theory could endorse the characteristic trend in their phonon +behavior. Furthermore, we employed the local stacking AF +values of reconstructed structures in BOLS expression to in- +still an alternate estimation of phonon frequencies in an at- +tempt to authenticate our classification method, as explained +hereon. We analyzed the phonon behavior of global TBG +their error % with actual βTBG and obtained the relative er- +ror comparison or times improvement with respect to actual +βTBG values. The mi values in Table II show significant times +improvement on considering our estimated AF values of re- +constructed structures. The similitude between global βTBG +and weighted βTBG using local AFs signify that the physical +attributes of local regions in a TBG structure directly correlate +with the global vibrational comportment. Besides, this analy- +sis shows another evidence that our stacking classification is +an effective method for wide-ranging θ and strain magnitudes, +which is shown to detect reconstruction in these structures. + +2. Comparison of BOLS-estimated phonon frequencies with +experimental Raman to validate sub-domain area fraction +measure +To authenticate our reconstructed AF measures with DFT- +based phonon calculations and AFs driven BOLS theory, we +first calculated the phonon spectra of strained TBGs using +DFT simulations followed by calculating Raman frequencies +using BOLS (see Supplementary). Figure 6(c) shows the op- +tical phonon branches of TBG (θ = 6°) including tensile and +compressive uniaxial heterostrain. We have considered Ra- +man G band frequency in this study, which can be obtained +at Γ point in high symmetry Brillouin Zone path55,59. We ob- +served strain-induced phonon band splitting due to inequiv- +alent strain present in both the layers59–62 (supplementary +section IV). This phenomenon is observed in Raman spec- +troscopy as represented by the schematic of G-band Raman +peaks in hetero-strained TBGs (Fig. 6(d)). Due to weak inter- +layer vdW interaction in TBGs, their interlayer shear strength +is negligible which results in slippage between the layers. +Hence, the bottom layer remains mostly unstrained when +structure based on two approaches, the first being βTBG cal- +straining the top layer62,63. The Raman spectra of heteros- +trained TBG show significant individual peaks of unstrained +culated directly from BOLS expression. For the other ap- +proach, we have taken a weighted average of β values of in- +bottom layer (p1 +) and strained top layer (p2′ +± ). The +dividual stacking with their reconstructed AF as the weights, +i.e., βTBG = AFAAβAA + AFABβAB + AFSPβSP. On comparing +the actual and weighted βTBG, i.e., eactual = (βTBG(weighted) +βTBG(actual))/βTBG(actual), we observed that they align very +peak of strained layer redshifts or blueshifts depending on the +nature of strain. Also, for the case of graphene, an increase in +the magnitude of strain further splits the G-band peaks cor- +responding to the doubly degenerate E+ and E2 +− +g phonons +well with a small error %, including for strained systems (Ta- +ble II). However, given the seemingly small difference in β +2′′ +ε=±1% in Fig 6(d)-(f))8,64. +AF values of reconstructed systems +values of the structures, it may be argued that these small er- +rors are not much intriguing. Therefore, we have additionally +We then used the local +in BOLS expression to estimate Raman G-band frequencies +for comparison with experiments. and establish a connec- +(p +Stacking +θ = 1.08° +θ = 6° +θ = 13.2° +AA +3.084 +3.198 +3.418 +AB +3.126 +3.294 +3.450 +SP +3.180 +3.376 +3.491 +TBG(βBOLS) +3.135 +3.306 +3.474 +TBG(βweighted ) +3.141 +3.292 +3.466 + + +An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +9 + +TABLE II: Error table for BOLS-estimated β pre-factors based on actual and weighted βTBG, for systems with and without +strain. + + +Strain (%) + +eactual +θ = 1.08° + +mi + +eactual +θ = 6° + +mi + +eactual +θ = 13.2° + +mi +0 +0.38 +6 +0.27 + +5 +0.22 +5 +0.2% +- + +- +0.42 + +5 +0.29 +4 +0.5% +- + +- +0.60 + +5 +0.35 +4 +0.7% +- + +- +0.51 + +5 +0.49 +4 +1% +- + +- +0.69 +5 +0.45 +3 + +FIG. 6: Phonon behavior of TBGs with respect to its local domains. (a) Optical phonon modes of TBLG (θ = 6°) and its +individual counterparts. (b) Longitudinal optical (LO) phonon frequency difference with respect to TBG system, (c) Phonon +band splitting with heterostrain (tension and compression) (d) Schematic of a typical Raman G-peak splitting with inequivalent +strain employed in a bilayer system. Comparison of G-band frequencies for (e) θ = 6° with uniaxial compression and (f) θ = +13.2° with uniaxial tension. Solid lines in (e), (f) denote the Raman G-peak data obtained from DFT-based phonon calculations. +Heterostrain-assisted peak splitting of top and bottom layer (as shown in the schematic) is also denoted. Sub-figures(e)-(f) also +shows the close alignment of Bond Order Length Strength (BOLS)-estimated data using reconstructed AFs with +DFT-calculated and experimental data (reported by Peña et. al.65) as compared to that using rigid TBG AFs. + + +tion between global and local vibrational behavior. We first +extracted the G-band frequency (ωG) from DFT-simulated +phonon spectra for both unstrained and strained structures. +Figure 6(e) and 6(f) respectively shows the variation of ωG +for 6° and 13.2° with strain. To demonstrate both directions +of uniaxial strain, we showed the case of compression for 6° +and tension for 13.2°. In both cases, we observed that ωG at +zero strain is 1588 cm−1, which changes negligibly for the +unstrained bottom layer. In Fig. 6(e) due to compression, we +observed blueshift in ωG and redshift for tensile strain in Fig. +6(f) (see Supplementary section V). On comparing our results +for 6° and 13.2° systems with the experimental data reported +by Pena et. al.65 and Gao et. al.8 respectively, we found a +good agreement between them (magenta data points in Fig. + +6(e) and (f)). Finally, to achieve an experimental validation of +our stacking identification method as well as to highlight that +the global behavior such as Raman scattering is tied to local +structural configurations, we used our calculated AFs of re- +constructed TBGs in BOLS to predict the Raman G-band fre- +quencies of heterostrained systems (see Supplementary sec- +tion I for details). +We found a qualitative agreement between BOLS estimated +and DFT calculated ωG Raman peaks shown in Fig. 6(e) and +Fig. 6(f) (green dots). It must be noted that since BOLS ap- +proach encompasses mathematical interpolation for project- +ing the phonon frequencies, it can not resolve the further band +splitting of the strained top layer. We have also used the rigid +TBG AFs to check how it compares with the estimated G- + +20 +1650 +LO +SP stacking +1650 +(cm +ABstacking +(cm +10 +AAstacking +Phonon frequency +TO +Phonon frequency +1500 +1500 +AOLO +1350 +1350 +TBG(0=6°) +AB stacking +-10 +Bottom layer (g=0%) +AAstacking +Top layer (=+1%) +SP stacking +1200 +1200 +- Top layer (c=-1%) +K +M +-20 +K +M +K +M +BOLSo.(reconstructedAF) +1600 +Tension +Compression +BOLS . (rigidAF) +Bottomlayerp +1650 +Experimental o(Pena et. al.) +Top layer p +1575 +Top layer p +Intensity +(cm +Top layer p +1550 +p=0 +1600 +(Bottomlayer) +BOLSo.(reconstructedAF) +=-1% +Bottomlayerp +Experimental o. (Gao et. al.) +=1% +8-1% +p--1%p +1525 +(Top layer)(Top layer) +(Toplayer)(Toplayer) +1575 +Ramano +peak frequencies +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +Uniaxial compressive strain (%) +Uniaxial tensile strain (%)An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle +10 + +band frequencies. We observed a distinct misalignment of +BOLS-estimated Raman data using rigid AFs with that us- +ing reconstructed AFs and experimentally obtained data as +well. Hence, our analysis clearly demonstrates the difference +in vibrational behavior of reconstructed and rigid structures +and also shows that the reconstructed systems align closely +with the experimentally obtained measurements. This cer- +tainly implies that the physical behavior of TBGs such as +their vibrational properties is governed by their reconstructed +phases even for a large θ system and hence establishes an +additional validation on the presence of moiré reconstruc- +tion in their structures. Moreover, an agreement between the +AF utilized BOLS-estimated Raman data and DFT-calculated +phonon shows a theoretical approach to calculate Raman fre- +quencies at a comparatively lower computational cost. We +have calculated the G-band data for the heterostrained 1.08° +system using BOLS formulation (Fig. S9). As a whole uti- +lizing our stacking classification method and analyzing their +Raman signature using BOLS, we established a precise au- +thentication about reconstruction in high twist angles and also +demonstrated a connection of the global phonon shift of a +TBG system with changes in its local atomic registries. + +IV. Conclusion +Using atomistic simulations, we studied the characteristics +of locally stacked domains in TBG moiré patterns and demon- +strated a comprehensive approach to study atomic reconstruc- +tion phenomena in these structures, including the presence of +heterostrain. We proposed a way to classify TBGs into their +stacking types (AA, AB, and SP) and calculated their area +fractions to track structural evolution as a function of θ and +strain. Our classification scheme allowed us to exhibit the +existence of moiré reconstruction even for larger twist angle +(>2°) TBG systems, which is difficult to detect experimen- +tally. We showed how the moiré patterns of these large-angle +TBGs can be distorted by applying strain. Besides, the atomic +reconstruction in the presence of strain (in terms of area frac- +tion change of commensurate domain) can be manipulated by +an amount between 55% to 73% (for θ = 6°) with an applied +strain of only 0.5%, opening up a massive opportunity for +large angle TBGs to be used in strain engineering applica- +tions. +We studied the extent of reconstruction over a wide range +of θ and realized how it evolves in the presence of strain. +To further analyze this finding and validate the AF measure, +we utilized DFT-based phonon calculations and a theoretical +approach (BOLS theory) to deduce Raman frequencies and +compare them with experimental data. Using BOLS theory, +we discovered that global phonon behavior is directly related +to the physical features of local regions. Further, we real- +ized that the Raman data using reconstructed AFs in BOLS +aligns closely with DFT-calculated as well as experimental +data. Moreover, on comparing the Raman data with rigid AFs, +our results show a clear difference with that using the recon- +structed sub-domains and hence imply that the latter governs +the physical behavior in TBGs even for higher angles. Hence, +our study shows a self-consistent approach to characterize lo- +cal regions in TBGs and utilize them to examine as well as + +validate moiré reconstruction phenomena, based on physical +measures. Our findings on the presence of reconstruction in +large θ TBGs might open up an interesting research outlook +in twistronics. Moreover, our methodologies can be utilized to +identify stacking types and perform similar analyses in other +twisted vdW systems, especially in the presence of strain. + +Acknowledgments +We wish to acknowledge the support from the National +Science Foundation (OMA-1936250) and National Science +Foundation Graduate Research Fellowship Program (DGE- +1939268). + +Data Availability Statement +The data that support the findings of this study are available +from the corresponding author upon reasonable request. + +References + +1X.-J. Zhao, H. Hou, X.-T. Fan, Y. Wang, Y.-M. Liu, C. Tang, S.-H. Liu, P.- +P. Ding, J. Cheng, D.-H. Lin, et al., “Molecular bilayer graphene,” Nature +communications 10, 1–7 (2019). +2K. Lee, B. Fallahazad, J. Xue, D. C. Dillen, K. Kim, T. Taniguchi, +K. Watanabe, and E. Tutuc, “Chemical potential and quantum hall fer- +romagnetism in bilayer graphene,” Science 345, 58–61 (2014). +3T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. 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Nimbalkar and H. Kim, “Opportunities and challenges in twisted bilayer +graphene: a review,” Nano-Micro Letters 12, 1–20 (2020). + +1 + +An atomistic insight to moir´e reconstruction in Twisted Bilayer Graphene beyond +magic angle +Aditya Dey,1, a) Shoieb Ahmed Chowdhury,1, a) Tara Pen˜a,2 Sobhit Singh,1 Stephen M. +Wu,2, b) and Hesam Askari1 +1)Department of Mechanical Engineering, University of Rochester, +New York +2)Department of Electrical and Computer Engineering, University of Rochester, +Rochester, New York + +Supplementary information + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +a)These authors contributed equally to this work +b)Department of Physics and Astronomy, University of Rochester, Rochester, New York + +2 + +I. +COMPUTATIONAL AND THEORETICAL METHODS + +A. +DFT calculations + +The real space lattices of TBG systems were constructed using ATOMISTIX TOOLKIT +(QuantumATK) commercial package. All the first principles simulations were conducted +with generalized gradient approximation (GGA)1,2 assimilated in Quantum Espresso open +source package. The Perdew-Burke-Ernzerhof (PBE) form along with GGA has been used +as the exchange-correlation functional3. Ion-electron interactions for carbon atoms in TBGs +have been described by ultrasoft pseudopotentials. The vdW interaction has been incor- +porated as well using the semi-empirical Grimme functional4. Wavefunctions are expanded +using a plane wave basis set with an energy cutoff and charge density of 55 Ry and 450 Ry +respectively. We used 14 × 14 × 1 k-point grid within Monkhorst-Pack5,6 scheme to sample +the reciprocal space Brillouin zone. The structures were optimized until all the atomic forces +were less than 0.01 eV/˚A. The in-plane lattice constants were relaxed including the non- +periodic out-of plane lattice (25 ˚A space) to elude interactions in that direction. Phonon +dispersion spectra of all TBG structures were simulated using self-consistent density func- +tional perturbation theory (DFPT)7,8. The dynamical matrices were first computed on an +adequate q-point grid. The inter-atomic constants used in computing the phonon dispersion +were obtained from the Fourier interpolation of these dynamical matrices. + + +B. MS simulations + +Molecular statics simulations were done using LAMMPS open source software. The +unstrained, DFT-relaxed TBG moir´e lattice was transformed into an orthogonal cell with +approximate dimensions of 32 nm × 20 nm for all the TBG structures. The number of MPs +generated in each structure is dependent on the twist angle, for example θ = 6◦has 72 MPs, +and θ = 13.2◦has 288 MPs respectively. A vacuum space of 50 ˚A is inserted along the out- +plane-direction to avoid interactions with the periodic images. Hydrogen passivation was +done along the free surfaces to obtain the most stable structure. The TBG structures were +minimized using a conjugate gradient energy minimization method to have minimum energy +configurations. A reactive empirical bond order (REBO) potential was used for the intra- +lyer covalent bonds9 and for the interlayer van der Waals interaction a registry-dependent + +3 + +8z +C += +Kolmogorov-Crespi (KC) potential10 was selected. As TBG contains different local stacking +configurations, an interatomic potential that considers registry different than equilibrium +minimum energy stacking is needed11,12. Subsequently, we loaded the structure with con- +stant incremental strain to the top layer. We limit the magnitude of applied strain to 1% +for impeding our analysis within the contended boundaries of the experimental capability of +straining such systems13,14. Between each loading step, the atoms of the top layer were kept +stationary at the applied strain level and energy minimization was performed. The snap- +shots of the structure at different strain magnitudes were taken in Ovito open visualization +tool15. + +C. +BOLS formulation + +The BOLS notion explains the bond contraction and bond strengthening phenomena +using the following expressions16: +dz += Cz +b +2 += 1 + exp[ 12−z ] + +(1) + + +Eb +z +m +(2) +z +Here, the subscripts z and b respectively represent the coordination number (CN) of +a particular atomic structure and its bulk counterpart as a standard. The terms d and +E denote bond length and bond energy respectively. Cz represents the bond contraction +coefficient that varies with atomic structures having different z. The bond nature index +is denoted by m which is 2.56 for carbon bonds17. Since we are dealing with graphitic +structures in this study, we consider the bulk counterpart as diamond. Using the bond +length of the diamond (db = 1.54˚A) and bond lengths dz for each stacking configuration, we +can calculate Cz and z for each configuration using equations (1) and (2). Again using the +relation given in equation (2), we can calculate the bond energy for each individual stacking. +For diamond, the single C-C bond energy can be obtained from its total cohesive energy, +which is known to us, i.e., Eb = 0.614 eV17. Having known z, dz and Ez, we calculate the +β pre-factor values for each stacking using equation (2). The relation stated in equation 1 +in the main text can be derived by equating the vibrational energy of a harmonic system to +the first-order approximated Taylor series of its interatomic potential as16: +d +E + +4 + +TBG +G +ref +ref +G +ref +TBG +− ω +β ϵ +TBG +i +| | +i +dz +µ += +i +1µ(∆ω)2x2 ∼= 1 δu(r) x2 ∝ 1 Ez x2 +(3) +2 +2 δr2 +2 dz2 +⇒ ∆ω ∝ z / +Ez + + +The BOLS correlation is also used to estimate the phonon frequencies pertaining to +Raman G-band peaks. To achieve this, we perform some steps of mathematical interpolation +for equation (3). We can write the equation as ∆ωG = ωG +− ωref = k (β), where ωG is +the G band frequency of any reference material. Now we can calculate ωG +for each TBG +system with respect to their bulk counterpart (diamond) by comparing respective β pre- +factors as, +G +TBG − ωref += βTBG . After obtaining ωG , we can exercise ωG,ϵ=0 (ωG +ωdiamond − ωG +βdiamond +ref +TBG +TBG +at zero strain) and β pre-factors of strained and unstrained TBG systems to estimate their +ωG,ϵ=0 − ωG +βϵ=0 +G-band frequency in strained configuration (ωG,ϵ ), as +TBG +G,ϵ +TBG +ref +G +ref += T BG . Operating +TBG +this individually for top and bottom layers, we can obtain their G-peak frequencies for both +directions and various magnitudes of applied strain. The βϵ +values for the strained top +layer are listed in Table SIV. Since the bottom layer remains unstrained, we observe negligible +differences between their β pre-factor values for strained and unstrained configurations. + +II. +GEOMETRIC ANALYSIS OF STRAINED MSCS + +We deduce the expressions of their reciprocal lattice (⃗q ) vectors to quantify the structural +changes in strained MSCs18,19. The reciprocal lattice vectors of TBG moir´e lattices20 (⃗q) is +given as ⃗q = b⃗′ − ⃗b, where b⃗′ and ⃗b denote the reciprocal lattice vectors of the rotated top +layer and bottom layer in a TBG structure respectively. The length of moir´e pattern (MP), +4π +Lm can be derived using the magnitude of ⃗q vector as Lm = √3 ⃗q . When strain is applied +to the top layer, the mathematical expression of its reciprocal lattice vector19 (b⃗ε) can be +written as b⃗ε = (I⃗ + S⃗ ) +−1 +b⃗′ , where I⃗ is the identity matrix and S⃗ denotes the strain tensor +i +i +which can be written as the following for the case of uniaxial tension, +S⃗ = +ε +0 + +0 −νε +Here, ε is the nominal strain applied and ν denotes the Poisson’s ratio. So, the reciprocal +lattice vector of TBG with heterostrain can be expressed as ⃗ε +b⃗ε − b⃗i. As shown in +ω +ω +q + +5 + +Fig 1(e) in the main text, the boundaries of MPs resemble a hexagon and we can draw a +triangle (∆ABC) with A⃗B and B⃗C as the MP lattice vectors and α being the angle between +them (α = 60°, ϕ = 120°). The variation of α and ϕ with the applied strain is shown in +Fig. S2. With uniaxial tension, we see a monotonic decrease in these angles and vice-versa +for uniaxial compression. The changes in expressions of ⃗q vectors are associated with the +geometrical changes enforced upon hetero-straining these systems. + + +III. +EXPLANATION OF STACKING IDENTIFICATION METHOD (FOR +UNSTRAINED AND STRAINED SYSTEMS) +Firstly, we performed the identification of atoms that should be classified as ’AA’ type +using ILS. As observed in main text Fig. 1(c) and (d), the spacing between two layers of +TBG varies due to out-of-plane displacements of atoms. The ILS of equilibrium structures +follows this trend: AA > SP > AB. Hence, in a TBG system, the maximum ILS (dmax) +corresponds to AA region and the minimum distance (dmin) represents AB region. It is +observed that dmax and dmin vary with increasing twist angle up to 21°, after which we +noticed a plateaued regime21. This results due to the depletion of perfectly stacked AA and +AB configurations, as the length of the MPs, reduces with increasing θ. We obtained the +maximum and minimum magnitudes of dmax (3.589˚A and 3.475˚A) and dmin (3.456˚A and +3.338˚A). Using the lower bound of dmax for all the twist angles, i.e., 3.475˚A, we classified +the atoms with local ILS greater than 3.475˚A as ’AA’ stacking type. On the other hand, +considering the upper bound of dmin and identifying the regions with ILS below that value +as AB stacking can lead to the misclassification of AB and SP types. For the wide range of +twist angle considered in this study, the ILS alone cannot provide a margin of separation for +classifying AB and SP stacked atoms. To address this issue, we considered interlayer energy +or ILE (per atom) in the structure. Perusing the ILE contour plot, we observed that the +center of MPs has the highest energy followed by the SP segments. The AB (or BA) has +the lowest energy corresponding to the ground state configuration of BLG. But, being a per +atom quantity, the C atoms in AB stacking that are present directly on top of a C atom on +the other layer show the highest ILE value as shown in main text Fig. 2(c). +To obtain the same measure of energy for AB stacked atoms whether they are located +at the center of a lattice hexagon or at the corner, we calculated the difference of interlayer + +6 + +energy of each atom with its three bonded neighbors and consider their average. The +interlayer energy difference with neighboring atoms allows us to easily classify AB stacked +atoms as they have the highest fluctuation of energy with neighbors compared to AA or SP +stacked regions where the quantity is quite uniform. To obtain a classification threshold of +interlayer energy difference for AB stacking, we first calculated the soliton width of different +TBG systems, i.e., the width of SP regions similarly as explained by Gargiulo et al21. On +analyzing the path from the center of AB domain to the center of another AB (or BA) +region, we traverse across the SP segment. Calculating the ILS and plotting it along the +centers of triangular (AB) regions, we observed a small peak (Fig S3). This peak corresponds +to the SP region and its full width at half maxima (FWHM) gives us the soliton width21. +Considering this soliton width (varies with twist angle), we obtained the interlayer energy +difference value at the boundary of SP domains. This process is repeated for different twist +angles to establish a unique threshold that can be applied to any TBG system. The energy +difference threshold lies in a diminutive range, 8.22-8.31 meV for the angles considered (Fig +2(e) in main text). On averaging these magnitudes, we defined a ∆EILE threshold of 8.24 +meV/atom, above which an atom is classified as AB stacking type. The contour plot of TBG +(θ = 6◦) system in main Fig 2(f) shows the outcome of applying the method where each atom +has been classified as belonging to either AA or AB or SP stacked. We utilized the same +approach for classifying the local domains in strained systems. Since the ILS parameter +defines the out-of-plane distancing of pristine structures, it is not affected by an in-plane +applied strain. However, the interlayer energy of the structure is expected to change because +an externally applied strain disturbs the interlayer interactions. But since the mechanical +deformation is applied globally, the local regions will experience a similar change in ILE +with respect to their nearest neighbors and hence ∆EILE remains approximately unchanged +(see Table S1). + + +IV. +STACKING IDENTIFICATION OF RIGID STRUCTURES + +We followed the same approach used for reconstructed or relaxed systems to classify local +regions in rigid structures. The atomistic structure of rigid TBGs (R-TBGs) is different +from reconstructed systems. Since they are created by simply employing a rigid twist to a +Bernal stacked bilayer graphene, they do not have a variation of interlayer spacing, which is + +7 + +present in reconstructed TBGs pertaining to the formation of local stackings in the structure. +When a R-TBG is modeled from Bernal stacked (or AB) graphene, it has an ILS equal to +that of AB stacked graphene throughout its structure. Hence to account for this we defined +their uniform ILS, which is different from their initial geometry. We first considered their +relaxed structure and obtained an average ILS value considering all the interlayer distances +throughout the structure. Then, we re-modeled the rigid TBG structure by adjusting the +layers with respect to the average ILS value. Since different structures have varying fractions +of local interlayer regions, this average ILS changes for systems with certain twist angles. +It must be noted that we have not utilized this average ILS to define any threshold to +classify local atoms, rather it is used only to define the respective rigid structures. Further, +following the same method as relaxed systems we obtained their interlayer energy followed by +calculating the ILE difference (∆EILE) per atom. Now to classify the individual stackings, +we referred back to the ILS and ∆EILE thresholds obtained for relaxed systems. Having +known the ILS threshold for AA region (3.475 ˚A ) , we then identified the ∆EILE value at the +location corresponding to that ILS value by traversing along path PQ (Fig. 2(a) main text). +Then, we employed this value in ∆EILE calculation for R-TBG and specified atoms above +that threshold (6.88 meV/atom) as AA. For identifying AB type, we have considered the +∆EILE threshold (8.24 meV/atom) corresponding to its location on the path PQ. Similarly, +we then used that location to detect ∆EILE threshold for AB type in R-TBG structure +(5.92 meV/atom, so it lies between 5.92 and 6.88 meV/atom). After classifying AB and +AA, we have assigned the remaining atoms as SP. Further, we have used this same method +to identify the local stackings in rigid structures of strained configurations. To model rigid +systems of strained TBGs in a way that physically makes sense, we first considered the +relaxed or reconstructed structure of pristine TBG. Now the top layer is stretched such +that an unrelaxed hetero-strained TBG system is generated, which is referred to as the +rigid structure in the presence of strain. Relaxing this strained structure results in a fully +optimized system, pertaining to the reconstructed TBG configuration with strain. + +8 + +V. +PHONON DISPERSION SPECTRA OF TBG AND ITS LOCAL +DOMAINS +The simulations for phonon dispersion spectra were performed for θ = 6° and 13.2° +sys- tems. Due to the computational cost of DFT-based phonon simulations for large +MPs, we computed phonon spectra only for θ > 4.41° systems. We discussed an approach +using BOLS correlation to predict the Raman peaks pertaining to optical phonon modes +for larger TBG systems. As described by Cocemasov et al, TBGs contain hybrid folded +phonon branches that require to be unfolded onto the single layer first BZ22. Using the +PhononUnfolding package23, we simplified the phonon spectra of TBGs along Γ-K-M-Γ +high symmetry path (Fig S7 shows unfolded spectra of θ = 6°). To obtain the phonon +spectra of local sub- domains, we first identified the atomic positions of each local +stacking as defined by our identification method and extract the data from the main +structure. Then, we calculated the average bond length lavg of each configuration and +deduce their respective lattice con- +stant as a +stacking = √3l +avg . With the calculated unit cell parameters, we have computed +their phonon spectrum. + + +VI. +PHONON BAND SPLITTING WITH HETEROSTRAIN + +A combination of Molecular statics and first principles simulations has been used to +compute phonon dispersion spectra of TBGs with heterostrain. By freezing the obtained +configuration from LAMMPS, we have extracted the atomic data of strained periodic moir´e +lattice and further minimized the supercell in DFT to obtain first-principles-level fidelity, +followed by phonon spectra calculations. We observed strain-induced phonon band splitting +due to inequivalent strain present in both layers. With tension, the atomic bonds in a crystal +are stretched relative to their unstrained condition. When the bond length is increased, +and the force constant remains unchanged, as a result, the vibrational frequency decreases. +Conversely for compression, the bond length reduces which leads to an increase in vibrational +frequency. That is why we observe redshift and blueshift in phonon frequencies for tensile +and compressive strain respectively24. The redshift and blueshift of Raman G-band for θ = +1.08°, shown in Fig. S9 is a good demonstration of this phenomenon. + +9 + + + + +FIG. 1: (a) Relaxed atomistic structure and (b) interlayer spacing contour plot of θ= 6° +TBG system under 1% uniaxial compressive strain. + + + + +TABLE I: Average ∆EILE threshold value considering five representative TBG systems (θ += 1.1°, 3.48°, 4.41°, 6° and 7.34°) in the presence of strain. + +Strain (%) ∆EILE (meV/atom) +0 +8.24 ++0.5 +8.223 +-0.5 +8.21 ++1 +8.23 +-1 +8.207 + +3.6 +Spacing (A) +鞋 +鞋 +3.5 +cocal Interlayer +鞋 +3.45 +3.4 +.3510 + + + + +FIG. 2: Variation of angles α and ϕ with strain demonstrating the deformation of moir´e +patterns (for TBG system θ= 6°) + + + + +TABLE II: Evolution of area fractions f of local stacking domains with uniaxial tension +and compression applied to the top layer + + + + +Strain (%) + +θ = 1.1° + +θ = 6° + +θ = 13.2° +fAA +fAB +fSP +fAA +fAB +fSP +fAA +fAB +fSP +0 +0.135 +0.474 +0.391 +0.25 +0.39 +0.36 +0.272 +0.379 +0.349 ++0.2 +- +- +- +0.261 +0.376 +0.363 +0.293 +0.338 +0.369 +-0.2 +- +- +- +0.239 +0.407 +0.354 +0.257 +0.399 +0.344 ++0.5 +- +- +- +0.274 +0.356 +0.37 +0.309 +0.317 +0.374 +-0.5 +- +- +- +0.218 +0.432 +0.35 +0.239 +0.419 +0.342 ++0.7 +- +- +- +0.289 +0.339 +0.372 +0.322 +0.301 +0.377 +-0.7 +- +- +- +0.2 +0.455 +0.345 +0.218 +0.443 +0.339 ++1 +- +- +- +0.302 +0.319 +0.379 +0.330 +0.291 +0.379 +-1 +- +- +- +0.188 +0.471 +0.341 +0.2 +0.462 +0.338 + +EUniaxial tension +EUniaxialtension +64 +124 +G Uniaxial compression +G Uniaxial compression +62 +122 +(c)0 +(o) +Angle +60 +e +120 +58 +118 +56 +116 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +Strain(%) +Strain(%)11 + +θ = 1.1° +θ = 6° +θ = 13.2° + + + +FIG. 3: Normalized spatial interlayer spacing difference (∆d) profiles traversing between +centers of moir´e pattern, i.e., path PQ in Fig. 2(a) (for TBG system θ= 6°) + + + + +TABLE III: Parameters for calculating βBOLS pre-factors for TBGs and their respective +sub-domains. + + + +Parameters TBG +AA +AB +SP +TBG +AA +AB +SP +TBG +AA +AB +SP +dz (˚A ) +1.406 1.40 1.405 1.411 1.424 1.417 1.423 1.43 +1.438 1.431 1.437 1.441 +z +5.008 4.88 4.987 5.12 +5.43 5.185 5.409 5.612 5.851 5.67 5.792 5.911 +Cz +0.913 0.909 0.912 0.916 0.926 0.918 0.924 0.929 0.933 0.929 0.933 0.935 +Ez (eV) +0.775 0.783 0.776 0.768 0.752 0.764 0.751 0.741 +0.73 0.743 0.733 0.727 + +AA +AA +3.6 +3.55 +3.5 +3.45 +SP +3.4 +AB +AB +3.35 +P +Q12 + + + +FIG. 4: Variation of area fractions of individual stacking domain with respect to +heterostrain (tension) for different twist angles + +0.5 +0.5 +0 = 3.48° +AA +0 = 4.410 +AA +■SP +SP +AB +AB +0.4 +0.4 +0.3 +Area +0.2 +0.1 +0.1 +0% strain +0.5% strain +1%strain +0% strain +0.5% strain +1% strain +(a) +(b) +0.5 +0.5 +0 = 5.08° +AA +0 = 7.34° +AA +■SP +SP +AB +AB +0.4 +0.4 +uo +0.3 +0.1 +0.1 +0% strain +0.5% strain +1% strain +0%strain +0.5% strain +1%strain +(c) +(d) +0.5 +0.5 +0 = 9.349 +AA +0 = 13.10 +AA +-SP +■SP +AB +AB +0.4 +0.4 +ion +on +cti +0.3 +fra +0.1 +0.1 +0% strain +0.5%strain +1%strain +0%strain +0.5%strain +1%strain +(e) +(f)13 + + + + +FIG. 5: Variation of area fractions of individual stacking domain with respect to +heterostrain (compression) for θ = 3.48°, 6° and 13.2° + + + + + + +FIG. 6: Interlayer energy or vdW stacking energy for rigid and relaxed TBG systems. The +ILE of relaxed TBG system is always lower than rigid TBG even for larger twist angles. + +0.5 +0.5 +0.5 +0=3.48° +AA +0 = 60 +AA +0 = 13.29 +AA +■SP +SP +SP +0.4 +AB +0.4 +AB +0.4 +AB +fract +0.1 +0.1 +0% strain +-0.5% strain +-1% strain +0% strain +0.5% strain +-1% strain +0% strain +-0.5% strain +-1% strain-10 +ILE (meV/atom) +-15 +Interlayer energy, 1 +20 +25 +Rigid TBG +Relaxed TBG +-30 +0 +2 +4 +6 +8 +10 +12 +14 +Twist angle (0o)14 + + + + + +FIG. 7: Unfolded phonon spectra of TBG system θ = 6° along high symmetry points of its +Brillouin zone. Phonon dispersion spectra of Bernal stacked BLG is also shown for +comparison. + +1500 +(cm +1200 +Phonon frequency ( +900 +600 +300 +Bernal stackedpristine BLG +TBG (0 = 6) +K +M15 + + + + +FIG. 8: Transverse optical (TO) phonon frequency difference with respect to TBG system +θ = 6° + +20 +10 +-10 +AA stacking +AB stacking +SP stacking +-20 +K +M16 + + + +FIG. 9: BOLS predicted Raman G band frequencies of θ = 1.1° TBG system as a function +of applied heterostrain (tension and compression) + +1610 +Bottom layer (unstrained) + Top layer (tension) +1600 +Top layer (compression) +1590 +(cm +1580 +1570 +1560 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Strain(%)17 + +TBG +TABLE IV: Calculated βϵ +pre-factor values of strained top layer using BOLS +parameters with respect to strain + + +Strain (%) +θ = 1.1° +θ = 6° +θ = 13.2° +0 +3.135 +3.306 + +3.466 ++0.2 +3.207 +3.355 + +3.527 +-0.2 +3.078 +3.214 + +3.421 ++0.5 +3.311 +3.451 + +3.619 +-0.5 +2.988 +3.064 + +3.356 ++0.7 +3.398 +3.506 + +3.674 +-0.7 +2.732 +2.961 + +3.312 ++1 +3.475 +3.592 + +3.773 +-1 +2.602 +2.795 + +3.248 + + +18 + +REFERENCES + +1A. 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Dar, “An extensive investigation of structural, electronic, +thermoelectric and optical properties of bi-based half-huesler alloys by first principles cal- +culations,” Materials Today Communications 25, 101647 (2020). + diff --git a/QdAzT4oBgHgl3EQfW_xa/content/tmp_files/load_file.txt b/QdAzT4oBgHgl3EQfW_xa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b639c02af25aaab87692e2c0eefaaa91c89704c9 --- /dev/null +++ b/QdAzT4oBgHgl3EQfW_xa/content/tmp_files/load_file.txt @@ -0,0 +1,1671 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf,len=1670 +page_content='An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle Aditya Dey∗,1, a) Shoieb Ahmed Chowdhury,1, a) Tara Peña,2 Sobhit Singh,1 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Wu,2, b) and Hesam Askari1 1) Department of Mechanical Engineering, University of Rochester, New York 2) Department of Electrical and Computer Engineering, University of Rochester, Rochester, New York (*Electronic mail: adey2@ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='rochester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='edu) Twisted bilayer graphene exhibits electronic properties that are highly correlated with the size and arrangement of moiré patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' While rigid rotation of two layers creates the topology of moiré patterns, local rearrangements of the atoms due to interlayer van der Waals interactions result in atomic reconstruction within the moiré cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The ability to manipulate these patterns by controlling twist angle and/or externally applied strain provides a promising route to tune their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' While this phenomenon has been extensively studied for angles close to or smaller than the magic angle (θm=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1°), its extent for higher angles and how it evolves with strain is unknown and is believed to be mostly absent at high angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We use theoretical and numerical analyses to resolve reconstruction in angles above θm using interpretive and fundamental physical measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' In addition, we propose a method to identify local regions within moiré cells and track their evolution with strain for a range of representative high twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Our results show that reconstruction is actively present beyond the magic angle and its contribution to the evolution of the moiré cells is major.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Our theoretical method to correlate local and global phonon behavior provides further validation on the role of reconstruction at higher angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Our findings provide a better understanding of moiré reconstruction in large twist angles and the evolution of moiré cells in the presence of strain, that might be very crucial for twistronics-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Introduction Engineering two-dimensional (2D) materials by control- ling the stacking orientation of atomic layers have emerged as a powerful technique to manipulate their mechanical and opto-electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Bilayer graphene (BLG) is one of the simplest van der Waals (vdW) structures that display di- verse physical properties such as contrasting electronic struc- tures depending on the stacking arrangement1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Introduc- ing a relative rotation between the layers forms the Twisted Bilayer Graphene (TBG) in which the atoms create a peri- odic hexagonal superlattice called ‘moiré pattern’ (MP)5–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Emergence of this pattern is due to the atoms occupying dif- ferent relative interlayer positions compared to BLG with a global size that is inversely correlated with the twist angle (θ ) as Lm = a/(2 sin(θ /2)) where a is the lattice constant of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Application of other mechanical stimuli such as in- equivalent strain to the individual layers of TBG can further manipulate the shape of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Thus, the combination of hetero-straining process and twist provides a promising out- look for creating unique shapes and geometries of MPs for exciting opto-electronic applications8–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The atomic arrangements within MPs are influenced by the interlayer vdW forces between the atoms that consider- ably influence the atomic arrangement landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To manifest this influence, we can consider a hypothetical intermediate configuration where atoms are rigidly twisted in their plane and consequently, the well-defined BLG stacking configura- tions of AA, AB and SP types with their spatial variations will emerge11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Upon allowing atomic reconfiguration, an a)These authors contributed equally to this work b)Department of Physics and Astronomy, University of Rochester, Rochester, New York atomic-scale reconstruction occurs and local stacked regions evolve to their true minimum local energy configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This process is known as moiré reconstruction13,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Previous stud- ies have reported this phenomenon for low angle TBGs, es- pecially in the vicinity of or below the ’magic angle’ (θm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1°)15,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As the size of MP shrinks with an increase in θ and leaves less space for reconfiguration of atoms, experimental observation of moiré reconstruction becomes a challenge and is generally assumed to be absent for θ > 2°15,17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since the large angle TBGs contain the same atomic registry but only over a smaller region compared to the small twist angles, it is unreasonable to expect moiré reconstruction should suddenly become absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The interplay between the in-plane elastic en- ergy and interlayer vdW energy is still expected to contribute to reconstruction at higher angles due to the same fundamen- tal physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Nevertheless, its extent remains unknown due to the current limitations of experimental methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Recent experimental studies have demonstrated the abil- ity to control TBGs with and without strain and characterize moiré reconstruction for smaller θ systems8,14,19–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Imag- ing techniques such as STM and TEM become challenging when feature size becomes comparable to their resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As the size of MP decreases with increasing twist, imaging for θ > 2° systems become unfeasible11,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Therefore, the cur- rent understanding of reconstruction through experimental vi- sualization is limited to low angle twists and largely based on image analysis techniques rather than physical measurable quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Optical procedures such as Raman spectroscopy offer an expedient method to characterize TBGs irrespective of their size and twist angle25–28 but such methods predomi- nantly extract the collective behavior of TBGs spanning nu- merous MPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Therefore the global vibrational behavior ob- tained by Raman cannot be readily used to infer stacking and the extent of reconstruction without an interrelation of phonon − i An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1: Atomistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Relaxed atomistic structures illustrate how periodic moiré superlattice is formed and how its shape evolves with strain (a & b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Unlike BLG where a single interlayer distancing is expected, twist results in spatial variations of interlayer distancing with as shown for (c) unstrained and (d) strained TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Data presented for the twist angle of θ = 6° and uniaxial strain of 1% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (e) Real space geometric analysis demonstrating the distortion of MPs with applied uniaxial tension to the top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' behavior between local sub-domains and the bulk of TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Atomistic analyses offer an alternative tool to study atomic ar- rangements locally with a fine resolution and allow for track- ing of atomistic evolution with varying twist angle11,20,29–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Current works are heavily concentrated on or below the magic angle and do not explain the correlation of local and global be- havior of TBGs and moreover, have not studied the evolution of MPs with strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As a result, there remains an outstanding question about the viability and the role of reconstruction at higher angles and how local and global vibrational properties are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' In this work, we utilized a combination of first-principles and molecular statics atomistic simulations to examine the lo- cal domains in TBGs and how global vibrational behavior is tied to changes in local atomic registries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Based on physi- cal parameters that include interlayer spacing and interlayer energy, our method associates each atom to known stacking types of the constituent bi-layer graphene and calculates their resultant area fraction and traces the evolution of local sub- domains, and demonstrates evidence of moiré reconstruction for larger θ TBG systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This paper presents an effective set of criteria for the identification of local stacking and recon- struction phenomena in TBGs that are valid with or without the application of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' In addition, we demonstrate the cor- relation between local and global vibrational characteristics of TBGs and how it validates our results on reconstructed struc- tures, especially at higher angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The methods presented in this paper are devised for graphene but further adaptations are possible for other 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Methods A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='Atomistic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' All the TBG structures are constructed by rotating the top layer of Bernal stacked bilayer graphene with respect to its bottom layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The moiré lattice is created by identifying a common periodic lattice for the two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Using the TBG commensurability conditions, we have modeled their real and reciprocal space lattice parameters32,33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The ⃗q vector or re- ciprocal lattice parameter of TBG moirlattice is given as ⃗q = ⃗b′ ⃗b, where ⃗b and ⃗b′ denote the reciprocal lattice vectors of the bottom layer and rotated top layer respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' When heterostrain is applied, the strained ⃗q vector is expressed as q⃗ε = b⃗ε −⃗b, where b⃗ε denotes the strained top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The mathematical expressions of b⃗ε are deduced in Supplementary section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' All the atomistic models are relaxed using density functional theory (DFT) simulations, except for θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Because of a large moiré lattice for this structure (11164 atoms), DFT becomes forbiddingly inefficient and thus, we use force-field potentials for relaxing this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The real space lattices of TBG systems were constructed us- ing ATOMISTIX TOOLKIT (QuantumATK) package34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' All the first-principles simulations were conducted with gener- alized gradient approximation (GGA) assimilated in Quan- tum Espresso open source package35,36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The Perdew-Burke- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 鞋 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='45 鞋 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='35± ± An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2: Local stacking identification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (a) Path PQ along the center of one moiré pattern to the other (θ = 6°) (b) Illustration of interlayer energy (ILE) which is the energy contribution of vdW interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (c) ILE contour plot for unstrained θ = 6° system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (d) Variation of interlayer spacing (ILS) with respect to moiré twist angles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Horizontal dotted line (magenta) shows the minima of maximum ILS (dmax) obtained throughout a span of low and high angles TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (e) Variation of ILE difference for five representative θ (the dotted line shows the energy difference at soliton width boundary) (f) Contour plot demonstrating individual stacking type locally, obtained after implementing classification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Ernzerhof (PBE) form along with GGA has been used as the exchange-correlation functional37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Ion-electron interactions for carbon atoms in TBGs have been described by ultrasoft pseudopotentials38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' All technical details about DFT parame- ters are given in Supplementary information-Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='MS simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Molecular statics simulations were done using LAMMPS open source software39,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The unstrained, DFT relaxed TBG moiré lattice was transformed into an orthogonal cell for per- forming MS simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The simulation box is considered with free surface boundary conditions allowing us to account for the aperiodic crystal geometry (or moiré lattice mismatch) due to strain applied to one of the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The uniaxial strain was incremented by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1% up to the final strain magnitude of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The snapshots of the structure at different strain mag- nitudes were taken in Ovito open visualization tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Further computational details are mentioned in Supplementary section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Results and Discussions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Global structural analysis of pristine and strained TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We have studied a number of TBG systems between θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° to perform our analysis on MPs close to θm as well as outside the limit of small angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' For simplic- ity, most of the presented data include three representative TBG systems θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08°, 6° and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The MP geometries are modeled using the well-defined commensurability con- ditions of TBG systems and relaxed using first-principles or force field optimization techniques (see Methods) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since the local domains in TBG evolve through high symme- try BLG stacking, we can observe topographical variation in the structure41,42 represented by interlayer spacing (ILS) con- tour plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The centers of hexagonal MPs have re- gions of atoms where AA stacking exists13,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' These central regions are surrounded by two domains, AB and BA stack- ing, which are energetically degenerate but topologically in- equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since both of these stacking represent the Bernal graphene, they can be categorized as one44,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The boundaries of these AB/BA regions are separated by segments referred as strain solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The shear strain which generates due to two inequivalent stacking domains facing each other is con- fined within those segments with characteristic width referred to as the soliton width43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The atomic structure in the soliton regions corresponds to SP stacking which is an intermediate configuration between AB (or BA) and AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' A TBG system displays an out-of-plane corrugation in its structure caused by local ILS variation with AA regions having the highest spac- ing followed by SP and AB regions11,12,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On employing heterostrain, we observed a similar topo- graphical feature with distorted MPs due to the inequiva- lence of strain in each layer that resulted in an oblique moiré arrangement8 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1(b), (d) for tension and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S1 for com- (LE (meV/atom) 28 17 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='6 dmax (AA stacking) 12 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 dmin (AB stacking) 8 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='48° 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='410 d max 60 d min 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='34° 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3 4 8 12 16 20 24 28 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='8 Twist angle (00) Normalized distance| | | | | | ̸ | | An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 4 pression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' A geometric analysis is represented to explicate the angular change due to distortion and rigid rotation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1(e)) by deducing the expressions of their reciprocal lattice (⃗q) vectors (see Supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The change in ⃗q vector with uniaxial strain triggers the distortion in MPs27,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1(e), the boundaries of MPs resemble a hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On connecting the centers of adjacent MPs, we can draw a tri- angle (∆ABC) with A⃗B and B⃗C as the moiré lattice vectors and α being the angle between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' In unstrained condition, the magnitude of vectors A⃗B = B⃗C = Lm (Lm = Length of MP) and the angles are α = 60°, φ = 120°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As the ⃗q vector changes with uniaxial heterostrain, ∆ABC transforms to ∆A′B ′C′ such that A⃗′B′ = B⃗′C′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The deformed moiré lattice can be quantified with a change in α with the applied strain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The expressions of moiré reciprocal lattice vectors, show the geometrical changes enforced upon hetero-straining these systems (Supplementary section II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Classification method to identify local domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The deformation of MP with strain gives rise to changes in their local sub-domains and it is important to examine them for quantifying their contribution to global physical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Traversing along a diagonal of MP (path PQ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2(a)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', from the center of one moiré pattern to the center of its second nearest neighbor, we expect to cross all the locally stacked regions: AA, AB, SP, BA, and AA11,43,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since we aim to develop a criteria to classify each atom into one of these stacking, we first examined the atoms along the path PQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To perform the stacking identification, we initially used the ILS parameter d because the local domains in TBGs vary in in- terlayer distancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since pristine BLG stacking follows an increasing ILS trend from AB to SP and finally the AA re- gion, dmax (maximum ILS) and dmin (minimum ILS) in TBGs can be respectively understood as the ILS of AA and AB re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' By examining the range of ILS (dmax and dmin) over different possible twist angles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2(d)) we identify the min- imum value of dmax (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='475Å) and classify atoms above this ILS threshold as AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It should be noted that this does not mis- classify AB and SP because this threshold is quite above the ILS of pristine AB (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='33Å) and SP (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='38Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Due to the small ILS difference between AB and SP, the same ILS parameter cannot be used to identify the rest of the stackings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We introduced another parameter called ′interlayer energy′ (ILE) to distinguish between AB and SP according to their energy, rather than ILS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The ILE is a physical measure of vdW interaction between atoms in two different layers, as il- lustrated by the schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It is obtained by com- puting the vdW part of the total potential energy between C- atoms in different layers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since these local domains have indistinguishable and strong in-plane covalent bonds, their total potential energy is predominantly sourced from the in-plane interactions, which show little variance risen from their interlayer configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Moreover, with applied strain, the changes in total potential energy due to stretching and compressing of the in-plane bonds are orders of magnitude higher than their interlayer vdw counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This motivates the use of vdW interaction energy and its variations for iden- tification purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' However, being a per-atom quantity there are a lot of fluctuations in ILE magnitudes, most prominently observed in AB regions (2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Moreover, if the average ILE magnitude is used with respect to their bonded neighbors, it will result in an insignificant difference between AB and SP sub-domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence to account for this, we calculated the av- erage ILE difference (∆EILE ) of each atom with its bonded neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Although it can be difficult to separate AA and SP regions since they have minimal fluctuations in ILE, this parameter easily allows to classify AB stacked atoms as they have the highest variations in energy with neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Based on the ∆EILE analysis for five representative TBGs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2e), we have identified the ∆EILE threshold at the soliton bound- ary (SP width) and classified atoms above that threshold as AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The infinitesimal difference in these thresholds allowed us to define a θ -independent ∆EILE value for identifying the two stackings (see Supplementary for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It is important to note that the same approach can be used for classification in the presence of strain because the physical parameters used do not depend on strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Although the magnitude of inter- layer energy can be expected to vary, we observed a negligible change in ∆EILE threshold with strain (see Supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Thus using these criterion based on ILS and ILE, we could classify atoms into their local stacking as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2(f), which applies to TBGs with any twist angle and strain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On implementing the classification method, we obtained area fractions (AF) of each sub-domains present in a TBG structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Using this measure to monitor the evolution of lo- cal domains in the presence of strain, we observed that the sub-domains’ AF remain almost unchanged (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 3(b), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S4 for tension and Fig S5 for compression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It demonstrates a characteristic tendency of these local regions to retain their registry with an external strain applied globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The varia- tion of AF as a function of twist angle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 3(c)) shows that area fraction of AB (AFAB) and SP (AFSP) increases whereas that of AA (AFAA) decreases with decreasing θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This can be attributed to the potential energy of soliton (SP) regions contributing to in-plane forces, that displace atoms to max- imize the area of AB/BA (most stable BLG-stacking) local domains30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Such observations are well-interpreted in exper- iments, particularly for systems close to θm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence we compared our theoretically estimated AF for θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° (and additional θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='21°, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='37°) systems with experimen- tally interpreted area fractions from graphical analysis of STM images19, as marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The close similitude be- tween these sets of area fraction values provides a valida- tion of our stacking classification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We believe our approach interprets the physical behavior of sub-domains at atomic-level and with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Besides, as our method is based on physical parameters such as energy, it directly en- capsulates the underlying physics while in contrast, the previ- ously reported data rely on a graphical interpretation of gradi- ent in image intensity and contrast from experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence, our methodology is more accurate and able to resolve atom- istic insights even at a higher twist angle where the moiré cell size shrinks drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 3: Evolution of local regions with twist angle and strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (a) Contour plot demonstrating local stacking type for heterostrained θ = 6° system (1% tension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Area fractions of individual stacking domain with respect to (b) strain (tension) and (c) twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The red markings in (c) are extracted from reported work by Kazmierczak et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='19 to compare our results with data obtained by analyzing experimental measurements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Detecting moiré reconstruction in high twist angle TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We further utilized this method to study the extent of atomic reconstruction in TBG systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Moiré reconstruction can be studied by examining local regions in rigidly twisted (R-TBG) structure and comparing with their relaxed geometry13–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The rigidly twisted TBG refers to its unrelaxed geometry, con- sidered in a conceptual intermediate configuration, in which the layers of BLG are twisted by a certain angle but the atoms are not allowed to reconfigure to form their true equi- librium structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' During reconstruction, local sites in the structure prefer to diverge from energetically unfavorable AA stacking by atomic displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This is achieved by rear- rangement of the atoms to minimize vdW energy and obtain- ing the nearly commensurate Bernal-stacked (AB/BA) BLG structure partitioned by the SP segments after reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The emergence of soliton (SP) domains is one of the pre- dominant features of reconstruction phenomena in 2D mate- rials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Previous studies have attributed the minor atomic dis- placements of large θ relaxed TBGs to insignificant change in atomic registry of local domains indicating the absence of reconstruction15–17,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' However, examining TBG systems with an atomistic insight and employing our sub-domain iden- tification method, we show considerable changes in the local registries for larger θ TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We utilized the area fraction measure to capture the structural changes in local domains of relaxed and unrelaxed geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The stacking identification assessment of R-TBG is conducted similarly to the relaxed TBG (see Supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' For θ = 6° structure (4(a)-(c)), the AA regions shrink upon relaxation and conversely, the AB/BA regions expand to approximate triangular domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Undoubt- edly, this structural change was expected and prominently ob- served for θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° system (4(d)-(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' But we encountered a similar observation for a large θ structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence, contrary to the general idea that reconstruction diminishes at higher angles, we show clear evidence demonstrating moiré recon- struction in higher θ (>2°) TBG systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This observation indicates that irrespective of how small the atomic displace- ments are, the change in AF of local domains for higher θ TBGs show pronounced variation in atomic registries upon relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 AA SP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 AB fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0%strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5%strain 1%strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 Area fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 -→AB -CAA --SP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 AB(Kazmierczaket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=') AA(Kazmierczaket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=') SP(Kazmierczaket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=') 0 0 2 4 6 8 10 12 Twist Angle (0)AFrigid An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 4: Demonstration of moiré reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Stacking contour plot for rigid (a) θ = 6°, (d) θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° and relaxed (b) θ = 6°, (e) θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° TBG systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (c), (f) Comparison of area fractions for each stacking , showing the change in local atomic registries before and after relaxation that signifies the extent of reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Analyzing extent of reconstruction in strained and unstrained TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Using this approach, we have also studied the extent of moiré reconstruction in high angle TBGs in the presence of heterostrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Lattice deformation due to heterostrain in- duces distortion in MPs, which is minimized by sustaining the formed domain-wall-like boundary lines (SP regions) due to superlattice reconstruction15,23,48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Similar to the unstrained case, we have compared the local AF of rigid and relaxed systems under heterostrain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The rigid system for strained TBGs refers to its unrelaxed structure obtained af- ter employing strain to the relaxed geometry of pristine TBG structure (see Supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We observed that our assess- ment could capture the variations in local atomic registry of strained TBGs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 5(a)-(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The substantial change in AF of AA and AB regions and perpetual of SP domains, signifies the tendency of preserving the SP boundaries with change in local atomic registry of AA and AB domains, thus indicating the presence of atomic reconstruction in large θ strained TBG systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To assess the extent of change in local registries, we have calculated the percentage change in local AF upon re- laxing the structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', ∆AF(%) = ( AFrelaxed−AFrigid ) × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On examining the variation of ∆AF over unstrained (Fig 5(c)) and strained (tensile Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 5(e) and compressive Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 5(f)) TBGs spanning a wide range of twist angles, it is observed that ∆AF for all local stackings monotonically decreases with increasing θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Although this implies that, as expected, the ef- fect of reconstruction reduces with increasing twist angle, AFs data shows that it can not be disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It is noticed that for both pristine and strained cases, the AB stacked domains show ample variation in rigid and relaxed configurations even for higher angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This variation rapidly decreases for AA and SP regions, especially at very high twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Nonetheless, this analysis reveals the existence of local atomic reconstruc- tion for both unstrained and strained large θ TBG systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It has been previously argued that for a large twist angle, the gaining vdW energy cannot compensate for the losing in- tralayer elastic energy15,17,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This results in no change of vdW stacking energy between rigid and relaxed structures, ul- timately indicating the absence of reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' However, our analysis of ILE over different θ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S6) clearly shows a small but relatively significant difference between the rigid and relaxed structures of higher θ TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Although we ob- served a quick increase and gradual decrease in energies of re- laxed and R-TBG respectively with increasing θ , the relaxed (or reconstructed) system has the lower energy throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Thus, even for large twist angles the reconstructed structure formed as a consequence of local atomic changes is their en- ergetically favorable configuration, which directly establishes the presence of reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It is not surprising that such minor changes in atomic registries for large twist angles are challenging to capture in experiments given the length scale limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' But based on our results, reconstruction should not be neglected for higher angles and motivate the study of the implications of reconstruction for large θ TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 RigidTBLG RelaxedTBLG Area Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 AA AB SP Stackings RigidTBLG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 RelaxedTBLG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 AA AB SP Stackings− An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 5: Moiré reconstruction in hetero-strained TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Stacking contour plot of (a) rigid and (b) relaxed θ = 6° structure in the presence of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% uniaxial tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (c) Change in local stacking area fractions before and after relaxation for the strained structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Percentage change in local AF of rigid and relaxed θ = 6° structures (∆AF) with respect to twist angle for (d) pristine (unstrained), (e) 1% strained (uniaxial tension) and (f) -1% strained (uniaxial compression) TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Positive and negative values of ∆AF (%) respectively indicates increase and decrease in respective local AFs E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Mapping local and global physical property (phonon behavior) to changes in local atomic registry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Further validation on the presence of reconstruction at high angles lies within an interrelation of local stacking domains and global vibrational properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To accomplish this, we have studied their phonon behavior that can be directly trans- lated to Raman scattering frequencies, which is an efficient experimental technique for examining these systems, espe- cially under strain49–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We have examined the phonon dis- persion spectra of TBGs and their local domains with ab-inito simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Initially, we obtained the phonon spectra of un- strained TBG systems using DFT (See Methods and Supple- mentary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As compared to phonon spectrum of BLG, the difference in phonon modes for TBG is quite small due to weaker interlayer interaction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Although we noticed some differences in low-frequency acoustic phonons, the ef- fect is substantially feeble for optical modes that correspond to the experimentally observed Raman peaks53,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Pertaining to our goal of probing Raman spectra of TBGs, we analyzed the high frequency optical (Longitudinal (LO) and Transverse (TO)) branches of its phonon spectra55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We have indepen- dently computed the phonon behavior of each sub-domain for comparing them to the global optical vibrational behavior (see Supplementary) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To analyze the minute difference between phonon frequencies of all the structures, we have plotted the optical phonon frequency difference (∆ω) of each stacking with respect to the whole TBG structure, ∆ω = ωTBG ωstacking (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6(b) shows ∆ω for LO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We ob- served that the phonon frequency magnitude of AA and AB regions are smaller than TBG, whereas larger for SP region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' A similar trend is observed while comparing the TO phonon modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The optical phonon behavior of AB stacking is the closest to that of TBG which indicates that AB-stacked domains predominantly control the overall phonon behavior in TBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This is because unfolded phonon branches of TBG exhibit an infinitesimal difference when compared to that of Bernal stacked BLG49,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The correlation of AF measure with local and global phonon behavior is discussed in the following sub-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Correlating local area fraction measure and phonon behavior using Bond-Order-Length-Strength theory To further establish a connection between the optical phonons modes of TBG and phonon frequencies of its sub- domains with individual stacking AF, we utilized the Bond Order Length Strength (BOLS) theory56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' BOLS can correlate Raman peaks and their shifts in terms of constitutive struc- tural parameters such as bond length and bond energy56–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It explains that the intrinsic association of bonds with their physical properties can describe the extrinsic process of opti- cal electron scattering captured by their phonon spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This theory provides an independent method of calculating phonon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 Rigid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 Relaxed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 AE AA AB SP 100 100 100 &XX 0% &XX -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% 50 50 50 AAF (%) AAF (%) △AF (%) 0 0 -50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='·AA -50 AA -50 -AA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='--AB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='--AB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='--AB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='-SP +SP -100 1 -1005 -100 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Twist angle (00) Twist angle (00) Twist angle (00)dz ε=0 ε= 1% 2g √ − An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 8 frequencies of TBG based on the AFs of each sub-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' TABLE I: List of β (eV 1/2Å−1) pre-factor values Therefore, the comparison of the results from BOLS theory and ab-initio phonon frequencies of TBG can further validate the accuracy of our sub-domain categorizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The details of BOLS formulation and the parameters involved are explained in Supplementary section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To obtain the vibrational prop- erties of various structures using BOLS correlation, we can deduce the phonon frequency shift based on bond length (dz), bond energy (Ez), reduced mass (µ) and atomic coordination number (z) using the following relation: ∆ω ∝ z Ez (1) analyzed both the βTBG values using a times improvement ba- sis (mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Using this we compared the weighted βTBG, first by dz µ taking our calculated local reconstructed AF as the weights and second by randomly assigning equal AF (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='33% weight for three regions) to each individual stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We calculated ∆ω = k z Ez (2) ∆ω = ωstructure −ωbulk = k (β ) (3) where, k is the proportionality constant in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1 (µ is con- stant because we have only carbon-based systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' ∆ω is the difference of the optical phonon frequency of a system and a reference material considered in bulk form (see Supplemen- tary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence ∆ω = kβ , where β is the pre-factor containing the variable parameters, such that β = Ez(z/dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The mag- nitude of this pre-factor directly relates to the optical phonon frequency of a structure ωstructure, and thus can help in calcu- lating its phonon behavior in terms of the associated physical parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', z, dz and Ez).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence, we have utilized this BOLS theory based pre-factor β to study the phonon behav- ior of TBGs and their local domains, including their strained configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The calculated β magnitudes for global TBG structure (βTBG) and its sub-domains are listed in Table I and values of all the parameters such as, d, z and E are listed in Table SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Although the β magnitudes are numerically close, they follow a trend as βSP > βTBG > βAB > βAA, on careful in- spection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This trend also aligns with the observation made while comparing the optical frequencies of these structures (6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Interestingly, this shows how effectively the BOLS theory could endorse the characteristic trend in their phonon behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Furthermore, we employed the local stacking AF values of reconstructed structures in BOLS expression to in- still an alternate estimation of phonon frequencies in an at- tempt to authenticate our classification method, as explained hereon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We analyzed the phonon behavior of global TBG their error % with actual βTBG and obtained the relative er- ror comparison or times improvement with respect to actual βTBG values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The mi values in Table II show significant times improvement on considering our estimated AF values of re- constructed structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The similitude between global βTBG and weighted βTBG using local AFs signify that the physical attributes of local regions in a TBG structure directly correlate with the global vibrational comportment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Besides, this analy- sis shows another evidence that our stacking classification is an effective method for wide-ranging θ and strain magnitudes, which is shown to detect reconstruction in these structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Comparison of BOLS-estimated phonon frequencies with experimental Raman to validate sub-domain area fraction measure To authenticate our reconstructed AF measures with DFT- based phonon calculations and AFs driven BOLS theory, we first calculated the phonon spectra of strained TBGs using DFT simulations followed by calculating Raman frequencies using BOLS (see Supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Figure 6(c) shows the op- tical phonon branches of TBG (θ = 6°) including tensile and compressive uniaxial heterostrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We have considered Ra- man G band frequency in this study, which can be obtained at Γ point in high symmetry Brillouin Zone path55,59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We ob- served strain-induced phonon band splitting due to inequiv- alent strain present in both the layers59–62 (supplementary section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This phenomenon is observed in Raman spec- troscopy as represented by the schematic of G-band Raman peaks in hetero-strained TBGs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Due to weak inter- layer vdW interaction in TBGs, their interlayer shear strength is negligible which results in slippage between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence, the bottom layer remains mostly unstrained when structure based on two approaches, the first being βTBG cal- straining the top layer62,63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The Raman spectra of heteros- trained TBG show significant individual peaks of unstrained culated directly from BOLS expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' For the other ap- proach, we have taken a weighted average of β values of in- bottom layer (p1 ) and strained top layer (p2′ ± ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The dividual stacking with their reconstructed AF as the weights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', βTBG = AFAAβAA + AFABβAB + AFSPβSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On comparing the actual and weighted βTBG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', eactual = (βTBG(weighted) βTBG(actual))/βTBG(actual), we observed that they align very peak of strained layer redshifts or blueshifts depending on the nature of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Also, for the case of graphene, an increase in the magnitude of strain further splits the G-band peaks cor- responding to the doubly degenerate E+ and E2 − g phonons well with a small error %, including for strained systems (Ta- ble II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' However, given the seemingly small difference in β 2′′ ε=±1% in Fig 6(d)-(f))8,64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' AF values of reconstructed systems values of the structures, it may be argued that these small er- rors are not much intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Therefore, we have additionally We then used the local in BOLS expression to estimate Raman G-band frequencies for comparison with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' and establish a connec- (p Stacking θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° θ = 6° θ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° AA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='084 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='198 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='418 AB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='126 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='294 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='450 SP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='180 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='376 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='491 TBG(βBOLS) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='135 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='306 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='474 TBG(βweighted ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='141 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='292 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='466 An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 9 TABLE II: Error table for BOLS-estimated β pre-factors based on actual and weighted βTBG, for systems with and without strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Strain (%) eactual θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° mi eactual θ = 6° mi eactual θ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° mi 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='38 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='27 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='22 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='42 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='29 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='60 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='35 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='51 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='49 4 1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='69 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='45 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6: Phonon behavior of TBGs with respect to its local domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (a) Optical phonon modes of TBLG (θ = 6°) and its individual counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (b) Longitudinal optical (LO) phonon frequency difference with respect to TBG system, (c) Phonon band splitting with heterostrain (tension and compression) (d) Schematic of a typical Raman G-peak splitting with inequivalent strain employed in a bilayer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Comparison of G-band frequencies for (e) θ = 6° with uniaxial compression and (f) θ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° with uniaxial tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Solid lines in (e), (f) denote the Raman G-peak data obtained from DFT-based phonon calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Heterostrain-assisted peak splitting of top and bottom layer (as shown in the schematic) is also denoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Sub-figures(e)-(f) also shows the close alignment of Bond Order Length Strength (BOLS)-estimated data using reconstructed AFs with DFT-calculated and experimental data (reported by Peña et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='65) as compared to that using rigid TBG AFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' tion between global and local vibrational behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We first extracted the G-band frequency (ωG) from DFT-simulated phonon spectra for both unstrained and strained structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Figure 6(e) and 6(f) respectively shows the variation of ωG for 6° and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° with strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To demonstrate both directions of uniaxial strain, we showed the case of compression for 6° and tension for 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' In both cases, we observed that ωG at zero strain is 1588 cm−1, which changes negligibly for the unstrained bottom layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6(e) due to compression, we observed blueshift in ωG and redshift for tensile strain in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6(f) (see Supplementary section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On comparing our results for 6° and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° systems with the experimental data reported by Pena et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='65 and Gao et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='8 respectively, we found a good agreement between them (magenta data points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6(e) and (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Finally, to achieve an experimental validation of our stacking identification method as well as to highlight that the global behavior such as Raman scattering is tied to local structural configurations, we used our calculated AFs of re- constructed TBGs in BOLS to predict the Raman G-band fre- quencies of heterostrained systems (see Supplementary sec- tion I for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We found a qualitative agreement between BOLS estimated and DFT calculated ωG Raman peaks shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6(e) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6(f) (green dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It must be noted that since BOLS ap- proach encompasses mathematical interpolation for project- ing the phonon frequencies, it can not resolve the further band splitting of the strained top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We have also used the rigid TBG AFs to check how it compares with the estimated G- 20 1650 LO SP stacking 1650 (cm ABstacking (cm 10 AAstacking Phonon frequency TO Phonon frequency 1500 1500 AOLO 1350 1350 TBG(0=6°) AB stacking -10 Bottom layer (g=0%) AAstacking Top layer (=+1%) SP stacking 1200 1200 - Top layer (c=-1%) K M -20 K M K M BOLSo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (reconstructedAF) 1600 Tension Compression BOLS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (rigidAF) Bottomlayerp 1650 Experimental o(Pena et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=') Top layer p 1575 Top layer p Intensity (cm Top layer p 1550 p=0 1600 (Bottomlayer) BOLSo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (reconstructedAF) =-1% Bottomlayerp Experimental o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' (Gao et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=') =1% 8-1% p--1%p 1525 (Top layer)(Top layer) (Toplayer)(Toplayer) 1575 Ramano peak frequencies 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='8 Uniaxial compressive strain (%) Uniaxial tensile strain (%)An atomistic insight to moiré reconstruction in Twisted Bilayer Graphene beyond magic angle 10 band frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We observed a distinct misalignment of BOLS-estimated Raman data using rigid AFs with that us- ing reconstructed AFs and experimentally obtained data as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence, our analysis clearly demonstrates the difference in vibrational behavior of reconstructed and rigid structures and also shows that the reconstructed systems align closely with the experimentally obtained measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This cer- tainly implies that the physical behavior of TBGs such as their vibrational properties is governed by their reconstructed phases even for a large θ system and hence establishes an additional validation on the presence of moiré reconstruc- tion in their structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Moreover, an agreement between the AF utilized BOLS-estimated Raman data and DFT-calculated phonon shows a theoretical approach to calculate Raman fre- quencies at a comparatively lower computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We have calculated the G-band data for the heterostrained 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° system using BOLS formulation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As a whole uti- lizing our stacking classification method and analyzing their Raman signature using BOLS, we established a precise au- thentication about reconstruction in high twist angles and also demonstrated a connection of the global phonon shift of a TBG system with changes in its local atomic registries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Conclusion Using atomistic simulations, we studied the characteristics of locally stacked domains in TBG moiré patterns and demon- strated a comprehensive approach to study atomic reconstruc- tion phenomena in these structures, including the presence of heterostrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We proposed a way to classify TBGs into their stacking types (AA, AB, and SP) and calculated their area fractions to track structural evolution as a function of θ and strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Our classification scheme allowed us to exhibit the existence of moiré reconstruction even for larger twist angle (>2°) TBG systems, which is difficult to detect experimen- tally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We showed how the moiré patterns of these large-angle TBGs can be distorted by applying strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Besides, the atomic reconstruction in the presence of strain (in terms of area frac- tion change of commensurate domain) can be manipulated by an amount between 55% to 73% (for θ = 6°) with an applied strain of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5%, opening up a massive opportunity for large angle TBGs to be used in strain engineering applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We studied the extent of reconstruction over a wide range of θ and realized how it evolves in the presence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To further analyze this finding and validate the AF measure, we utilized DFT-based phonon calculations and a theoretical approach (BOLS theory) to deduce Raman frequencies and compare them with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Using BOLS theory, we discovered that global phonon behavior is directly related to the physical features of local regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Further, we real- ized that the Raman data using reconstructed AFs in BOLS aligns closely with DFT-calculated as well as experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Moreover, on comparing the Raman data with rigid AFs, our results show a clear difference with that using the recon- structed sub-domains and hence imply that the latter governs the physical behavior in TBGs even for higher angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence, our study shows a self-consistent approach to characterize lo- cal regions in TBGs and utilize them to examine as well as validate moiré reconstruction phenomena, based on physical measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Our findings on the presence of reconstruction in large θ TBGs might open up an interesting research outlook in twistronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Moreover, our methodologies can be utilized to identify stacking types and perform similar analyses in other twisted vdW systems, especially in the presence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Acknowledgments We wish to acknowledge the support from the National Science Foundation (OMA-1936250) and National Science Foundation Graduate Research Fellowship Program (DGE- 1939268).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Data Availability Statement The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' References 1X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Zhao, H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Fang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Taniguchi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Kaxiras, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Jarillo-Herrero, “Unconventional superconductivity in magic-angle graphene superlattices,” Nature 556, 43–50 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 67A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Nimbalkar and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Kim, “Opportunities and challenges in twisted bilayer graphene: a review,” Nano-Micro Letters 12, 1–20 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1 An atomistic insight to moir´e reconstruction in Twisted Bilayer Graphene beyond magic angle Aditya Dey,1, a) Shoieb Ahmed Chowdhury,1, a) Tara Pen˜a,2 Sobhit Singh,1 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Wu,2, b) and Hesam Askari1 1)Department of Mechanical Engineering, University of Rochester, New York 2)Department of Electrical and Computer Engineering, University of Rochester, Rochester, New York Supplementary information a)These authors contributed equally to this work b)Department of Physics and Astronomy, University of Rochester, Rochester, New York 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' COMPUTATIONAL AND THEORETICAL METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' DFT calculations The real space lattices of TBG systems were constructed using ATOMISTIX TOOLKIT (QuantumATK) commercial package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' All the first principles simulations were conducted with generalized gradient approximation (GGA)1,2 assimilated in Quantum Espresso open source package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The Perdew-Burke-Ernzerhof (PBE) form along with GGA has been used as the exchange-correlation functional3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Ion-electron interactions for carbon atoms in TBGs have been described by ultrasoft pseudopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The vdW interaction has been incor- porated as well using the semi-empirical Grimme functional4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Wavefunctions are expanded using a plane wave basis set with an energy cutoff and charge density of 55 Ry and 450 Ry respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We used 14 × 14 × 1 k-point grid within Monkhorst-Pack5,6 scheme to sample the reciprocal space Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The structures were optimized until all the atomic forces were less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='01 eV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The in-plane lattice constants were relaxed including the non- periodic out-of plane lattice (25 ˚A space) to elude interactions in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Phonon dispersion spectra of all TBG structures were simulated using self-consistent density func- tional perturbation theory (DFPT)7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The dynamical matrices were first computed on an adequate q-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The inter-atomic constants used in computing the phonon dispersion were obtained from the Fourier interpolation of these dynamical matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' MS simulations Molecular statics simulations were done using LAMMPS open source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The unstrained, DFT-relaxed TBG moir´e lattice was transformed into an orthogonal cell with approximate dimensions of 32 nm × 20 nm for all the TBG structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The number of MPs generated in each structure is dependent on the twist angle, for example θ = 6◦has 72 MPs, and θ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2◦has 288 MPs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' A vacuum space of 50 ˚A is inserted along the out- plane-direction to avoid interactions with the periodic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hydrogen passivation was done along the free surfaces to obtain the most stable structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The TBG structures were minimized using a conjugate gradient energy minimization method to have minimum energy configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' A reactive empirical bond order (REBO) potential was used for the intra- lyer covalent bonds9 and for the interlayer van der Waals interaction a registry-dependent 3 8z C = Kolmogorov-Crespi (KC) potential10 was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As TBG contains different local stacking configurations, an interatomic potential that considers registry different than equilibrium minimum energy stacking is needed11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Subsequently, we loaded the structure with con- stant incremental strain to the top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We limit the magnitude of applied strain to 1% for impeding our analysis within the contended boundaries of the experimental capability of straining such systems13,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Between each loading step, the atoms of the top layer were kept stationary at the applied strain level and energy minimization was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The snap- shots of the structure at different strain magnitudes were taken in Ovito open visualization tool15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' BOLS formulation The BOLS notion explains the bond contraction and bond strengthening phenomena using the following expressions16: dz = Cz b 2 = 1 + exp[ 12−z ] (1) Eb z m (2) z Here, the subscripts z and b respectively represent the coordination number (CN) of a particular atomic structure and its bulk counterpart as a standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The terms d and E denote bond length and bond energy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Cz represents the bond contraction coefficient that varies with atomic structures having different z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The bond nature index is denoted by m which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='56 for carbon bonds17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since we are dealing with graphitic structures in this study, we consider the bulk counterpart as diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Using the bond length of the diamond (db = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='54˚A) and bond lengths dz for each stacking configuration, we can calculate Cz and z for each configuration using equations (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Again using the relation given in equation (2), we can calculate the bond energy for each individual stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' For diamond, the single C-C bond energy can be obtained from its total cohesive energy, which is known to us, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', Eb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='614 eV17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Having known z, dz and Ez, we calculate the β pre-factor values for each stacking using equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The relation stated in equation 1 in the main text can be derived by equating the vibrational energy of a harmonic system to the first-order approximated Taylor series of its interatomic potential as16: d E 4 TBG G ref ref G ref TBG − ω β ϵ TBG i | | i dz µ = i 1µ(∆ω)2x2 ∼= 1 δu(r) x2 ∝ 1 Ez x2 (3) 2 2 δr2 2 dz2 ⇒ ∆ω ∝ z / Ez The BOLS correlation is also used to estimate the phonon frequencies pertaining to Raman G-band peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To achieve this, we perform some steps of mathematical interpolation for equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We can write the equation as ∆ωG = ωG − ωref = k (β), where ωG is the G band frequency of any reference material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Now we can calculate ωG for each TBG system with respect to their bulk counterpart (diamond) by comparing respective β pre- factors as, G TBG − ωref = βTBG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' After obtaining ωG , we can exercise ωG,ϵ=0 (ωG ωdiamond − ωG βdiamond ref TBG TBG at zero strain) and β pre-factors of strained and unstrained TBG systems to estimate their ωG,ϵ=0 − ωG βϵ=0 G-band frequency in strained configuration (ωG,ϵ ), as TBG G,ϵ TBG ref G ref = T BG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Operating TBG this individually for top and bottom layers, we can obtain their G-peak frequencies for both directions and various magnitudes of applied strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The βϵ values for the strained top layer are listed in Table SIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since the bottom layer remains unstrained, we observe negligible differences between their β pre-factor values for strained and unstrained configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' GEOMETRIC ANALYSIS OF STRAINED MSCS We deduce the expressions of their reciprocal lattice (⃗q ) vectors to quantify the structural changes in strained MSCs18,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The reciprocal lattice vectors of TBG moir´e lattices20 (⃗q) is given as ⃗q = b⃗′ − ⃗b, where b⃗′ and ⃗b denote the reciprocal lattice vectors of the rotated top layer and bottom layer in a TBG structure respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The length of moir´e pattern (MP), 4π Lm can be derived using the magnitude of ⃗q vector as Lm = √3 ⃗q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' When strain is applied to the top layer, the mathematical expression of its reciprocal lattice vector19 (b⃗ε) can be written as b⃗ε = (I⃗ + S⃗ ) −1 b⃗′ , where I⃗ is the identity matrix and S⃗ denotes the strain tensor i i which can be written as the following for the case of uniaxial tension, S⃗ = ε 0 0 −νε Here, ε is the nominal strain applied and ν denotes the Poisson’s ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' So, the reciprocal lattice vector of TBG with heterostrain can be expressed as ⃗ε b⃗ε − b⃗i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As shown in ω ω q 5 Fig 1(e) in the main text, the boundaries of MPs resemble a hexagon and we can draw a triangle (∆ABC) with A⃗B and B⃗C as the MP lattice vectors and α being the angle between them (α = 60°, ϕ = 120°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The variation of α and ϕ with the applied strain is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' With uniaxial tension, we see a monotonic decrease in these angles and vice-versa for uniaxial compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The changes in expressions of ⃗q vectors are associated with the geometrical changes enforced upon hetero-straining these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' EXPLANATION OF STACKING IDENTIFICATION METHOD (FOR UNSTRAINED AND STRAINED SYSTEMS) Firstly, we performed the identification of atoms that should be classified as ’AA’ type using ILS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As observed in main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1(c) and (d), the spacing between two layers of TBG varies due to out-of-plane displacements of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The ILS of equilibrium structures follows this trend: AA > SP > AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence, in a TBG system, the maximum ILS (dmax) corresponds to AA region and the minimum distance (dmin) represents AB region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It is observed that dmax and dmin vary with increasing twist angle up to 21°, after which we noticed a plateaued regime21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This results due to the depletion of perfectly stacked AA and AB configurations, as the length of the MPs, reduces with increasing θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We obtained the maximum and minimum magnitudes of dmax (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='589˚A and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='475˚A) and dmin (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='456˚A and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='338˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Using the lower bound of dmax for all the twist angles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='475˚A, we classified the atoms with local ILS greater than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='475˚A as ’AA’ stacking type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On the other hand, considering the upper bound of dmin and identifying the regions with ILS below that value as AB stacking can lead to the misclassification of AB and SP types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' For the wide range of twist angle considered in this study, the ILS alone cannot provide a margin of separation for classifying AB and SP stacked atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To address this issue, we considered interlayer energy or ILE (per atom) in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Perusing the ILE contour plot, we observed that the center of MPs has the highest energy followed by the SP segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The AB (or BA) has the lowest energy corresponding to the ground state configuration of BLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' But, being a per atom quantity, the C atoms in AB stacking that are present directly on top of a C atom on the other layer show the highest ILE value as shown in main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To obtain the same measure of energy for AB stacked atoms whether they are located at the center of a lattice hexagon or at the corner, we calculated the difference of interlayer 6 energy of each atom with its three bonded neighbors and consider their average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The interlayer energy difference with neighboring atoms allows us to easily classify AB stacked atoms as they have the highest fluctuation of energy with neighbors compared to AA or SP stacked regions where the quantity is quite uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To obtain a classification threshold of interlayer energy difference for AB stacking, we first calculated the soliton width of different TBG systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', the width of SP regions similarly as explained by Gargiulo et al21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On analyzing the path from the center of AB domain to the center of another AB (or BA) region, we traverse across the SP segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Calculating the ILS and plotting it along the centers of triangular (AB) regions, we observed a small peak (Fig S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This peak corresponds to the SP region and its full width at half maxima (FWHM) gives us the soliton width21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Considering this soliton width (varies with twist angle), we obtained the interlayer energy difference value at the boundary of SP domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' This process is repeated for different twist angles to establish a unique threshold that can be applied to any TBG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The energy difference threshold lies in a diminutive range, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='22-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='31 meV for the angles considered (Fig 2(e) in main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' On averaging these magnitudes, we defined a ∆EILE threshold of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='24 meV/atom, above which an atom is classified as AB stacking type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The contour plot of TBG (θ = 6◦) system in main Fig 2(f) shows the outcome of applying the method where each atom has been classified as belonging to either AA or AB or SP stacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We utilized the same approach for classifying the local domains in strained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since the ILS parameter defines the out-of-plane distancing of pristine structures, it is not affected by an in-plane applied strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' However, the interlayer energy of the structure is expected to change because an externally applied strain disturbs the interlayer interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' But since the mechanical deformation is applied globally, the local regions will experience a similar change in ILE with respect to their nearest neighbors and hence ∆EILE remains approximately unchanged (see Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' STACKING IDENTIFICATION OF RIGID STRUCTURES We followed the same approach used for reconstructed or relaxed systems to classify local regions in rigid structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The atomistic structure of rigid TBGs (R-TBGs) is different from reconstructed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since they are created by simply employing a rigid twist to a Bernal stacked bilayer graphene, they do not have a variation of interlayer spacing, which is 7 present in reconstructed TBGs pertaining to the formation of local stackings in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' When a R-TBG is modeled from Bernal stacked (or AB) graphene, it has an ILS equal to that of AB stacked graphene throughout its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Hence to account for this we defined their uniform ILS, which is different from their initial geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We first considered their relaxed structure and obtained an average ILS value considering all the interlayer distances throughout the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Then, we re-modeled the rigid TBG structure by adjusting the layers with respect to the average ILS value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Since different structures have varying fractions of local interlayer regions, this average ILS changes for systems with certain twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' It must be noted that we have not utilized this average ILS to define any threshold to classify local atoms, rather it is used only to define the respective rigid structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Further, following the same method as relaxed systems we obtained their interlayer energy followed by calculating the ILE difference (∆EILE) per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Now to classify the individual stackings, we referred back to the ILS and ∆EILE thresholds obtained for relaxed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Having known the ILS threshold for AA region (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='475 ˚A ) , we then identified the ∆EILE value at the location corresponding to that ILS value by traversing along path PQ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2(a) main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Then, we employed this value in ∆EILE calculation for R-TBG and specified atoms above that threshold (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='88 meV/atom) as AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' For identifying AB type, we have considered the ∆EILE threshold (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='24 meV/atom) corresponding to its location on the path PQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Similarly, we then used that location to detect ∆EILE threshold for AB type in R-TBG structure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='92 meV/atom, so it lies between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='92 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='88 meV/atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' After classifying AB and AA, we have assigned the remaining atoms as SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Further, we have used this same method to identify the local stackings in rigid structures of strained configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To model rigid systems of strained TBGs in a way that physically makes sense, we first considered the relaxed or reconstructed structure of pristine TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Now the top layer is stretched such that an unrelaxed hetero-strained TBG system is generated, which is referred to as the rigid structure in the presence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Relaxing this strained structure results in a fully optimized system, pertaining to the reconstructed TBG configuration with strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' PHONON DISPERSION SPECTRA OF TBG AND ITS LOCAL DOMAINS The simulations for phonon dispersion spectra were performed for θ = 6° and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Due to the computational cost of DFT-based phonon simulations for large MPs, we computed phonon spectra only for θ > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='41° systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We discussed an approach using BOLS correlation to predict the Raman peaks pertaining to optical phonon modes for larger TBG systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' As described by Cocemasov et al, TBGs contain hybrid folded phonon branches that require to be unfolded onto the single layer first BZ22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Using the PhononUnfolding package23, we simplified the phonon spectra of TBGs along Γ-K-M-Γ high symmetry path (Fig S7 shows unfolded spectra of θ = 6°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' To obtain the phonon spectra of local sub- domains, we first identified the atomic positions of each local stacking as defined by our identification method and extract the data from the main structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Then, we calculated the average bond length lavg of each configuration and deduce their respective lattice con- stant as a stacking = √3l avg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' With the calculated unit cell parameters, we have computed their phonon spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' PHONON BAND SPLITTING WITH HETEROSTRAIN A combination of Molecular statics and first principles simulations has been used to compute phonon dispersion spectra of TBGs with heterostrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' By freezing the obtained configuration from LAMMPS, we have extracted the atomic data of strained periodic moir´e lattice and further minimized the supercell in DFT to obtain first-principles-level fidelity, followed by phonon spectra calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' We observed strain-induced phonon band splitting due to inequivalent strain present in both layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' With tension, the atomic bonds in a crystal are stretched relative to their unstrained condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' When the bond length is increased, and the force constant remains unchanged, as a result, the vibrational frequency decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Conversely for compression, the bond length reduces which leads to an increase in vibrational frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' That is why we observe redshift and blueshift in phonon frequencies for tensile and compressive strain respectively24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The redshift and blueshift of Raman G-band for θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08°, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' S9 is a good demonstration of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1: (a) Relaxed atomistic structure and (b) interlayer spacing contour plot of θ= 6° TBG system under 1% uniaxial compressive strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' TABLE I: Average ∆EILE threshold value considering five representative TBG systems (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1°, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='48°, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='41°, 6° and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='34°) in the presence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Strain (%) ∆EILE (meV/atom) 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='24 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='223 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='21 +1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='23 -1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='207 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='6 Spacing (A) 鞋 鞋 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 cocal Interlayer 鞋 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3510 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2: Variation of angles α and ϕ with strain demonstrating the deformation of moir´e patterns (for TBG system θ= 6°) TABLE II: Evolution of area fractions f of local stacking domains with uniaxial tension and compression applied to the top layer Strain (%) θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1° θ = 6° θ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° fAA fAB fSP fAA fAB fSP fAA fAB fSP 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='25 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='338 EUniaxial tension EUniaxialtension 64 124 G Uniaxial compression G Uniaxial compression 62 122 (c)0 (o) Angle 60 e 120 58 118 56 116 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='8 1 Strain(%) Strain(%)11 θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1° θ = 6° θ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 3: Normalized spatial interlayer spacing difference (∆d) profiles traversing between centers of moir´e pattern, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=', path PQ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 2(a) (for TBG system θ= 6°) TABLE III: Parameters for calculating βBOLS pre-factors for TBGs and their respective sub-domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Parameters TBG AA AB SP TBG AA AB SP TBG AA AB SP dz (˚A ) 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='743 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='727 AA AA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='45 SP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 AB AB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='35 P Q12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 4: Variation of area fractions of individual stacking domain with respect to heterostrain (tension) for different twist angles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='48° AA 0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='410 AA ■SP SP AB AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3 Area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0% strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% strain 1%strain 0% strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% strain 1% strain (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='08° AA 0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='34° AA ■SP SP AB AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 uo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0% strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% strain 1% strain 0%strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% strain 1%strain (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='349 AA 0 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='10 AA SP ■SP AB AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 ion on cti 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3 fra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0% strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5%strain 1%strain 0%strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5%strain 1%strain (e) (f)13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 5: Variation of area fractions of individual stacking domain with respect to heterostrain (compression) for θ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='48°, 6° and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 6: Interlayer energy or vdW stacking energy for rigid and relaxed TBG systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' The ILE of relaxed TBG system is always lower than rigid TBG even for larger twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 0=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='48° AA 0 = 60 AA 0 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='29 AA ■SP SP SP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 AB fract 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0% strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% strain 1% strain 0% strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% strain 1% strain 0% strain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5% strain 1% strain 10 ILE (meV/atom) 15 Interlayer energy, 1 20 25 Rigid TBG Relaxed TBG 30 0 2 4 6 8 10 12 14 Twist angle (0o)14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 7: Unfolded phonon spectra of TBG system θ = 6° along high symmetry points of its Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Phonon dispersion spectra of Bernal stacked BLG is also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 1500 (cm 1200 Phonon frequency ( 900 600 300 Bernal stackedpristine BLG TBG (0 = 6) K M15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 8: Transverse optical (TO) phonon frequency difference with respect to TBG system θ = 6° 20 10 10 AA stacking AB stacking SP stacking 20 K M16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' 9: BOLS predicted Raman G band frequencies of θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1° TBG system as a function of applied heterostrain (tension and compression) 1610 Bottom layer (unstrained) Top layer (tension) 1600 Top layer (compression) 1590 (cm 1580 1570 1560 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='5 Strain(%)17 TBG TABLE IV: Calculated βϵ pre-factor values of strained top layer using BOLS parameters with respect to strain Strain (%) θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='1° θ = 6° θ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2° 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='135 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='306 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='466 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='207 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='355 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='398 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='506 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='732 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='961 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='312 +1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='475 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='592 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='773 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='602 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='795 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content='248 18 REFERENCES 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfW_xa/content/2301.01310v1.pdf'} +page_content=' Dey, R.' metadata={'source': 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b/QtE3T4oBgHgl3EQfDAk3/content/tmp_files/2301.04281v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4380015fffb50e016ed566abe246d2aeee74c15 --- /dev/null +++ b/QtE3T4oBgHgl3EQfDAk3/content/tmp_files/2301.04281v1.pdf.txt @@ -0,0 +1,3385 @@ +3D photophoretic aircraft made from ultralight porous +materials can carry kg-scale payloads in the mesosphere +Thomas Celenza, Andy Eskenazi and Igor Bargatin + + +We show that photophoretic aircraft would greatly benefit from a three-dimensional (3D) hollow geometry +that pumps ambient air through sidewalls to create a high-speed jet. To identify optimal geometries, we +developed a theoretical expression for the lift force based on both Stokes (low-Re) and momentum (high- +Re) theory and validated it using finite-element fluid-dynamics simulations. We then systematically varied +geometric parameters, including Knudsen pump porosity, to minimize the operating altitude or maximize +the payload. Assuming that the large vehicles can be made from previously demonstrated nanocardboard +material, the minimum altitude is 55 km while the payload can reach 1 kilogram for 3D structures with 10- +meter diameter at 80 km altitude. In all cases, the maximum areal density of the sidewalls cannot exceed a +few grams per square meter, demonstrating the need for ultralight porous materials. + + +For centuries, humans have been exploring Earth’s atmosphere and outer space, a quest that has +led to discoveries in fields ranging from aerodynamics to astronomy and climate modeling [1-3]. However, +the study of certain regions of the atmosphere is hindered by available propulsion technologies. For instance, +in Earth’s mesosphere, anthropogenic emissions of carbon dioxide are counterintuitively producing rapid +cooling [4]. The shrinking of the atmosphere resulting from this cooling [5] can be problematic, given that +a contracting mesosphere can result in reduced satellite drag, which could translate into a greater +accumulation of space debris [6]. Unfortunately, uncertainties in calculations of these effects are currently +large because experimental observations within the mesosphere are challenging [7], given that this region, +extending from fifty to eighty kilometers above the surface of Earth, has air pressures too low to sustain +planes or balloons and too high for orbiting satellites. + +Another region of significant interest is the Martian atmosphere, where most recently the +helicopter Ingenuity achieved near-surface flight [8]. Even with this milestone, sustained flight at high +altitudes in Mars, e.g., from Olympus Mons, is not yet possible due to decreasing atmospheric density +[9,10]. Like the study of Earth’s mesosphere, the exploration of Mars’ atmosphere at high altitudes is +limited by the lack of long-duration methods of flight and propulsion at ambient pressures below ~1 mbar +(100 Pa). As a result, developing an airborne platform that can operate in a very thin atmosphere, both on +Mars and on Earth, would be extremely useful in helping collect valuable and atmospheric data related to +wind patterns, temperature and pressure variations, as well as the concentrations of atmospheric gases. + +One promising concept, based on the lightweight light-powered centimeter-scale microflyers +developed by Cortes et al. [11], can potentially overcome the issues faced by the current propulsion +mechanisms and achieve sustained flight in Earth’s mesosphere and the Martian atmosphere. These devices, +composed of porous plates, can levitate due to photophoresis, a light-driven propulsion mechanism where +a jet is created using Knudsen pumping of ambient gas [12]. Knudsen pumps have no moving parts and +instead exploit temperature gradients to induce gas flows through these plates. Known as “nanocardboard”, +these ultralight porous plates are composed of nanometer-thick (25–400 nm) aluminum oxide face sheets +that are connected by channels with micrometer-scale width and height. They offer an areal density of only +~1 g/m2 and a bending stiffness orders of magnitude higher relative to solid plates of the same mass [13]. + +Photophoretic levitation is typically enabled by a difference in physical properties between the top +and bottom of the plate. For instance, in the study performed by Cortes et al. [12], the bottom side of the +nanocardboard was coated with carbon nanotubes (CNTs), which absorbed the incident light and +subsequently increased in temperature relative to the top side. This difference in temperatures caused the +Knudsen pumping, which pushed air down through the channels of nanocardboard from the cold to the hot +side and thus creating a downward jet below the nanocardboard that levitated plates with centimeter-scale +sizes [11]. This mechanism works best in low pressure environments (1-100 Pa) [14], such as in Earth’s +mesosphere or near the top of Olympus Mons on Mars [15]. If the lift forces are large enough to carry tiny +“smart dust” sensor payloads [16], many such microflyers can be deployed on Earth or on Mars to record +data in these regions of the atmosphere. + +In this work, we propose much larger photophoretic vehicles, which are many meters in diameter, +three-dimensional rather than planar, and use porous sidewalls that push air into an inner chamber and out + +of a small nozzle (Fig. 1). Using the nozzle increases the speed of the air jet, and such 3D photophoretic +vehicles can not only increase the resulting lift force but also widen the range of operating pressures. +Combining design concepts from the previously demonstrated photophoretic levitation of planar +nanocardboard [11] and analytical tools we used for solid mylar-CNT composite disks [17], we analyzed +3D geometries with porous alumina nanocardboard walls and CNTs deposited on their inner side. Because +alumina is transparent, CNTs on the inside of the structure would absorb the incident light, inducing the +Knudsen pumping of air from the outside into the interior chamber through the pores and then out of the +chamber through the exit nozzle, producing a jet as illustrated in Fig. 1. + + +Figure 1: A hollow sphere with porous alumina-CNT composite walls flying in Earth’s mesosphere (a) and over the +top of Olympus Mons in Mars (b). The cross-sectional view (c) of the sphere shows the air flow in, with velocity 𝑣𝑓𝑡, +due to Knudsen pumping (across the nanocardboard walls, as seen on the zoomed-in view) and out as a jet through +the exit nozzle, with velocity 𝑣𝑗𝑒𝑡. As depicted in (c), A is the nanocardboard channel width, L the nanocardboard +channel height, and r the structure’s outlet radius, while D the structure’s overall size dimension. Background Earth +and Mars Image Credits: NASA. + +To identify the optimal 3D geometry that maximized payload, we considered three representative +geometries (a sphere, a cone, and a rocket), and performed a series of simulations to determine the +parameters that would yield the greatest lift forces. However, first, it was necessary to develop an analytical +expression that predicted the lift forces produced by such structures across a wide range of Reynold +numbers. To determine this expression, we modeled these 3D structures with outlet jet velocities as small +as 10-6 m/s to as large as ~100 m/s and at various atmospheric altitudes up to 80 km using computational +fluid dynamics simulations in ANSYS Fluent, as detailed in the supplementary information. For each fluid- +flow simulation, we found the reaction forces induced from the air flow (equal and opposite to the lift force), +and then fitted the collected data using the equation + + +𝐹 = 𝐶18𝜇𝐷𝑣𝑓𝑡 + 𝐶2𝜌𝐴𝑣jet +2 . +(1) + + +a +Exterior +Vft +Vft +Interior +Vft +Cross- +sectional +viewHere, 𝜇 corresponded to the fluid viscosity, 𝜌 to the density, 𝐴 = 𝜋𝑟2 is the area of a nozzle with radius r, +D is the geometry’s characteristic (i.e., largest) dimension, while 𝑣𝑓𝑡 is the flow-through velocity of the +fluid flow through the porous material and 𝑣𝑗𝑒𝑡 is the velocity of the fluid exiting the structure through the +small nozzle. As outlined in the supplementary information, 𝑣𝑓𝑡 depends on the light intensity, I, the +altitude dependent air pressure, P, and the geometric parameters of the nanocardboard. The upper limit of +the flow-through velocity typically scales as 𝑣𝑓𝑡 ≈ 0.03 𝐼/𝑃 (see supplementary information), resulting in +velocities of less than 1 mm/s under natural sunlight (~1000 W/m2) and standard atmospheric pressure (105 +Pa) but increasing by many orders of magnitude as the pressure drops at higher altitudes. + +In Eqn. (1), the first term is based on Stokes’ drag on a disk, obtained from a linearization of the +steady-state Navier-Stokes equations in the case of dominating viscous forces, i.e., in the low-Re limit. +Cortes et al. previously showed that at vanishingly low air flow speeds, the lift of a stationary +nanocardboard plate with air flowing through it was equal to the Stokes drag for a solid disk [11]. In contrast, +at high jet speeds, the inertial terms dominate, and the lift is mostly dependent on the velocity of the jet +exiting the nozzle. The helicopter-momentum theory equation, which can be derived from a simple +application of Reynolds Transport Theorem and represents the second term in Eqn. (1), can model the lift +in this high-Re limit. Summing both terms results in a simple interpolation between the two operating +regimes that gives an estimate for the lift force at all pressures and velocities (and, therefore, all values of +Re). Table 1 summarizes the average fitted C1 and C2 parameters, both on the order of 1, obtained from +fitting the results for 27 ANSYS Fluent simulations using 3 different altitudes (0 km, 40 km and 70 km), 3 +geometry types (sphere, cone, and rocket), and 3 different structure sizes (1cm, 5cm and 10cm). + +Fitting Parameters for Each Geometry +Geometry +Cone +Sphere +Rocket +C1 +1.2 +1.3 +1.4 +C2 +0.9 +0.9 +0.4 + +Table 1: Fitting parameters for the three geometries in addition to key dimensions. Notice that these ANSYS +simulations were performed assuming a 100% porosity along each one of these structures’ walls. + +After determining the coefficients C1 and C2, we proceeded to numerically optimize the various +parameters controlling the overall 3D shape and nanocardboard porous microstructure to maximize the +payload capabilities. The developed MATLAB code [18] was based on the photophoretic levitation theory +for nanocardboard [11] adapted to axisymmetric 3D structures, as detailed in the supplementary +information. The code also took into account how temperature and pressure depend on the altitude in the +atmosphere, employing empirical relations developed from standard atmospheric data [19]. Our +optimization sought the combination of A (nanocardboard channel width), L (nanocardboard channel +height), and r (the structure’s outlet/nozzle radius) that resulted in the highest payload or achieved flight at +the lowest altitude as a function of the overall aircraft size D (diameter for sphere and cone, and length for +the rocket). All these geometric parameters are illustrated in Fig. 1c. + +Our numerical optimizations revealed that the optimal nanocardboard porosity parameters A and +L were of the same order of magnitude across all geometries and dimensions D. When optimized for +achieving flight at the minimum altitude (55 km with zero-payload), A and L were ≈ 0.20 mm and ≈ 0.21 +mm, respectively. When optimized for maximum payload (achieved at 80 km altitude), A and L were 0.90 +mm and 0.91 mm, or about a factor of 4 greater. Because these parameters are of the same order of +magnitude despite the approximately 40-fold change in ambient pressure at the minimum possible altitude +of 55 km and the max payload altitude of 80 km, we can make structures that simultaneously achieve +levitation at low altitudes while carrying significant payload at higher altitudes. + +The maximum areal densities, i.e., the maximum lift force divided by specific gravity g and the +area of nanocardboard, were also comparable for all structures. Table 2 shows that the typical value of +maximum areal density was ≈ 7.1 g/m2 (grams per square meter) for small aircraft (D = 10 cm) compared +to ≈ 5.5 g/m2 for large aircraft (D = 10 m). Both these densities are in the same order of magnitude as the +theoretical upper limit derived for the high-Re case in the supplementary information, of 0.016 𝐼/ +(𝑣𝑎𝑣𝑔𝑔) ≈ 0.004 kg/m2 = 4 g/m2. Here, 𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇/𝜋 ≈ 400 m/s is average speed of air +molecules at 55-80 km altitudes, while 𝑅𝑎𝑖𝑟 = 𝑅𝑢/𝑀𝑎𝑖𝑟 = 287 𝐽/(𝑘𝑔 ∙ 𝐾) is the gas-specific ideal constant +of air, equal to the universal gas constant 𝑅𝑢 divided by the average molar mass of air 𝑀𝑎𝑖𝑟. Fig 2a shows +how the maximum areal densities varies with aircraft size D and, therefore, the airflow’s Reynolds number. + +The permissible areal densities of each structure decrease with increasing size and Re and stabilize at ~5.5 +g/m2 for larger aircraft that carry payloads of 1 gram or more. + +Areal Densities and Areas Ratio +Geometry +Cone +Sphere +Rocket +D = 10 cm +D = 10 m +D = 10 cm +D = 10 m +D = 10 cm +D = 10 m +Max Areal +Density +For Max. +Payload +6.6 g/m2 +5.4 g/m2 +7.8 g/m2 +5.5 g/m2 +6.9 g/m2 +5.7 g/m2 +Area +Ratios +For Min. +Altitude +18 +26 +26 +27 +23 +25 +For Max. +Payload +5 +5 +5 +6 +6 +6 + +Table 2: Summary of the parametric studies results for the Cone, Sphere and Rocket, for values of D = 10 cm and D += 10 m (full data for all the probed values of D can be found in the supplementary information section). Here, the area +ratio refers to the 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio, of the structure’s total surface area to its outlet area. + +Figure 2: Areal Density versus Characteristic Size (a) and Maximum Payload versus Surface Area (b) for the three +considered 3D geometries at 80-km altitude. Each data point corresponds to the optimized geometry at each of the +probed values of the parameter D. The overlap between the curves, in particular starting at surface areas larger than +0.01 m2, suggests that the geometries have similar areal densities and maximum payload capabilities. + +Plotting maximum payloads against the structure surface area in Fig. 2b revealed that, for a given +surface area, the maximum payload was very similar across all three geometries. While the sphere +outperformed at smallest sizes, all three shapes (cone, sphere, and rocket) offered essentially the same +performance at the largest sizes, i.e., for sizes that maximize the payload and are most promising for +practical applications. Fig. 3 below illustrates optimized shapes for the 10-meter cone (a), sphere (b) and +rocket (c), which could carry 780, 540, and 1020 grams of payload, respectively. This is sufficient capacity +to carry modern communication devices [20] and similar to the payload of a typical CubeSat [21]. +a +b +𝑅𝑒 = 𝜌𝑣𝑓𝑡𝐷 +𝜇 + + +D=10m +a +b +c +D= 10m +D= +10m +r=4.97m +r=3.67m +r=4.97m +Payload: 780 g +Payload: 540 g +Payload: 1020 gMax.Payload againstGeometrySurfaceArea +100 +Max. Payload (kg) +0 +10° +Sphere +Cone +Rocket +10-4 +10~2 +100 +102 +Surface Area (m3)Max.Areal Density against characteristic D +ReynoldsNumber +100 +10l +102 +103 +25 +Sphere +Cone +20 +Rocket +15 +10 +10-2 +10~1 +100 +10l +D (m) +Figure 3: Geometrically optimized cone (a), sphere (b) and rocket (c) for maximum payload capabilities with a fixed +characteristic dimension of D = 10 meters. D represents the cone and sphere diameter, and the rocket length. Achieving +a payload of 1kg required a D of 11.5 and 14 m for the cone and sphere, respectively. + +Finally, as demonstrated in Table 2 and the supplementary information section, we noticed that +the 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio, of the total surface area to the outlet area, was approximately constant for the optimal +geometries. For the minimum altitude case, this ratio ranged from 17 to 42, averaging ≈ 23 across the three +geometries and sizes. For the maximum payload case, the typical value of this ratio was approximately 6, +resulting in relative nozzle sizes shown in Fig. 3. Due to mass conservation, the outlet jet speed needs to +be larger than the flow-through velocity by the same factor as precisely the 𝐴𝑖𝑛/𝐴𝑜𝑢𝑡 area ratio. Therefore, +recalling the 𝑣𝑓𝑡 ≈ 0.03 𝐼/𝑃 relationship, at the maximum payload altitude of 80 km, we can approximate +𝑣𝑗𝑒𝑡 = 𝑣𝑓𝑡𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ≈ 0.18 𝐼/𝑃 ≈ 0.18 × 1300 𝑊 𝑚−2/1 𝑃𝑎 = 234 𝑚/𝑠 , while at the minimum +altitude of 55 km (i.e., for zero payload), 𝑣𝑗𝑒𝑡 = 𝑣𝑓𝑡𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ≈ 0.70 𝐼/𝑃 ≈ 0.70 × 1200 𝑊 𝑚−2/ +10 𝑃𝑎 = 84 𝑚/𝑠. Notice that for the payload altitude of 80 km, the jet speed approaches but remains below +the speed of sound, given by 𝑣𝑠𝑜𝑢𝑛𝑑 = √𝛾𝑅𝑎𝑖𝑟𝑇80𝑘𝑚 ≈ √1.4 × 287 𝐽/(𝑘𝑔 𝐾) × 200 𝐾 ≈ 280 m/s, +where 𝛾 is the adiabatic constant of air, while 𝑇80𝑘𝑚 ≈ 200 𝐾 is the air temperature at 80 km altitude. +Achieving kg-scale payloads in the mesosphere will therefore require building 10m-scale photophoretic +aircraft out of ultralight materials that simultaneously possess low areal densities (≈ 1 g/m2) and sufficient +structural integrity. However, these aircraft do not necessarily have to be rigid; instead, it is possible to +make use of flexible parachute or balloon-like structures with overall dimensions similar to those shown in +Fig. 3. + +In the calculations above, we assumed that all surfaces are illuminated with 1000 W/m2 light +intensity, which is not always realistic. The direct sunlight intensity in the mesosphere is similar to that in +outer space, ~1360 W/m2. Additional ~500 W/m2 of sunlight will be reflected from the clouds and Earth +below the aircraft due to Earth’s planetary albedo of approximately 0.3. Depending on the elevation of the +Sun in the sky and the orientation of the surface, it may be exposed to anywhere between essentially zero +and almost 2000 W/m2 of combined direct and reflected sunlight. If the aircraft ends up rotating as balloons +often do, all walls will experience an average flux on the order of 1000 W/ m2 or slightly less. For reference, +we also performed simulations at a reduced intensity of 500 W/m2, which results in payloads ~4 times lower +than those shown above. One last important aspect to note about these photophoretic aircraft is that they +only create lift when exposed to light (i.e., during the day), limiting the steady operation to ~12 hours at +most latitudes, after which the aircraft will start to descend to the ground. However, near the poles, the +polar day can last many months and extended operations of up to several months may be possible. + +To conclude, we show that 3D photophoretic aircraft with porous walls made of ultralight, +ultrathin materials are capable of carrying kg-scale payloads, comparable to those of typical CubeSats. The +results presented above can be easily generalized for high-altitude operation on Mars using a Martian +atmospheric model [22]. This work opens the way to creating persistent, low-cost, sensor-carrying aircraft +in the previously inaccessible atmospheric regions at 55-80 km altitudes on Earth and 20-40 km altitudes +on Mars, enabling a greater understanding of our planet and the worlds beyond. + + +References + +[1] Baum, S. D. (2009). Cost–benefit analysis of space exploration: Some ethical considerations. Space +Policy, 25(2), 75-80. 8 + +[2] Bainbridge, W. S. (2009). Motivations for space exploration. Futures, 41(8), 514-522. + +[3] Lindgren, E. A., Sheshadri, A., Podglajen, A., & Carver, R. W. (2020). Seasonal and latitudinal +variability of the gravity wave spectrum in the lower stratosphere. Journal of Geophysical Research: +Atmospheres, 125(18), e2020JD032850. + +[4] Laštovička, J. (2017). A review of recent progress in trends in the upper atmosphere. Journal of +Atmospheric and Solar-Terrestrial Physics, 163, 2-13. + + +[5] Bailey, S. M., Thurairajah, B., Hervig, M. E., Siskind, D. E., Russell III, J. M., & Gordley, L. L. (2021). +Trends in the polar summer mesosphere temperature and pressure altitude from satellite observations. +Journal of Atmospheric and Solar-Terrestrial Physics, 220, 105650. + +[6] Tran, L. “NASA Satellites See Upper Atmosphere Cooling and Contracting.” NASA. Goddard Space +Flight Center, June 28, 2021. https://www.nasa.gov/feature/goddard/2021/nasa-satellites-see-upper- +atmosphere-cooling-contracting-climate-change. + +[7] Goessling, H. F., & Bathiany, S. (2016). Why CO 2 cools the middle atmosphere–a consolidating model +perspective. Earth System Dynamics, 7(3), 697-715. + +[8] Gohd, C. “Mars Helicopter Ingenuity: First Aircraft to Fly on Red Planet.” Space.com. Space, May 22, +2021. https://www.space.com/ingenuity-mars-helicopter-perseverance-rover. + +[9] Squyres, S. W., Arvidson, R. E., Baumgartner, E. T., Bell III, J. F., Christensen, P. R., Gorevan, S., ... +& Romero, R. A. (2003). Athena Mars rover science investigation. 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KONA Powder and Particle Journal, 31, 181- +199. + +[15] +European +Space +Agency. +“Facts +about +Mars.” +https://www.esa.int/Science_Exploration/Space_Science/Mars_Express/Facts_about_Mars. + +[16] Niccolai, L., Bassetto, M., Quarta, A. A., & Mengali, G. (2019). A review of Smart Dust architecture, +dynamics, and mission applications. Progress in Aerospace Sciences, 106, 1-14. + +[17] Azadi, M., Popov, G. A., Lu, Z., Eskenazi, A. G., Bang, A. J. W., Campbell, M. F., ... & Bargatin, I. +(2021). Controlled levitation of nanostructured thin films for sun-powered near-space flight. Science +Advances, 7(7), eabe1127. + +[18] Eskenazi, A., Celenza, T., & Bargatin, I. (2022). MATLAB-fluid-flow-parametric-studies. +https://github.com/andyeske/MATLAB-fluidflow-parametric-studies + +[19] +Engineering +ToolBox. +(2003). U.S. +Standard +Atmosphere +vs. +Altitude. +https://www.engineeringtoolbox.com/standard-atmosphere-d_604.html + +[20] Saeed, N., Elzanaty, A., Almorad, H., Dahrouj, H., Al-Naffouri, T. Y., & Alouini, M. S. (2020). +Cubesat communications: Recent advances and future challenges. IEEE Communications Surveys & +Tutorials, 22(3), 1839-1862. + +[21] NASA. (2017). CubeSat 101: Basic Concepts and Processes for First-Time CubeSat Developers. +https://www.nasa.gov/sites/default/files/atoms/files/nasa_csli_cubesat_101_508.pdf + +[22] Justh, H. L., Cianciolo, A. D., & Hoffman, J. (2021). Mars Global Reference Atmospheric Model +(Mars-GRAM): User Guide (No. NASA/TM-20210023957). + + + +Page +1 +3D photophoretic aircraft made from ultralight porous +materials can carry kg-scale payloads in the mesosphere +Supplementary Information +Thomas Celenza, Andy Eskenazi and Igor Bargatin + +In this document, we present and expand on the computational and theoretical framework behind our work. +The first section is devoted to the ANSYS Fluent simulations, covering the solver set-up and the theory +behind the force calculations. The second section of this document focuses on the MATLAB code, +specifically the derivation of the equations used in the optimization of the geometrical and channel +parameters of the 3D geometries, including the rocket, cone and sphere. + +1. ANSYS Fluent Simulations + +The goal of the ANSYS Fluent simulations was to determine an analytical expression to estimate the lift +forces produced by various types of 3D structures. Because we sought geometries that operated across a +wide range of velocities and altitudes (and thus air pressures, densities, temperatures and viscosities), the +expression for the lift force needed to be valid across a wide range of Reynolds (Re) numbers as well. In +particular, this equation needed to reasonably accurately model the transition between the low-Re (Stokes) +regime to the high-Re regime. As the main paper argues, an appropriate expression is + + +𝐹 = 𝐶18𝜇𝐷𝑣𝑓𝑡 + 𝐶2𝜌𝐴𝑣jet +2 . +(S1) + +Here, 𝜇 corresponds to the fluid viscosity, 𝜌 to the density, 𝐴 = 𝜋𝑟2 is the area of a nozzle with radius r, D +is the geometry’s characteristic (usually largest) dimension, while 𝑣𝑓𝑡 is the flow-through velocity of the +fluid through the porous material and 𝑣𝑗𝑒𝑡 is the velocity of the fluid exiting the structure through the small +nozzle. The fitting parameters 𝐶1 and 𝐶2 depended on the geometry and were determined using ANSYS +simulations. In this work, we considered three geometries, a cone, sphere, and rocket, shown in Figure S1. +Figure S1: Main geometric parameters for the cone (a), rocket (b) and sphere (c). Notice that here, the variable D +serves as an overall indicator of the size of the geometry, while the variable r controls the outlet radii of the nozzle. + +Isometric +View +Isometric +View +Side +View +121 +h +a +Side +View +Isometric +View +Side +View +2r +bPage +2 +Through the ANSYS Simulations, we determined the average 𝐶1 and 𝐶2 coefficients for each structure and +examined how these would evolve with overall size of the structure or the altitude. We performed 9 sets of +simulations for each geometry, where we varied three different inlet/outlet area ratios at three different +altitudes, resulting in flow-through velocities as small as 10-6 m/s or as large as 1 m/s. + +Figure S2 shows boundary conditions +employed in our simulations using a sphere +as an example. To make our simulations +computationally more efficient, we took +advantage of the axial symmetry of our +three geometries and thus constructed our +models +in +a +2D, +axisymmetric +environment, which allowed us to only +simulate fluid flow on the top half of each +structure. We formed these geometries +using ANSYS’ “Design Modeler” module, +and they were essentially composed of +three spaces: an outer air box, and inner air +box, and the nanocardboard geometry itself +(whose interior was “subtracted” from the +inner air box, as seen in Figure S2). + +The next step was to specify mesh +elements, shown in Figure S3. Plot (a) +shows the larger, outer air box with coarser +mesh elements, while plot (b) is a zoomed- +in view into the smaller, inner air box, +containing smaller mesh elements. By +dividing the air box into these two regions, we optimized the overall number of mesh elements in the +simulation by providing a higher resolution just in the area close to the geometry. We created the mesh by +selecting edges and dividing them into a discrete number of points; to enforce a uniform grid pattern, we +used the quadrilaterals face meshing command. For the sphere, this resulted in 184,180 elements (185,408 +nodes); for the cone, 194,322 elements (195,865 nodes); for the rocket, 293,053 elements (294,616 nodes). +These were the final numbers of mesh elements obtained as a result of performing a convergence analysis +until observing negligible changes in the computed lift forces. + + +The final step was to establish Fluent’s “set-up” module parameters. For the model, we chose the viscous +k-omega, with the low-Re (viscous) corrections feature enabled. Next, we fixed the boundary conditions as +described by Figure S2, and manually modified operating conditions (environment pressure, fluid density +and fluid viscosity) matching the chosen altitude. Since our fluid was air, we extracted its properties as +tabulated in altitude-dependent standard atmospheric tables, summarized in Table 1 below for 0 km, 40 km +and 70 km (our probed altitudes). Last, we specified the inlet velocity as a variable parameter, since that +Figure S2: ANSYS Simulations boundary conditions. As the +illustration shows, the inner wall of the geometry (red) was chosen +as the flow-velocity inlet (inducing the air to flow from the into the +structure), while the outer wall (violet) was selected as the outlet +(mass outflow in Fluent, inducing the air to pass through the +structure’s walls). For the purposes of these simulations, we are +assuming we have 100% porous walls through which the air flows +at velocity 𝑣𝑓𝑡 (an idealization of the actual nanocardboard +geometry). +b +a +Figure S3: Sample meshing of the axisymmetric sphere simulation in ANSYS Fluent. Here, plot (a) provides an +overall picture of the air box (which is more than ten times larger than the geometry in question in each dimension), +while plot (b) shows a zoomed-in image of the area immediately surrounding the sphere. The size of the outer air +box was not arbitrary, but rather resulted from a series of simulations that gradually increased its dimensions until +force values converged. + + +OuterAir BoX +(coarsemeshelements +-InerAirBox +Cfimrmcsh clements +-AirOut +Airiln +-Axis of SymmetryPage +3 +allowed us to sweep through values ranging from 10-6 m/s to 1 m/s in 7 logarithmically equally spaced +points. + +Summary of Altitude-Dependent Atmospheric Properties +Altitude +0 km +40 km +70 km +Atmospheric Pressure (Pa) +101300 +275.47 +4.66 +Atmospheric Temperature (K) +288 +251 +220 +Air Density (kg/m3) +1.23 +3.83*10-3 +7.38*10-5 +Air Viscosity (Pa * s) +1.796*10-5 +1.610*10-5 +1.447*10-5 + +Table 1: Tabulated altitude-dependent atmospheric conditions for 0 km, 40 km and 70 km. These values were manually +inputted for each simulation set into the Fluent solver. + +We repeated this process 36 times, to construct 18 simulations for the cone, 9 for the sphere and 9 for the +rocket, using operating conditions corresponding to 3 different altitudes (0 km, 40 km and 70 km) and 3 +different geometry sizes. In each case, we computed the reaction force in the axisymmetric direction using +a line integral along the walls of the outer air box, resulting in the force values shown in Figures S4–S7. +This computation made use of the fact that under steady-state operation, the reaction force is equal to the +lift force. The 𝐶1 and 𝐶2 coefficients were then determined by performing a non-linear fitting in MATLAB +to equation (S1), resulting in the values that are shown in the same figures and tabulated in Tables 2-5. In +general, most curves of Figures S4–S7 (in the logarithmic scale) show a transition from the viscous, low- +Re regime to the high-Re regime that is manifested through a change in the slopes of the force curves. +However, at 70 km in altitude, the lift force stayed in the Stokes (low-Re) regime and the high-Re 𝐶2 +coefficients remained uncertain at this particular altitude. Thus, when computing the overall average 𝐶1 and +𝐶2, we did not incorporate the 𝐶2 corresponding to the 70 km altitude. + + +Fitting Parameters for the Rocket, Dia. = 2 cm +Altitude +Length = 1 cm +Length = 5 cm +Length = 10 cm +C1 +C2 +C1 +C2 +C1 +C2 +0 km +2.0 +(1.6–2.4) +1.1 +(0.9–1.3) +1.0 +(0.8–1.2) +1.1 +(0.9–1.2) +0.9 +(0.7–1.1) +1.1 +(0.9–1.2) +40 km +2.24 +(2.12–2.38) +0.73 +(0.62–0.85) +1.1 +(1.0–1.3) +0.8 +(0.6–1.0) +1.0 +(0.9–1.2) +0.8 +(0.6–1.0) +70 km +2.361 +(2.353–2.368) + +1.20 +(1.20–1.20) + +1.08 +(1.08–1.10) + +Average +2.22 +0.91 +1.12 +0.92 +1.00 +0.95 + +Table 2: 𝐶1 and 𝐶2 coefficients computed for the rocket geometry of different lengths (1 cm, 5 cm and 10 cm), alongside +the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry). + + + + + + + +a +b +c +Figure S4: Results from the altitude-dependent rocket simulations in ANSYS Fluent; each data point corresponds +to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) correspond to +different rocket lengths. + +Reactionforcesforvariousflow-throughvelocities +RocketGeometry:Dia.=2cm,Len.=1cm +102 +ANSYS Force (Altitude: 0 km) +Fit:C1-2.00,C2=1.10 +104 +ANSYSForce (Altitude: 40km) +Fit:C1=2.24,C2=0.73 +ANSYSForce (Altitude:70km) +Force (N) +Fit:C1=2.36,C2=0.06 +10-6 +10-8 +10-10 +10-12 +10~6 +104 +102 +100Reactionforcesforvariousflow-throughvelocities +100 +RocketGeometry:Dia.=2cm,Len.5cm +ANSYSForce (Altitude:0km) +Fit:C1-1.04,C2=1.10 +ANSYSForce (Altitude:40km) +Fit:C1=1.14,C2=0.77 +ANSYSForce(Altitude:70km) +Force (N) +Fit:C1=1.20,C2=0.17 +10-5 +10-10 +10-6 +104 +102 +100 +V, (m/s)Reactionforcesforvariousflow-throughvelocities +100 +RocketGeometry:Dia=2cm,Len.=10cm +ANSYSForce (Altitude:0km) +Fit:C1-0.90,C2=1.08 +ANSYSForce(Altitude:40km) +Fit:C1=1.02,C2=0.82 +ANSYSForce(Altitude:70km) +Force (N) +Fit:C1=1.08,C2=0.19 +105 +10-10 +10-6 +104 +102 +V, (m/s) +100Page +4 + +Fitting Parameters for the Sphere, Dia. = 2 cm +Altitude +rout = 0.1 cm +rout = 0.5 cm +rout = 1 cm +C1 +C2 +C1 +C2 +C1 +C2 +0 km +1.4 +(0.7–2.0) +0.29 +(0.21–0.37) +1.5 +(1.3–1.7) +1.06 +(0.95–1.18) +0.9 +(0.8–1.0) +1.5 +(1.4–1.7) +40 km +1.4 +(1.0–1.9) +0.6 +(0.4–0.8) +1.5 +(1.3–1.6) +0.9 +(0.7–1.0) +0.91 +(0.89–0.93) +0.99 +(0.91–1.08) +70 km +1.65 +(1.63–1.67) + +1.58 +(1.52–1.64) + +0.95 +(0.94–0.96) + +Average +1.48 +0.45 +1.50 +0.97 +0.91 +1.26 + +Table 3: 𝐶1 and 𝐶2 coefficients computed for the sphere geometry of different outlet radii (0.1 cm, 0.5 cm and 1 cm), +alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry). + + + + + +Fitting Parameters for the Cone, Dia. = 2 cm +Altitude +Length = 2 cm +Length = 5 cm +Length = 10 cm +C1 +C2 +C1 +C2 +C1 +C2 +0 km +0.7 +(0.5–1.0) +0.9 +(0.7–1.1) +0.7 +(0.5–0.9) +0.9 +(0.8–1.1) +0.7 +(0.4–1.0) +0.9 +(0.7–1.1) +40 km +1.0 +(0.8–1.2) +0.6 +(0.3–0.8) +0.9 +(0.7–1.1) +0.6 +(0.4–0.8) +0.8 +(0.7–1.0) +0.7 +(0.6–0.9) +70 km +1.07 +(0.98–1.16) + +1.01 +(0.94–1.07) + +0.98 +(0.95–1.02) + +Average +0.94 +0.72 +0.88 +0.76 +0.84 +0.82 + +Table 4: 𝐶1 and 𝐶2 coefficients computed for the cone geometry (2 cm diameter) of different lengths (2 cm, 5 cm and +10 cm), alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry). + +a +b +c +Figure S5: Results from the altitude-dependent sphere simulations in ANSYS Fluent; each data point corresponds +to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) correspond to +different sphere outlet radii. +a +b +c +Figure S6: Results from the altitude-dependent cone (2 cm diameter) simulations in ANSYS Fluent; each data point +corresponds to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) +correspond to different cone lengths. + +Reactionforcesfor various flow-through velocities +Sphere Geometry:Dia.=2 cm,r +out +=0.1 cm +100 +ANSYSForce (Altitude:0km) +Fit:C1-1.35,C2=0.29 +ANSYSForce (Altitude; 40km) +Fit:CI-1.43,C2=0.62 +ANSYS Force (Altitude: 70 km) +Fit:C1 1.65,C20.08 +10-5 +Force +1o-to +10-6 +10-4 +102 +100Reactionforcesforvariousflow-throughvelocities +SphereGeometry:Dia.=2cm,r +=0.5 cm +100 +ou +ANSYSForce (Altitude:0km) +Fit:C1=1.46,C2-1.06 +ANSYSForce (Altitude:40km) +Fit:C1=1.47,C2=0.87 +ANSYS Force (Altitude: 70km) +Force (N) +Fit:C1 1.58,C2.0.41 +10-s +10-10 +10~6 +104 +102 +100 +Va(m/s)Reactionforcesforvariousflow-throughvelocities +SphereGeometry:Dia,=2 cm,r +=1 cm +102 +out +ANSYSForce (Altitude:0km) +Fit:C1=0.88,C2=1.54 +104 +ANSYSForce (Altitude:40km) +Fit:C1=0.91,C2=0.99 +" +ANSYSForce (Altitude:70km) +Force (N) +10*6 +Fit:CI=0.95,C2=0.47 +10-8 +10-10 +10~/2 +10-6 +104 +10-2 +100Reaction forcesfor various flow-through velocities +100 +Cone Geometry:Dia.=2 cm, Len.=2cm +ANSYSForce(Altitude:0km) +Fit: C1 =0.74, C2=0.87 +ANSYSForce(Altitude:40km) +Fit:C11.00,C2-0.56 +ANSYSForce(Altitude:70km) +Fit:C1-1.07.C2-0.01 +Force( +10-5 +10-10 +10-6 +104 +10-2 +100Reactionforcesfor various flow-through velocities +100 +Cone Geometry:Dia.=2 cm, Len.=5cm +ANSYSForce (Altitude:0km) +Fit:C1=0.71,C2=0.92 +ANSYSForce(Altitude:40km) +Fit:C10.93,C2-0.60 +ANSYSForce(Altitude:70km) +Fit:C1-1.01.C2-0.01 +Force( +10-5 +10-10 +10-6 +104 +10-2 +100 +Va (m/s)Reactionforcesfor various flow-through velocities +ConeGeometry:Dia.=2cm,l/日@Q价 +100 +ANSYSForce(Altitude:0km) +FitCI=0.69,C2=0.90 +ANSYSForce (Altitude:40km) +Fit:C10.84,C2-0.74 +ANSYSForce(Altitude:70km) +Fit:C10.98,C2-0.10 +Force( +10-5 +10-lo +106 +104 +102 +100Page +5 + +Fitting Parameters for the Cone, Dia. = 4 cm +Altitude +Length = 2 cm +Length = 5 cm +Length = 10 cm +C1 +C2 +C1 +C2 +C1 +C2 +0 km +0.9 +(0.7–1.1) +1.0 +(0.8–1.2) +1.0 +(0.8–1.2) +1.0 +(0.8–1.1) +1.0 +(0.7–1.3) +1.0 +(0.8–1.1) +40 km +1.4 +(1.1–1.7) +0.6 +(0.4–0.9) +1.2 +(1.0–1.3) +0.7 +(0.6–0.9) +1.1 +(1.0–1.2) +0.8 +(0.7–1.0) +70 km +1.5 +(1.3–1.6) + +1.24 +(1.22–1.25) + +1.19 +(1.18–1.20) + +Average +1.27 +0.82 +1.13 +0.86 +1.09 +0.89 + +Table 5: 𝐶1 and 𝐶2 coefficients computed for the cone geometry (4 cm diameter) of different lengths (2 cm, 5 cm and +10 cm), alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry). + + +As we increased in altitude, the value of the 𝐶1 parameter increased while that of 𝐶2 decreased. All in all, +Table 6 below summarizes the average 𝐶1 and 𝐶2 coefficients obtained for each geometry. In all cases, the +coefficients are on the order of 1. + +Average Fitting Parameters for Each Geometry +Geometry +Cone +Sphere +Rocket +D = 2 cm +D = 4 cm +D = 2 cm +D = 2 cm +C1 +0.9 +1.2 +1.3 +1.4 +C2 +0.8 +0.9 +0.9 +0.9 + +Table 6: Fitting parameters for the analytical theory for standard atmospheric conditions on Earth, for each geometry. + +To verify our simulations were based on realistic boundary conditions, we examined the streamline plots +generated in ANSYS Fluent’s results module, a sample of which is shown in Figure S8 below. + +a +b +c +d +Figure S8: Velocity streamlines corresponding to the cone (a, c) and rocket (b, d) geometries simulations in +ANSYS, for a flow-through velocity of 1 m/s and atmospheric conditions corresponding to 0 km in altitude. Both +the cone and rocket have a characteristic dimension (D) of 5 cm. (c) and (d) denote a zoomed-in view of plots (a) +and (b), respectively. +a +b +c +Figure S7: Results from the altitude-dependent cone (4 cm diameter) simulations in ANSYS Fluent; each data +point corresponds to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) +correspond to different cone lengths. + +Velocity +24.059 +18.044 +12.029 +6.015 +0.000 +[ms^-1]19:076 +14307 +4.769 +0.000 +[ms>-1]Reactionforcesforvariousflow-throughvelocities +100 +ConeGeometry:Dia.=4cm, Len.=2cm +ANSYSForce(Altitude:0km) +FitC1=0.90,C2=0.99 +ANSYSForce (Altitude:40km) +Fit:CI=1.42,C2=0.64 +ANSYSForce (Altitude:70km) +Force (N) +Fit:C1-1.48.C2-0.05 +10'5 +10-10 +10-6 +104 +102 +100 +Va(m/s)Reactionforcesforvariousflow-throughvelocities +ConeGeometry:Dia.=4cm,Len.=5cm +100 +ANSYSForce(Altitude:0km) +Fit:CI=0.98,C20.98 +ANSYSForce(Altitude:40km) +FitC=1.16,C2=0.73 +ANSYSForee (Altitude:70km) +Fit:C1=1.24,C2=0.20 +10-5 +Force +10-lo +10-6 +104 +102 +100 +Va (m/s)Reactionforcesforvariousflow-throughvelocities +ConeGeometry:Dia,=4 cm,Len,=10cm +100 +ANSYSForce(Altitude:0km) +Fit:C1-0.97,C2=0.95 +ANSYSForce (Altitude:40km) +Fit:C1=1.11C2=0.83 +ANSYSForce(Altitude:70km) +Fit:CI-1.19,C2-0.22 +10~5 +Force( +10-lo +106 +104 +102 +100Page +6 +As expected, a jet of high-speed air exited the geometries as a result of the air flowing in through the porous +structures. Once the air left the geometry, it interacted with the walls of the outer air box by forming large +vortices, as anticipated for a fluid circulating in a contained box. + +The next section of this document takes the force fitting parameters found from the ANSYS Fluent +simulations and focuses on MATLAB-based parametric optimization of our three different geometries. + + +2. MATLAB Code and Extension of Theoretical Framework + +In this section of the supplementary information, we present the extension to 3D structures of the original +nanocardboard fluid mechanic theory developed by [R3]. The equations derived below were implemented +in a MATLAB code to perform a series of parametric studies that seek to optimize the geometric and porous +parameters of our three study geometries, a cone, a sphere and a rocket. More information about our code +can be found in our publicly available repository [R4]. + +2.1. Derivation of Equations + +2.1.1 General Overview + +For a general 3D porous structure, conservation of mass establishes that + + +𝐴𝑡𝑜𝑡𝑎𝑙𝑣𝑓𝑡 = 𝐴𝑜𝑢𝑡𝑣𝑜𝑢𝑡 . +(S2) + +Here, 𝐴𝑡𝑜𝑡𝑎𝑙 represents the total surface area of the structure (as if the structure had no pores/channels) and +𝑣𝑓𝑡 is the flow-through velocity of the fluid across this surface. Similarly, 𝐴𝑜𝑢𝑡 corresponds to the area +covered by the outlet, while 𝑣𝑜𝑢𝑡 is the exit velocity of the fluid out of the structure. Adding Bernoulli’s +equation, we get the relationship that + + +𝑃𝑖𝑛 − 𝑃𝑜𝑢𝑡 +𝜌 += ∆𝑃 +𝜌 = 𝑣𝑜𝑢𝑡 +2 − 𝑣𝑓𝑡 +2 +2 + . +(S3) + +In (S5), 𝑃𝑖𝑛 is the pressure right at the inlet of the structure, 𝑃𝑜𝑢𝑡 is the pressure right as the jet of fluid is +leaving the structure, located around the space close to 𝐴𝑜𝑢𝑡, while 𝜌 is the fluid density. This equation can +be rearranged to yield an expression for the pressure difference across both ends of the structure, resulting +in + + +∆𝑃 = 𝜌(𝑣𝑜𝑢𝑡 +2 − 𝑣𝑓𝑡 +2) +2 + . +(S4) + +Assuming that the porosity of the 3D structure originates from using the nanocardboard geometry +developed by [R3] as the wall material, then we can model the mass flow rate of the fluid across one of the +structure’s pores (or more properly said, channels) using the following equation + + +𝑚̇ = −𝛼 ∗ ∆𝑃 + 𝛾 ∗ ∆𝑇 . +(S5) + +In (S5), 𝛼 and 𝛾 represent two constants, which take the following forms1: + + +𝛼 = (𝛿 +6 + 1) (1 + 0.25 +√𝛿 +) 𝐴2𝐵𝛽∗ +𝐿 + , +(S6) + +and + +𝛾 = ( +1.1 +1.5 + 𝛿) 𝐴2𝐵𝑃∗𝛽∗ +𝑇∗𝐿 + . +(S7) + + +1 The variables 𝛼 and 𝛾 come from curve-fitting the data from by [R7] and transforming the non-dimensional flow rate equation into +a dimensional form again, with both pressure and temperature contributions. For more information, please see [R2]. + +Page +7 +Here, the variable 𝑃∗ denotes the average pressure2 between the two sides of the structure’s nanocardboard +wall, 𝑇∗ analogously describes the average temperature between both sides of the wall’s surface, while 𝛽∗ +is an inverse velocity parameter. Specifically, this last one is given by + + +𝛽∗ = √ + 𝑚 +2𝑘𝐵𝑇∗ + , +(S8) + +where 𝑘𝐵 is the Boltzmann constant (equal to 1.38 * 10-23 J/K), and m is the mass of an air molecule3. Lastly, +the parameter 𝛿 is the gas rarefaction coefficient, which [R7] defines as + + +𝛿 = √𝜋𝐴 +2𝜆 = √𝜋 +2𝐾𝑛 . +(S9) + +In this expression, 𝜆 is the molecular mean free path, defined as the average distance traveled by a molecule +between collisions with other molecules, and Kn is the Knudsen number, which is characterized in terms +the of channel width. In essence, higher values of the 𝛿 parameter designates flows in the continuum regime, +while smaller values indicate flows taking place in the free molecular regime. As for the molecular mean +free path, mathematically it is usually expressed as + + +𝜆 = 𝜇(𝑇) +𝑃(𝑇) √𝜋𝑘𝐵𝑇 +2𝑚 = 𝜇(𝑇) +𝑃(𝑇) √𝜋𝑅𝑎𝑖𝑟𝑇 +2 + , +(S10) + +where 𝜇(𝑇) is the fluid’s viscosity and P(T) is the operating pressure, both given as a function of T, the +operating temperature. In addition, from equation (S9), we see the Knudsen number is defined as + + +𝐾𝑛 = 𝜆 +𝐴 . +(S11) + +Additionally, as seen in Figure S9 below, the variables A and B characterize the nanocardboard channel’s +width and length, respectively, yielding a cross-sectional area of A x B. In addition, L denotes the channel’s +height. Note that in [R3], A is assumed to be much smaller than B. + +After defining these variables and introducing the expression for the mass flow rate, 𝑚̇ , across one of +nanocardboard’s channels, then an equation can be derived for the average flow-through velocity across +the structure’s surface, which is simply described by + + +𝑣𝑓𝑡 = 𝜑𝑚̇ +𝜌𝐴𝐵 = 𝜑(−𝛼∆𝑃 + 𝛾∆𝑇) +𝜌𝐴𝐵 + . +(S12) + +Here, 𝑚̇ /𝜌 is no other than the volumetric flow rate 𝑉̇ , while the term 𝜑 denotes the geometric fill factor, +which is defined in terms of 𝐴𝑖𝑛 (porous area) and 𝐴𝑡𝑜𝑡𝑎𝑙4, or the channel parameters, and takes the form + + +𝜑 = +𝐴𝑖𝑛 +𝐴𝑡𝑜𝑡𝑎𝑙 += +𝐴𝐵𝑋 +(𝐴𝐵𝑋 + 𝑆𝐵𝑋) = +𝐴 +(𝐴 + 𝑆) . +(S13) + +The latter two equivalencies in (S13) originates from analyzing a single nanocardboard unit cell as opposed +to the full 3D structure. Indeed, as Figure S9 shows, the total cross-sectional area of the cell (if no channels +were present) is given by + + +𝐴𝑐𝑒𝑙𝑙 = (𝐴𝐵𝑋 + 𝑆𝐵𝑋) = (𝐴 + 𝑆)𝐵𝑋 , +(S14) + +where the variable X is just the number of channels in a unit cell. + +2 The value of this variable may be found from performing CFD simulations but will be simply approximated as the operating pressure. +3 The molar mass of air is 0.02896 kg/mol, so then the approximated mass of an air molecule would be 0.02896/(6.022*1023 ) +(Avogadro’s number), or 4.8089 * 10-26 kg. +4 This area is essentially the total 3D structure wall area if there were no channels present. This is analogous to 𝐴𝑐𝑒𝑙𝑙 in the single +nanocardboard unit cell. + +Page +8 + + +However, this number (X) is not arbitrarily chosen, and is dictated by A, B and S in the following way + + +𝑋 = 𝐵 − 𝑆 +𝑆 + 𝐴 . +(S15) + +This expression considers the channel width A and spacing S as a unit, and tries to fit as many of those A ++S units into the channel length B. Nonetheless, we need to consider an additional S for spacing against the +perpendicular channels. This can be seen more clearly in Figure S10 below, where the yellow bars represent +the A +S units, and as drawn, five of these fit in the length of B, after subtracting one S. + +Overall, the flow-through velocity expression provided in (S12) is a step closer towards calculating the lift +force that a 3D structure could generate for a given combination of geometric and channel parameters. +However, computing lift will not be possible until we solve for 𝑣𝑜𝑢𝑡. Therefore, (S12) can be rearranged to +instead solve for another unknown, ∆𝑃 , and obtain + + +∆𝑃 = 𝛾∆𝑇 +𝛼 +− 𝑣𝑓𝑡𝜌𝐴𝐵 +𝛼𝜑 + . +(S16) + +Since both (S16) and (S4) from above provide two distinct expressions for the pressure difference, it is +possible to equate them, giving rise to yet another relationship between 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡, giving + +Figure S9: Main nanocardboard channel parameters. +Figure S10: Illustration of equation (S15), with the yellow bars showing the A + S units fitted into the channel length B. + +Top +Isometric +View +View +Key +Pi, T1 +Air +A:ChannelWidth +Trapped +Side +B:Channel Length +A +Air +View +S:Channel Spacing +L: Channel Height +P2, T2 +Air +t: Alumina ThicknessTop +View +Key +A: Channel Width +B: Channel Length +S:Channel SpacingPage +9 + +𝜌(𝑣𝑜𝑢𝑡 +2 − 𝑣𝑓𝑡 +2) +2 + = ∆𝑃 = 𝛾∆𝑇 +𝛼 +− 𝑣𝑓𝑡𝜌𝐴𝐵 +𝛼𝜑 + . +(S17) + +Rearranging this expression further, we get + + +𝑣𝑜𝑢𝑡 +2 = 2 +𝜌 (𝛾∆𝑇 +𝛼 +− 𝑣𝑓𝑡𝜌𝐴𝐵 +𝛼𝜑 +) + 𝑣𝑓𝑡 +2 . +(S18) + +Now, recalling the conservation of mass relationship provided in (S2), it is possible to write 𝑣𝑓𝑡, the flow- +through velocity across the channels, in terms of 𝑣𝑜𝑢𝑡 + + +𝑣𝑓𝑡 = 𝐴𝑜𝑢𝑡 +𝐴𝑡𝑜𝑡𝑎𝑙 +𝑣𝑜𝑢𝑡 = 𝜑𝐴𝑜𝑢𝑡 +𝐴𝑖𝑛 +𝑣𝑜𝑢𝑡. +(S19) + +Thus, (S19) can replace the 𝑣𝑓𝑡 term in (S18), leaving everything in terms of just 𝑣𝑜𝑢𝑡 + + +𝑣𝑜𝑢𝑡 +2 = 2 +𝜌 (𝛾∆𝑇 +𝛼 +− 𝐴𝑜𝑢𝑡𝑣𝑜𝑢𝑡𝜌𝐴𝐵 +𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑 +) + ( 𝐴𝑜𝑢𝑡 +𝐴𝑡𝑜𝑡𝑎𝑙 +) +2 +𝑣𝑜𝑢𝑡 +2. +(S20) + +Further manipulating (S20), we get the following quadratic + + +𝑣𝑜𝑢𝑡 +2 (1− ( 𝐴𝑜𝑢𝑡 +𝐴𝑡𝑜𝑡𝑎𝑙 +) +2 +) + 𝑣𝑜𝑢𝑡 (2𝐴𝑜𝑢𝑡𝐴𝐵 +𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑) − 2𝛾∆𝑇 +𝜌𝛼 += 0 , +(S21) + +which has precisely 𝑣𝑜𝑢𝑡 as its only unknown. The coefficients of this polynomial are + + +𝑎 = 1− ( 𝐴𝑜𝑢𝑡 +𝐴𝑡𝑜𝑡𝑎𝑙 +) +2 +, +𝑏 = 2𝐴𝑜𝑢𝑡𝐴𝐵 +𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑 , +𝑐 = − 2𝛾∆𝑇 +𝜌𝛼 , +(S22) + +making it a fairly straightforward process to solve for the roots of the equation, provided by + + +𝑣𝑜𝑢𝑡 = +− (2𝐴𝑜𝑢𝑡𝐴𝐵 +𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑) ± √(2𝐴𝑜𝑢𝑡𝐴𝐵 +𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑) +2 ++ 8𝛾∆𝑇 +𝜌𝛼 (1− ( 𝐴𝑜𝑢𝑡 +𝐴𝑡𝑜𝑡𝑎𝑙) +2 +) + 2 (1− ( 𝐴𝑜𝑢𝑡 +𝐴𝑡𝑜𝑡𝑎𝑙) +2 +) + . +(S23) + +One underlying advantage of this derivation was that it removed the need to know the pressure difference, +∆𝑃, while providing us with enough information to solve for 𝑣𝑜𝑢𝑡 and 𝑣𝑓𝑡. In the following sub-section, we +deliver more details on the heat conduction modeling across the nanocardboard’s thickness, which enabled +obtaining an expression for the temperature difference, ∆𝑇, necessary to solve for 𝑣𝑜𝑢𝑡 in (S23). + +2.1.2 Heat Conduction Modeling + +2.1.2.1 Full Analytical Derivation for ∆𝑻 + +In order to compute ∆𝑇, the temperature difference between both sides of the structure’s walls, we needed +to model the heat conduction across the structure’s thickness. We performed a heat energy balance that +considered heat transfer across three distinct cross-sectional areas: the channel’s column of air, across the +alumina thickness of the channel, and across the air trapped within the structure, as shown in Figure S11 +below. As a result, we can let 𝑄𝑡, the total heat transfer, be + + +𝑄𝑡 = ∆𝑇 +𝑅𝑡1 ++ ∆𝑇 +𝑅𝑡2 ++ ∆𝑇 +𝑅𝑡3 + , +(S24) + +where the 𝑅𝑡1, 𝑅𝑡2 and 𝑅𝑡3 represent the thermal resistances under the three scenarios detailed above. + +Page 10 + +For the first of these areas (A1), the column of air in the channel, we define its thermal resistance as + + +𝑅𝑡1 = +𝐿 +𝑘𝑎𝑖𝑟𝐴1𝑋 = +𝐿 +𝑘𝑎𝑖𝑟𝐴𝐵𝑋 , +(S25) + +where 𝑘𝑎𝑖𝑟 is the thermal conductivity of air, L is as usual the channel height, and 𝐴𝐵𝑋 is the cross-sectional +area of the individual channels multiplied by the number of channels in a unit cell, as shown in Figure S9 +above. Notice that 𝜅𝑎𝑖𝑟 is both temperature and pressure dependent, as the equation developed by [R10] +captures, specifically for the small MEMS scale: + + +𝜅𝑎𝑖𝑟 = +𝜅0 +(1 + 0.00076𝑇 +𝑃𝐿 +) + . +(S26) + +In this expression, 𝜅0 is the air conductivity at standard atmospheric conditions, normally quoted as 𝜅0 = +0.024 +𝑊 +𝑚 𝐾. Another comparable and slightly more succinct model for the conductivity of air is from [R8]: + + +𝜅𝑎𝑖𝑟 = +𝜅0 +(1 + 3.116𝜆 +𝐿 +) + +(S27) + +As the pressure decrease, the mean free path eventually becomes comparable to the channel length, and the +effective conductivity starts to decrease below the continuum value. Both equations (S26) and (S27) yielded +very similar values for the conductivity of air as a function of the channel thickness L, although we used +Eq. S27 in the simulations. + +Continuing with the heat conduction modeling, the corresponding expression for the thermal resistance +across the alumina thickness on the channels (area A2 in Figure S11) is given by + + +𝑅𝑡2 = +𝐿 +𝑘𝑎𝑙𝑑𝐴2𝑋 = +𝐿 +𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 , +(S28) + +where [(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 is the cross-sectional area occupied by the alumina thickness of the +channels, which is denoted as 𝑡. In (S28), 𝑘𝑎𝑙𝑑 is the thermal conductivity of alumina, which has a constant +value of 1.8 +𝑊 +𝑚 𝐾 [R2]. Lastly, the thermal resistance of the air trapped within the structure (area A3) is + + +𝑅𝑡3 = 𝐿 − 2𝑡 +𝑘𝑎𝑖𝑟𝐴3 += +𝐿 − 2𝑡 +𝑘𝑎𝑖𝑟 [𝐴𝐵 +𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋 + , +(S29) + +Figure S11: Main nanocardboard cross-sectional areas for which thermal resistance is calculated. + +Isometric +View +Top +View +Key +Ai:Channelcross-sectionalarea +A2:ChannelAluminathicknesscross-sectionalarea +SectionCut +As:Cross-sectional area oftrappedairwithinnanocardboardPage 11 +where recall from (S13) that +𝐴𝐵𝑋 +𝜑 is the full area of the cell, from which we subtract the combined cross- +sectional area of the channels with thickness 𝑡 of alumina. Now, performing an energy balance, the heat +flow through the structure’s walls must be equal to that from the absorbed irradiation of the sun, which in +this case is given by + +𝑄𝑡 = 𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 +𝜑 ) (1 − 𝜑) . +(S30) + +In equation (S30), 𝜀 denotes the absorption coefficient (approximated to 0.9 based-off the measurements +from [R3]), 𝜓 the proportion of absorbed optical flux dissipated upward through the nanocardboard (which +is assumed to be 0.5 or 50%), and 𝐼𝑠𝑢𝑛 the intensity of the sun at a particular altitude. In particular, this last +term can be modeled using the following equation + + +𝐼𝑠𝑢𝑛 = 1000 + 3.8ℎ , +(S31) + +where the variable h refers to the elevation above sea level in kilometers. Notice that this expression returns +the sun’s intensity in units of Watts per meter square. Furthermore, in equation (S30), +(𝐴𝐵𝑋/𝜑)(1 − 𝜑) corresponds to the solid area of the nanocardboard, 𝐴𝑠𝑜𝑙𝑖𝑑, where the sun’s irradiation +is absorbed. In any case, (S24) through (S31) were combined to write a general expression for ∆𝑇, which +is summarized by + + +∆𝑇 = 𝑇2 − 𝑇1 = +𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 +𝜑 ) (1 − 𝜑) +1 +𝑅𝑡1 + 1 +𝑅𝑡2 + 1 +𝑅𝑡3 += + += +𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 +𝜑 ) (1 − 𝜑) +𝑘𝑎𝑖𝑟𝐴𝐵𝑋 +𝐿 ++ 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 +𝐿 ++ +𝑘𝑎𝑖𝑟 [𝐴𝐵 +𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋 +𝐿 − 2𝑡 + . + +(S32) + +In (S32), 𝑇1 and 𝑇2 represent the average temperatures outside and inside the 3D structure, respectively. +However, these might not necessarily be known beforehand, reason why calculating ∆𝑇 or 𝑇∗, the average +temperature between both sides of the surface, may not be as trivial. In particular, to compute 𝑇∗, we make +use of the fact that we know what ∆𝑇 is from (S32) and take the following expression + + +𝑇∗ = 𝑇1 + 𝑇2 +2 += (𝑇2 − 𝑇1) + 2 ∗ 𝑇1 +2 += ∆𝑇 + 2 ∗ 𝑇1 +2 + . +(S33) + +Here, notice that 𝑇1 is simply equal to the temperature corresponding to the particular operating conditions +(altitude, pressure, density) of the fluid. Overall, ∆𝑇 allows us to solve for 𝑇∗ (which is needed to compute +𝛾 and 𝛽∗ in (S7) and (S9), respectively) and the last part of the puzzle in (S23), the 𝑣𝑜𝑢𝑡 expression. + +2.1.2.2 Simplified Expression for ∆𝑻 in the limit of zero alumina thickness + +Beyond the derivation provided in 1.2.1, notice that one could potentially also approximate ∆𝑇 through a +more simplified expression given by + + +∆𝑇~ 𝐿𝐼𝑠𝑢𝑛(1 − 𝜑) +2𝜅𝑎𝑖𝑟 + . +(S34) + +The origin of (S34) comes from taking the limit as t, the alumina thickness, approaches zero, in equation +(S32). Indeed, + + +lim +𝑡→0 +𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 +𝜑 ) (1 − 𝜑) +𝑘𝑎𝑖𝑟𝐴𝐵𝑋 +𝐿 ++ 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 +𝐿 ++ +𝑘𝑎𝑖𝑟 [𝐴𝐵 +𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋 +𝐿 − 2𝑡 + + +(S35) + +Page 12 += lim +𝑡→0 +𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 +𝜑 ) (1 − 𝜑) +𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 + 𝑘𝑎𝑖𝑟 [𝐴𝐵 +𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋 + + += lim +𝑡→0 +𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 +𝜑 ) (1 − 𝜑) +𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑙𝑑[𝐴𝐵 − 𝐴𝐵]𝑋 + 𝑘𝑎𝑖𝑟 [𝐴𝐵 +𝜑 − 𝐴𝐵] 𝑋 + + += lim +𝑡→0 +𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 +𝜑 ) (1 − 𝜑) +𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑖𝑟 𝐴𝐵𝑋 +𝜑 +− 𝑘𝑎𝑖𝑟𝐴𝐵𝑋 += +𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 +𝜑 ) (1 − 𝜑) +𝑘𝑎𝑖𝑟 𝐴𝐵𝑋 +𝜑 + + += 𝐿𝜀𝜓𝐼𝑠𝑢𝑛(1 − 𝜑) +𝑘𝑎𝑖𝑟 + . + + +Furthermore, letting 𝜀 = 1 and 𝜓 = 0.5, then (S37) indeed becomes equation (S34) from above. As +evidenced by its compressed form, using (S36) to approximate ∆𝑇 simplifies the process of solving for the +flow-through velocity, 𝑣𝑓𝑡. This is especially true if we were to also neglect the pressure term, assuming its +contribution is negligible. As a result, the mass flow rate from (S5) can be re-written as + + +𝑚̇ ~𝛾 ∗ ∆𝑇 . +(S36) + +This helps reduce the flow-through velocity expression to + + +𝑣𝑓𝑡 = 𝜑𝑚̇ +𝜌𝐴𝐵 = 𝜑 𝛾∆𝑇 +𝜌𝐴𝐵 = 𝜑 𝛾 +𝜌𝐴𝐵 +𝐿𝐼𝑠𝑢𝑛(1 − 𝜑) +2𝜅𝑎𝑖𝑟 + . +(S37) + +Even this expression can be further simplified by reducing the 𝛾 term from (S7) to + + +𝛾~ 1.1𝐴2𝐵𝑃∗𝛽∗ +𝛿𝑇∗𝐿 += 1.1𝐴2𝐵𝑃𝛽∗ +𝑇𝐿𝐴√𝜋/(2𝜆) = 2.2𝜆𝐴𝐵𝑃 +√𝜋𝑇𝐿 +√ + 𝑚 +2𝑘𝐵𝑇 . +(S38) + +From the ideal gas law, we have that 𝑃 = 𝜌𝑅𝑎𝑖𝑟𝑇, so the pressure term can be replaced in (S38) to obtain + + +𝛾~ 2.2𝜆𝐴𝐵𝜌𝑅𝑎𝑖𝑟𝑇 +√𝜋𝑇𝐿 +√ + 𝑚 +2𝑘𝐵𝑇 = 2.2𝜆𝐴𝐵𝜌𝑅𝑎𝑖𝑟 +√𝜋𝐿 +√ + 𝑚 +2𝑘𝐵𝑇 . +(S39) + +Combining equations (S37) and (S39), we resultant expression turns out as + + +𝑣𝑓𝑡 = 𝜑 +𝜌𝐴𝐵 +𝐿𝐼(1 − 𝜑) +2𝜅𝑎𝑖𝑟 +2.2𝜆𝐴𝐵𝜌𝑅𝑎𝑖𝑟 +√𝜋𝐿 +√ + 𝑚 +2𝑘𝐵𝑇 = 1.1𝜑𝐼(1 − 𝜑)𝜆𝑅𝑎𝑖𝑟 +𝜅𝑎𝑖𝑟 +√ + 𝑚 +2𝑘𝐵𝑇𝜋 +(S40) + +Now, recall that the average molecular velocity is equal to + + +𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇 +𝜋 + , +(S41) + +and the relationship between viscosity and velocity, as provided by [R6], is equal to + + +𝜇 = 𝜆𝜌𝑣𝑎𝑣𝑔 +2 + . +(S42) + + +Page 13 +Hence, combining both (S41) and (S42) and solving for 𝜆, we obtain an expression which can be +incorporated in (S40) to yield + + +𝑣𝑓𝑡 = 1.1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 +𝜅𝑎𝑖𝑟 +𝜇 +𝑃 √𝜋𝑘𝐵𝑇 +2𝑚 √ + 𝑚 +2𝑘𝐵𝑇𝜋 = 1.1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 +𝜅𝑎𝑖𝑟 +𝜇 +𝑃 √ 𝑚𝜋𝑘𝐵𝑇 +4𝑘𝐵𝑇𝜋𝑚 + += 1.1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 +𝜅𝑎𝑖𝑟 +𝜇 +𝑃 √ 1 +4 = 1.1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 +2𝜅𝑎𝑖𝑟 +𝜇 +𝑃 . +(S43) + +Now, according to [R6], the conductivity of air can be often approximated as 𝜅𝑎𝑖𝑟 = +2𝜇𝐶𝑣′ +𝑀 += 2𝜇𝐶𝑣, where +M is the molar mass of air and 𝐶𝑣′ is the specific heat capacity at constant volume, in units of J/k mol. Thus, +equation (S43) can further simplify into + + +𝑣𝑓𝑡 = 1.1𝜑𝐼(1 − 𝜑)𝑀𝑅𝑎𝑖𝑟 +4𝜇𝐶𝑣′ +𝜇 +𝑃 = 1.1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 +4𝑃𝐶𝑣 += 1.1𝜑(1 − 𝜑)𝑅𝑎𝑖𝑟 +4𝐶𝑣 +𝐼 +𝑃 = 𝐶 𝐼 +𝑃 . +(S44) + +where the constant C is simply given by + + +𝐶 = 1.1𝜑(1 − 𝜑)𝑅𝑎𝑖𝑟 +4𝐶𝑣 += 1.1 ∗ 0.5 ∗ (1 − 0.5) ∗ 0.287 +4 ∗ 0.718 += 0.0275 . +(S45) + +Hence, what these series of derivations shows is that it is possible to approximate and obtain order-of- +magnitude estimations of the flow-through velocity by using + + +𝑣𝑓𝑡 = 0.0275 𝐼 +𝑃 . +(S46) + +2.2. Lift-Force Calculations and Temperature-dependencies + +Once we knew how to calculate 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡 using the equations derived above (whether it is in the +simplified or full analytical form), we used the following equation to calculate the lift forces produced by +each geometry, as outlined in the ANSYS simulations section at the beginning of this document: + + +∑𝐹 = 𝐶1 ∗ 8 ∗ 𝜇 ∗ 𝐷 ∗ 𝑣𝑓𝑡 + 𝐶2 ∗ 𝜌 ∗ 𝐴𝑜𝑢𝑡 ∗ 𝑣𝑜𝑢𝑡 +2 . +(S47) + +Here, R is the characteristic radius of the geometry (usually the inlet radius), while 𝜇 is the viscosity and 𝜌 +the fluid density. In addition, C1 and C2 are the geometry dependent coefficients summarized in Table 6. + +As the derivation of equations above evidences, all of the geometric (𝐴𝑡𝑜𝑡𝑎𝑙 and 𝐴𝑜𝑢𝑡) and channel (A, B, +L, S, t) variables are present in (S23), meaning that it was possible to construct parametric studies exploring +the dependency of 𝑣𝑓𝑡, and consequentially lift, on all of these. Notice, all of these variables were largely +independent of each other, making it possible to modify each separately. However, some other parameters +within (S23), such as 𝐼𝑠𝑢𝑛, density 𝜌, and air viscosity 𝜇, were actually dependent on temperature, which +in turn was also altitude dependent. As a result, in order to accurately calculate the flow-through velocities +𝑣𝑓𝑡 experienced by a 3D geometry in a range of altitudes, we needed to derive expressions for +approximating the air temperature, air pressure, air viscosity and air density as a function of altitude itself. + +2.2.1 Temperature-dependent Relations + +We developed the relations characterizing the dependency between temperature and the fluid variable in +question by using standard atmospheric5 empirical data and fitting equations to it. For instance, for the data +describing the dependency between air temperature and altitude, we fit both a 6th, 10th and 15th order + +5 The specific standard atmospheric data was taken from the following three websites: +https://www.engineeringtoolbox.com/standard-atmosphere-d_604.html | https://www.pdas.com/atmosTable1SI.html +https://www.pdas.com/bigtables.html + +Page 14 +polynomial, as Figure S12 to the below shows. Overall, the 15th order polynomial provided the best +empirical fit, which was why we decided to use it for the rest of this analysis. However, one interesting +aspect of this fit was that we actually fitted at the inverse of the temperature, the reason for which will +become clearer in the derivation of the altitude-pressure dependency. In any case, equation (S48) below +shows this explicit relation, with h (the altitude) being in kilometers, and all terms in the column added. + + +𝑇−1(ℎ) = +−4.592 ∗ 10−29 +4.023 ∗ 10−27 +1.491 ∗ 10−23 +−7.942 ∗ 10−21 +2.021 ∗ 10−18 +−3.152 ∗ 10−16 +3.271 ∗ 10−14 +−2.332 ∗ 10−12 +1.150 ∗ 10−10 +−3.862 ∗ 10−09 +8.525 ∗ 10−08 +−1.150 ∗ 10−06 +8.154 ∗ 10−06 +−2.283 ∗ 10−05 +9.912 ∗ 10−05 +3.473 ∗ 10−03 +∙ +ℎ15 +ℎ14 +ℎ13 +ℎ12 +ℎ11 +ℎ10 +ℎ9 +ℎ8 +ℎ7 +ℎ6 +ℎ5 +ℎ4 +ℎ3 +ℎ2 +ℎ1 +1 . + +(S48) + +Having derived the empirical relation between temperature (its inverse) and altitude, it was possible to +determine a similar expression for pressure. In essence, the differential equation describing the pressure- +altitude relationship is given by + + +𝑑𝑃(ℎ) = −𝑔 ∗ 𝜌(ℎ) ∗ 𝑑ℎ , +(S49) + +where 𝑔 is the gravitational constant on earth, and 𝜌(ℎ) the density of air at a particular altitude h. Using +the ideal gas law, 𝜌(ℎ) can be substituted to yield the following expression for the above differential in +equation (S49) + + +𝑑𝑃(ℎ) = −𝑔 ∗ +𝑃(ℎ) +𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) ∗ 𝑑ℎ , +(S50) + +where now 𝑅𝑎𝑖𝑟 is the ideal gas constant of air and is equal to 287 𝐽/𝑘𝑔 ∗ 𝑚3. Easily enough, one can +utilize the technique of separation of variables to obtain that + + +𝑑𝑃(ℎ) +𝑃(ℎ) = +−𝑔 +𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) ∗ 𝑑ℎ , +(S51) + +which leaves all of the pressure terms on one side, and the rest on the other. As a result, it is possible to see +with more clarity why the above polynomial fit was done for the inverse of temperature. Indeed, equation +(S51) can be equivalently written as + + +𝑑𝑃(ℎ) +𝑃(ℎ) = −𝑔 ∗ 𝑇−1(ℎ) +𝑅𝑎𝑖𝑟 +∗ 𝑑ℎ . +(S52) + +This expression can be easily integrated to obtain the following logarithm: + + +ln(𝑃) = −𝑔 +𝑅𝑎𝑖𝑟 +∗ ∫ 𝑇−1(ℎ) ∗ 𝑑ℎ . +(S53) + +Letting 𝜁(ℎ) = ∫ 𝑇−1(ℎ) ∗ 𝑑ℎ be a placeholder for the integral of the inverse temperature polynomial and +C be simply a constant of integration, we obtain that + + +ln(𝑃(ℎ)) = −𝑔 +𝑅𝑎𝑖𝑟 +∗ 𝜁(ℎ) + 𝐶 . +(S54) + +Figure S12: Modeled temperature dependency on altitude. + +Fitsto altitude-dependentT +5.5+10~3 +5 +4.5 +4 +3.5 +Data +3 +Order6polynomial +Order10polynomial +Order15polynomial +2.5 +0 +20 +40 +60 +80 +100 +120 +A/titude(km)Page 15 +Figure S13: Modeled pressure dependency on altitude. +Now, in order to remove the logarithm from the pressure, we can raise both sides of the expression to the +Euler’s number power, and get + +𝑃(ℎ) = 𝑒 +−𝑔 +𝑅𝑎𝑖𝑟∗𝜁(ℎ)+𝐶 . +(S55) + +After applying exponent rules, (S55) decomposes into the product given by + + +𝑃(ℎ) = 𝑒𝐶 ∗ 𝑒 +−𝑔 +𝑅𝑎𝑖𝑟∗𝜁(ℎ) , +(S56) + +and can be further simplified, upon application of boundary conditions, into + + +𝑃(ℎ) = 101300 𝑃𝑎 ∗ 𝑒 +−𝑔 +𝑅𝑎𝑖𝑟∗𝜁(ℎ) , + +(S57) + +which takes the following full form: + +𝑃(ℎ) = 101300 𝑃𝑎 ∗ exp +[ + + + + + + + + + + + + + + + +−𝑔 +𝑅𝑎𝑖𝑟 +∗ +( + + + + + + + + + + + + + +−2.870 ∗ 10−30 +2.682 ∗ 10−28 +1.064 ∗ 10−24 +−6.109 ∗ 10−22 +1.684 ∗ 10−19 +−2.865 ∗ 10−17 +3.271 ∗ 10−15 +−2.591 ∗ 10−13 +1.437 ∗ 10−11 +−5.518 ∗ 10−10 +1.421 ∗ 10−08 +−2.299 ∗ 10−07 +2.0385 ∗ 10−06 +−7.608 ∗ 10−06 +4.955 ∗ 10−05 +3.473 ∗ 10−03 +∙ +ℎ16 +ℎ15 +ℎ14 +ℎ13 +ℎ12 +ℎ11 +ℎ10 +ℎ9 +ℎ8 +ℎ7 +ℎ6 +ℎ5 +ℎ4 +ℎ3 +ℎ2 +ℎ1 ) + + + + + + + + + + + + + +] + + + + + + + + + + + + + + + + . +(S58) + + + +As Figure S13 above shows, the agreement of this equation with the empirical data is very reasonable, +especially below 80 km altitude. Above 80 km, the atmosphere is no longer well mixed, has increasing +concentrations of atomic oxygen, and the simple ideal gas law we used above no longer applies. For this +reason, the results that will be presented below correspond to altitudes below 80 km. + +The next step was modelling the air density dependency on altitude. With expressions for T(h) and P(h) +above, we could use the ideal gas law to write + + +Finally, the last dependency that remained to be defined was the air viscosity and altitude relation. To that +end, we could use Sutherland’s law, which relates viscosity and temperature through the following equation: + + +𝜇(ℎ) = 𝜇𝑟𝑒𝑓 ∗ (𝑇(ℎ) +𝑇𝑟𝑒𝑓 +) +1.5 +∗ ( +𝑇𝑟𝑒𝑓 + 𝑆 +𝑇(ℎ) + 𝑆) , +(S60) + +where 𝜇𝑟𝑒𝑓 is the reference dynamic viscosity and 𝑇𝑟𝑒𝑓 the reference temperature. In this work, for air, at +𝑇𝑟𝑒𝑓 = 20 𝐶, we have that 𝜇𝑟𝑒𝑓 = 0.000018205 𝑃𝑎 ∗ 𝑠. Finally, S is a constant, known as Sutherland’s +temperature, which is given by 110.4 K. + + + + + + +𝜌(ℎ) = +𝑃(ℎ) +𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) . +(S59) + +DerivedEquationforPressure +106 +Data +DerivedEquation +104 +(Pa) +Pressure +100 +102 +104 +0 +20 +40 +60 +80 +100 +120 +A/titude (km)Page 16 +2.2.2 Payload Calculations + +Once all of the required equations and relationships were derived, it was possible to calculate 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡 +for a specific set of geometric and channel parameters defining unique 3D structures. By calculating these +velocities, we determined the total force produced by each geometry, as outlined by equation (S47), from +which it was possible to perform some payload estimates. However, in order to obtain the payload estimates, +it was paramount to first determine the surface areas of each one of the 3D geometries in question, the +reason being that density of these structure was defined in areal terms as opposed to volumetric terms. As +was mentioned in the main paper, this work considered a truncated cone, truncated sphere, and a rocket, +and their defining equations are shown in Table 7 below. + +Main Geometrical Area Definitions +Area +Truncated Cone +Truncated Sphere +Rocket +𝐴𝑡𝑜𝑡𝑎𝑙 +𝜋 (𝐷 +2) +2 ++ 𝜋 (𝐷 +2)ℎ2 − 𝜋𝑟(ℎ2 − ℎ1) +𝜋(𝐷2 − 2𝑟ℎ) +2𝜋𝑟(𝑟 + 𝐷) +𝐴𝑜𝑢𝑡 +𝜋𝑟2 +𝐴𝑖𝑛 +𝜑𝐴𝑡𝑜𝑡𝑎𝑙 +𝐴𝑠𝑜𝑙𝑖𝑑 +(1 − 𝜑)𝐴𝑡𝑜𝑡𝑎𝑙 +Special +Variables +ℎ1 = √(𝐷 +2 − 𝑟) +2 ++ 𝐷2 +ℎ2 = √(𝐷 +2) +2 ++ ℎ3 +ℎ3 = +𝐷2 +(𝐷 − 2𝑟) +ℎ = (𝐷 +2) − √(𝐷 +2) +2 +− 𝑟2 +N/A + +Table 7: Area definitions used across this work for the cone, sphere and rocket. Notice that here, the variable ℎ3 +follows from using similar triangles analysis, and letting ℎ3/(D/2) = D/(D/2 – r). For all three geometries, the variable +D represents the overall scale of the structure while r their outlet radius. Notice that 𝐴𝑖𝑛 is the porous area, while +𝐴𝑠𝑜𝑙𝑖𝑑 is the solid area in which the sun’s irradiance is absorbed, and it follows that 𝐴𝑡𝑜𝑡𝑎𝑙 = 𝐴𝑠𝑜𝑙𝑖𝑑 + 𝐴𝑖𝑛. + +As a result, having defined these surface areas (using the parameters established in Figure S1), we +calculated the mass of our three 3D structures. In particular, since the cross-sectional area of a channel is +simply 𝐴𝐵, then one can define the number of channels as the following integer floor: + + +𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 = ⌊𝐴𝑖𝑛 +𝐴𝐵⌋ . +(S61) + +The number of channels, 𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠, is an important parameter, given that now it is possible to calculate the +volume of the structure that is occupied by the deposited alumina around each channel, which has thickness +t and relatively high density 𝜌𝑎𝑙𝑑 of 3950 kg/m3 [R9]. Indeed, similarly to equation (S28) above, we can +define this volume as + + +𝑉𝑎𝑙𝑑,𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 = 𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠(𝐿 − 2𝑡)[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵] . +(S62) + +Experimentally, it has already been found that the areal density of nanocardboard, 𝜎𝑛𝑐𝑏, is about 1 g/m2 +[R5], but this corresponds to a value of L (nanocardboard thickness) equal to 50 μm. However, in our +parametric studies, as we sweep through various values of L, especially those that are larger than 50 μm, +this areal density alone is not enough to estimate the weight of the structure. As a result, calculating the +volume of alumina around each of the channels is paramount, since the structure naturally becomes heavier +with increasing thickness. Hence, the overall mass of any one of these geometries will be given by + + +𝑚𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 = 𝜎𝑔𝑒𝑜𝑚(𝐴𝑠𝑜𝑙𝑖𝑑 − 𝐴𝑖𝑛) + 𝜌𝑎𝑙𝑑𝑉𝑎𝑙𝑑,𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 , +(S63) + +where this expression accounts both for the areal density (𝜎𝑔𝑒𝑜𝑚) and the increases in the amount of the +deposited alumina as a result of changes in the wall thickness L. Thus, the net lift produced by the geometry +is simply given by subtracting the structure’s weight from the force expression in (S47), or + + +Page 17 + +𝐿𝑖𝑓𝑡𝑛𝑒𝑡 = 𝐹 − 𝑔𝑚𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 . +(S64) + +While we know from simulations what 𝜎𝑔𝑒𝑜𝑚 is, notice that it is also possible to use our equations and a +series of approximations to obtain a theoretical upper bound for this value. In essence, we can start by +letting the force be equal to the expression below + + +𝐹 = 𝑚̇ 𝑣𝑜𝑢𝑡 = (𝐴𝑖𝑛𝜌𝑎𝑖𝑟𝑣𝑓𝑡)𝑣𝑜𝑢𝑡 = (𝐴𝑖𝑛 +𝑃 +𝑅𝑎𝑖𝑟𝑇 𝑣𝑓𝑡)𝑣𝑜𝑢𝑡 , +(S65) + +which incorporates mass flow rate and the ideal gas law. Now, recall that equation (S4) provides an +expression relating 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡, while (S46) provides a simplified approximation for 𝑣𝑓𝑡. As a result, +taking a conservative approach that lets 𝑣𝑜𝑢𝑡 = 0.2𝑣𝑎𝑣𝑔, a fifth of the average molecular velocity of a gas, +shown in (S41) above, and incorporating (S2) and (S46), it is possible to re-write (S68) to obtain + + +𝐹 = 𝐴𝑖𝑛 +𝑃 +𝑅𝑎𝑖𝑟𝑇 𝑣𝑓𝑡0.2√8𝑅𝑎𝑖𝑟𝑇 +𝜋 += += 0.0055𝐴𝑖𝑛 +𝑃 +𝑅𝑎𝑖𝑟𝑇 +𝐼 +𝑃 √8𝑅𝑎𝑖𝑟𝑇 +𝜋 += 0.0055𝐴𝑖𝑛 +𝐼 +𝑅𝑎𝑖𝑟𝑇 √8𝑅𝑎𝑖𝑟𝑇 +𝜋 + . + +(S66) + +Upon further simplification, equation (S69) reduces to + + +𝐹 = 0.0055𝐴𝑖𝑛 +𝐼 +𝑅𝑎𝑖𝑟𝑇 √8𝑅𝑎𝑖𝑟𝑇 +𝜋 += 0.0055√8 +𝜋 𝐴𝑖𝑛𝐼√ +1 +𝑅𝑎𝑖𝑟𝑇 . +(S67) + +Thus, the maximum areal density that can be entertained by these 3D structures can be approximated by + + +𝜎𝑔𝑒𝑜𝑚 = +𝐹 +𝐴𝑖𝑛𝑔 = 0.0055√8 +𝜋 +𝐼 +𝑔 √ +1 +𝑅𝑎𝑖𝑟𝑇 = 𝐾𝐼√ +1 +𝑅𝑎𝑖𝑟𝑇 = 0.016 +𝐼 +𝑣𝑎𝑣𝑔𝑔 , +(S68) + +where 𝐾 = +0.0055 +𝑔 +√ +8 +𝜋 = 0.0009 and 𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇/𝜋 ≈ 400 m/s is the average velocity of air +molecules. Upon inserting the parameters, we find that 𝜎𝑔𝑒𝑜𝑚 can have an average value of 0.004 kg/m2, +four times of what the areal density of nanocardboard typically is in experiments. The main paper provides +additional areal density calculations based off from the parametric studies (detailed below) as well as cloud +plots denoting the maximum areal density for each of the study geometries. They are generally of the same +order of magnitude as the estimate (S68). + +2.2.3 Parametric Studies + +In this section, we provide four tables that accompany the presentation of the results shown in the main +paper. In essence, Table 8 both summarizes the chosen optimization ranges and discretization for the +variables that were varied (A, L and r) and specifies the values that the remaining variables (B, N, X, S and +t) took. Similarly, Table 9 through Table 11 present the results for the performed parametric optimization +on the three geometries, detailing the specific combination of A, L and r that first, yielded the maximum +payload capabilities and second, achieved flight at the lower altitude. In addition, Table 9 through Table +11 also provide the areal density of each structure for when the maximum payload was achieved. Notice +that this process was repeated for multiple values of D, as to explore the dependency of the overall +optimization results with the scale of the geometries. + +Parametric Optimization Variables +Variable +Range +Truncated Cone +Truncated Sphere +Rocket +Discretization +𝐴 +Min. +10 nm +80 equally spaced points +(log scale) +Max. +5 mm +𝐿 +Min. +1 μm +80 equally spaced points + +Page 18 + +Table 8: Main values used across the various variables during the parametric optimization. As can be seen, the search +range for the optimal A, L and r was discretized in all three cases in 100 points, following a log scale. Changing the +granularity of the discretization or the bounds of the search range did not significantly modify the results seen in Table +9 through Table 11 below. + + +Table 9: Combinations of A, L and r that returned the spheres capable of carrying the greatest payload and achieving +flight at the lowest altitude, for various values of D, as specified in Figure S1. + + +Table 10: Combinations of A, L and r that returned the cones capable of carrying the greatest payload and achieving +flight at the lowest altitude, for various values of D, as specified in Figure S1. + + +Max. +1 cm +(log scale) +𝑟 +Min. +rmin = D/20 (see Table 9 through Table 12) +80 equally spaced points +(log scale) +Max. +rmax = D/2.01 (see Table 9 through Table 12) +Altitude +Min. +0 km +17 equally spaced points +(5 km intervals) +Max. +80 km + +𝐵 +10𝐴 +𝑁 +1 sun +𝑋 +𝐵 − 𝑆 +𝑆 + 𝐴 +𝑆 +𝐴 +𝑡 +50 nm +Parametric Optimization Results – Various Sphere Sizes +Variable +Case +D = 2 cm +D = 0.1 m +D = 0.5 m +D = 1 m +D = 2 m +D = 5 m +rmin = D/20, rmax = D/2.01, with a discretization of 80 points (log scale) +𝐴 +Max. Payload +0.90 mm +0.90 mm +0.90 mm +0.90 mm +0.90 mm +0.90 mm +Min. Altitude +0.13 mm +0.13 mm +0.20 mm +0.20 mm +0.20 mm +0.20 mm +𝐿 +Max. Payload +0.91 mm +0.91 mm +0.91 mm +0.91 mm +0.91 mm +0.91 mm +Min. Altitude +0.14 mm +0.14 mm +0.21 mm +0.21 mm +0.21 mm +0.21 mm +𝑟 +Max. Payload +9.95 mm +4.07 cm +19.05 cm +36.85 cm +73.70 cm +1.84 m +Min. Altitude +4.05 mm +1.89 cm +10.82 cm +21.63 cm +43.27 cm +1.08 m + +Max. +Payload +Payload (mg) +8.34 +79.11 +1 445 +5 526 +21 612 +133 242 +Altitude (km) +80 +80 +80 +80 +80 +80 +A. Density (g/m2) +25.48 +7.81 +5.91 +5.64 +5.54 +5.49 +Sphere Area (m2) +0.0007 +0.025 +0.64 +2.63 +10.52 +65.82 +𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio +2.22 +4.77 +5.68 +6.17 +6.17 +6.17 + +Min. +Altitude +Payload (mg) +0.24 +0.58 +223.94 +872.33 +3 442 +21 339 +Altitude (km) +55 +55 +60 +60 +60 +60 +𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio +23.34 +26.96 +20.30 +20.32 +20.31 +20.38 +Parametric Optimization Results – Various Cone Sizes +Variable +Case +D = 2 cm +D = 0.1 m +D = 0.5 m +D = 1 m +D = 2 m +D = 5 m +rmin = D/20, rmax = D/2.01, with a discretization of 80 points (log scale) +𝐴 +Max. Payload +0.90 mm +0.90 mm +0.90 mm +0.90 mm +0.90 mm +0.90 mm +Min. Altitude +0.13 mm +0.35 mm +0.35 mm +0.35 mm +0.35 mm +0.35 mm +𝐿 +Max. Payload +0.91 mm +0.91 mm +0.91 mm +0.91 mm +0.91 mm +0.91 mm +Min. Altitude +0.14 mm +0.36 mm +0.36 mm +0.36 mm +0.36 mm +0.36 mm +𝑟 +Max. Payload +9.95 mm +4.97 cm +24.86 cm +49.73 cm +99.45 cm +2.49 m +Min. Altitude +4.05 mm +2.39 cm +11.56 cm +23.12 cm +46.25 cm +1.16 m + +Max. +Payload +Payload (mg) +7.96 +101.26 +2 043 +7 929 +31 228 +193 348 +Altitude (km) +80 +80 +80 +80 +80 +80 +A. Density (g/m2) +11.59 +6.61 +5.61 +5.48 +5.42 +5.38 +Cone Area (m2) +0.0016 +0.039 +0.98 +3.92 +15.67 +97.97 +𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio +5.04 +5.05 +5.05 +5.04 +5.04 +5.03 + +Min. +Altitude +Payload (mg) +0.18 +10.12 +208.65 +812.97 +3 209 + 19 892 +Altitude (km) +55 +60 +60 +60 +60 +60 +𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio +23.97 +17.75 +18.84 +18.84 +18.83 +18.72 + +Page 19 + +Table 11: Combinations of A, L and r that returned the rockets capable of carrying the greatest payload and achieving +flight at the lowest altitude, for various values of D, as specified in Figure S1. + +The results from these tables are discussed in greater detail in the main paper. However, there are four +important points to highlight. First, changing D (the scaling of the overall geometries) did not affect +significantly the optimal channel parameters A and L that yielded the maximum payload capabilities and +achieved flight at the lowest altitude. Secondly, the obtained maximum areal densities were similar across +the three geometries (as seen in Figure S14 (a) below) and had average values of 9.31 g/m2, 6.68 g/m2 and +6.96 g/m2, for the sphere, cone, and rocket, respectively. Notice that these are above the theoretical order- +of-magnitude estimation for the upper limit of 4 g/m2 in (S71). Thirdly, the optimized 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratios +for the three geometries were relatively invariant across the various values of D and the two missions (max. +payload and minimum altitude). For instance, for the maximum payload optimization, 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 +averaged 5.20, 5.04, and 6.02 for the sphere, cone and rocket, respectively, while for the minimum altitude +case, this ratio averaged 21.94, 19.49 and 28.65, respectively. Lastly, for a given surface area, the amount +of payload that each geometry could carry was comparable, as can be seen in Figure S14 (b) below. As a +result, 1 m2 of a porous and geometrically optimized cone has a similar maximum payload capability than +1 m2 of an optimized rocket and sphere. +Finally, Figure S15 through Figure S20 present cloud plots that permit visualizing the results from the +parametric studies, in particular how different combinations of A, L and r enabled geometries with various +altitude (a), payload (b) and areal density (c) capabilities. These plots correspond to the D = 10 cm and D += 10 m cone, sphere and rocket, and are accompanied with illustrations of the optimized geometries that +achieved flight at minimum altitude (d) and carried the most payload (e). These figures were generated by +discretizing the search ranges of A, L and r in 500 equally spaced, and the results from the optimized +geometries are shown in Table 12 through Table 14). Despite the increase in discretization points (from 80 +to 500) in each dimension, the results were comparable. +Parametric Optimization Results – Various Rocket Sizes +Variable +Case +D = 2 cm +D = 10 cm +D = 0.5 m +D = 1 m +D = 2 m +D = 5 m +rmin = D/20, rmax = D/2.01, with a discretization of 80 points (log scale) +𝐴 +Max. Payload +0.90 mm +0.90 mm +0.90 mm +0.90 mm +0.90 mm +0.90 mm +Min. Altitude +0.092 mm +0.13 mm +0.13 mm +0.13 mm +0.13 mm +0.13 mm +𝐿 +Max. Payload +0.91 mm +0.91 mm +0.91 mm +0.91 mm +0.91 mm +0.91 mm +Min. Altitude +0.094 mm +0.14 mm +0.14 mm +0.14 mm +0.14 mm +0.14 mm +𝑟 +Max. Payload +9.95 mm +4.97 cm +24.86 cm +49.73 cm +99.45 cm +2.49 m +Min. Altitude +1.00 mm +0.94 cm +4.12 cm +8.24 cm +15.94 cm +0.40 m + +Max. +Payload +Payload (mg) +9.51 +127.59 +2 639 +10 281 +40 573 +251 516 +Altitude (km) +80 +80 +80 +80 +80 +80 +A. Density (g/m2) +11.60 +6.89 +5.95 +5.83 +5.77 +5.74 +Rocket Area (m2) +0.0019 +0.047 +1.17 +4.68 +18.71 +117.12 +𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio +6.02 +6.02 +6.02 +6.02 +6.02 +6.02 + +Min. +Altitude +Payload (mg) +0.03 +1.54 +8.29 +18.57 +45.37 +175.67 +Altitude (km) +45 +55 +55 +55 +55 +55 +𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio +42 +23.28 +26.27 +26.27 +27.09 +27 +Figure S14: D against Areal Density (a) and Surface Area against Payload (b) for the 3D geometries. +a +b + +Max.PayloadagainstGeometrySurfaceArea +100 +Max.Payload (kg) +10 +Sphere +-Cone +Rocket +9-01 +10-4 +102 +100 +102Max.Areal Density against characteristic D +25 +Sphere +Cone +20 +Rocket +15 +10 +102 +10-1 +100 +101 +D (m)Page 20 + + +Comparison of D = 10 cm and D = 10 m Cone Geometries +Case +A +L +r +Surface +Area (m2) +𝑨𝒕𝒐𝒕𝒂𝒍/ +𝑨𝒐𝒖𝒕 ratio +Payload +(mg) +Altitude +(km) +Discretization of 500 points +D = +10 cm +Min. Altitude +0.15 mm +0.16 mm +1.94 cm +0.03 +25.92 +0.52 +55 +Max. Payload +1.24 mm +1.25 mm +4.97 cm +0.04 +5.05 +102.31 +80 +D = +10 m +Min. Altitude +0.21 mm +0.22 mm +2.36 m +317.52 +18.16 +95 288 +60 +Max. Payload +1.24 mm +1.25 mm +4.97 m +391.56 +5.05 +780 408 +80 + +Table 12: Combinations of A, L and r that returned the optimal cone geometries described in Figure S15 and Figure +S16 above. + +Figure S15: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Cone +Geometry. Here, the geometry that was able to levitate payload at minimum altitude (0.52 mg at 55 km) is shown in +(d), while that which was able to levitate the maximum payload (102.31 mg at 80 km) is shown in (e). + +a +c +b +d +e +Figure S16: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Cone +Geometry. Here, the geometry that was able to levitate payload at minimum altitude (95 288 mg at 60 km) is shown +in (d), while that which was able to levitate the maximum payload (780 408 mg at 80 km) is shown in (e). + +a +c +b +d +e + +0.045 +0.04 +0.035 +0.03 +日 +Aerial Densities: Cone Geometry +0.025 +5.8g/m²99%percentile density +0.02 +4.49g/m~190%percentile density +2.84g/m150%percentile density +2.35g/m|25%percentiledensity +-01 +10-3 +102 +A [m] +104 +L [m]0.04 +0.03 +[m] +MinimumAltitudes:ConeGeometry +0.02 +55km +60 km +65km +70km +10~2 +104 +103 +102 +A [m] +104 +L[m]0.045 +0.04 +0.035 +0.03 +Maximum Payloads:Cone Geometry +101.29mg/99%max.payload +0.025 +92.08 mg/90% max.payload +51.15 mg/50% max.payload +30.69mg|30%max.payload +0.02 +10~ +104 +103 +102 +A [m] +L[m]r=1.94cm +r=4.97cmAerial Densities:ConeGeometry +4.85g/m199%percentile density +2 +3.88g/m²90%percentile density +2.58g/m50%percentiledensity +2.19g/m25%percentile density +102 +10-3 +103 +10-2 +A[m] +o1 +104 +L [m]3 +[u] +MinimumAltitudes:Cone Geometry +2 +60km +65 km +70km +75km +102 +102 +A [m] +-01 +104 +L[m]4.5 +4. +3.5 +3 +MaximumPayloads:ConeGeometry +772603.92mg/99%max.payload +2.53 +702367.2mg/90%max.payload +390204mg/50%max.payload +2 +234122.4mg/30%max.payload +102 +10-3 +104 +103 +102 +A[m] +104 +L [m]r=2.36m +r=4.97mPage 21 + +Comparison of D = 10 cm and D = 10 m Rocket Geometries +Case +A +L +r +Surface +Area (m2) +𝑨𝒕𝒐𝒕𝒂𝒍/ +𝑨𝒐𝒖𝒕 ratio +Payload +(mg) +Altitude +(km) +Discretization of 500 points +D = +10 cm +Min. Altitude +0.11 mm +0.12 mm +0.50 cm +0.001 +>100 +0.01 +50 +Max. Payload +1.24 mm +1.25 mm +4.97 cm +0.05 +6.02 +129.56 +80 +D = +10 m +Min. Altitude +0.15 mm +0.16 mm +0.87 m +59.39 +24.98 +2132.57 +55 +Max. Payload +1.24 mm +1.25 mm +4.97 m +467.23 +6.02 +1021162 +80 + +Table 13: Combinations of A, L and r that returned the optimal rocket geometries described in Figure S17 and Figure +S18 above. +Figure S17: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Rocket +Geometry. Here, the geometry that was able to levitate payload at minimum altitude (0.01 mg at 50 km) is shown in (d), +while that which was able to levitate the maximum payload (129.56 mg at 80 km) is shown in (e). + +a +c +b +d +r = 5.00 mm +e +r = 4.97 cm +Figure S18: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Rocket +Geometry. Here, the geometry that was able to levitate payload at minimum altitude (2 132.57 mg at 55 km) is shown +in (d), while that which was able to levitate the maximum payload (1 021 162 mg at 80 km) is shown in (e). + +a +c +b +r = 0.87 m +r = 4.97 m +d +e + +0.04 +0.03 +0.02 +AerialDensities:RocketGeometry +6.64g/m²99%percentiledensity +0.01 +5.78g/m|90%percentiledensity +4.33g/m/50%percentiledensity +3.36g/m²|25%percentile density +102 +A [m] +104 +103 +102 +104 +L [m]0.04 +0.03 +0.02 +[m] +MinimumAltitudes:Rocket Geometry +50km +0.01 +55km +60km +65km +102 +104 +102 +A[m] +104 +L[m]0.045 +0.04 +0.035 +0.03 +MaximumPayloads:RocketGeometry +128.26mg|99%max.payload +0.025 +116.6mg/90%max.payload +64.78mg|50% max.payload +38.87mg|30%max.payload +0.02 +102 +103 +10~ +10-3 +10~2 +A [m] +104 +L[m]41 +3 +2 +[u] +Aerial Densities:RocketGeometry +5.18g/m²199%percentile density +1. +4.15g/m90%percentiledensity +2.78g/m150%percentiledensity +2.37g/m²25%percentiledensity +102 +10-3 +104 +10-3 +10-2 +A [m] +104 +L[m]4 +3 +2 +宜 +MinimumAltitudes:Rocket Geometry +55km +1 +60km +65km +70km +102 +10-4 +102 +A [m] +-01 +L [m]4.5 +43 +3.5, +MaximumPayloads:RocketGeometry +3 +1010951.02mg|99%max.payload +919046.38mg|90%max.payload +2.53 +510581.32mg|50%max.payload +306348.79mg30%max.payload +102 +103 +104 +10-3 +102 +A[m] +104 +L [m]Page 22 + +Comparison of D = 10 cm and D = 10 m Sphere Geometries +Case +A +L +r +Surface +Area (m2) +𝑨𝒕𝒐𝒕𝒂𝒍/ +𝑨𝒐𝒖𝒕 ratio +Payload +(mg) +Altitude +(km) +Discretization of 500 points +D = +10 cm +Min. Altitude +0.15 mm +0.16 mm +1.93 cm +0.03 +25.81 +1.41 +55 +Max. Payload +1.03 mm +1.04 mm +4.02 cm +0.03 +4.93 +79.86 +80 +D = +10 m +Min. Altitude +0.15 mm +0.16 mm +1.90 m +302.22 +26.66 +831.92 +55 +Max. Payload +1.24 mm +1.25 mm +3.67 m +263.63 +6.23 +540 528 +80 + +Table 14: Combinations of A, L and r that returned the optimal sphere geometries described in Figure S19 and Figure +S20 above. +Figure S19: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Sphere +Geometry. Here, the geometry that was able to levitate payload at minimum altitude (1.41 mg at 55 km) is shown in +(d), while that which was able to levitate the maximum payload (79.86 mg at 80 km) is shown in (e). + +a +c +b +d +e +Figure S20: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Sphere +Geometry. Here, the geometry that was able to levitate payload at minimum altitude (831.92 mg at 55 km) is shown +in (d), while that which was able to levitate the maximum payload (540 528 mg at 80 km) is shown in (e). + +a +c +b +d +e + +0.043 +0.03 +AerialDensities:SphereGeometry +7.05g/m²|99% percentile density +0.02 +5.13g/m²190%percentile density +3.01g/m/50%percentiledensity +2.43g/m²25%percentiledensity +102 +103 +102 +A [m] +10~4 +104 +10-3 +L[m]0.043 +0.03 +[m] +MinimumAltitudes:SphereGeometry +0.02 +55km +60km +65km +70km +102 +102 +A[m] +-01 +104 +L [m]0.04 +MaximumPayloads:SphereGeometry +79.06mg/99%max.payload +71.87mg90%max.payload +0.02 +39.93mg/50%max.payload +23.96mg|30%max.payload +102 +103 +102 +A [m] +104 +104 +10-3 +L[m]r=1.90 +r=3.67 +m +u4.5 +4. +3.5 +3 +AerialDensities:SphereGeometry +2.5 +5.03g/m199%percentile density +2 +3.99g/m90%percentiledensity +2.61g/m|50%percentile density +2.2g/m²25%percentiledensity +10~2 +10-3 +104 +10-3 +102 +A [m] +10-4 +L[m]43 +m +[m] +MinimumAltitudes: Sphere Geometry +2 +55km +60km +65km +70km +102 +102 +A [m] +104 +104 +L[m]4.5 +4 +3.5 +E +3 +MaximumPavloads: Sphere Geometry +2.53 +535122.95mg/99%max:payload +486475.41mg/90%max.payload +23 +270264.11mg/50%max.payload +162158.47mg/30%max.payload +102 +10- +104 +103 +102 +A [m] +104 +L [m]r=1.93 +r=4.02 +cm +cmPage 23 +References + +[R1] Azadi, Mohsen, George A. Popov, Zhipeng Lu, Andy G. Eskenazi, Avery Ji Won Bang, Matthew F. +Campbell, Howard Hu, and Igor Bargatin. "Controlled levitation of nanostructured thin films for sun- +powered near-space flight." Science Advances 7, no. 7 (2021): eabe1127. + +[R2] Cappella, Andrea, Jean‐Luc Battaglia, Vincent Schick, Andrzej Kusiak, Alessio Lamperti, Claudia +Wiemer, and Bruno Hay. "High Temperature Thermal Conductivity of Amorphous Al2 O 3 Thin Films +Grown by Low Temperature ALD." Advanced Engineering Materials 15, no. 11 (2013): 1046-1050. + +[R3] Cortes, John, Christopher Stanczak, Mohsen Azadi, Maanav Narula, Samuel M. Nicaise, Howard Hu, +and Igor Bargatin. "Photophoretic levitation of macroscopic nanocardboard plates." Advanced Materials 32, +no. 16 (2020): 1906878. + +[R4] Eskenazi, Andy, Tom Celenza, and Igor Bargatin. “MATLAB-fluid-flow-parametric-studies.” (2022) +https://github.com/andyeske/MATLAB-fluidflow-parametric-studies + +[R5] Lin, Chen, Samuel M. Nicaise, Drew E. Lilley, Joan Cortes, Pengcheng Jiao, Jaspreet Singh, Mohsen +Azadi et al. "Nanocardboard as a nanoscale analog of hollow sandwich plates." Nature communications 9, +no. 1 (2018): 1-8. + +[R6] O'Neal Jr, Cleveland, and Richard S. Brokaw. "Relation between thermal conductivity and viscosity +for some nonpolar gases." The Physics of Fluids 5, no. 5 (1962): 567-574. + +[R7] Sharipov, Felix, and Vladimir Seleznev. "Data on internal rarefied gas flows." Journal of Physical +and Chemical Reference Data 27, no. 3 (1998): 657-706. + +[R8] Teagan, William P., and George S. Springer. "Heat‐Transfer and Density‐Distribution Measurements +between Parallel Plates in the Transition Regime." The Physics of Fluids 11, no. 3 (1968): 497-506. + +[R9] Wagiman, Abdullah, Mohammad Sukri Mustapa, Mohd Amri Lajis, Shazarel Shamsudin, Mahmod +Abd Hakim, and Rosli Asmawi. "Effect of Thermally Formed Alumina on Density of AlMgSi Alloys +Extrudate Recycled Via Solid State Technique." Journal of Advanced Research in Fluid Mechanics and +Thermal Sciences 87, no. 2 (2021): 137-144. + +[R10] Wu, H., S. Grabarnik, A. Emadi, G. De Graaf, and R. F. Wolffenbuttel. "Characterization of thermal +cross-talk in a MEMS-based thermopile detector array." Journal of Micromechanics and +Microengineering 19, no. 7 (2009): 074022. + diff --git a/QtE3T4oBgHgl3EQfDAk3/content/tmp_files/load_file.txt b/QtE3T4oBgHgl3EQfDAk3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..15f89868cc8c7a61bd464618c099a5c6407508b0 --- /dev/null +++ b/QtE3T4oBgHgl3EQfDAk3/content/tmp_files/load_file.txt @@ -0,0 +1,1590 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf,len=1589 +page_content='3D photophoretic aircraft made from ultralight porous materials can carry kg-scale payloads in the mesosphere Thomas Celenza, Andy Eskenazi and Igor Bargatin We show that photophoretic aircraft would greatly benefit from a three-dimensional (3D) hollow geometry that pumps ambient air through sidewalls to create a high-speed jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' To identify optimal geometries, we developed a theoretical expression for the lift force based on both Stokes (low-Re) and momentum (high- Re) theory and validated it using finite-element fluid-dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' We then systematically varied geometric parameters, including Knudsen pump porosity, to minimize the operating altitude or maximize the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Assuming that the large vehicles can be made from previously demonstrated nanocardboard material, the minimum altitude is 55 km while the payload can reach 1 kilogram for 3D structures with 10- meter diameter at 80 km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In all cases, the maximum areal density of the sidewalls cannot exceed a few grams per square meter, demonstrating the need for ultralight porous materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For centuries, humans have been exploring Earth’s atmosphere and outer space, a quest that has led to discoveries in fields ranging from aerodynamics to astronomy and climate modeling [1-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, the study of certain regions of the atmosphere is hindered by available propulsion technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For instance, in Earth’s mesosphere, anthropogenic emissions of carbon dioxide are counterintuitively producing rapid cooling [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The shrinking of the atmosphere resulting from this cooling [5] can be problematic, given that a contracting mesosphere can result in reduced satellite drag, which could translate into a greater accumulation of space debris [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Unfortunately, uncertainties in calculations of these effects are currently large because experimental observations within the mesosphere are challenging [7], given that this region, extending from fifty to eighty kilometers above the surface of Earth, has air pressures too low to sustain planes or balloons and too high for orbiting satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Another region of significant interest is the Martian atmosphere, where most recently the helicopter Ingenuity achieved near-surface flight [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Even with this milestone, sustained flight at high altitudes in Mars, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', from Olympus Mons, is not yet possible due to decreasing atmospheric density [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Like the study of Earth’s mesosphere, the exploration of Mars’ atmosphere at high altitudes is limited by the lack of long-duration methods of flight and propulsion at ambient pressures below ~1 mbar (100 Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, developing an airborne platform that can operate in a very thin atmosphere, both on Mars and on Earth, would be extremely useful in helping collect valuable and atmospheric data related to wind patterns, temperature and pressure variations, as well as the concentrations of atmospheric gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' One promising concept, based on the lightweight light-powered centimeter-scale microflyers developed by Cortes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [11], can potentially overcome the issues faced by the current propulsion mechanisms and achieve sustained flight in Earth’s mesosphere and the Martian atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' These devices, composed of porous plates, can levitate due to photophoresis, a light-driven propulsion mechanism where a jet is created using Knudsen pumping of ambient gas [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Knudsen pumps have no moving parts and instead exploit temperature gradients to induce gas flows through these plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Known as “nanocardboard”, these ultralight porous plates are composed of nanometer-thick (25–400 nm) aluminum oxide face sheets that are connected by channels with micrometer-scale width and height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' They offer an areal density of only ~1 g/m2 and a bending stiffness orders of magnitude higher relative to solid plates of the same mass [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Photophoretic levitation is typically enabled by a difference in physical properties between the top and bottom of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For instance, in the study performed by Cortes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [12], the bottom side of the nanocardboard was coated with carbon nanotubes (CNTs), which absorbed the incident light and subsequently increased in temperature relative to the top side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This difference in temperatures caused the Knudsen pumping, which pushed air down through the channels of nanocardboard from the cold to the hot side and thus creating a downward jet below the nanocardboard that levitated plates with centimeter-scale sizes [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This mechanism works best in low pressure environments (1-100 Pa) [14], such as in Earth’s mesosphere or near the top of Olympus Mons on Mars [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' If the lift forces are large enough to carry tiny “smart dust” sensor payloads [16], many such microflyers can be deployed on Earth or on Mars to record data in these regions of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In this work, we propose much larger photophoretic vehicles, which are many meters in diameter, three-dimensional rather than planar, and use porous sidewalls that push air into an inner chamber and out of a small nozzle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Using the nozzle increases the speed of the air jet, and such 3D photophoretic vehicles can not only increase the resulting lift force but also widen the range of operating pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Combining design concepts from the previously demonstrated photophoretic levitation of planar nanocardboard [11] and analytical tools we used for solid mylar-CNT composite disks [17], we analyzed 3D geometries with porous alumina nanocardboard walls and CNTs deposited on their inner side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Because alumina is transparent, CNTs on the inside of the structure would absorb the incident light, inducing the Knudsen pumping of air from the outside into the interior chamber through the pores and then out of the chamber through the exit nozzle, producing a jet as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Figure 1: A hollow sphere with porous alumina-CNT composite walls flying in Earth’s mesosphere (a) and over the top of Olympus Mons in Mars (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The cross-sectional view (c) of the sphere shows the air flow in, with velocity 𝑣𝑓𝑡, due to Knudsen pumping (across the nanocardboard walls, as seen on the zoomed-in view) and out as a jet through the exit nozzle, with velocity 𝑣𝑗𝑒𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As depicted in (c), A is the nanocardboard channel width, L the nanocardboard channel height, and r the structure’s outlet radius, while D the structure’s overall size dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Background Earth and Mars Image Credits: NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' To identify the optimal 3D geometry that maximized payload, we considered three representative geometries (a sphere, a cone, and a rocket), and performed a series of simulations to determine the parameters that would yield the greatest lift forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, first, it was necessary to develop an analytical expression that predicted the lift forces produced by such structures across a wide range of Reynold numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' To determine this expression, we modeled these 3D structures with outlet jet velocities as small as 10-6 m/s to as large as ~100 m/s and at various atmospheric altitudes up to 80 km using computational fluid dynamics simulations in ANSYS Fluent, as detailed in the supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For each fluid- flow simulation, we found the reaction forces induced from the air flow (equal and opposite to the lift force), and then fitted the collected data using the equation 𝐹 = 𝐶18𝜇𝐷𝑣𝑓𝑡 + 𝐶2𝜌𝐴𝑣jet 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (1) a Exterior Vft Vft Interior Vft Cross- sectional viewHere, 𝜇 corresponded to the fluid viscosity, 𝜌 to the density, 𝐴 = 𝜋𝑟2 is the area of a nozzle with radius r, D is the geometry’s characteristic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', largest) dimension, while 𝑣𝑓𝑡 is the flow-through velocity of the fluid flow through the porous material and 𝑣𝑗𝑒𝑡 is the velocity of the fluid exiting the structure through the small nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As outlined in the supplementary information, 𝑣𝑓𝑡 depends on the light intensity, I, the altitude dependent air pressure, P, and the geometric parameters of the nanocardboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The upper limit of the flow-through velocity typically scales as 𝑣𝑓𝑡 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 𝐼/𝑃 (see supplementary information), resulting in velocities of less than 1 mm/s under natural sunlight (~1000 W/m2) and standard atmospheric pressure (105 Pa) but increasing by many orders of magnitude as the pressure drops at higher altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (1), the first term is based on Stokes’ drag on a disk, obtained from a linearization of the steady-state Navier-Stokes equations in the case of dominating viscous forces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', in the low-Re limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Cortes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' previously showed that at vanishingly low air flow speeds, the lift of a stationary nanocardboard plate with air flowing through it was equal to the Stokes drag for a solid disk [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In contrast, at high jet speeds, the inertial terms dominate, and the lift is mostly dependent on the velocity of the jet exiting the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The helicopter-momentum theory equation, which can be derived from a simple application of Reynolds Transport Theorem and represents the second term in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (1), can model the lift in this high-Re limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Summing both terms results in a simple interpolation between the two operating regimes that gives an estimate for the lift force at all pressures and velocities (and, therefore, all values of Re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Table 1 summarizes the average fitted C1 and C2 parameters, both on the order of 1, obtained from fitting the results for 27 ANSYS Fluent simulations using 3 different altitudes (0 km, 40 km and 70 km), 3 geometry types (sphere, cone, and rocket), and 3 different structure sizes (1cm, 5cm and 10cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Fitting Parameters for Each Geometry Geometry Cone Sphere Rocket C1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 C2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 Table 1: Fitting parameters for the three geometries in addition to key dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that these ANSYS simulations were performed assuming a 100% porosity along each one of these structures’ walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' After determining the coefficients C1 and C2, we proceeded to numerically optimize the various parameters controlling the overall 3D shape and nanocardboard porous microstructure to maximize the payload capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The developed MATLAB code [18] was based on the photophoretic levitation theory for nanocardboard [11] adapted to axisymmetric 3D structures, as detailed in the supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The code also took into account how temperature and pressure depend on the altitude in the atmosphere, employing empirical relations developed from standard atmospheric data [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Our optimization sought the combination of A (nanocardboard channel width), L (nanocardboard channel height), and r (the structure’s outlet/nozzle radius) that resulted in the highest payload or achieved flight at the lowest altitude as a function of the overall aircraft size D (diameter for sphere and cone, and length for the rocket).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' All these geometric parameters are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Our numerical optimizations revealed that the optimal nanocardboard porosity parameters A and L were of the same order of magnitude across all geometries and dimensions D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' When optimized for achieving flight at the minimum altitude (55 km with zero-payload), A and L were ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='20 mm and ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='21 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' When optimized for maximum payload (achieved at 80 km altitude), A and L were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm, or about a factor of 4 greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Because these parameters are of the same order of magnitude despite the approximately 40-fold change in ambient pressure at the minimum possible altitude of 55 km and the max payload altitude of 80 km, we can make structures that simultaneously achieve levitation at low altitudes while carrying significant payload at higher altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The maximum areal densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', the maximum lift force divided by specific gravity g and the area of nanocardboard, were also comparable for all structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Table 2 shows that the typical value of maximum areal density was ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 g/m2 (grams per square meter) for small aircraft (D = 10 cm) compared to ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 g/m2 for large aircraft (D = 10 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Both these densities are in the same order of magnitude as the theoretical upper limit derived for the high-Re case in the supplementary information, of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='016 𝐼/ (𝑣𝑎𝑣𝑔𝑔) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='004 kg/m2 = 4 g/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, 𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇/𝜋 ≈ 400 m/s is average speed of air molecules at 55-80 km altitudes, while 𝑅𝑎𝑖𝑟 = 𝑅𝑢/𝑀𝑎𝑖𝑟 = 287 𝐽/(𝑘𝑔 ∙ 𝐾) is the gas-specific ideal constant of air, equal to the universal gas constant 𝑅𝑢 divided by the average molar mass of air 𝑀𝑎𝑖𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Fig 2a shows how the maximum areal densities varies with aircraft size D and, therefore, the airflow’s Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The permissible areal densities of each structure decrease with increasing size and Re and stabilize at ~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 g/m2 for larger aircraft that carry payloads of 1 gram or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Areal Densities and Areas Ratio Geometry Cone Sphere Rocket D = 10 cm D = 10 m D = 10 cm D = 10 m D = 10 cm D = 10 m Max Areal Density For Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6 g/m2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 g/m2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8 g/m2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 g/m2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 g/m2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7 g/m2 Area Ratios For Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 18 26 26 27 23 25 For Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 5 5 5 6 6 6 Table 2: Summary of the parametric studies results for the Cone, Sphere and Rocket, for values of D = 10 cm and D = 10 m (full data for all the probed values of D can be found in the supplementary information section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, the area ratio refers to the 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio, of the structure’s total surface area to its outlet area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Figure 2: Areal Density versus Characteristic Size (a) and Maximum Payload versus Surface Area (b) for the three considered 3D geometries at 80-km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Each data point corresponds to the optimized geometry at each of the probed values of the parameter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The overlap between the curves, in particular starting at surface areas larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 m2, suggests that the geometries have similar areal densities and maximum payload capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Plotting maximum payloads against the structure surface area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2b revealed that, for a given surface area, the maximum payload was very similar across all three geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' While the sphere outperformed at smallest sizes, all three shapes (cone, sphere, and rocket) offered essentially the same performance at the largest sizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', for sizes that maximize the payload and are most promising for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 3 below illustrates optimized shapes for the 10-meter cone (a), sphere (b) and rocket (c), which could carry 780, 540, and 1020 grams of payload, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This is sufficient capacity to carry modern communication devices [20] and similar to the payload of a typical CubeSat [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a b 𝑅𝑒 = 𝜌𝑣𝑓𝑡𝐷 𝜇 D=10m a b c D= 10m D= 10m r=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97m r=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='67m r=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97m Payload: 780 g Payload: 540 g Payload: 1020 gMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='Payload againstGeometrySurfaceArea 100 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload (kg) 0 10° Sphere Cone Rocket 10-4 10~2 100 102 Surface Area (m3)Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='Areal Density against characteristic D ReynoldsNumber 100 10l 102 103 25 Sphere Cone 20 Rocket 15 10 10-2 10~1 100 10l D (m) Figure 3: Geometrically optimized cone (a), sphere (b) and rocket (c) for maximum payload capabilities with a fixed characteristic dimension of D = 10 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' D represents the cone and sphere diameter, and the rocket length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Achieving a payload of 1kg required a D of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 and 14 m for the cone and sphere, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Finally, as demonstrated in Table 2 and the supplementary information section, we noticed that the 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio, of the total surface area to the outlet area, was approximately constant for the optimal geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For the minimum altitude case, this ratio ranged from 17 to 42, averaging ≈ 23 across the three geometries and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For the maximum payload case, the typical value of this ratio was approximately 6, resulting in relative nozzle sizes shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Due to mass conservation, the outlet jet speed needs to be larger than the flow-through velocity by the same factor as precisely the 𝐴𝑖𝑛/𝐴𝑜𝑢𝑡 area ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Therefore, recalling the 𝑣𝑓𝑡 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 𝐼/𝑃 relationship, at the maximum payload altitude of 80 km, we can approximate 𝑣𝑗𝑒𝑡 = 𝑣𝑓𝑡𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='18 𝐼/𝑃 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='18 × 1300 𝑊 𝑚−2/1 𝑃𝑎 = 234 𝑚/𝑠 , while at the minimum altitude of 55 km (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', for zero payload), 𝑣𝑗𝑒𝑡 = 𝑣𝑓𝑡𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='70 𝐼/𝑃 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='70 × 1200 𝑊 𝑚−2/ 10 𝑃𝑎 = 84 𝑚/𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that for the payload altitude of 80 km, the jet speed approaches but remains below the speed of sound, given by 𝑣𝑠𝑜𝑢𝑛𝑑 = √𝛾𝑅𝑎𝑖𝑟𝑇80𝑘𝑚 ≈ √1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 × 287 𝐽/(𝑘𝑔 𝐾) × 200 𝐾 ≈ 280 m/s, where 𝛾 is the adiabatic constant of air, while 𝑇80𝑘𝑚 ≈ 200 𝐾 is the air temperature at 80 km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Achieving kg-scale payloads in the mesosphere will therefore require building 10m-scale photophoretic aircraft out of ultralight materials that simultaneously possess low areal densities (≈ 1 g/m2) and sufficient structural integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, these aircraft do not necessarily have to be rigid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' instead, it is possible to make use of flexible parachute or balloon-like structures with overall dimensions similar to those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In the calculations above, we assumed that all surfaces are illuminated with 1000 W/m2 light intensity, which is not always realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The direct sunlight intensity in the mesosphere is similar to that in outer space, ~1360 W/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Additional ~500 W/m2 of sunlight will be reflected from the clouds and Earth below the aircraft due to Earth’s planetary albedo of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Depending on the elevation of the Sun in the sky and the orientation of the surface, it may be exposed to anywhere between essentially zero and almost 2000 W/m2 of combined direct and reflected sunlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' If the aircraft ends up rotating as balloons often do, all walls will experience an average flux on the order of 1000 W/ m2 or slightly less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For reference, we also performed simulations at a reduced intensity of 500 W/m2, which results in payloads ~4 times lower than those shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' One last important aspect to note about these photophoretic aircraft is that they only create lift when exposed to light (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', during the day), limiting the steady operation to ~12 hours at most latitudes, after which the aircraft will start to descend to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, near the poles, the polar day can last many months and extended operations of up to several months may be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' To conclude, we show that 3D photophoretic aircraft with porous walls made of ultralight, ultrathin materials are capable of carrying kg-scale payloads, comparable to those of typical CubeSats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The results presented above can be easily generalized for high-altitude operation on Mars using a Martian atmospheric model [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This work opens the way to creating persistent, low-cost, sensor-carrying aircraft in the previously inaccessible atmospheric regions at 55-80 km altitudes on Earth and 20-40 km altitudes on Mars, enabling a greater understanding of our planet and the worlds beyond.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='gov/sites/default/files/atoms/files/nasa_csli_cubesat_101_508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='pdf [22] Justh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', Cianciolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', & Hoffman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Mars Global Reference Atmospheric Model (Mars-GRAM): User Guide (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' NASA/TM-20210023957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Page 1 3D photophoretic aircraft made from ultralight porous materials can carry kg-scale payloads in the mesosphere Supplementary Information Thomas Celenza, Andy Eskenazi and Igor Bargatin In this document, we present and expand on the computational and theoretical framework behind our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The first section is devoted to the ANSYS Fluent simulations, covering the solver set-up and the theory behind the force calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The second section of this document focuses on the MATLAB code, specifically the derivation of the equations used in the optimization of the geometrical and channel parameters of the 3D geometries, including the rocket, cone and sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' ANSYS Fluent Simulations The goal of the ANSYS Fluent simulations was to determine an analytical expression to estimate the lift forces produced by various types of 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Because we sought geometries that operated across a wide range of velocities and altitudes (and thus air pressures, densities, temperatures and viscosities), the expression for the lift force needed to be valid across a wide range of Reynolds (Re) numbers as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In particular, this equation needed to reasonably accurately model the transition between the low-Re (Stokes) regime to the high-Re regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As the main paper argues, an appropriate expression is 𝐹 = 𝐶18𝜇𝐷𝑣𝑓𝑡 + 𝐶2𝜌𝐴𝑣jet 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S1) Here, 𝜇 corresponds to the fluid viscosity, 𝜌 to the density, 𝐴 = 𝜋𝑟2 is the area of a nozzle with radius r, D is the geometry’s characteristic (usually largest) dimension, while 𝑣𝑓𝑡 is the flow-through velocity of the fluid through the porous material and 𝑣𝑗𝑒𝑡 is the velocity of the fluid exiting the structure through the small nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The fitting parameters 𝐶1 and 𝐶2 depended on the geometry and were determined using ANSYS simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In this work, we considered three geometries, a cone, sphere, and rocket, shown in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Figure S1: Main geometric parameters for the cone (a), rocket (b) and sphere (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that here, the variable D serves as an overall indicator of the size of the geometry, while the variable r controls the outlet radii of the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Isometric View Isometric View Side View 121 h a Side View Isometric View Side View 2r bPage 2 Through the ANSYS Simulations, we determined the average 𝐶1 and 𝐶2 coefficients for each structure and examined how these would evolve with overall size of the structure or the altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' We performed 9 sets of simulations for each geometry, where we varied three different inlet/outlet area ratios at three different altitudes, resulting in flow-through velocities as small as 10-6 m/s or as large as 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Figure S2 shows boundary conditions employed in our simulations using a sphere as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' To make our simulations computationally more efficient, we took advantage of the axial symmetry of our three geometries and thus constructed our models in a 2D, axisymmetric environment, which allowed us to only simulate fluid flow on the top half of each structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' We formed these geometries using ANSYS’ “Design Modeler” module, and they were essentially composed of three spaces: an outer air box, and inner air box, and the nanocardboard geometry itself (whose interior was “subtracted” from the inner air box, as seen in Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The next step was to specify mesh elements, shown in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Plot (a) shows the larger, outer air box with coarser mesh elements, while plot (b) is a zoomed- in view into the smaller, inner air box, containing smaller mesh elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' By dividing the air box into these two regions, we optimized the overall number of mesh elements in the simulation by providing a higher resolution just in the area close to the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' We created the mesh by selecting edges and dividing them into a discrete number of points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' to enforce a uniform grid pattern, we used the quadrilaterals face meshing command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For the sphere, this resulted in 184,180 elements (185,408 nodes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' for the cone, 194,322 elements (195,865 nodes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' for the rocket, 293,053 elements (294,616 nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' These were the final numbers of mesh elements obtained as a result of performing a convergence analysis until observing negligible changes in the computed lift forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The final step was to establish Fluent’s “set-up” module parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For the model, we chose the viscous k-omega, with the low-Re (viscous) corrections feature enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Next, we fixed the boundary conditions as described by Figure S2, and manually modified operating conditions (environment pressure, fluid density and fluid viscosity) matching the chosen altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Since our fluid was air, we extracted its properties as tabulated in altitude-dependent standard atmospheric tables, summarized in Table 1 below for 0 km, 40 km and 70 km (our probed altitudes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Last, we specified the inlet velocity as a variable parameter, since that Figure S2: ANSYS Simulations boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As the illustration shows, the inner wall of the geometry (red) was chosen as the flow-velocity inlet (inducing the air to flow from the into the structure), while the outer wall (violet) was selected as the outlet (mass outflow in Fluent, inducing the air to pass through the structure’s walls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For the purposes of these simulations, we are assuming we have 100% porous walls through which the air flows at velocity 𝑣𝑓𝑡 (an idealization of the actual nanocardboard geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' b a Figure S3: Sample meshing of the axisymmetric sphere simulation in ANSYS Fluent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, plot (a) provides an overall picture of the air box (which is more than ten times larger than the geometry in question in each dimension), while plot (b) shows a zoomed-in image of the area immediately surrounding the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The size of the outer air box was not arbitrary, but rather resulted from a series of simulations that gradually increased its dimensions until force values converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' OuterAir BoX (coarsemeshelements -InerAirBox Cfimrmcsh clements -AirOut Airiln -Axis of SymmetryPage 3 allowed us to sweep through values ranging from 10-6 m/s to 1 m/s in 7 logarithmically equally spaced points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Summary of Altitude Dependent Atmospheric Properties Altitude 0 km 40 km 70 km Atmospheric Pressure (Pa) 101300 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='66 Atmospheric Temperature (K) 288 251 220 Air Density (kg/m3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='83 10 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='38 10 5 Air Viscosity (Pa s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='796 10 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='610 10 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='447 10 5 Table 1: Tabulated altitude-dependent atmospheric conditions for 0 km, 40 km and 70 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' These values were manually inputted for each simulation set into the Fluent solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' We repeated this process 36 times, to construct 18 simulations for the cone, 9 for the sphere and 9 for the rocket, using operating conditions corresponding to 3 different altitudes (0 km, 40 km and 70 km) and 3 different geometry sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In each case, we computed the reaction force in the axisymmetric direction using a line integral along the walls of the outer air box, resulting in the force values shown in Figures S4–S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This computation made use of the fact that under steady-state operation, the reaction force is equal to the lift force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The 𝐶1 and 𝐶2 coefficients were then determined by performing a non-linear fitting in MATLAB to equation (S1), resulting in the values that are shown in the same figures and tabulated in Tables 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In general, most curves of Figures S4–S7 (in the logarithmic scale) show a transition from the viscous, low- Re regime to the high-Re regime that is manifested through a change in the slopes of the force curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, at 70 km in altitude, the lift force stayed in the Stokes (low-Re) regime and the high-Re 𝐶2 coefficients remained uncertain at this particular altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Thus, when computing the overall average 𝐶1 and 𝐶2, we did not incorporate the 𝐶2 corresponding to the 70 km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Fitting Parameters for the Rocket, Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' = 2 cm Altitude Length = 1 cm Length = 5 cm Length = 10 cm C1 C2 C1 C2 C1 C2 0 km 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2) 40 km 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='12–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='38) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='73 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='62–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='85) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 70 km 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='361 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='353–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='368) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='20 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='20–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='20) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='08 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='08–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='10) Average 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95 Table 2: 𝐶1 and 𝐶2 coefficients computed for the rocket geometry of different lengths (1 cm, 5 cm and 10 cm), alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a b c Figure S4: Results from the altitude-dependent rocket simulations in ANSYS Fluent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' each data point corresponds to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) correspond to different rocket lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Reactionforcesforvariousflow-throughvelocities RocketGeometry:Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2cm,Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=1cm 102 ANSYS Force (Altitude: 0 km) Fit:C1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='00,C2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='10 104 ANSYSForce (Altitude: 40km) Fit:C1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='73 ANSYSForce (Altitude:70km) Force (N) Fit:C1=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='06 10-6 10-8 10-10 10-12 10~6 104 102 100Reactionforcesforvariousflow-throughvelocities 100 RocketGeometry:Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2cm,Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5cm ANSYSForce (Altitude:0km) Fit:C1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04,C2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='10 ANSYSForce (Altitude:40km) Fit:C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='77 ANSYSForce(Altitude:70km) Force (N) Fit:C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='20,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='17 10-5 10-10 10-6 104 102 100 V, (m/s)Reactionforcesforvariousflow-throughvelocities 100 RocketGeometry:Dia=2cm,Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=10cm ANSYSForce (Altitude:0km) Fit:C1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90,C2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='08 ANSYSForce(Altitude:40km) Fit:C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='82 ANSYSForce(Altitude:70km) Force (N) Fit:C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='08,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='19 105 10-10 10-6 104 102 V, (m/s) 100Page 4 Fitting Parameters for the Sphere, Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' = 2 cm Altitude rout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 cm rout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 cm rout = 1 cm C1 C2 C1 C2 C1 C2 0 km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='29 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='21–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='37) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7) 40 km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='89–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='93) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='99 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='08) 70 km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='65 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='63–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='67) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='58 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='52–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='64) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='96) Average 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='26 Table 3: 𝐶1 and 𝐶2 coefficients computed for the sphere geometry of different outlet radii (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 cm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 cm and 1 cm), alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Fitting Parameters for the Cone, Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' = 2 cm Altitude Length = 2 cm Length = 5 cm Length = 10 cm C1 C2 C1 C2 C1 C2 0 km 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1) 40 km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9) 70 km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='07 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='98–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='16) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='98 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02) Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='82 Table 4: 𝐶1 and 𝐶2 coefficients computed for the cone geometry (2 cm diameter) of different lengths (2 cm, 5 cm and 10 cm), alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a b c Figure S5: Results from the altitude-dependent sphere simulations in ANSYS Fluent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' each data point corresponds to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) correspond to different sphere outlet radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a b c Figure S6: Results from the altitude-dependent cone (2 cm diameter) simulations in ANSYS Fluent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' each data point corresponds to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) correspond to different cone lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Reactionforcesfor various flow-through velocities Sphere Geometry:Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2 cm,r out =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 cm 100 ANSYSForce (Altitude:0km) Fit:C1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='35,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='29 ANSYSForce (Altitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 40km) Fit:CI-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='43,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='62 ANSYS Force (Altitude: 70 km) Fit:C1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='65,C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='08 10-5 Force 1o-to 10-6 10-4 102 100Reactionforcesforvariousflow-throughvelocities SphereGeometry:Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2cm,r =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 cm 100 ou ANSYSForce (Altitude:0km) Fit:C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='46,C2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='06 ANSYSForce (Altitude:40km) Fit:C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='47,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='87 ANSYS Force (Altitude: 70km) Force (N) Fit:C1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='58,C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='41 10-s 10-10 10~6 104 102 100 Va(m/s)Reactionforcesforvariousflow-throughvelocities SphereGeometry:Dia,=2 cm,r =1 cm 102 out ANSYSForce (Altitude:0km) Fit:C1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='88,C2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='54 104 ANSYSForce (Altitude:40km) Fit:C1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='99 " ANSYSForce (Altitude:70km) Force (N) 10*6 Fit:CI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='47 10-8 10-10 10~/2 10-6 104 10-2 100Reaction forcesfor various flow-through velocities 100 Cone Geometry:Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2 cm, Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2cm ANSYSForce(Altitude:0km) Fit: C1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='74, C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='87 ANSYSForce(Altitude:40km) Fit:C11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='00,C2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='56 ANSYSForce(Altitude:70km) Fit:C1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='C2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 Force( 10-5 10-10 10-6 104 10-2 100Reactionforcesfor various flow-through velocities 100 Cone Geometry:Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2 cm, Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=5cm ANSYSForce (Altitude:0km) Fit:C1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='71,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='92 ANSYSForce(Altitude:40km) Fit:C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='93,C2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='60 ANSYSForce(Altitude:70km) Fit:C1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='C2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 Force( 10-5 10-10 10-6 104 10-2 100 Va (m/s)Reactionforcesfor various flow-through velocities ConeGeometry:Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2cm,l/日@Q价 100 ANSYSForce(Altitude:0km) FitCI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='69,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 ANSYSForce (Altitude:40km) Fit:C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='84,C2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='74 ANSYSForce(Altitude:70km) Fit:C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='98,C2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='10 Force( 10-5 10-lo 106 104 102 100Page 5 Fitting Parameters for the Cone, Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' = 4 cm Altitude Length = 2 cm Length = 5 cm Length = 10 cm C1 C2 C1 C2 C1 C2 0 km 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1) 40 km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0) 70 km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='22–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='25) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='19 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='18–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='20) Average 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='89 Table 5: 𝐶1 and 𝐶2 coefficients computed for the cone geometry (4 cm diameter) of different lengths (2 cm, 5 cm and 10 cm), alongside the 66% confidence intervals for each fitting parameter (tabulated below each coefficient entry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As we increased in altitude, the value of the 𝐶1 parameter increased while that of 𝐶2 decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' All in all, Table 6 below summarizes the average 𝐶1 and 𝐶2 coefficients obtained for each geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In all cases, the coefficients are on the order of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Average Fitting Parameters for Each Geometry Geometry Cone Sphere Rocket D = 2 cm D = 4 cm D = 2 cm D = 2 cm C1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 C2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 Table 6: Fitting parameters for the analytical theory for standard atmospheric conditions on Earth, for each geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' To verify our simulations were based on realistic boundary conditions, we examined the streamline plots generated in ANSYS Fluent’s results module, a sample of which is shown in Figure S8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a b c d Figure S8: Velocity streamlines corresponding to the cone (a, c) and rocket (b, d) geometries simulations in ANSYS, for a flow-through velocity of 1 m/s and atmospheric conditions corresponding to 0 km in altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Both the cone and rocket have a characteristic dimension (D) of 5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (c) and (d) denote a zoomed-in view of plots (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a b c Figure S7: Results from the altitude-dependent cone (4 cm diameter) simulations in ANSYS Fluent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' each data point corresponds to a different flow-through velocity, ranging from 10-6 m/s to 1 m/s, while plots (a), (b) and (c) correspond to different cone lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Velocity 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='059 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='044 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='029 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='000 [ms^-1]19:076 14307 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='000 [ms>-1]Reactionforcesforvariousflow-throughvelocities 100 ConeGeometry:Dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=4cm, Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=2cm ANSYSForce(Altitude:0km) FitC1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='99 ANSYSForce (Altitude:40km) Fit:CI=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='42,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='64 ANSYSForce (Altitude:70km) Force (N) Fit:C1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='C2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content="05 10'5 10-10 10-6 104 102 100 Va(m/s)Reactionforcesforvariousflow-throughvelocities ConeGeometry:Dia." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=4cm,Len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='=5cm 100 ANSYSForce(Altitude:0km) Fit:CI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='98,C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='98 ANSYSForce(Altitude:40km) FitC=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='16,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='73 ANSYSForee (Altitude:70km) Fit:C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='20 10-5 Force 10-lo 10-6 104 102 100 Va (m/s)Reactionforcesforvariousflow-throughvelocities ConeGeometry:Dia,=4 cm,Len,=10cm 100 ANSYSForce(Altitude:0km) Fit:C1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97,C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95 ANSYSForce (Altitude:40km) Fit:C1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='11C2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='83 ANSYSForce(Altitude:70km) Fit:CI-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='19,C2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='22 10~5 Force( 10-lo 106 104 102 100Page 6 As expected, a jet of high-speed air exited the geometries as a result of the air flowing in through the porous structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Once the air left the geometry, it interacted with the walls of the outer air box by forming large vortices, as anticipated for a fluid circulating in a contained box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The next section of this document takes the force fitting parameters found from the ANSYS Fluent simulations and focuses on MATLAB-based parametric optimization of our three different geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' MATLAB Code and Extension of Theoretical Framework In this section of the supplementary information, we present the extension to 3D structures of the original nanocardboard fluid mechanic theory developed by [R3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The equations derived below were implemented in a MATLAB code to perform a series of parametric studies that seek to optimize the geometric and porous parameters of our three study geometries, a cone, a sphere and a rocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' More information about our code can be found in our publicly available repository [R4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Derivation of Equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 General Overview For a general 3D porous structure, conservation of mass establishes that 𝐴𝑡𝑜𝑡𝑎𝑙𝑣𝑓𝑡 = 𝐴𝑜𝑢𝑡𝑣𝑜𝑢𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S2) Here, 𝐴𝑡𝑜𝑡𝑎𝑙 represents the total surface area of the structure (as if the structure had no pores/channels) and 𝑣𝑓𝑡 is the flow-through velocity of the fluid across this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Similarly, 𝐴𝑜𝑢𝑡 corresponds to the area covered by the outlet, while 𝑣𝑜𝑢𝑡 is the exit velocity of the fluid out of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Adding Bernoulli’s equation, we get the relationship that 𝑃𝑖𝑛 − 𝑃𝑜𝑢𝑡 𝜌 = ∆𝑃 𝜌 = 𝑣𝑜𝑢𝑡 2 − 𝑣𝑓𝑡 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S3) In (S5), 𝑃𝑖𝑛 is the pressure right at the inlet of the structure, 𝑃𝑜𝑢𝑡 is the pressure right as the jet of fluid is leaving the structure, located around the space close to 𝐴𝑜𝑢𝑡, while 𝜌 is the fluid density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This equation can be rearranged to yield an expression for the pressure difference across both ends of the structure, resulting in ∆𝑃 = 𝜌(𝑣𝑜𝑢𝑡 2 − 𝑣𝑓𝑡 2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S4) Assuming that the porosity of the 3D structure originates from using the nanocardboard geometry developed by [R3] as the wall material, then we can model the mass flow rate of the fluid across one of the structure’s pores (or more properly said, channels) using the following equation 𝑚̇ = −𝛼 ∗ ∆𝑃 + 𝛾 ∗ ∆𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S5) In (S5), 𝛼 and 𝛾 represent two constants, which take the following forms1: 𝛼 = (𝛿 6 + 1) (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='25 √𝛿 ) 𝐴2𝐵𝛽∗ 𝐿 , (S6) and 𝛾 = ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 + 𝛿) 𝐴2𝐵𝑃∗𝛽∗ 𝑇∗𝐿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S7) 1 The variables 𝛼 and 𝛾 come from curve-fitting the data from by [R7] and transforming the non-dimensional flow rate equation into a dimensional form again, with both pressure and temperature contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For more information, please see [R2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Page 7 Here, the variable 𝑃∗ denotes the average pressure2 between the two sides of the structure’s nanocardboard wall, 𝑇∗ analogously describes the average temperature between both sides of the wall’s surface, while 𝛽∗ is an inverse velocity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Specifically, this last one is given by 𝛽∗ = √ 𝑚 2𝑘𝐵𝑇∗ , (S8) where 𝑘𝐵 is the Boltzmann constant (equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='38 * 10-23 J/K), and m is the mass of an air molecule3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Lastly, the parameter 𝛿 is the gas rarefaction coefficient, which [R7] defines as 𝛿 = √𝜋𝐴 2𝜆 = √𝜋 2𝐾𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S9) In this expression, 𝜆 is the molecular mean free path, defined as the average distance traveled by a molecule between collisions with other molecules, and Kn is the Knudsen number, which is characterized in terms the of channel width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In essence, higher values of the 𝛿 parameter designates flows in the continuum regime, while smaller values indicate flows taking place in the free molecular regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As for the molecular mean free path, mathematically it is usually expressed as 𝜆 = 𝜇(𝑇) 𝑃(𝑇) √𝜋𝑘𝐵𝑇 2𝑚 = 𝜇(𝑇) 𝑃(𝑇) √𝜋𝑅𝑎𝑖𝑟𝑇 2 , (S10) where 𝜇(𝑇) is the fluid’s viscosity and P(T) is the operating pressure, both given as a function of T, the operating temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In addition, from equation (S9), we see the Knudsen number is defined as 𝐾𝑛 = 𝜆 𝐴 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S11) Additionally, as seen in Figure S9 below, the variables A and B characterize the nanocardboard channel’s width and length, respectively, yielding a cross-sectional area of A x B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In addition, L denotes the channel’s height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Note that in [R3], A is assumed to be much smaller than B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' After defining these variables and introducing the expression for the mass flow rate, 𝑚̇ , across one of nanocardboard’s channels, then an equation can be derived for the average flow-through velocity across the structure’s surface, which is simply described by 𝑣𝑓𝑡 = 𝜑𝑚̇ 𝜌𝐴𝐵 = 𝜑(−𝛼∆𝑃 + 𝛾∆𝑇) 𝜌𝐴𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S12) Here, 𝑚̇ /𝜌 is no other than the volumetric flow rate 𝑉̇ , while the term 𝜑 denotes the geometric fill factor, which is defined in terms of 𝐴𝑖𝑛 (porous area) and 𝐴𝑡𝑜𝑡𝑎𝑙4, or the channel parameters, and takes the form 𝜑 = 𝐴𝑖𝑛 𝐴𝑡𝑜𝑡𝑎𝑙 = 𝐴𝐵𝑋 (𝐴𝐵𝑋 + 𝑆𝐵𝑋) = 𝐴 (𝐴 + 𝑆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S13) The latter two equivalencies in (S13) originates from analyzing a single nanocardboard unit cell as opposed to the full 3D structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Indeed, as Figure S9 shows, the total cross-sectional area of the cell (if no channels were present) is given by 𝐴𝑐𝑒𝑙𝑙 = (𝐴𝐵𝑋 + 𝑆𝐵𝑋) = (𝐴 + 𝑆)𝐵𝑋 , (S14) where the variable X is just the number of channels in a unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2 The value of this variable may be found from performing CFD simulations but will be simply approximated as the operating pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 3 The molar mass of air is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02896 kg/mol, so then the approximated mass of an air molecule would be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02896/(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='022*1023 ) (Avogadro’s number), or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8089 * 10-26 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 4 This area is essentially the total 3D structure wall area if there were no channels present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This is analogous to 𝐴𝑐𝑒𝑙𝑙 in the single nanocardboard unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Page 8 However, this number (X) is not arbitrarily chosen, and is dictated by A, B and S in the following way 𝑋 = 𝐵 − 𝑆 𝑆 + 𝐴 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S15) This expression considers the channel width A and spacing S as a unit, and tries to fit as many of those A +S units into the channel length B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Nonetheless, we need to consider an additional S for spacing against the perpendicular channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This can be seen more clearly in Figure S10 below, where the yellow bars represent the A +S units, and as drawn, five of these fit in the length of B, after subtracting one S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Overall, the flow-through velocity expression provided in (S12) is a step closer towards calculating the lift force that a 3D structure could generate for a given combination of geometric and channel parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, computing lift will not be possible until we solve for 𝑣𝑜𝑢𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Therefore, (S12) can be rearranged to instead solve for another unknown, ∆𝑃 , and obtain ∆𝑃 = 𝛾∆𝑇 𝛼 − 𝑣𝑓𝑡𝜌𝐴𝐵 𝛼𝜑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S16) Since both (S16) and (S4) from above provide two distinct expressions for the pressure difference, it is possible to equate them, giving rise to yet another relationship between 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡, giving Figure S9: Main nanocardboard channel parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Figure S10: Illustration of equation (S15), with the yellow bars showing the A + S units fitted into the channel length B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Top Isometric View View Key Pi, T1 Air A:ChannelWidth Trapped Side B:Channel Length A Air View S:Channel Spacing L: Channel Height P2, T2 Air t: Alumina ThicknessTop View Key A: Channel Width B: Channel Length S:Channel SpacingPage 9 𝜌(𝑣𝑜𝑢𝑡 2 − 𝑣𝑓𝑡 2) 2 = ∆𝑃 = 𝛾∆𝑇 𝛼 − 𝑣𝑓𝑡𝜌𝐴𝐵 𝛼𝜑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S17) Rearranging this expression further, we get 𝑣𝑜𝑢𝑡 2 = 2 𝜌 (𝛾∆𝑇 𝛼 − 𝑣𝑓𝑡𝜌𝐴𝐵 𝛼𝜑 ) + 𝑣𝑓𝑡 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S18) Now, recalling the conservation of mass relationship provided in (S2), it is possible to write 𝑣𝑓𝑡, the flow- through velocity across the channels, in terms of 𝑣𝑜𝑢𝑡 𝑣𝑓𝑡 = 𝐴𝑜𝑢𝑡 𝐴𝑡𝑜𝑡𝑎𝑙 𝑣𝑜𝑢𝑡 = 𝜑𝐴𝑜𝑢𝑡 𝐴𝑖𝑛 𝑣𝑜𝑢𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S19) Thus, (S19) can replace the 𝑣𝑓𝑡 term in (S18), leaving everything in terms of just 𝑣𝑜𝑢𝑡 𝑣𝑜𝑢𝑡 2 = 2 𝜌 (𝛾∆𝑇 𝛼 − 𝐴𝑜𝑢𝑡𝑣𝑜𝑢𝑡𝜌𝐴𝐵 𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑 ) + ( 𝐴𝑜𝑢𝑡 𝐴𝑡𝑜𝑡𝑎𝑙 ) 2 𝑣𝑜𝑢𝑡 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S20) Further manipulating (S20), we get the following quadratic 𝑣𝑜𝑢𝑡 2 (1− ( 𝐴𝑜𝑢𝑡 𝐴𝑡𝑜𝑡𝑎𝑙 ) 2 ) + 𝑣𝑜𝑢𝑡 (2𝐴𝑜𝑢𝑡𝐴𝐵 𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑) − 2𝛾∆𝑇 𝜌𝛼 = 0 , (S21) which has precisely 𝑣𝑜𝑢𝑡 as its only unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The coefficients of this polynomial are 𝑎 = 1− ( 𝐴𝑜𝑢𝑡 𝐴𝑡𝑜𝑡𝑎𝑙 ) 2 , 𝑏 = 2𝐴𝑜𝑢𝑡𝐴𝐵 𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑 , 𝑐 = − 2𝛾∆𝑇 𝜌𝛼 , (S22) making it a fairly straightforward process to solve for the roots of the equation, provided by 𝑣𝑜𝑢𝑡 = − (2𝐴𝑜𝑢𝑡𝐴𝐵 𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑) ± √(2𝐴𝑜𝑢𝑡𝐴𝐵 𝐴𝑡𝑜𝑡𝑎𝑙𝛼𝜑) 2 + 8𝛾∆𝑇 𝜌𝛼 (1− ( 𝐴𝑜𝑢𝑡 𝐴𝑡𝑜𝑡𝑎𝑙) 2 ) 2 (1− ( 𝐴𝑜𝑢𝑡 𝐴𝑡𝑜𝑡𝑎𝑙) 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S23) One underlying advantage of this derivation was that it removed the need to know the pressure difference, ∆𝑃, while providing us with enough information to solve for 𝑣𝑜𝑢𝑡 and 𝑣𝑓𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In the following sub-section, we deliver more details on the heat conduction modeling across the nanocardboard’s thickness, which enabled obtaining an expression for the temperature difference, ∆𝑇, necessary to solve for 𝑣𝑜𝑢𝑡 in (S23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2 Heat Conduction Modeling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 Full Analytical Derivation for ∆𝑻 In order to compute ∆𝑇, the temperature difference between both sides of the structure’s walls, we needed to model the heat conduction across the structure’s thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' We performed a heat energy balance that considered heat transfer across three distinct cross-sectional areas: the channel’s column of air, across the alumina thickness of the channel, and across the air trapped within the structure, as shown in Figure S11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, we can let 𝑄𝑡, the total heat transfer, be 𝑄𝑡 = ∆𝑇 𝑅𝑡1 + ∆𝑇 𝑅𝑡2 + ∆𝑇 𝑅𝑡3 , (S24) where the 𝑅𝑡1, 𝑅𝑡2 and 𝑅𝑡3 represent the thermal resistances under the three scenarios detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Page 10 For the first of these areas (A1), the column of air in the channel, we define its thermal resistance as 𝑅𝑡1 = 𝐿 𝑘𝑎𝑖𝑟𝐴1𝑋 = 𝐿 𝑘𝑎𝑖𝑟𝐴𝐵𝑋 , (S25) where 𝑘𝑎𝑖𝑟 is the thermal conductivity of air, L is as usual the channel height, and 𝐴𝐵𝑋 is the cross-sectional area of the individual channels multiplied by the number of channels in a unit cell, as shown in Figure S9 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that 𝜅𝑎𝑖𝑟 is both temperature and pressure dependent, as the equation developed by [R10] captures, specifically for the small MEMS scale: 𝜅𝑎𝑖𝑟 = 𝜅0 (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='00076𝑇 𝑃𝐿 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S26) In this expression, 𝜅0 is the air conductivity at standard atmospheric conditions, normally quoted as 𝜅0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='024 𝑊 𝑚 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Another comparable and slightly more succinct model for the conductivity of air is from [R8]: 𝜅𝑎𝑖𝑟 = 𝜅0 (1 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='116𝜆 𝐿 ) (S27) As the pressure decrease, the mean free path eventually becomes comparable to the channel length, and the effective conductivity starts to decrease below the continuum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Both equations (S26) and (S27) yielded very similar values for the conductivity of air as a function of the channel thickness L, although we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' S27 in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Continuing with the heat conduction modeling, the corresponding expression for the thermal resistance across the alumina thickness on the channels (area A2 in Figure S11) is given by 𝑅𝑡2 = 𝐿 𝑘𝑎𝑙𝑑𝐴2𝑋 = 𝐿 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 , (S28) where [(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 is the cross-sectional area occupied by the alumina thickness of the channels, which is denoted as 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In (S28), 𝑘𝑎𝑙𝑑 is the thermal conductivity of alumina, which has a constant value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8 𝑊 𝑚 𝐾 [R2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Lastly, the thermal resistance of the air trapped within the structure (area A3) is 𝑅𝑡3 = 𝐿 − 2𝑡 𝑘𝑎𝑖𝑟𝐴3 = 𝐿 − 2𝑡 𝑘𝑎𝑖𝑟 [𝐴𝐵 𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋 , (S29) Figure S11: Main nanocardboard cross-sectional areas for which thermal resistance is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Isometric View Top View Key Ai:Channelcross-sectionalarea A2:ChannelAluminathicknesscross-sectionalarea SectionCut As:Cross-sectional area oftrappedairwithinnanocardboardPage 11 where recall from (S13) that 𝐴𝐵𝑋 𝜑 is the full area of the cell, from which we subtract the combined cross- sectional area of the channels with thickness 𝑡 of alumina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Now, performing an energy balance, the heat flow through the structure’s walls must be equal to that from the absorbed irradiation of the sun, which in this case is given by 𝑄𝑡 = 𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 𝜑 ) (1 − 𝜑) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S30) In equation (S30), 𝜀 denotes the absorption coefficient (approximated to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='9 based-off the measurements from [R3]), 𝜓 the proportion of absorbed optical flux dissipated upward through the nanocardboard (which is assumed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 or 50%), and 𝐼𝑠𝑢𝑛 the intensity of the sun at a particular altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In particular, this last term can be modeled using the following equation 𝐼𝑠𝑢𝑛 = 1000 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8ℎ , (S31) where the variable h refers to the elevation above sea level in kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that this expression returns the sun’s intensity in units of Watts per meter square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Furthermore, in equation (S30), (𝐴𝐵𝑋/𝜑)(1 − 𝜑) corresponds to the solid area of the nanocardboard, 𝐴𝑠𝑜𝑙𝑖𝑑, where the sun’s irradiation is absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In any case, (S24) through (S31) were combined to write a general expression for ∆𝑇, which is summarized by ∆𝑇 = 𝑇2 − 𝑇1 = 𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 𝜑 ) (1 − 𝜑) 1 𝑅𝑡1 + 1 𝑅𝑡2 + 1 𝑅𝑡3 = = 𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 𝜑 ) (1 − 𝜑) 𝑘𝑎𝑖𝑟𝐴𝐵𝑋 𝐿 + 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 𝐿 + 𝑘𝑎𝑖𝑟 [𝐴𝐵 𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋 𝐿 − 2𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S32) In (S32), 𝑇1 and 𝑇2 represent the average temperatures outside and inside the 3D structure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, these might not necessarily be known beforehand, reason why calculating ∆𝑇 or 𝑇∗, the average temperature between both sides of the surface, may not be as trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In particular, to compute 𝑇∗, we make use of the fact that we know what ∆𝑇 is from (S32) and take the following expression 𝑇∗ = 𝑇1 + 𝑇2 2 = (𝑇2 − 𝑇1) + 2 ∗ 𝑇1 2 = ∆𝑇 + 2 ∗ 𝑇1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S33) Here, notice that 𝑇1 is simply equal to the temperature corresponding to the particular operating conditions (altitude, pressure, density) of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Overall, ∆𝑇 allows us to solve for 𝑇∗ (which is needed to compute 𝛾 and 𝛽∗ in (S7) and (S9), respectively) and the last part of the puzzle in (S23), the 𝑣𝑜𝑢𝑡 expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2 Simplified Expression for ∆𝑻 in the limit of zero alumina thickness Beyond the derivation provided in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1, notice that one could potentially also approximate ∆𝑇 through a more simplified expression given by ∆𝑇~ 𝐿𝐼𝑠𝑢𝑛(1 − 𝜑) 2𝜅𝑎𝑖𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S34) The origin of (S34) comes from taking the limit as t, the alumina thickness, approaches zero, in equation (S32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Indeed, lim 𝑡→0 𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 𝜑 ) (1 − 𝜑) 𝑘𝑎𝑖𝑟𝐴𝐵𝑋 𝐿 + 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 𝐿 + 𝑘𝑎𝑖𝑟 [𝐴𝐵 𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋 𝐿 − 2𝑡 (S35) Page 12 = lim 𝑡→0 𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 𝜑 ) (1 − 𝜑) 𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑙𝑑[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵]𝑋 + 𝑘𝑎𝑖𝑟 [𝐴𝐵 𝜑 − (𝐴 + 2𝑡)(𝐵 + 2𝑡)] 𝑋 = lim 𝑡→0 𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 𝜑 ) (1 − 𝜑) 𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑙𝑑[𝐴𝐵 − 𝐴𝐵]𝑋 + 𝑘𝑎𝑖𝑟 [𝐴𝐵 𝜑 − 𝐴𝐵] 𝑋 = lim 𝑡→0 𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 𝜑 ) (1 − 𝜑) 𝑘𝑎𝑖𝑟𝐴𝐵𝑋 + 𝑘𝑎𝑖𝑟 𝐴𝐵𝑋 𝜑 − 𝑘𝑎𝑖𝑟𝐴𝐵𝑋 = 𝐿𝜀𝜓𝐼𝑠𝑢𝑛 (𝐴𝐵𝑋 𝜑 ) (1 − 𝜑) 𝑘𝑎𝑖𝑟 𝐴𝐵𝑋 𝜑 = 𝐿𝜀𝜓𝐼𝑠𝑢𝑛(1 − 𝜑) 𝑘𝑎𝑖𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Furthermore, letting 𝜀 = 1 and 𝜓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5, then (S37) indeed becomes equation (S34) from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As evidenced by its compressed form, using (S36) to approximate ∆𝑇 simplifies the process of solving for the flow-through velocity, 𝑣𝑓𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' This is especially true if we were to also neglect the pressure term, assuming its contribution is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, the mass flow rate from (S5) can be re-written as 𝑚̇ ~𝛾 ∗ ∆𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S36) This helps reduce the flow-through velocity expression to 𝑣𝑓𝑡 = 𝜑𝑚̇ 𝜌𝐴𝐵 = 𝜑 𝛾∆𝑇 𝜌𝐴𝐵 = 𝜑 𝛾 𝜌𝐴𝐵 𝐿𝐼𝑠𝑢𝑛(1 − 𝜑) 2𝜅𝑎𝑖𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S37) Even this expression can be further simplified by reducing the 𝛾 term from (S7) to 𝛾~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝐴2𝐵𝑃∗𝛽∗ 𝛿𝑇∗𝐿 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝐴2𝐵𝑃𝛽∗ 𝑇𝐿𝐴√𝜋/(2𝜆) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2𝜆𝐴𝐵𝑃 √𝜋𝑇𝐿 √ 𝑚 2𝑘𝐵𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S38) From the ideal gas law, we have that 𝑃 = 𝜌𝑅𝑎𝑖𝑟𝑇, so the pressure term can be replaced in (S38) to obtain 𝛾~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2𝜆𝐴𝐵𝜌𝑅𝑎𝑖𝑟𝑇 √𝜋𝑇𝐿 √ 𝑚 2𝑘𝐵𝑇 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2𝜆𝐴𝐵𝜌𝑅𝑎𝑖𝑟 √𝜋𝐿 √ 𝑚 2𝑘𝐵𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S39) Combining equations (S37) and (S39), we resultant expression turns out as 𝑣𝑓𝑡 = 𝜑 𝜌𝐴𝐵 𝐿𝐼(1 − 𝜑) 2𝜅𝑎𝑖𝑟 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2𝜆𝐴𝐵𝜌𝑅𝑎𝑖𝑟 √𝜋𝐿 √ 𝑚 2𝑘𝐵𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑𝐼(1 − 𝜑)𝜆𝑅𝑎𝑖𝑟 𝜅𝑎𝑖𝑟 √ 𝑚 2𝑘𝐵𝑇𝜋 (S40) Now, recall that the average molecular velocity is equal to 𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇 𝜋 , (S41) and the relationship between viscosity and velocity, as provided by [R6], is equal to 𝜇 = 𝜆𝜌𝑣𝑎𝑣𝑔 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S42) Page 13 Hence, combining both (S41) and (S42) and solving for 𝜆, we obtain an expression which can be incorporated in (S40) to yield 𝑣𝑓𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 𝜅𝑎𝑖𝑟 𝜇 𝑃 √𝜋𝑘𝐵𝑇 2𝑚 √ 𝑚 2𝑘𝐵𝑇𝜋 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 𝜅𝑎𝑖𝑟 𝜇 𝑃 √ 𝑚𝜋𝑘𝐵𝑇 4𝑘𝐵𝑇𝜋𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 𝜅𝑎𝑖𝑟 𝜇 𝑃 √ 1 4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 2𝜅𝑎𝑖𝑟 𝜇 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S43) Now, according to [R6], the conductivity of air can be often approximated as 𝜅𝑎𝑖𝑟 = 2𝜇𝐶𝑣′ 𝑀 = 2𝜇𝐶𝑣, where M is the molar mass of air and 𝐶𝑣′ is the specific heat capacity at constant volume, in units of J/k mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Thus, equation (S43) can further simplify into 𝑣𝑓𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑𝐼(1 − 𝜑)𝑀𝑅𝑎𝑖𝑟 4𝜇𝐶𝑣′ 𝜇 𝑃 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑𝐼(1 − 𝜑)𝑅𝑎𝑖𝑟 4𝑃𝐶𝑣 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑(1 − 𝜑)𝑅𝑎𝑖𝑟 4𝐶𝑣 𝐼 𝑃 = 𝐶 𝐼 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S44) where the constant C is simply given by 𝐶 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1𝜑(1 − 𝜑)𝑅𝑎𝑖𝑟 4𝐶𝑣 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 ∗ (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5) ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='287 4 ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='718 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0275 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S45) Hence, what these series of derivations shows is that it is possible to approximate and obtain order-of- magnitude estimations of the flow-through velocity by using 𝑣𝑓𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0275 𝐼 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S46) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Lift Force Calculations and Temperature dependencies Once we knew how to calculate 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡 using the equations derived above (whether it is in the simplified or full analytical form), we used the following equation to calculate the lift forces produced by each geometry, as outlined in the ANSYS simulations section at the beginning of this document: ∑𝐹 = 𝐶1 ∗ 8 ∗ 𝜇 ∗ 𝐷 ∗ 𝑣𝑓𝑡 + 𝐶2 ∗ 𝜌 ∗ 𝐴𝑜𝑢𝑡 ∗ 𝑣𝑜𝑢𝑡 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S47) Here, R is the characteristic radius of the geometry (usually the inlet radius), while 𝜇 is the viscosity and 𝜌 the fluid density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In addition, C1 and C2 are the geometry dependent coefficients summarized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As the derivation of equations above evidences, all of the geometric (𝐴𝑡𝑜𝑡𝑎𝑙 and 𝐴𝑜𝑢𝑡) and channel (A, B, L, S, t) variables are present in (S23), meaning that it was possible to construct parametric studies exploring the dependency of 𝑣𝑓𝑡, and consequentially lift, on all of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice, all of these variables were largely independent of each other, making it possible to modify each separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, some other parameters within (S23), such as 𝐼𝑠𝑢𝑛, density 𝜌, and air viscosity 𝜇, were actually dependent on temperature, which in turn was also altitude dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, in order to accurately calculate the flow-through velocities 𝑣𝑓𝑡 experienced by a 3D geometry in a range of altitudes, we needed to derive expressions for approximating the air temperature, air pressure, air viscosity and air density as a function of altitude itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 Temperature dependent Relations We developed the relations characterizing the dependency between temperature and the fluid variable in question by using standard atmospheric5 empirical data and fitting equations to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For instance, for the data describing the dependency between air temperature and altitude, we fit both a 6th, 10th and 15th order 5 The specific standard atmospheric data was taken from the following three websites: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='engineeringtoolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='com/standard-atmosphere-d_604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='html | https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='pdas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='com/atmosTable1SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='html https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='pdas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='com/bigtables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='html Page 14 polynomial, as Figure S12 to the below shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Overall, the 15th order polynomial provided the best empirical fit, which was why we decided to use it for the rest of this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, one interesting aspect of this fit was that we actually fitted at the inverse of the temperature, the reason for which will become clearer in the derivation of the altitude-pressure dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In any case, equation (S48) below shows this explicit relation, with h (the altitude) being in kilometers, and all terms in the column added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 𝑇−1(ℎ) = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='592 ∗ 10−29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='023 ∗ 10−27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='491 ∗ 10−23 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='942 ∗ 10−21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='021 ∗ 10−18 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='152 ∗ 10−16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='271 ∗ 10−14 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='332 ∗ 10−12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='150 ∗ 10−10 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='862 ∗ 10−09 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='525 ∗ 10−08 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='150 ∗ 10−06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='154 ∗ 10−06 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='283 ∗ 10−05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='912 ∗ 10−05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='473 ∗ 10−03 ℎ15 ℎ14 ℎ13 ℎ12 ℎ11 ℎ10 ℎ9 ℎ8 ℎ7 ℎ6 ℎ5 ℎ4 ℎ3 ℎ2 ℎ1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S48) Having derived the empirical relation between temperature (its inverse) and altitude, it was possible to determine a similar expression for pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In essence, the differential equation describing the pressure- altitude relationship is given by 𝑑𝑃(ℎ) = −𝑔 ∗ 𝜌(ℎ) ∗ 𝑑ℎ , (S49) where 𝑔 is the gravitational constant on earth, and 𝜌(ℎ) the density of air at a particular altitude h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Using the ideal gas law, 𝜌(ℎ) can be substituted to yield the following expression for the above differential in equation (S49) 𝑑𝑃(ℎ) = −𝑔 ∗ 𝑃(ℎ) 𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) ∗ 𝑑ℎ , (S50) where now 𝑅𝑎𝑖𝑟 is the ideal gas constant of air and is equal to 287 𝐽/𝑘𝑔 ∗ 𝑚3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Easily enough, one can utilize the technique of separation of variables to obtain that 𝑑𝑃(ℎ) 𝑃(ℎ) = −𝑔 𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) ∗ 𝑑ℎ , (S51) which leaves all of the pressure terms on one side, and the rest on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, it is possible to see with more clarity why the above polynomial fit was done for the inverse of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Indeed, equation (S51) can be equivalently written as 𝑑𝑃(ℎ) 𝑃(ℎ) = −𝑔 ∗ 𝑇−1(ℎ) 𝑅𝑎𝑖𝑟 ∗ 𝑑ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S52) This expression can be easily integrated to obtain the following logarithm: ln(𝑃) = −𝑔 𝑅𝑎𝑖𝑟 ∗ ∫ 𝑇−1(ℎ) ∗ 𝑑ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S53) Letting 𝜁(ℎ) = ∫ 𝑇−1(ℎ) ∗ 𝑑ℎ be a placeholder for the integral of the inverse temperature polynomial and C be simply a constant of integration, we obtain that ln(𝑃(ℎ)) = −𝑔 𝑅𝑎𝑖𝑟 ∗ 𝜁(ℎ) + 𝐶 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S54) Figure S12: Modeled temperature dependency on altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Fitsto altitude-dependentT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5+10~3 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 Data 3 Order6polynomial Order10polynomial Order15polynomial 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 0 20 40 60 80 100 120 A/titude(km)Page 15 Figure S13: Modeled pressure dependency on altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Now, in order to remove the logarithm from the pressure, we can raise both sides of the expression to the Euler’s number power, and get 𝑃(ℎ) = 𝑒 −𝑔 𝑅𝑎𝑖𝑟∗𝜁(ℎ)+𝐶 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S55) After applying exponent rules, (S55) decomposes into the product given by 𝑃(ℎ) = 𝑒𝐶 ∗ 𝑒 −𝑔 𝑅𝑎𝑖𝑟∗𝜁(ℎ) , (S56) and can be further simplified, upon application of boundary conditions, into 𝑃(ℎ) = 101300 𝑃𝑎 ∗ 𝑒 −𝑔 𝑅𝑎𝑖𝑟∗𝜁(ℎ) , (S57) which takes the following full form: 𝑃(ℎ) = 101300 𝑃𝑎 ∗ exp [ −𝑔 𝑅𝑎𝑖𝑟 ∗ ( −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='870 ∗ 10−30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='682 ∗ 10−28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='064 ∗ 10−24 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='109 ∗ 10−22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='684 ∗ 10−19 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='865 ∗ 10−17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='271 ∗ 10−15 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='591 ∗ 10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='437 ∗ 10−11 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='518 ∗ 10−10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='421 ∗ 10−08 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='299 ∗ 10−07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0385 ∗ 10−06 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='608 ∗ 10−06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='955 ∗ 10−05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='473 ∗ 10−03 ℎ16 ℎ15 ℎ14 ℎ13 ℎ12 ℎ11 ℎ10 ℎ9 ℎ8 ℎ7 ℎ6 ℎ5 ℎ4 ℎ3 ℎ2 ℎ1 ) ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S58) As Figure S13 above shows, the agreement of this equation with the empirical data is very reasonable, especially below 80 km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Above 80 km, the atmosphere is no longer well mixed, has increasing concentrations of atomic oxygen, and the simple ideal gas law we used above no longer applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For this reason, the results that will be presented below correspond to altitudes below 80 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The next step was modelling the air density dependency on altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' With expressions for T(h) and P(h) above, we could use the ideal gas law to write Finally, the last dependency that remained to be defined was the air viscosity and altitude relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' To that end, we could use Sutherland’s law, which relates viscosity and temperature through the following equation: 𝜇(ℎ) = 𝜇𝑟𝑒𝑓 ∗ (𝑇(ℎ) 𝑇𝑟𝑒𝑓 ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 ∗ ( 𝑇𝑟𝑒𝑓 + 𝑆 𝑇(ℎ) + 𝑆) , (S60) where 𝜇𝑟𝑒𝑓 is the reference dynamic viscosity and 𝑇𝑟𝑒𝑓 the reference temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In this work, for air, at 𝑇𝑟𝑒𝑓 = 20 𝐶, we have that 𝜇𝑟𝑒𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='000018205 𝑃𝑎 ∗ 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Finally, S is a constant, known as Sutherland’s temperature, which is given by 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 𝜌(ℎ) = 𝑃(ℎ) 𝑅𝑎𝑖𝑟 ∗ 𝑇(ℎ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S59) DerivedEquationforPressure 106 Data DerivedEquation 104 (Pa) Pressure 100 102 104 0 20 40 60 80 100 120 A/titude (km)Page 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2 Payload Calculations Once all of the required equations and relationships were derived, it was possible to calculate 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡 for a specific set of geometric and channel parameters defining unique 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' By calculating these velocities, we determined the total force produced by each geometry, as outlined by equation (S47), from which it was possible to perform some payload estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, in order to obtain the payload estimates, it was paramount to first determine the surface areas of each one of the 3D geometries in question, the reason being that density of these structure was defined in areal terms as opposed to volumetric terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As was mentioned in the main paper, this work considered a truncated cone, truncated sphere, and a rocket, and their defining equations are shown in Table 7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Main Geometrical Area Definitions Area Truncated Cone Truncated Sphere Rocket 𝐴𝑡𝑜𝑡𝑎𝑙 𝜋 (𝐷 2) 2 + 𝜋 (𝐷 2)ℎ2 − 𝜋𝑟(ℎ2 − ℎ1) 𝜋(𝐷2 − 2𝑟ℎ) 2𝜋𝑟(𝑟 + 𝐷) 𝐴𝑜𝑢𝑡 𝜋𝑟2 𝐴𝑖𝑛 𝜑𝐴𝑡𝑜𝑡𝑎𝑙 𝐴𝑠𝑜𝑙𝑖𝑑 (1 − 𝜑)𝐴𝑡𝑜𝑡𝑎𝑙 Special Variables ℎ1 = √(𝐷 2 − 𝑟) 2 + 𝐷2 ℎ2 = √(𝐷 2) 2 + ℎ3 ℎ3 = 𝐷2 (𝐷 − 2𝑟) ℎ = (𝐷 2) − √(𝐷 2) 2 − 𝑟2 N/A Table 7: Area definitions used across this work for the cone, sphere and rocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that here, the variable ℎ3 follows from using similar triangles analysis, and letting ℎ3/(D/2) = D/(D/2 – r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For all three geometries, the variable D represents the overall scale of the structure while r their outlet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that 𝐴𝑖𝑛 is the porous area, while 𝐴𝑠𝑜𝑙𝑖𝑑 is the solid area in which the sun’s irradiance is absorbed, and it follows that 𝐴𝑡𝑜𝑡𝑎𝑙 = 𝐴𝑠𝑜𝑙𝑖𝑑 + 𝐴𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, having defined these surface areas (using the parameters established in Figure S1), we calculated the mass of our three 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In particular, since the cross-sectional area of a channel is simply 𝐴𝐵, then one can define the number of channels as the following integer floor: 𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 = ⌊𝐴𝑖𝑛 𝐴𝐵⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S61) The number of channels, 𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠, is an important parameter, given that now it is possible to calculate the volume of the structure that is occupied by the deposited alumina around each channel, which has thickness t and relatively high density 𝜌𝑎𝑙𝑑 of 3950 kg/m3 [R9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Indeed, similarly to equation (S28) above, we can define this volume as 𝑉𝑎𝑙𝑑,𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 = 𝑛𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠(𝐿 − 2𝑡)[(𝐴 + 2𝑡)(𝐵 + 2𝑡) − 𝐴𝐵] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S62) Experimentally, it has already been found that the areal density of nanocardboard, 𝜎𝑛𝑐𝑏, is about 1 g/m2 [R5], but this corresponds to a value of L (nanocardboard thickness) equal to 50 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, in our parametric studies, as we sweep through various values of L, especially those that are larger than 50 μm, this areal density alone is not enough to estimate the weight of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, calculating the volume of alumina around each of the channels is paramount, since the structure naturally becomes heavier with increasing thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Hence, the overall mass of any one of these geometries will be given by 𝑚𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 = 𝜎𝑔𝑒𝑜𝑚(𝐴𝑠𝑜𝑙𝑖𝑑 − 𝐴𝑖𝑛) + 𝜌𝑎𝑙𝑑𝑉𝑎𝑙𝑑,𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠 , (S63) where this expression accounts both for the areal density (𝜎𝑔𝑒𝑜𝑚) and the increases in the amount of the deposited alumina as a result of changes in the wall thickness L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Thus, the net lift produced by the geometry is simply given by subtracting the structure’s weight from the force expression in (S47), or Page 17 𝐿𝑖𝑓𝑡𝑛𝑒𝑡 = 𝐹 − 𝑔𝑚𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S64) While we know from simulations what 𝜎𝑔𝑒𝑜𝑚 is, notice that it is also possible to use our equations and a series of approximations to obtain a theoretical upper bound for this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In essence, we can start by letting the force be equal to the expression below 𝐹 = 𝑚̇ 𝑣𝑜𝑢𝑡 = (𝐴𝑖𝑛𝜌𝑎𝑖𝑟𝑣𝑓𝑡)𝑣𝑜𝑢𝑡 = (𝐴𝑖𝑛 𝑃 𝑅𝑎𝑖𝑟𝑇 𝑣𝑓𝑡)𝑣𝑜𝑢𝑡 , (S65) which incorporates mass flow rate and the ideal gas law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Now, recall that equation (S4) provides an expression relating 𝑣𝑓𝑡 and 𝑣𝑜𝑢𝑡, while (S46) provides a simplified approximation for 𝑣𝑓𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, taking a conservative approach that lets 𝑣𝑜𝑢𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2𝑣𝑎𝑣𝑔, a fifth of the average molecular velocity of a gas, shown in (S41) above, and incorporating (S2) and (S46), it is possible to re-write (S68) to obtain 𝐹 = 𝐴𝑖𝑛 𝑃 𝑅𝑎𝑖𝑟𝑇 𝑣𝑓𝑡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2√8𝑅𝑎𝑖𝑟𝑇 𝜋 = = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0055𝐴𝑖𝑛 𝑃 𝑅𝑎𝑖𝑟𝑇 𝐼 𝑃 √8𝑅𝑎𝑖𝑟𝑇 𝜋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0055𝐴𝑖𝑛 𝐼 𝑅𝑎𝑖𝑟𝑇 √8𝑅𝑎𝑖𝑟𝑇 𝜋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S66) Upon further simplification, equation (S69) reduces to 𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0055𝐴𝑖𝑛 𝐼 𝑅𝑎𝑖𝑟𝑇 √8𝑅𝑎𝑖𝑟𝑇 𝜋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0055√8 𝜋 𝐴𝑖𝑛𝐼√ 1 𝑅𝑎𝑖𝑟𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' (S67) Thus, the maximum areal density that can be entertained by these 3D structures can be approximated by 𝜎𝑔𝑒𝑜𝑚 = 𝐹 𝐴𝑖𝑛𝑔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0055√8 𝜋 𝐼 𝑔 √ 1 𝑅𝑎𝑖𝑟𝑇 = 𝐾𝐼√ 1 𝑅𝑎𝑖𝑟𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='016 𝐼 𝑣𝑎𝑣𝑔𝑔 , (S68) where 𝐾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0055 𝑔 √ 8 𝜋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0009 and 𝑣𝑎𝑣𝑔 = √8𝑅𝑎𝑖𝑟𝑇/𝜋 ≈ 400 m/s is the average velocity of air molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Upon inserting the parameters, we find that 𝜎𝑔𝑒𝑜𝑚 can have an average value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='004 kg/m2, four times of what the areal density of nanocardboard typically is in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The main paper provides additional areal density calculations based off from the parametric studies (detailed below) as well as cloud plots denoting the maximum areal density for each of the study geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' They are generally of the same order of magnitude as the estimate (S68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='3 Parametric Studies In this section, we provide four tables that accompany the presentation of the results shown in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In essence, Table 8 both summarizes the chosen optimization ranges and discretization for the variables that were varied (A, L and r) and specifies the values that the remaining variables (B, N, X, S and t) took.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Similarly, Table 9 through Table 11 present the results for the performed parametric optimization on the three geometries, detailing the specific combination of A, L and r that first, yielded the maximum payload capabilities and second, achieved flight at the lower altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' In addition, Table 9 through Table 11 also provide the areal density of each structure for when the maximum payload was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that this process was repeated for multiple values of D, as to explore the dependency of the overall optimization results with the scale of the geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Parametric Optimization Variables Variable Range Truncated Cone Truncated Sphere Rocket Discretization 𝐴 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 10 nm 80 equally spaced points (log scale) Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 5 mm 𝐿 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 1 μm 80 equally spaced points Page 18 Table 8: Main values used across the various variables during the parametric optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As can be seen, the search range for the optimal A, L and r was discretized in all three cases in 100 points, following a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Changing the granularity of the discretization or the bounds of the search range did not significantly modify the results seen in Table 9 through Table 11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Table 9: Combinations of A, L and r that returned the spheres capable of carrying the greatest payload and achieving flight at the lowest altitude, for various values of D, as specified in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Table 10: Combinations of A, L and r that returned the cones capable of carrying the greatest payload and achieving flight at the lowest altitude, for various values of D, as specified in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 1 cm (log scale) 𝑟 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' rmin = D/20 (see Table 9 through Table 12) 80 equally spaced points (log scale) Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' rmax = D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 (see Table 9 through Table 12) Altitude Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 0 km 17 equally spaced points (5 km intervals) Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 80 km 𝐵 10𝐴 𝑁 1 sun 𝑋 𝐵 − 𝑆 𝑆 + 𝐴 𝑆 𝐴 𝑡 50 nm Parametric Optimization Results – Various Sphere Sizes Variable Case D = 2 cm D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 m D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 m D = 1 m D = 2 m D = 5 m rmin = D/20, rmax = D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01, with a discretization of 80 points (log scale) 𝐴 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='21 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='21 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='21 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='21 mm 𝑟 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='07 cm 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05 cm 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='85 cm 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='70 cm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='84 m Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='89 cm 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='82 cm 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='63 cm 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='27 cm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='08 m Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload Payload (mg) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='34 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='11 1 445 5 526 21 612 133 242 Altitude (km) 80 80 80 80 80 80 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Density (g/m2) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='48 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='81 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='64 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='49 Sphere Area (m2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='63 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='52 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='82 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='68 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='17 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude Payload (mg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='58 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94 872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='33 3 442 21 339 Altitude (km) 55 55 60 60 60 60 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='34 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='96 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='30 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='32 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='31 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='38 Parametric Optimization Results – Various Cone Sizes Variable Case D = 2 cm D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='1 m D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 m D = 1 m D = 2 m D = 5 m rmin = D/20, rmax = D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01, with a discretization of 80 points (log scale) 𝐴 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='13 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='35 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='35 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='35 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='35 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='35 mm 𝐿 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36 mm 𝑟 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 cm 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='86 cm 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='73 cm 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='45 cm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='49 m Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05 mm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='39 cm 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='56 cm 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='12 cm 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='25 cm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='16 m Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload Payload (mg) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='96 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='26 2 043 7 929 31 228 193 348 Altitude (km) 80 80 80 80 80 80 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Density (g/m2) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='59 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='61 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='61 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='38 Cone Area (m2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='92 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='67 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude Payload (mg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='12 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='65 812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 3 209 19 892 Altitude (km) 55 60 60 60 60 60 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='75 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='84 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='84 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='83 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='72 Page 19 Table 11: Combinations of A, L and r that returned the rockets capable of carrying the greatest payload and achieving flight at the lowest altitude, for various values of D, as specified in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' The results from these tables are discussed in greater detail in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' However, there are four important points to highlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' First, changing D (the scaling of the overall geometries) did not affect significantly the optimal channel parameters A and L that yielded the maximum payload capabilities and achieved flight at the lowest altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Secondly, the obtained maximum areal densities were similar across the three geometries (as seen in Figure S14 (a) below) and had average values of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='31 g/m2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='68 g/m2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='96 g/m2, for the sphere, cone, and rocket, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Notice that these are above the theoretical order- of-magnitude estimation for the upper limit of 4 g/m2 in (S71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Thirdly, the optimized 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratios for the three geometries were relatively invariant across the various values of D and the two missions (max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' payload and minimum altitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' For instance, for the maximum payload optimization, 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 averaged 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='20, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 for the sphere, cone and rocket, respectively, while for the minimum altitude case, this ratio averaged 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='49 and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='65, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Lastly, for a given surface area, the amount of payload that each geometry could carry was comparable, as can be seen in Figure S14 (b) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' As a result, 1 m2 of a porous and geometrically optimized cone has a similar maximum payload capability than 1 m2 of an optimized rocket and sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Finally, Figure S15 through Figure S20 present cloud plots that permit visualizing the results from the parametric studies, in particular how different combinations of A, L and r enabled geometries with various altitude (a), payload (b) and areal density (c) capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' These plots correspond to the D = 10 cm and D = 10 m cone, sphere and rocket, and are accompanied with illustrations of the optimized geometries that achieved flight at minimum altitude (d) and carried the most payload (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' These figures were generated by discretizing the search ranges of A, L and r in 500 equally spaced, and the results from the optimized geometries are shown in Table 12 through Table 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Despite the increase in discretization points (from 80 to 500) in each dimension, the results were comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Parametric Optimization Results – Various Rocket Sizes Variable Case D = 2 cm D = 10 cm D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 m D = 1 m D = 2 m D = 5 m rmin = D/20, rmax = D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01, with a discretization of 80 points (log scale) 𝐴 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 mm Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='092 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='13 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='13 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='13 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='13 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='13 mm 𝐿 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='91 mm Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='094 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='14 mm 𝑟 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 cm 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='86 cm 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='73 cm 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='45 cm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='49 m Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='00 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94 cm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='12 cm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 cm 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='40 m Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload Payload (mg) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='51 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='59 2 639 10 281 40 573 251 516 Altitude (km) 80 80 80 80 80 80 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Density (g/m2) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='60 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='89 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='74 Rocket Area (m2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='0019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='047 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='68 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='71 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='12 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude Payload (mg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='54 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='29 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='57 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='37 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='67 Altitude (km) 45 55 55 55 55 55 𝐴𝑡𝑜𝑡𝑎𝑙/𝐴𝑜𝑢𝑡 ratio 42 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='28 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='27 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='27 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='09 27 Figure S14: D against Areal Density (a) and Surface Area against Payload (b) for the 3D geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a b Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='PayloadagainstGeometrySurfaceArea 100 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='Payload (kg) 10 Sphere -Cone Rocket 9-01 10-4 102 100 102Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='Areal Density against characteristic D 25 Sphere Cone 20 Rocket 15 10 102 10-1 100 101 D (m)Page 20 Comparison of D = 10 cm and D = 10 m Cone Geometries Case A L r Surface Area (m2) 𝑨𝒕𝒐𝒕𝒂𝒍/ 𝑨𝒐𝒖𝒕 ratio Payload (mg) Altitude (km) Discretization of 500 points D = 10 cm Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='15 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='16 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='52 55 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='25 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='31 80 D = 10 m Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='21 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='22 mm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36 m 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='52 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='16 95 288 60 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='25 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 m 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='56 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05 780 408 80 Table 12: Combinations of A, L and r that returned the optimal cone geometries described in Figure S15 and Figure S16 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Figure S15: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Cone Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, the geometry that was able to levitate payload at minimum altitude (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='52 mg at 55 km) is shown in (d), while that which was able to levitate the maximum payload (102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='31 mg at 80 km) is shown in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a c b d e Figure S16: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Cone Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, the geometry that was able to levitate payload at minimum altitude (95 288 mg at 60 km) is shown in (d), while that which was able to levitate the maximum payload (780 408 mg at 80 km) is shown in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a c b d e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 日 Aerial Densities: Cone Geometry 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='025 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='8g/m²99%percentile density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='49g/m~190%percentile density 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='84g/m150%percentile density 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='35g/m|25%percentiledensity 01 10 3 102 A [m] 104 L [m]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 [m] MinimumAltitudes:ConeGeometry 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 55km 60 km 65km 70km 10~2 104 103 102 A [m] 104 L[m]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 Maximum Payloads:Cone Geometry 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='29mg/99%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='025 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='08 mg/90% max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='15 mg/50% max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='69mg|30%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 10~ 104 103 102 A [m] L[m]r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='94cm r=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97cmAerial Densities:ConeGeometry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='85g/m199%percentile density 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='88g/m²90%percentile density 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='58g/m50%percentiledensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='19g/m25%percentile density 102 10 3 103 10 2 A[m] o1 104 L [m]3 [u] MinimumAltitudes:Cone Geometry 2 60km 65 km 70km 75km 102 102 A [m] 01 104 L[m]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 3 MaximumPayloads:ConeGeometry 772603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='92mg/99%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='53 702367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2mg/90%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 390204mg/50%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 2 234122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='4mg/30%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 102 10 3 104 103 102 A[m] 104 L [m]r=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36m r=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97mPage 21 Comparison of D = 10 cm and D = 10 m Rocket Geometries Case A L r Surface Area (m2) 𝑨𝒕𝒐𝒕𝒂𝒍/ 𝑨𝒐𝒖𝒕 ratio Payload (mg) Altitude (km) Discretization of 500 points D = 10 cm Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='11 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='12 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='50 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='001 >100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 50 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='25 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='56 80 D = 10 m Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='15 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='16 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='87 m 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='39 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='98 2132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='57 55 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='25 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 m 467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 1021162 80 Table 13: Combinations of A, L and r that returned the optimal rocket geometries described in Figure S17 and Figure S18 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Figure S17: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Rocket Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, the geometry that was able to levitate payload at minimum altitude (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 mg at 50 km) is shown in (d), while that which was able to levitate the maximum payload (129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='56 mg at 80 km) is shown in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a c b d r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='00 mm e r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 cm Figure S18: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Rocket Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, the geometry that was able to levitate payload at minimum altitude (2 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='57 mg at 55 km) is shown in (d), while that which was able to levitate the maximum payload (1 021 162 mg at 80 km) is shown in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a c b r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='87 m r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='97 m d e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 AerialDensities:RocketGeometry 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='64g/m²99%percentiledensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='78g/m|90%percentiledensity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='33g/m/50%percentiledensity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='36g/m²|25%percentile density 102 A [m] 104 103 102 104 L [m]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 [m] MinimumAltitudes:Rocket Geometry 50km 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01 55km 60km 65km 102 104 102 A[m] 104 L[m]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 MaximumPayloads:RocketGeometry 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='26mg|99%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='025 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='6mg/90%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='78mg|50% max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='87mg|30%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 102 103 10~ 10 3 10~2 A [m] 104 L[m]41 3 2 [u] Aerial Densities:RocketGeometry 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='18g/m²199%percentile density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='15g/m90%percentiledensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='78g/m150%percentiledensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='37g/m²25%percentiledensity 102 10 3 104 10 3 10 2 A [m] 104 L[m]4 3 2 宜 MinimumAltitudes:Rocket Geometry 55km 1 60km 65km 70km 102 10 4 102 A [m] 01 L [m]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5, MaximumPayloads:RocketGeometry 3 1010951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02mg|99%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 919046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='38mg|90%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='53 510581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='32mg|50%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 306348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='79mg30%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 102 103 104 10 3 102 A[m] 104 L [m]Page 22 Comparison of D = 10 cm and D = 10 m Sphere Geometries Case A L r Surface Area (m2) 𝑨𝒕𝒐𝒕𝒂𝒍/ 𝑨𝒐𝒖𝒕 ratio Payload (mg) Altitude (km) Discretization of 500 points D = 10 cm Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='15 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='16 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='93 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='41 55 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='93 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='86 80 D = 10 m Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Altitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='15 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='16 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 m 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='22 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='66 831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='92 55 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Payload 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='24 mm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='25 mm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='67 m 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='63 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='23 540 528 80 Table 14: Combinations of A, L and r that returned the optimal sphere geometries described in Figure S19 and Figure S20 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Figure S19: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 cm Sphere Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, the geometry that was able to levitate payload at minimum altitude (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='41 mg at 55 km) is shown in (d), while that which was able to levitate the maximum payload (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='86 mg at 80 km) is shown in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a c b d e Figure S20: Minimum Altitude (a), Maximum Payload (b) and Areal Density (c) plots for the D = 10 m Sphere Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Here, the geometry that was able to levitate payload at minimum altitude (831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='92 mg at 55 km) is shown in (d), while that which was able to levitate the maximum payload (540 528 mg at 80 km) is shown in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' a c b d e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 AerialDensities:SphereGeometry 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='05g/m²|99% percentile density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='13g/m²190%percentile density 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='01g/m/50%percentiledensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='43g/m²25%percentiledensity 102 103 102 A [m] 10~4 104 10 3 L[m]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03 [m] MinimumAltitudes:SphereGeometry 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 55km 60km 65km 70km 102 102 A[m] 01 104 L [m]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='04 MaximumPayloads:SphereGeometry 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='06mg/99%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='87mg90%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='93mg/50%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='96mg|30%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 102 103 102 A [m] 104 104 10 3 L[m]r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='90 r=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='67 m u4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 3 AerialDensities:SphereGeometry 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='03g/m199%percentile density 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='99g/m90%percentiledensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='61g/m|50%percentile density 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='2g/m²25%percentiledensity 10~2 10 3 104 10 3 102 A [m] 10 4 L[m]43 m [m] MinimumAltitudes: Sphere Geometry 2 55km 60km 65km 70km 102 102 A [m] 104 104 L[m]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='5 E 3 MaximumPavloads: Sphere Geometry 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='53 535122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='95mg/99%max:payload 486475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='41mg/90%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 23 270264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='11mg/50%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 162158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='47mg/30%max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='payload 102 10 104 103 102 A [m] 104 L [m]r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='93 r=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='02 cm cmPage 23 References [R1] Azadi, Mohsen, George A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Popov, Zhipeng Lu, Andy G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Eskenazi, Avery Ji Won Bang, Matthew F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Campbell, Howard Hu, and Igor Bargatin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "Controlled levitation of nanostructured thin films for sun- powered near-space flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" Science Advances 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 7 (2021): eabe1127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [R2] Cappella, Andrea, Jean‐Luc Battaglia, Vincent Schick, Andrzej Kusiak, Alessio Lamperti, Claudia Wiemer, and Bruno Hay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "High Temperature Thermal Conductivity of Amorphous Al2 O 3 Thin Films Grown by Low Temperature ALD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" Advanced Engineering Materials 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 11 (2013): 1046-1050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [R3] Cortes, John, Christopher Stanczak, Mohsen Azadi, Maanav Narula, Samuel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Nicaise, Howard Hu, and Igor Bargatin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "Photophoretic levitation of macroscopic nanocardboard plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" Advanced Materials 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 16 (2020): 1906878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [R4] Eskenazi, Andy, Tom Celenza, and Igor Bargatin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' “MATLAB-fluid-flow-parametric-studies.” (2022) https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='com/andyeske/MATLAB-fluidflow-parametric-studies [R5] Lin, Chen, Samuel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Nicaise, Drew E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Lilley, Joan Cortes, Pengcheng Jiao, Jaspreet Singh, Mohsen Azadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "Nanocardboard as a nanoscale analog of hollow sandwich plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" Nature communications 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 1 (2018): 1-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=" [R6] O'Neal Jr, Cleveland, and Richard S." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Brokaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "Relation between thermal conductivity and viscosity for some nonpolar gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" The Physics of Fluids 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 5 (1962): 567-574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [R7] Sharipov, Felix, and Vladimir Seleznev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "Data on internal rarefied gas flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" Journal of Physical and Chemical Reference Data 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 3 (1998): 657-706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [R8] Teagan, William P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', and George S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "Heat‐Transfer and Density‐Distribution Measurements between Parallel Plates in the Transition Regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" The Physics of Fluids 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 3 (1968): 497-506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [R9] Wagiman, Abdullah, Mohammad Sukri Mustapa, Mohd Amri Lajis, Shazarel Shamsudin, Mahmod Abd Hakim, and Rosli Asmawi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "Effect of Thermally Formed Alumina on Density of AlMgSi Alloys Extrudate Recycled Via Solid State Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 87, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 2 (2021): 137-144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' [R10] Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Grabarnik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Emadi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' De Graaf, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' Wolffenbuttel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' "Characterization of thermal cross-talk in a MEMS-based thermopile detector array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content='" Journal of Micromechanics and Microengineering 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} +page_content=' 7 (2009): 074022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE3T4oBgHgl3EQfDAk3/content/2301.04281v1.pdf'} diff --git a/SdAyT4oBgHgl3EQfuPmB/content/tmp_files/2301.00609v1.pdf.txt b/SdAyT4oBgHgl3EQfuPmB/content/tmp_files/2301.00609v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fcb20af36e89fc83c0a406dd38a07f8b627d796d --- /dev/null +++ b/SdAyT4oBgHgl3EQfuPmB/content/tmp_files/2301.00609v1.pdf.txt @@ -0,0 +1,2307 @@ +Generalized Uncertainty Principle Impact on Nonextensive Black Hole Thermodynamics +Ilim Çimdiker,1, ∗ Mariusz P. Da¸browski,1, 2, 3, † and Hussain Gohar1, ‡ +1Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland +2National Centre for Nuclear Research, Andrzeja Sołtana 7, 05-400 Otwock, Poland +3Copernicus Center for Interdisciplinary Studies, Szczepa´nska 1/5, 31-011 Kraków, Poland +(Dated: January 3, 2023) +The effect of the generalized uncertainty principle (GUP) on nonextensive thermodynamics ap- +plied to black holes, as well as the sparsity of radiation at different temperatures associated with +each nonextensive entropy, is investigated. We examine the Rényi, Tsallis-Cirto, Kaniadakis, Sharma +Mittal, and Barrow entropies, temperatures, and heat capacities and show that, in each case, due to +GUP corrections, the temperature and entropy have finite values, implying that the final state of the +black hole is a remnant at the end of the evaporation process and that the sparsity of the radiation at +each temperature depends on the mass of the black hole. We also find that GUP reduces the value of +the sparsity parameter for each case as compared to the sparsity parameter at Hawking temperature, +which is always constant throughout the evaporation. +I. +INTRODUCTION +Black holes emit radiation due to the Hawking evap- +oration process, and therefore, there is an established +concept of Hawking temperature [1] and Bekenstein +entropy [2] connected with the black hole horizon. +The black hole evaporation process operates within the +purview of quantum field theory, and one of its more in- +triguing aspects may be that it appears to indicate a non- +unitary evolution, which gives rise to the well-known is- +sue of the information loss paradox [3–5]. In this regard, +black holes behave like thermodynamic objects, and the +laws of black hole thermodynamics [6–10] are analogous +to the conventional thermodynamic laws. The thermo- +dynamics of black holes have been extensively studied +and used in a variety of cosmological and gravitational +applications [11–20]. +Entropy measures how difficult it is for an outside ob- +server to get information about the underlying structure +of the system. This is a clear reflection of the macro- +scopic features that result from the quantum statisti- +cal mechanics that govern the behavior of quantum mi- +crostates. For the case of black holes, there is no defi- +nition of Bekenstein entropy in quantum statistical me- +chanics and it only relies on Hawking’s area theorem +[21], therefore, it would be required to have a complete +theory of quantum gravity in order to fully comprehend +the origin of this entropy and the nature of microstates +in the case of black holes. Therefore, we rely on the defi- +nition of Bekenstein entropy for black holes. For the case +of a Schwarzschild black hole with mass M, the Hawk- +ing temperature TH and Bekenstein entropy SB are given +by [1, 2] +TH = +¯hκ +2πkBc , SB = kBc3A +4G¯h +, +(1) +∗ ilim.cimdiker@phd.usz.edu.pl +† mariusz.dabrowski@usz.edu.pl +‡ hussain.gohar@usz.edu.pl +where ¯h, G, kB, and c are the reduced Planck constant, +the Newton gravitational constant, the Boltzmann con- +stant, and the speed of light, respectively. The area A of +the event horizon is defined as A = 4πr2 +h in the above +equation (1), where rh = 2GM/c2 is the Schwarzschild +radius and κ = c4/4πGM is the surface gravity defined +on the event horizon of the Schwarzschild black hole. +The core assumption of Gibbs thermodynamics and +statistical mechanics is that entropy is extensive and ad- +ditive. Nonextensive statistical mechanics, such as Tsal- +lis nonextensive statistical mechanics [22–31], is the out- +come of removing this assumption. +The assumption +of the extensive nature of entropy is connected to ig- +noring the long-range forces between thermodynamic +sub-systems. Since the size of the system exceeds the +range of the interaction between the system’s compo- +nents, Gibbs thermodynamics ignores these long-range +forces. Because of this, the total entropy of a composite +system equals the sum of the entropies of the individ- +ual subsystems and entropy grows with the size of the +system. However, long-range forces are important in +various unique thermodynamic systems. For instance, +if we think of a black hole as a (3 + 1) dimensional ob- +ject, it is vital to note that Bekenstein entropy scales with +the area and is thus regarded as a nonextensive quan- +tity [32–38]. Furthermore, because of the area scaling, +Bekenstein entropy is nonadditive. +Therefore, Gibbs +thermodynamics or statistical mechanics may not be the +appropriate choice for studying the thermodynamics of +black holes. In order to understand the nonextensive +and nonadditive nature of Bekenstein entropy, several +extensions [22, 39–44] of standard Gibbs thermodynam- +ics have been applied to black holes and cosmologi- +cal horizons [45–70]. One of the main proposals is the +Tsallis-Cirto’s black hole entropy definition [32], which +makes the black entropy extensive and compatible with +the Legendre structure. Rényi entropy [39], being a mea- +sure of entanglement, is another definition of entropy +applied to black holes and cosmological horizons which +is nonextensive, but additive (by assumption). There +arXiv:2301.00609v1 [gr-qc] 2 Jan 2023 + +2 +have been some other nonextensive forms of entropy +suggested such as the Sharma-Mittal entropy [40, 41] +as a generalization of Rényi entropy, the Kaniadakis en- +tropy [42] which takes inspiration from Lorentz group +transformations and the Barrow entropy [44] which is +based on a hypothetical fractal structure of black hole +horizon as a result of quantum fluctuations. +Due to the prevalence of quantum gravity effects, it is +anticipated that the semiclassical technique would fail +during the last phases of Hawking evaporation. There is +currently no satisfactory theory of quantum gravity that +enables us to completely explain that regime, despite the +development of several quite diverse proposals [71–77]. +Investigating the phenomenological consequences of an +underlying theory of quantum gravity is one technique +to explore the quantum gravity effects at those scales. +The generalized uncertainty principle (GUP) [76–79] is +one approach that has the benefit of being sufficiently +generic to be compatible with several quantum gravity +theories. The Bekenstein entropy and Hawking temper- +ature of a black hole in its last phases of evaporation +are modified within this framework [73]. +Because of +these modifications, black holes do not entirely evapo- +rate during the evaporation process, and the final state +of the black hole is a remnant of the order of Planck mass +Sparsity [80–91] is an important feature of Hawking +radiation. It is defined as the average time between the +emission of successive quanta over the timescales set by +the energies of the emitted quanta. It was shown that +Hawking radiation is very sparse during the black hole +evaporation process [84], which is one of the key char- +acteristics that distinguish it from black-body radiation. +However, it has been found that when GUP corrections +are incorporated [87–89], the sparsity decreases toward +the late stages of evaporation. When nonextensivity is +considered in the context of Rényi temperature [90], the +Rényi radiation is initially not sparse, but as evaporation +progresses, it begins to become sparse and eventually +approaches the case of Hawking radiation. +In this paper, we are interested in exploring the GUP +modifications to the nonextensive entropies and corre- +sponding thermodynamic quantities in Rényi, Tsallis- +Cirto, Sharma-Mittal, Kaniadakis, and Barrow nonex- +tensive statistics. Furthermore, the sparsity of the radia- +tion is analyzed at different temperatures corresponding +to different nonextensive entropies. +The following is the outline of the paper. In Sec. II, we +introduce the notion of GUP and apply it to the case of +standard thermodynamic black hole quantities. In Sec. +III, we introduce nonextensive entropies and accompa- +nying nonextensive thermodynamic quantities, as well +as GUP modifications to nonextensive black hole ther- +modynamics. Finally, in Sec. IV, we summarize and +discuss our findings. +II. +GUP AND BLACK HOLE THERMODYNAMICS +A. +Generalized Uncertainty Principle +One common aspect of several quantum gravity the- +ories is that they all predict a minimum measurable +length [77, 92]. +For example, the notion of minimal +length is defined in string theory as the string length +[72, 93], in loop quantum gravity [74] it is the expec- +tation value of the length operator, and this notion +can also be developed by the phenomenological aspects +coming from black hole physics [77]. Because of the ap- +pearance of a minimum length at the Planck scale in var- +ious quantum gravity approaches, it has been proposed +that the Heisenberg Uncertainty Principle (HUP) +∆x0∆p ≥ ¯h, or ∆x0 ∼ +¯h +∆p +(2) +where ∆x0 and ∆p are position and momentum uncer- +tainties can be modified when gravitational interaction +is introduced. The simplest argument for the modifica- +tion of HUP within the framework of Newtonian theory +is that there is a gravitational acceleration⃗a of an electron +due to photon of mass E/c2 [73], where E is the pho- +ton energy and r is the photon-electron distance, which +reads +⃗a = ¨⃗r = − G(E/c2) +r2 +⃗r +r, +(3) +and the interaction takes place in a characteristic region +of length L ∼ r and in characteristic time t ∼ L/c. Then, +the velocity acquired by an electron ∆v is +∆v ∼ GE +c2r2 +L +c , +(4) +and the (extra due to gravity) distance ∆x1 it is shifted +reads +∆x1 ∼ GE +c2r2 +L2 +c2 ∼ G∆p +c3 += c∆p +4Fmax += l2 +p +∆p +¯h , +(5) +where lp = +√ +G¯h/c3 is the Planck length, and Fmax = +c4/4G is the maximum force [94–97]. Extra uncertainty (5) +adds to the standard HUP uncertainty of position ∆x0 as +in (2) giving +∆x = ∆x0 + ∆x1 ∼ +¯h +∆p + l2 +p +∆p +¯h , +(6) +leading to the generalized uncertainty principle (GUP) +∆x∆p ≥ ¯h +� +1 + +l2 +p +¯h2 (∆p)2 +� +. +(7) +Taking an algebraic point of view, GUP can be derived +from the deformed commutation relation between the + +3 +position operator ˆx and the momentum operator ˆp such +that +[ ˆx, ˆp] = i¯h f ( ˆp), +(8) +where f ( ˆp) is a general function of momentum operator +ˆp and there exist different proposed functions for f ( ˆp). +In order to make the function f ( ˆp) compatible with (7), +following the literature, we choose +f ( ˆp) = 1 + α +l2 +p +¯h2 ˆp2, +(9) +where we the introduce GUP parameter α – a dimen- +sionless parameter predicted to be an order of unity, but +in reality bounded by different experiments and obser- +vations to be much larger than that [98–102]. By intro- +ducing α, the equation (10), now, reads as +∆x∆p ≥ ¯h +� +1 + α +l2 +p +¯h2 (∆p)2 +� +. +(10) +1. +GUP Modified Hawking Temperature and Bekenstein Entropy +An interesting application of (10) to black hole physics +is the modification to the Hawking temperature, which +can be derived by solving it for ∆p, which gives +∆p = ∆x ¯h +αl2p +� +�1 ± +� +1 − +αl2p +(∆x)2 +� +� . +(11) +We consider the + sign in (11), following the discus- +sion in [87]. Considering the minimum position uncer- +tainty near the event horizon of the Schwarzschild black +hole as ∆x = 2lp = 4GM/c2, where lp is taken as the +Schwarzschild radius rh, the GUP modified Hawking +temperature TGUP reads +TGUP = +m2 +pc2 +8πkBM +� +��� +4 +2 + +� +4 − α +m2p +M2 +� +��� . +(12) +By introducing a correction term due to GUP, K(α, M), +TGUP can be written in terms of TH and K, such that +TGUP = TH(M)K(α, M), +(13) +where the GUP correction term is defined as +K(α, M) = +4 +2 + +� +4 − α +m2p +M2 +. +(14) +This provides us with a more compact form of TGUP, +which will be used in the next sections for GUP mod- +ifications to the thermodynamic quantities. Using the +Clausius relation, the GUP modified Bekenstein entropy +SGUP in terms of SB and the correction term K(α, M) can +be written as +SGUP = SB +K − απkB +2 +ln +� 4M +m0K +� +, +(15) +where m0 is a dimensionful constant of unit mass, which +is introduced in order to make the logarithm dimension- +less. Note that in the limit α → 0, the correction term K +goes to one, and hence TGUP and SGUP reduce to TH and +SB. The plots of (12) and (15) are given in Figs. 1 and 2. +It is important to mention that all the plots in the pa- +per, unless explicitly stated, are given in natural units +¯h = c = G = 1 and also with the GUP parameter α = 1. +TH +TGUP,α=1 +TGUP,α=-1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +M +T +Figure 1. Temperature vs mass for the Hawking temperature +TH and the GUP corrected temperature with positive and neg- +ative values of α. Threshold with positive α for mass lies at the +remnant mass M2r = (α/4)m2p (cf. formula (16)). +SB +SGUP,α=1 +SGUPα=-1 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +10 +20 +30 +40 +50 +M +S +Figure 2. Entropy vs mass for the Hawking temperature and +GUP corrected temperatures with positive and negative val- +ues of α. The threshold for mass lies at the remnant mass given +by M2r = (α/4)m2p. +It is interesting to note that, for real physical situa- +tions, the equation (14) gives a bound on the mass which +reads: M2 ≥ αm2 +p/4. This means that for positive values +of α, the black hole evaporation stops when the mass of +the black hole reaches some critical value of mass +Mr = +√αmp +2 += 2lp +√α +c2 +Fmax, +(16) +which is called the black hole remnant mass. Therefore, +we can say that the final state of the black hole evapora- +tion is a remnant having the mass Mr. In fact, without + +4 +a well-defined quantum gravity theory, we cannot pre- +dict what happens if the mass of a black hole is smaller +than this critical value. For the critical mass value Mr, +the formulas (12) and (15) for TGUP and SGUP, give the +temperature Tr and the entropy Sr for the remnant as +[90] +Tr = +mpc2 +2πkB +√α, Sr = παkB +2 +� +1 − ln +�√αmp +m0 +�� +, +(17) +provided that α > 0. For α < 0 in (14), we have a smooth +correction function defined for all black hole mass val- +ues. +In this case, the black hole continues to radiate +slowly and yields an infinite lifetime [89]. When M ap- +proaches zero, interestingly, the temperature is still fi- +nite, and for this case, in [103], it is referred to as a rem- +nant with zero rest mass. +2. +GUP Modified Heat Capacity +In order to investigate the GUP modifications to the +heat capacity of a black hole with mass M, we use the +definition of heat capacity C, which reads +C = −S′2(M) +S′′(M) , +(18) +where S is the black hole entropy and prime and dou- +ble prime denote the first and second derivative with +respect to the mass M. For the case of Schwarzschild +black hole, we have (denoting C as CSc) +CSc = −8πkB +M2 +m2p +, +(19) +and we can see that it is negative for all mass values. +This means that the Schwarzschild black hole is thermo- +dynamically unstable. In order to introduce GUP cor- +rections, we introduce the quantity +βGUP = +1 +kBTGUP +, +(20) +which after using (12) gives +S′ +GUP(M) +kBc2 += βGUP = β +K , +(21) +where β = 1/kBTH is the inverse Hawking tempera- +ture. Differentiating βGUP once more, and using equa- +tions (18) and (21), we obtain the GUP modified heat +capacity CGUP, which can be written as (cf. Fig. 3) +CGUP = CSc +�2 − K +K2 +� +. +(22) +This means that the GUP corrections still yield a nega- +tive heat capacity for M > Mr, and when the black hole +mass approaches the critical mass Mr, we have K = 2 +and interestingly, we get the zero heat capacity for the +remnant. In such a case, from the thermodynamic point +of view, a small amount of heat would then increase the +temperature of the remnant by an infinite amount. +CSc +CGUP,α=1 +CGUP,α=-1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-30 +-25 +-20 +-15 +-10 +-5 +0 +M +C +Figure 3. Specific heat capacity of the Hawking radiation for +GUP corrected black holes. For positive α, there is a remnant +with zero heat capacity. +3. +GUP Modified Sparsity of Hawking Radiation +One of the most important aspects of Hawking radia- +tion is that it is extremely sparse as compared to black- +body radiation. The sparsity can be defined by using the +parameter η [84, 87, 90], +η = C +g +� +λ2 +t +Ae f f +� +, +(23) +where C is a dimensionless constant associated with dif- +ferent physical cases [84], g is the spin degeneracy fac- +tor of the particle, λt = 2π¯hc/kBT is the thermal wave- +length in terms of the temperature T and Ae f f = 27A/4 +[80, 84] is the effective area with A being the horizon +area for the case of black holes. For the Schwarzschild +black hole, one can find the thermal wavelength λt by +taking T = TH = 1/kBβ as +λt = 2π¯hc +kBTH += 2π¯hcβ, +(24) +and the sparsity parameter for the Hawking radiation +reads [84] +ηH = 64π3 +27 +≈ 73.38, +(25) +which is constant and is much greater than one. Note +that for standard black body radiation, the value of η +is less than one. This implies that the sparsity param- +eter clearly differentiates the Hawking radiation from +blackbody radiation. One can obtain the GUP effects on +the sparsity by replacing the Hawking temperature with +the GUP corrected temperature TGUP given by (12) [87]. +However, it is assumed that GUP also modifies the black +hole horizon area [87, 90]. Thus, it is logical to take the +effective area that GUP modifies. In fact, the GUP mod- +ifications to A can be derived from the equation (15) by +writing it as +SGUP = kBc3AGUP +4¯hG +, +(26) + +5 +ηH +ηGUP,α=1 +ηGUP,α=-1 +0 +1 +2 +3 +4 +0 +20 +40 +60 +80 +100 +M +η +Figure 4. Sparsity of Hawking vs GUP corrected black holes in +natural units. For positive values of α, we observe that sparsity +decreases when a black hole is near the final evaporation state. +where the GUP modified area AGUP reads +AGUP = A +K − απl2 +p ln +� 16A +A0K2 +� +, +(27) +and A0 = 16πm2 +0G2/c4 is a constant having the dimen- +sion of area. Note that in [90], corrections are only in the +first order of α, while in the above equation (27) the area +is corrected to all orders in α. Now, the GUP modified +sparsity can be found by replacing T by TGUP and A by +AGUP in (23), which now reads +ηGUP = ηH +K2 +� +A +AGUP +� +. +(28) +Interestingly, GUP modified sparsity ηGUP, depends on +the mass of the black hole and the GUP parameter α. +For the negative values of α, the sparsity parameter in- +creases as M goes to zero. For the positive values of +α, the sparsity parameter decreases below the values of +sparsity for the Hawking radiation until it reaches the +critical mass Mr. In Fig. 4, we can see that the GUP +corrected sparsity is not a constant and it increases first +before M approaches Mr for α > 0 and then it decreases +to finite value when M approaches to Mr. For the case +of α < 0, first, it decreases, and then it goes to plus in- +finity when M approaches zero. It is due to the fact that +A/AGUP > 1 for α > 0 and ηH/K2 turns back the spar- +sity from a maximum value to a constant value, which +is less than ηH. Therefore, we can clearly see the effects +of GUP on sparsity due to TGUP and AGUP as depicted +in Fig. 4. Similarly, A/AGUP < 1 for α < o and K goes +to zero when M approaches zero, therefore, sparsity de- +creases first, and then it goes to infinity. Note that in +[89], the GUP corrected area is not taken into account, +therefore, there is no bump in the sparsity parameter. +III. +GUP AND NONEXTENSIVE BLACK HOLE +THERMODYNAMICS +A. +Tsallis Nonextensive Entropy +Entropy plays a significant role in Gibbs thermody- +namics or statistical mechanics. +It is extensive and +adheres to the additive composition rule. +However, +Gibbs statistical mechanics ignores long-range forces. +On the other hand, there are some physical systems for +which Gibbs thermodynamics cannot be the appropri- +ate choice to apply [24] since they are subject to long- +range forces. +Important examples are the some self- +gravitating systems such as black holes, since for them +long-range forces play significant role. For that reason +Constantino Tsallis in Refs. [22, 24] generalized the con- +ventional Gibbs entropy for nonextensive systems in or- +der to encompass and address this issue. Tsallis entropy +ST was one of the earliest proposals to extend Gibbs en- +tropy and the suggested new form of it reads +ST = −kB ∑ +i +[p(i)]q lnq p(i), +(29) +where p(i) is the probability distribution defined on a +set of microstates Ω, with the parameter q determining +the degree of nonextensivity, and we consider it positive +to ensure the concavity of Sq. The q-logarithmic function +lnq p is given by +lnq p = p1−q − 1 +1 − q +, +(30) +where, in the limit q → 1, Tsallis entropy Sq given by +(29), reduces to Gibbs entropy SG +SG = −kB ∑ +i +p(i) ln p(i). +(31) +In fact, the Tsallis entropy (29) satisfies quite general, +nonadditive composition rule of the following form +ST 12 = ST 1 + ST 2 + λ +kB +ST 1ST 2, +(32) +for a composite system ”12”, made up of two subsys- +tems ”1” and ”2”. In above equation, we have defined a +new nonextensivity parameter λ = 1 − q. +B. +Rényi Entropy +The Rényi entropy [39], a measure of entanglement +in quantum information that is additive and preserves +event independence, is another important generaliza- +tion of the Gibbs-Shannon entropy. It is defined as +SR = kB +ln ∑i pq(i) +1 − q +. +(33) + +6 +It is important that SR can be written in terms of ST by +using the formal logarithm approach [30], and both en- +tropies are related as follows +SR = kB +λ ln[1 + λ +kB +ST ]. +(34) +It is interesting to mention here that SR is the equilib- +rium entropy which corresponds to an equilibrium tem- +perature TR defined from the equilibrium condition by +maximizing the Tsallis entropy (32), which is given by +[53] +TR = (1 + λ +kB +ST ) 1 +kBβ. +(35) +Here, kBβ = ∂ST /∂U, where U is the internal energy of +the nonextensive system. +1. +Rényi black hole Entropy and Temperature +For the case of a Schwarzschild black hole, assuming +that the Bekenstein entropy SB is just the Tsallis entropy +ST , and replacing internal energy U with the mass of +the black hole M in equations (34) and (35), the Rényi +entropy can be defined on the horizon of a black hole as +[33–37] +SR = kB +λ ln[1 + λ +kB +SB], +(36) +and the associated Rényi temperature reads +TR = (1 + λ +kB +SB)TH. +(37) +Furthermore, we can write down the GUP corrected +Rényi entropy using GUP corrected Bekenstein entropy +as follows [90] (cf. Fig. 5) +SRgup = kB +λ ln +� +1 + λ +kB +(SGUP) +� +, +(38) +and corresponding GUP modified Rényi temperature +TRgup can be written as (cf. Fig. 6) +TRgup = +� +1 + λ +kB +(SGUP) +� +KTH. +(39) +The Rényi entropy increases logarithmically (for 0 < +λ < 1), whereas the Bekenstein entropy (λ → 0) in- +creases quadratically, as shown in Fig. 5. Furthermore, +for the GUP corrections, the Rényi black holes do not +completely evaporate; rather, evaporation stops at the +critical mass Mr, leaving a remnant with finite entropy +and temperature as the Rényi black hole’s final state. +Using (37) and (39), we can write the inverse Rényi +temperature parameters, βR and βRgup, which will fur- +ther be used in calculating the heat capacities, such that +kBβR = S′ +B(M)/c2 +1 + λ +kB SB += +kBβ +1 + λ +kB SB +, +(40) +λ=0 +λ=0.5 +λ=1 +λ=0 +λ=0.5 +λ=1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +M +SR +Figure 5. +Rényi entropy SR of a black hole vs its mass M. +Dashed lines represent GUP corrected cases, λ → 0 limit is +the Bekenstein-Hawking case. +λ=0 +λ=0.5 +λ=1 +λ=0 +λ=0.5 +λ=1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.5 +1.0 +1.5 +2.0 +M +TR +Figure 6. Rényi temperature TR of a black hole vs its mass M. +Dashed lines represent GUP corrected cases, λ → 0 limit is the +Bekenstein-Hawking case. +and the GUP-corrected inverse Rényi temperature reads +kBβRgup = S′ +GUP(M)/c2 +1 + λ +kB SGUP += +kBβGUP +1 + λ +kB SGUP +. +(41) +One may determine the characteristic length scale LR +for λ [49, 50, 52], which reveals the impact of nonexten- +sive parameter λ in SR and SRgup, and in TR and TRgup. +As a result, it can be concluded that below this charac- +teristic length scale LR, the Rényi temperature behaves +like TH, and that above LR, the nonextensive effects in- +crease and TR grows linearly with M. The precise value +for the length scale is found in the following subsection. +2. +Heat Capacity for the Rényi black hole +In order to investigate the thermodynamic stability of +Rényi black holes, we define the heat capacity CR of the +Rényi black hole as +CR = −S′2 +R(M) +S′′ +R(M) . +(42) + +7 +Inserting (40) and (41) into (42), the heat capacity for the +non-GUP case reads +CR = +CSc +1 + λ +kB SB + λ +kB CSc +, +(43) +and for the GUP case, we have +CRgup = +CGUP +1 + λ +kB SGUP + λ +kB CGUP +. +(44) +We plot the heat capacity in Fig. +(7), where we can +λ=0 +λ=0.5 +λ=1 +λ=0 +λ=0.5 +λ=1 +0.0 +0.5 +1.0 +1.5 +2.0 +-10 +-5 +0 +5 +10 +M +CR +Figure 7. Heat capacity CR of a Rényi black hole vs its mass M. +Dashed lines represent GUP corrected cases, λ → 0 limit is the +Bekenstein-Hawking case. +see that L differentiates two regions for non-GUP and +GUP cases. In order to understand the behavior of CR in +both regions, we find LR in terms of λ from the singular +points of equation (43) for the case Schwarzschild black +hole. We find, for the non-GUP case +λ = − +kB +[SB + CSc] = +m2 +p +4πM2 , +(45) +and for the GUP case, we have +λ = − +kB +[SGUP + CGUP] +(46) +≈ +m2 +p +4πM2 + +3αm4 +p +64πM4 + +αm4 +p log +� +4M +mp +� +32πM4 +by ignoring the higher order terms in α. This means that +for the non-GUP case, we define the mass scale +Mc = +mp +2 +√ +πλ +, +(47) +which differentiates the two regions and can be further +used to define the characteristic length scale LR, which +can be written as +LR = 2lp +√ +πλ, +(48) +where we have defined LR = GMc/c2. For the GUP +case, we would expect the characteristic length scale +LRgup ≈ LR + α f (λ) by using equation (47), where f is +a function of the nonextensivity parameter λ. However, +we can not solve it exactly, and it again shows the effects +of α and λ for the values of M greater than the GUP cor- +rected mass scale. Interestingly, for the non-GUP case, +the heat capacity is positive for the values greater than +this scale, and below this scale, black holes have neg- +ative heat capacity. This means that black holes with +higher masses than Mc are thermodynamically stable +and with masses lower than Mc, they are unstable. Note +that, if we exclude quantum gravity effects, LR should +be greater than lp. This puts a numerical constraint on +the nonextensive parameter λ > 1/4π and this can also +be derived by considering Mc > mp by excluding the +quantum gravity effects. In [49, 50, 52], the authors de- +rived this constraint as λ > 1/π because they consid- +ered LR = 2GMc/c2 as characteristic length scale for λ, +where the extra 2 in LR is motivated by Schwarzschild +radius rh = 2GM/c2. We believe that the proper way to +introduce the length or mass scale for λ should be irre- +spective of the definition which is motivated by rh. +3. +Sparsity of the Rényi Radiation +In order to calculate the sparsity of Rényi radiation, +we replace T with TR in (23), and so the sparsity param- +eter ηR reads +ηR = +ηH +[1 + λ +kB SB]2 . +(49) +Replacing T with TRgup and using GUP modified area +AGUP in equation (23), the GUP modified sparsity pa- +rameter ηRgup reads +ηRgup = +ηGUP +[1 + λ +kB SGUP]2 . +(50) +From (49), we conclude that the sparsity parameter ηR +λ=0 +λ=0.5 +λ=1 +λ=0 +λ=0.5 +λ=1 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +20 +40 +60 +80 +M +ηR +Figure 8. +Sparsity ηR of a Rényi blackhole vs its mass M. +Dashed lines represent GUP corrected cases, λ → 0 limit is +the Bekenstein-Hawking case. +depends on both the mass of the black hole and the +nonextensivity parameter λ. +From Fig. +(8), we can +easily see that the radiation is not sparse initially and +then, at the final stages of the evaporation, the sparsity + +8 +grows, reaching the value of ηH, when M approaches to +zero. For the GUP case, initially, the behavior of spar- +sity is similar to the non-GUP case, however, when M +approaches Mr, it has a finite value which is much less +than the sparsity of Hawking radiation for the non-GUP +and GUP cases. Again, we can see the bump before M +reaches Mr, which is due to the effect of GUP correc- +tions to the Rényi temperature and GUP corrections to +the area. +C. +Tsallis-Cirto Black Hole Entropy +Tsallis-Cirto black hole entropy [32] is based on key +principles of Gibbs thermodynamics. First, the entropy +must be extensive and additive, and second, the entropy +and associated temperature for a thermodynamic sys- +tem must satisfy the Legendre structure. For the case +of black holes, if we rely on the definition of Beken- +stein entropy, then black holes are considered to be +two-dimensional thermodynamic objects since Beken- +stein entropy scales with area and Bekenstein entropy +and Hawking temperature fulfill the Legendre struc- +ture. However, if we consider a black hole as a (3 + 1) +dimensional thermodynamic object, then the Bekenstein +entropy is thought to be nonextensive due to its area +scaling and also because it follows a nonadditive com- +position rule S12 = S1 + S2 + 2√S1 +√S2 (see e.g. [90]), +whereas Gibbs statistical mechanics or thermodynam- +ics is based on the extensive and additive properties of +the entropy. This indicates that Bekenstein entropy vio- +lates a key principle of classical Gibbs thermodynamics +and that new definitions of entropy and temperature for +black holes are required in order to comply with the fun- +damental principles of thermodynamics in the case of +(3 + 1)-dimensional black holes. Therefore, Tsallis and +Cirto proposed the following entropy definition [32, 38]. +Sδ +kB += +�SB +kB +�δ +, +(51) +where δ > 0 is a real parameter and it follows the com- +position rule for a composite thermodynamic system, +which is given by +Sδ12 = kB +��Sδ1 +kB +�1/δ ++ +�Sδ2 +kB +�1/δ�δ +. +(52) +In this context, the SB is additive, and Sδ is nonadditive. +For δ = 3/2, Sδ is proportional to the volume for the +case of the Schwarzschild black hole, and so it is an ex- +tensive quantity. The corresponding Tsallis-Cirto tem- +perature can be written by using the Clausius relation +[53] +Tδ = TH +δ +�SB +kB +�1−δ +, +(53) +and it scales with 1/M2 for δ = 3/2, i.e., Tδ ∝ 1/M2, for +the case of Schwarzschild black hole. GUP corrections +to the Tsallis-Cirto black hole entropy can be obtained +by the GUP corrected Bekenstein entropy SGUP given +by (15) into (51), which results in +Sδgup +kB += +�SGUP +kB +�δ +, +(54) +and the corresponding GUP-modified Tsallis-Cirto tem- +perature can be derived from the Clausius relation, giv- +ing +Tδgup = TGUP +δ +�SGUP +kB +�1−δ +. +(55) +From the Figs. +(9) and (10), it shows that the evap- +δ=0.4 +δ=0.7 +δ=1.5 +δ=0.4 +δ=0.7 +δ=1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +5 +10 +15 +20 +M +Sδ +Figure 9. Tsallis-Cirto entropy ST of a black hole vs its mass +M. Dashed lines represent GUP-corrected cases in this figure +oration process stops at the critical value Mr for the +Tsallis-Cirto case when GUP corrections are included. +This means that the final state of the black hole for the +Tsallis-Cirto case is also a remnant with finite entropy +and temperature. Generally, for the non-GUP case, the +parameter δ plays a significant role. For δ > 1/2, the +Tsallis-Cirto entropy behaves similarly to Bekenstein en- +tropy and increases exponentially with mass, whereas +for δ < 1/2, it increases with mass sub-linearly. For +δ = 1/2, the entropy depends linearly on mass, and +in this case, Tsallis-Cirto temperature becomes constant. +Furthermore, the behavior of the Tsallis temperature is +similar to the Hawking temperature for δ > 1/2 while +for δ < 1/2, the behavior is completely different for the +non-GUP case and, interestingly, it behaves like Rényi +temperature for the GUP-corrected case. Note that, un- +like λ parameter of the Rényi entropy, δ is not associated +with the length scale for the non-GUP case. On the other +hand, introducing GUP corrections to Tsallis-Cirto en- +tropy, one can define a characteristic length scale for δ +as well. +1. +Heat Capacity for Tsallis-Cirto black holes +Following the previous subsection, the heat capacity +for the Tsallis-Cirto case can be written in terms of Csc, + +9 +δ=0.4 +δ=0.7 +δ=1.5 +δ=0.4 +δ=0.7 +δ=1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +M +Tδ +Figure 10. +Temperature Tδ vs the mass M for Tsallis-Cirto +black hole entropy. Dashed lines correspond to a GUP case. +and SB +Cδ = CSc +� +SB +SB − (δ − 1)CSc +� +, +(56) +where for the Schwarzschild black hole, we have CSc = +−2SB. For δ = 1/2, we have infinite heat capacity for +all masses. For δ < 1/2, we have positive heat capac- +ity values and negative heat capacity for δ > 1/2. This +means that black holes are thermodynamically stable for +δ < 1/2, and unstable for δ > 1/2. For the GUP correc- +δ=0.4 +δ=0.7 +δ=1.5 +δ=0.4 +δ=0.7 +δ=1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +-20 +-10 +0 +10 +20 +M +Cδ +Figure 11. Heat Capacity Cδ for Tsallis-Cirto black hole en- +tropy. Dashed lines correspond to a GUP case. +tions, we can write the GUP-corrected heat capacity as +Cδgup = CGUP +� +SGUP +SGUP − (δ − 1)CGUP +� +. +(57) +Note that from equations (15) and (22), we have +−2SGUP ̸= CGUP, therefore, we can find an associated +characteristic length scale Lδgup for the δ parameter, for +which, we have two regions, which corresponds to pos- +itive and negative values of GUP corrected heat capac- +ities. The length scale Lδgup can be found by using the +singular points of the above equation (57) for δ, which is +given by +δ = SGUP +CGUP ++ 1. +(58) +One could solve the above equation (58) for mass M, +which gives Lδgup as a function of δ. However, it is ana- +lytically not possible. One may use the perturbative ap- +proach to solve the equation for M and define the corre- +sponding length scale or mass scale. From the Figs. (9) +and (11), for δ < 1/2, and below Lδgup, the GUP cor- +rected Tsallis-Cirto entropy behaves like SR and it gives +positive GUP modified heat capacity for the GUP case. +For values δ > 1/2, Lδgup does not exist as (58) yields +imaginary numbers. Thus, it gives negative heat capac- +ity, implying that GUP-corrected Tsallis black holes are +thermodynamically stable for δ < 1/2, and unstable for +δ > 1/2. +2. +Sparsity of the Tsallis-Cirto Radiation +By following the previous subsection, and using the +Tsallis-Cirto temperature, we can write the sparsity pa- +rameter ηδ for Tsallis-Cirto radiation as +ηδ = ηHδ2 +�SB +kB +�2δ−2 +, +(59) +and the GUP-corrected sparsity ηδgup, by using (23) and +(55), it can be written as +ηδgup = ηGUPδ2 +�SGUP +kB +�2δ−2 +. +(60) +Fig. (12) depicts the sparsity vs. mass relationship. For +δ=0.8 +δ=1 +δ=1.1 +δ=0.8 +δ=1 +δ=1.1 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +50 +100 +150 +200 +M +ηδ +Figure 12. +Sparsity ηδ for Tsallis-Cirto black hole entropy. +Dashed lines correspond to a GUP case. +the Tsallis-Cirto temperature, the sparsity scales with +M4δ−4. Again, the value of δ, significantly changes the +behavior of the sparsity. It should be noted that the spar- +sity parameter is now affected by mass as well as δ and +the GUP-parameter α. In the non-GUP case, ηδ = ηH +for δ = 1. When δ > 1, the value of ηδ is initially very +high and approaches zero at the end of the black hole +evaporation. This means that, initially, the Tsallis-Cirto +radiation is highly sparse, and during the final stages of +evaporation, it is not sparse at all. In this way, for δ < 1, +Tsallis-Cirto radiation is initially not sparse, but at the +end of the evaporation, it is extremely sparse with the +sparsity parameter infinite. For the GUP case, initially, +the behavior is the same as for the non-GUP case, but + +10 +when the mass approaches the order of Planck mass, +i.e., the remnant mass Mr, the sparsity parameter de- +creases to some finite values for each case. Note that all +these finite values of sparsity parameters are less than +the standard sparsity parameter ηH. +D. +Sharma-Mittal Entropy +Sharma-Mittal (SM) is an entropic form [40, 104] that +generalizes the Rényi and Tsallis entropies. It is defined +as +SSM = 1 +R +� +� +� +W +∑ +i=1 +p1−λ +i +� R +λ +− 1 +� +� +(61) +where R is another free parameter that is introduced in +SM entropy. Under the equiprobability condition of the +states [69], the above equation (61) reduces to +SSM = kB +R +� +(1 + λ +kB +ST)R/λ − 1 +� +, +(62) +where R → λ limit yields the Tsallis entropy, and R → 0 +yields Rényi entropy. The Sharma-Mittal entropy obeys +the same general nonextensive composition rule (32). +Assuming that the Bekenstein entropy SB is the same +as the Tsallis entropy ST , we can write SSM for the case +of a Schwarzschild black hole as +SSM = kB +R +� +(1 + λ +kB +SB)R/λ − 1 +� +, +(63) +and replacing SGUP with ST in equation (62), the GUP +corrected SM entropy SSMgup reads as +SSMgup = kB +R +� +(1 + λ +kB +SGUP)R/λ − 1 +� +. +(64) +The corresponding temperatures can be found by using +the Clausius relation, as +TSM = TH(1 + λ +kB +SB)1− R +λ , +(65) +and the GUP corrected SM temperature TSMgup reads as +TSMgup = TGUP(1 + λ +kB +SGUP)1− R +λ . +(66) +We can now define the inverse temperature parameters +for GUP and non-GUP cases by using the above equa- +tions (65) and (66), which are given, for the non-GUP +case, as +βSM = S′ +SM +kBc2 = β(1 + λ +kB +SB) +R +λ −1, +(67) +and for the GUP case, as +βSMgup = +S′ +SMgup +kBc2 += βGUP(1 + λ +kB +SGUP) +R +λ −1. +(68) +R=0.2 +R=0.6 +R=0.9 +R=0.2 +R=0.6 +R=0.9 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +10 +20 +30 +40 +50 +M +SSM +Figure 13. +Plot of the Sharma-Mittal entropy for λ = 0.7. +Dashed lines correspond to a GUP case. +Since SM entropy is the generalization of the Tsallis and +Rényi entropy, the behavior of the temperature and the +entropy are similar to that of SB and SR and TH and TR +for different values of Sharma-Mittal parameter R. Also, +the black hole does not evaporate in this case as well, +and the evaporation process stops at Mr, leaving the fi- +nal state of the black hole as a remnant having finite en- +tropy and temperature. The plots of SM entropy and +temperature are given in Figs. 13 and 14. +R=0.2 +R=0.6 +R=0.9 +R=0.2 +R=0.6 +R=0.9 +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +M +TSM +Figure 14. Sharma-Mittal temperature for λ = 0.7. Dashed +lines correspond to a GUP case. +1. +Heat Capacity for Sharma-Mittal Black Holes +By following the previous subsections, we can calcu- +late the heat capacity CSM for the SM black holes as +CSM = +CSc(1 + λ +kB SB) +R +λ +(1 + λ +kB SB) − λ +kB CSc +� +R +λ − 1 +� , +(69) +and for the GUP SM black holes case, it reads as +CSMgup = +CGUP(1 + λ +kB SGUP) +R +λ +(1 + λ +kB SGUP) − λ +kB CGUP +� +R +λ − 1 +� . (70) +The plots of (69) and (70) are given in Fig. 15. Similarly +as for the Rényi case, we define the characteristic length +scale LSM in terms of λ and R by employing the singular + +11 +point of CSM. For the non-GUP case, we have such a +singular point for +λ = RCSc − kB +CSc + SB +. +(71) +From (71), we can easily define the following character- +istic relation by solving it for M, which reads +LSM = 2lp +� +π(λ − 2R), +(72) +where LSM = GMc/c2, and the mass scale Mc is defined +as +Mc = +mp +2 +� +π(λ − 2R) +. +(73) +Similarly, one can define LSMgup for the GUP case by +using the following singular point at +λ = RCGUP − kB +CGUP + SGUP +, +(74) +and solve it for M. Since the analytic solution is not pos- +sible, one could use a perturbative approach to find the +GUP corrections to LSM up to the first order in α. Note +that R → 0 limit yields the LR for the Rényi case. For +λ − 2R > 0 and M > Mc, the heat capacity is positive +for both non-GUP and GUP cases, and for M < Mc, +the heat capacity is negative for both non-GUP and GUP +cases. +R=0.2 +R=0.6 +R=0.9 +R=0.2 +R=0.6 +R=0.9 +0.0 +0.5 +1.0 +1.5 +2.0 +-30 +-20 +-10 +0 +10 +20 +30 +M +CSM +Figure 15. Heat capacity CSM for Sharma-Mittal entropy for +λ = 0.7. Dashed lines correspond to a GUP case. +2. +Sparsity of the Sharma-Mittal Radiation +The sparsity parameter ηSM can be derived by apply- +ing the Sharma-Mittal temperature to (23), and reads +ηSM = ηH(1 + λ +kB +SB)2( R +λ −1), +(75) +and for the GUP case, substituting equations (66) and +(27) in (23), the GUP modified sparsity parameter for the +Sharma-Mittal radiation reads as +ηSMgup = ηGUP(1 + λ +kB +SGUP)2( R +λ −1). +(76) +R=0.45 +R=0.5 +R=0.6 +R=0.45 +R=0.5 +R=0.6 +R=0.3 +R=0.3 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +100 +200 +300 +400 +500 +600 +M +ηSM +Figure 16. Sparsity for Sharma-Mittal entropy for λ = 0.4. +Dashed lines correspond to a GUP case. +The plots of the sparsity for SM (75) and SM GUP (76) +cases are given in Fig. 16. The behavior of the sparsity +parameter again depends on the Sharma-Mittal param- +eter R in addition to the nonextensive parameter λ and +also the GUP parameter α in the case of GUP corrections. +For the values of λ and R, which satisfy the inequality +λ + 2R > 0, the sparsity of the Sharma-Mittal radiation +behaves like the sparsity of the Rényi radiation for both +non-GUP and GUP cases. This means that, initially, the +Sharma-Mittal radiation is not sparse, and at the end +of the evaporation, its value approaches the value of +Hawking’s case, i.e., ηH, for the non-GUP case. For the +GUP case, when M approaches Mr, the Sharma-Mittal +sparsity parameter approaches some finite value, which +is less than ηH. For λ > R, initially, the Sharma-Mittal +sparsity parameter is higher than ηH and its value ex- +actly approaches ηH at the end of the evaporation, while +for the case of GUP, it approaches to some finite value +less than ηH. It is interesting to note that, for α > 0, the +GUP modified sparsity parameter is always less than the +standard Hawking case. +E. +Kaniadakis Entropy +Kaniadakis entropy [42, 70] is a type of nonextensive +entropy that results from the Lorentz transformation of +special relativity. It is a single parameter deformation of +Gibbs entropy in which The standard Gibbs entropy is +generalized to the relativistic regime with the help of a +new parameter K that is connected to the dimensionless +rest energy of the various parts of a multibody relativis- +tic system. The Kaniadakis entropy SK is defined as +SK = kB logK Ω +(77) +where +logK(Ω) = ΩK − Ω−K +2K +. +(78) +Considering SB = kB ln Ω, which means that the num- +ber of microstates Ω for a black hole is proportional to + +12 +eSB/kB, the above equation (77) can be written in the fol- +lowing form +SK = kB +K sinh +� +K SB +kB +� +, +(79) +where we have used equation (78) for the sinh x function +and used the relation Ω = eSB/kB. Replacing SB with +SGUP, the GUP modified Kaniadakis entropy SKGUP +reads as +SKGUP = kB +K sinh +� +K SGUP +kB +� +. +(80) +Note that, in the limit K → 0, SK reduces to Gibbs en- +K=0.1 +K=0.5 +K=0.9 +K=0.1 +K=0.5 +K=0.9 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +20 +40 +60 +80 +100 +M +SK +Figure 17. Kaniadakis Entropy SK vs mass M. Dashed lines +correspond to a GUP case. +tropy. In Fig. (17), one can see the characteristic form of +sine hyperbolic (sinh) function for different small val- +ues of K which shows the similar behaviour like the +Bekenstein entropy. +As expected, for the GUP case, +black holes do not evaporate completely and the final +state of the black hole is a remnant like for the case of +standard GUP modified Bekenstein-Hawking case. Fur- +thermore, as K increases, the entropy increases sharply. +By using the Clausius relation, the corresponding Kani- +adakis black black hole temperature TK reads as +TK = TH sech +� +K SB +kB +� +, +(81) +and the GUP modified Kaniadakis temperature TKGUP +can be written as +TKgup = TGUP sech +� +K SGUP +kB +� +. +(82) +By using (81) and (82), one can write the following in- +verse temperature parameters βK as follows +kBβK = kBβ cosh +� +K SB +kB +� +, +(83) +and for the GUP case, βKGUP reads +kBβKgup = kBβGUP cosh +� +K SGUP +kB +� +, +(84) +K=0.1 +K=0.5 +K=0.9 +K=0.1 +K=0.5 +K=0.9 +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.05 +0.10 +0.15 +0.20 +M +TK +Figure 18. Kaniadakis temprature TK vs mass. Dashed lines +correspond to a GUP case. +which can further be used to find the heat capacities +for Kaniadiakis black holes. Fig. (18) shows that Ka- +niadakis temperature behaves as Hawking temperature +with a slight change depending on the parameter K. For +the GUP case, it stops at some finite value, when M ap- +proaches to Mr during the final stages of the black hole +evaporation process. +1. +Heat capacity for Kaniadakis Black Holes +The heat capacities for Kaniadakis entropy can be +calculated by following the previous subsections. For +the non-GUP case, the heat capacity CK for Kaniadakis +black hole reads as +CK = CSc +cosh2[K SB +kB ] +cosh[K SB +kB ] − CSc sinh[K SB +kB ] +, +(85) +and for the GUP modified heat capacity, CKgup, it can +written as +CKgup = CGUP +cosh2[K SGUP +kB ] +cosh[K SGUP +kB ] − CGUP sinh[K SGUP +kB ] +. (86) +From Fig. (19), one can easily notice the negative heat +K=0.1 +K=0.5 +K=0.9 +K=0.1 +δ=0.5 +K=0.9 +0.0 +0.5 +1.0 +1.5 +2.0 +-100 +-80 +-60 +-40 +-20 +0 +M +CK +Figure 19. Kaniadakis heat capacity CK vs mass M. Dashed +lines correspond to a GUP case. +capacities for all values of K. +This means that Kani- +adakis black holes are thermodynamically unstable for +all M. + +13 +K=0.1 +K=0.5 +K=0.7 +K=0.1 +K=0.5 +K=0.7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0 +50 +100 +150 +200 +250 +300 +M +ηK +Figure 20. Sparsity ηK for Kaniadakis radiation vs mass M +of Kaniadakis black hole. Dashed lines correspond to a GUP +case. +2. +Sparsity of the Kaniadakis Radiation +The sparsity parameter ηK for the Kaniadakis radia- +tion can be derived by applying (81) into (23), and reads +ηK = ηH cosh2 +� +K SB +kB +� +, +(87) +and for the GUP modified sparsity parameter ηKGUP, we +apply (82) and (27) into (23), to obtain +ηKGUP = ηGUP cosh2 +� +K SGUP +kB +� +. +(88) +From Fig. +(20), the sparsity parameter for the Kani- +adakis case is always high from the beginning of the +evaporation process as compared to the standard Beken- +stein Hawking case. However, for the non-GUP case, ηK +approaches to the value of ηH at the end of the evapo- +ration. For the GUP case, again, it approaches to some +finite value of sparsity when M approaches Mr, which +is always less than the sparsity parameter ηH. Further- +more, we see that increasing value of K directly results +in sparser Kaniadakis radiation. +F. +Barrow entropy +Barrow entropy [44] is an entropic form that has no +statistical roots, but is closely tied to black hole hori- +zon geometry. +It is proposed to replace the smooth +black hole horizon with a fractal of spheres known as +a sphereflake. This structure is distinguished by its frac- +tal dimension d f , where 3 ≥ d f ≥ 2, and results in an +effective horizon area of r+d f , where r+ is the horizon +radius. As a result, in this scenario, the horizon area is +modified, yielding Barrow entropy as below SBarrow +SBarrow = kB +� A +Ap +�1+ ∆ +2 +(89) +where A is the horizon area, Ap is the Planck area, and +∆ is the parameter directly tied to the fractal dimension +d f through ∆ = d f − 2. In this form, ∆ can take values +between 0 and 1, and ∆ → 1 limit yields maximally frac- +tal structure, where the horizon area effectively behaves +like a 3−dimensional volume, while ∆ → 0 limit yields +the well-known Bekenstein area law where no fractal- +ization occurs. Although Barrow entropy offers a dif- +ferent picture in the geometrical sense, in its essence, +it has the same form as Tsallis-Cirto entropy. We can +see that they are equivalent by making the following +parametrization in Tsallis-Cirto entropy [105] +δ → 1 + ∆ +2 +(90) +Thus, qualitatively, both entropic forms yield the same +temperatures and heat capacities as a function of black +hole mass. Similarly, the Tsallis-Cirto entropy limit ∆ = +1 (δ = 3/2 for Sδ) yields an extensive, but still nonaddi- +tive entropy for black holes. +IV. +SUMMARY AND DISCUSSION +We have investigated the nonextensive thermody- +namics of black holes, the impact of the generalized +uncertainty principle on nonextensive thermodynamics +quantities, and the sparsity and GUP-modified sparsity +of the radiation in the nonextensive scenario. We have +found that all nonextensive black hole entropies and as- +sociated temperatures have finite values at the end of +the black hole evaporation process due to GUP modifi- +cations, indicating the existence of a remnant at the end +of the evaporation. This means that black holes do not +evaporate fully in the nonextensive setup as well. We +have also investigated the sparsity parameter in each +nonextensive configuration. Despite the fact that the be- +havior of the sparsity parameter varies for each nonex- +tensive scenario, GUP consistently lowers the radiation +sparsity in all circumstances toward the end of the evap- +oration process. +Even though multiple nonextensive +scenarios have the same temperatures and entropic pro- +files, we have demonstrated that the sparsity parameter +can be used to distinguish between them. +We have introduced GUP and GUP-corrected thermo- +dynamic parameters and have revised otherwise well- +known GUP corrected quantities to a better form in +which the two crucial limits - the extensivity limit for +λ → 0 and the HUP limit for α → 0 - are easily iden- +tified. Even though GUP corrections on Rényi entropy +in black hole thermodynamics have been researched in +the literature, we presented a full discussion of it in or- +der to help readers distinguish between various sorts of +nonextensive scenarios. Additionally, we have provided +non-perturbative results for each quantity, with a focus +on the Rényi sparsity parameter, which rises (as shown +by the "bump" in Fig. (8)) before the value of the rem- +nant mass. This is because it is assumed that the area +can change as a result of the GUP-modified Bekenstein +entropy, which is explicitly shown in (28). This indi- + +14 +cates that AGUP as well as TGUP have an impact on the +sparsity parameter. Furthermore, we have introduced +black hole mass scale Mc = mp/2 +√ +πλ for the nonexten- +sive parameter λ for the Rényi black hole quantities and +we defined corresponding characteristic length for λ in +terms of Mc, i.e. LR = GMc/c2 = 2lp +√ +πλ. We have +shown that, for M > Mc, the heat capacity is positive +and hence black holes in Rényi scenario are thermody- +namically stable, while for M < Mc, the heat capacity is +negative and SR and TR behave like Bekenstein entropy +SB and Hawking temperature TH, hence unstable black +holes. +Similarly, we have also analyzed the thermodynamic +black hole quantities associated with Tsallis-Cirto black +hole entropy. +Particularly, we have focused on GUP +corrections and the sparsity of the Tsallis-Cirto radia- +tion. We have shown that, when GUP corrections are +included, Tsallis-Cirto entropy and associated temper- +ature have a finite value, and this proves that the fi- +nal state of the black hole is also a remnant with finite +entropy and temperature. It is interesting to note that +the Tsallis-Cirto parameter δ plays a significant role. We +have found that, for δ > 1/2, Tsallis-Cirto entropy and +temperature behave similarly to Bekenstein entropy and +Hawking temperature, and hence have negative heat ca- +pacity. For the GUP case, Tsallis-Cirto temperature be- +haves like Rényi temperature and has positive heat ca- +pacity for δ < 1/2. This means that, in this framework, +we must have δ < 1/2 for thermodynamic stability of +black holes. In this way, we have shown that the Tsallis- +Cirto sparsity parameter is very high during the start of +the evaporation for δ > 1, but it approaches zero at the +the end of the black hole evaporation. On the contrary, +for δ < 1, we have shown that the Tsallis-Cirto radi- +ation is not sparse during the start of the evaporation, +but at the end of the evaporation, the sparsity parame- +ter becomes infinite and hence shows the highly sparse +Tsallis-Cirto radiation. The behavior of the GUP case is +initially the same as that of the non-GUP case, but as the +mass approaches the order of Planck mass, i.e., Mr, the +Tsallis-Cirto sparsity parameter for each case reduces to +some finite values. It should be noted that all of these fi- +nite sparsity parameter values are less than the sparsity +parameter ηH for the standard Hawking case. +We have also shown that the behavior of the tempera- +ture and the entropy for the Sharma-Mittal case is com- +parable to that of SB and SR and TH and TR for differ- +ent values of the Sharma-Mittal parameter R since the +Sharma-Mittal entropy is the extension of the Tsallis and +Rényi entropy. Also, in this instance, the black hole does +not evaporate, and the evaporation process stops at Mr, +leaving the black hole in its ultimate state as a remnant +of mass Mr with finite entropy and temperature. We +have analysed the sparsity of the Sharma-Mittal radia- +tion and compared it with the standard Hawking case. +We have found that the sparsity of the Sharma-Mittal ra- +diation behaves similarly to the Rényi radiation in both +non-GUP and GUP instances for values of λ and R that +fulfill the condition λ − 2R > 0. +This indicates that +the Sharma-Mittal radiation is initially not sparse and +that by the end of the evaporation, its value approaches +that of Hawking’s scenario, or ηH, for the non-GUP case. +When M approaches Mr for the GUP case, the Sharma- +Mittal sparsity parameter approaches a finite value that +is smaller than ηH. For the case, R > λ, we have shown +that the Sharma-Mittal sparsity parameter is initially +larger than ηH and its value exactly approaches ηH by +the end of the evaporation whereas for the case of GUP, +it approaches a finite value that is smaller than ηH. It is +noteworthy to notice that, for α > 0, the GUP modified +sparsity parameter is always lower than the standard +Hawking case. Moreover, we have also introduced the +characteristic mass scale, Mc = mp/2 +� +π(λ − 2R), for +the Sharma-Mittal scenario and also, defined the corre- +sponding characteristic length scale LSM = GMc/c2 = +2lp +� +π(λ − 2R). We have shown that, for M > Mc with +λ − 2R > 0, the black holes are thermodynamically sta- +ble in the Sharma-Mittal scenario for both GUP and non- +GUP cases, while for M < Mc, black holes are thermo- +dynamically unstable. +We have also examined the Kaniadakis thermody- +namic black hole quantities, and the results demonstrate +that, with a little variation depending on the parame- +ter K, Kaniadakis entropy and temperature behave sim- +ilarly to Bekenstein entropy and Hawking temperature. +In the case of the GUP, both quantities reach a finite +value as black hole mass approaches Mr during the late +stages of the black hole evaporation process. It results in +negative heat capacity for all values of K, indicating that +Kaniadakis black holes are thermodynamically unstable +for all values of black hole mass. Furthermore, in con- +trast to the typical Hawking example, the sparsity pa- +rameter for the Kaniadakis instance is consistently high +from the beginning of the evaporation process. For the +non-GUP example, however, ηK approaches the value of +ηH at the end of the evaporation. In the GUP situation, +it approaches some finite value of sparsity when M ap- +proaches Mr, which is always smaller than the sparsity +parameter ηH. Additionally, it is clear that a rise in the +value of K causes the Kaniadakis radiation to become +sparser. +Finally, our short look onto the Barrow entropy has +proven its equivalence (though in a restricted range of +parameters) to the Tsallis-Cirto entropy. In view of that, +all the discussion of termodynamical quantities for Bar- +row entropy should be the same as for Tsallis-Cirto. +ACKNOWLEDGMENTS +The work of I.C. and M.P.D. was supported by +the Polish National Science Centre grant No. +DEC- +2020/39/O/ST2/02323. + +15 +[1] S. W. 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B 810, 135805 +(2020), arXiv:2009.10133 [gr-qc]. + diff --git a/TNFAT4oBgHgl3EQf2R4K/content/tmp_files/load_file.txt b/TNFAT4oBgHgl3EQf2R4K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..30e6d1f97b273bc87953f758e8b4166a0c1e0a55 --- /dev/null +++ b/TNFAT4oBgHgl3EQf2R4K/content/tmp_files/load_file.txt @@ -0,0 +1,934 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf,len=933 +page_content='Massively Parallel Genetic Optimization through Asynchronous Propagation of Populations Oskar Taubert[0000−0002−3707−499X],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Marie Weiel[0000−0001−9648−4385],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Daniel Coquelin[0000−0001−8552−5153],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Anis Farshian[0000−0002−9888−0653],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Charlotte Debus[0000−0002−7156−2022],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Alexander Schug[0000−0002−0534−502X],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Achim Streit[0000−0002−5065−469X],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' and Markus G¨otz[0000−0002−2233−1041] Steinbuch Centre for Computing (SCC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Karlsruhe Institute of Technology (KIT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 76344 Eggenstein-Leopoldshafen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Germany,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' markus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='goetz@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We present Propulate, an evolutionary optimization algo- rithm and software package for global optimization and in particular hy- perparameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For efficient use of HPC resources, Propulate omits the synchronization after each generation as done in conventional genetic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Instead, it steers the search with the complete population present at time of breeding new individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We provide an MPI-based implementation of our algorithm, which features variants of selection, mutation, crossover, and migration and is easy to extend with custom functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We compare Propulate to the established optimization tool Optuna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We find that Propulate is up to three orders of magnitude faster without sacrificing solution accuracy, demonstrating the efficiency and efficacy of our lazy synchronization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Code and documentation are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='com/Helmholtz-AI-Energy/propulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Keywords: Genetic Optimization · AI · Parallelization · Evolutionary Algo- rithm 1 Introduction Machine learning (ML) algorithms are heavily used in almost every area of human life today, from medical diagnosis and critical infrastructure to trans- portation and food production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Almost all ML algorithms have non-learnable hyperparameters (HPs) that influence the training and in particular their pre- dictive capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' As evaluating a set of HPs involves at least a partial train- ing, state-free approaches to HP optimization (HPO), like grid and random search, often go beyond available compute resources [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To explore the high- dimensional HP spaces efficiently, information from previous evaluations must be leveraged to guide the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Such state-dependent strategies minimize the number of evaluations to find a useful model, reducing search times and thus the energy consumption of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Bayesian and bio-inspired optimizers are the most popular of these AutoML approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Among the latter, genetic algorithms (GAs) are versatile metaheuristics inspired by natural evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='08713v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='NE] 20 Jan 2023 2 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Taubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' solve a search-for-solutions problem, a population of candidate solutions (or in- dividuals) is evolved in an iterative interplay of selection and variation [23,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Although reaching the global optimum is not guaranteed, GAs often find near- optimal solutions with less computational effort than classical optimizers [9,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' They have become popular for various optimization problems, including HPO for ML and neural architecture search (NAS) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To take full advantage of the increasingly bigger models and datasets, de- signing scalable algorithms for high performance computing (HPC) has become a must [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' While Bayesian optimization is inherently serial, the structure of GAs renders them suitable for parallelization [34]: Since all candidates in each iteration are independent, they can be evaluated in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To breed the next generation, however, the previous one has to be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' As the computational expenses for evaluating different candidates vary, synchronizing the parallel evo- lutionary process affects the scalability by introducing a substantial bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Approaches to reducing the overall communication in parallel GAs like the island model (IM) [34] do not address the underlying synchronization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To solve the issues arising from explicit synchronization, we introduce Prop- ulate, a massively parallel genetic optimizer with asynchronous propagation of populations and migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Unlike classical GAs, Propulate maintains a contin- uous population of already evaluated individuals with a softened notion of the typically strictly separated, discrete generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Our contributions include: – A novel parallel genetic algorithm based on a fully asynchronous island model with independently processing workers, allowing to parallelize the optimiza- tion process and distribute the internal evaluation of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Massive parallelism by asynchronous propagation of continuous populations and migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – A prototypical implementation in Python using extremely efficient commu- nication via the message passing interface (MPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Optimal use of parallel hardware by minimizing idle times in HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We use Propulate to optimize various benchmark functions and the HPs of a deep neural network on a supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Comparing our results to those of the popu- lar HPO package Optuna, we find that Propulate is consistently drastically faster without sacrificing solution accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We further show that Propulate scales well to at least 100 processing elements (PEs) without relevant loss of efficiency, demonstrating the efficacy of our asynchronous evolutionary approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 2 Related Work Recent progress in ML has triggered heavy use of these techniques with Python as the de facto standard programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Tuning HPs requires solving high-dimensional optimization problems with ML algorithms as black boxes and model performance metrics as objective functions (OFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Most common are Bayesian optimizers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Optuna [2], Hyperopt [7], SMAC3 [24,27], Spearmint [32], GPyOpt [5], and MOE [38]) and bio-inspired methods such as swarm-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate 3 FLAPS [39]) and evolutionary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' DEAP [16], MENNDL [40]) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Below, we provide an overview of popular HP optimizers in Python, with a focus on state- dependent parallel algorithms and implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' A theoretical overview of parallel GAs can be found in surveys [12,4,3] and books [37,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Optuna adopts various algorithms for HP sampling and pruning of unpromis- ing trials, including tree-structured Parzen estimators (TPEs), Gaussian pro- cesses, and covariance matrix adaption evolution strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It enables parallel runs via a relational database server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In the parallel case, an Optuna candidate obtains information about previous candidates from and stores results to disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' SMAC3 (Sequential Model-based Algorithm Configuration) combines a ran- dom-forest based Bayesian approach with an aggressive racing mechanism [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Its parallel variant pSMAC uses multiple collaborating SMAC3 runs which share their evaluations through the file system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Spearmint, GPyOpt, and MOE are Gaussian-process based Bayesian optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Spearmint enables distributed HPO via Sun Grid Engine and MongoDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' GPyOpt is integrated into the Sherpa package [22], which provides implementations of recent HP optimizers along with the infrastructure to run them in parallel via a grid engine and a database server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' MOE (Metric Optimization Engine) uses a one-step Bayes-optimal algorithm to maximize the multi-points expected im- provement in a parallel setting [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Using a REST-based client-server model, it enables multi-level parallelism by distributing each evaluation and running multiple evaluations at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Nevergrad [31] and Autotune [25] provide gradient-free and evolutionary optimizers, including Bayesian, particle swarm, and one-shot optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In Nevergrad, parallel evaluations use several workers via an executor from Python’s concurrent module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Autotune enables concurrent global and local searches, cross-method sharing of evaluations, method hybridization, and multi-level par- allelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Open Source Vizier [33] is a Python interface for Google’s HPO ser- vice Vizier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It implements Gaussian process bandits [19] and enables dynamic optimizer switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' A central database server does the algorithmic proposal work, clients perform evaluations and communicate with the server via remote procedure calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Katib [18] is a cloud-native AutoML project based on the Kuber- netes container orchestration system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It integrates with Optuna and Hyperopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Tune [26] is built on the Ray distributed computing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It interfaces with Optuna, Hyperopt, and Nevergrad and leverages multi-level parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' DEAP (Distributed Evolutionary Algorithms in Python) [16] implements gen- eral GAs, evolution strategies, multi-objective optimization, and co-evolution of multi-populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It enables parallelization via Python’s multiprocessing or SCOOP module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' EvoTorch [36] is built on PyTorch and implements distribution- and population-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Using a Ray cluster, it can scale over mul- tiple CPUs, GPUs, and computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' MENNDL (Multi-node Evolutionary Neural Networks for Deep Learning) [40] is a closed-source MPI-parallelized HP op- timizer for automated network selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' A master node handles the genetic operations while evaluations are done on the remaining worker nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' However, global synchronization hinders optimal resource utilization [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Taubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Algorithm 1: Basic GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In each generation, the individuals are eval- uated in terms of the optimization problem’s OF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Genetic operators propagate them to the next generation: The selection operator chooses a portion of the current generation, where better individuals are usually preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To breed new individuals, the genes of two or more parent individuals from the selected pool are manipulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' While the crossover operator recombines the parents’ genes, the mutation operator alters them randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This is repeated until a stopping condition is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Input: Search-space limits, population size P, termination condition, selection policy, crossover probability, mutation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 1 Initialize population pop of P individuals within search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 2 while not termination condition do // OPTIMIZE 3 Evaluate individuals in pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' // EVALUATE 4 Choose parents from pop following selection policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' // SELECT 5 foreach individual in pop do // VARY 6 if random ≤ crossover probability then // RECOMBINE 7 Recombine individuals randomly chosen from parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 8 if random ≤ mutation probability then // MUTATE 9 Mutate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 10 Update individual in pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Result: Best individual found (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', with lowest OF value for minimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 3 Propulate Algorithm and Implementation To alleviate the bottleneck inherent to synchronized parallel genetic algorithms, our massively parallel genetic optimizer Propulate (propagate and populate) implements a fully asynchronous island model specifically designed for large- scale HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Unlike conventional GAs, Propulate maintains a continu- ous population of evaluated individuals with a softened notion of the typically strictly separated generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This enables asynchronous evaluation, variation, propagation, and migration of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate’s basic mechanism is that of Darwinian evolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', beneficial traits are selected, recombined, and mutated to breed more fit individuals (see Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' On a higher level, Propulate employs an IM, which combines inde- pendent evolution of self-contained subpopulations with intermittent exchange of selected individuals [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To coordinate the search globally, each island occasion- ally delegates migrants to be included in the target islands’ populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Islands communicate genetic information competitively, thus increasing diversity among the subpopulations compared to panmictic models [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For synchronous IMs, this exchange occurs simultaneously after fixed intervals, with no computation happening in that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The following hyperparameters characterize IMs: – Island number and subpopulation sizes – Migration (pollination) probability – Number of migrants (pollinators): How many individuals migrate from the source population at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Asynchronous propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Interaction of two workers on one island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Indi- viduals bred by worker 1 and 2 are shown in blue and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Their origins are given by a generation sub- and an island superscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Populations are depicted as round grey boxes, where most recent individuals have black outlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Varying evaluation times are represented by sharp boxes of different widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We illustrate the asynchronous propagation and intra-island synchronization of the population using the example of the blue individual indi1 g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This individual is bred by worker 1 in generation 3 by apply- ing the propagator (yellow) to the worker’s current population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' After evaluating indi1 g3, worker 1 sends it to all workers on its island and appends it to its population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' As no evaluated individuals dispatched by worker 2 await to be received, worker 1 proceeds with breeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Worker 2 receives the blue indi1 g3 only after finishing the evaluation of the red indi1 g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It then appends both to its population and breeds a new individual for generation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Migration (pollination) topology: Directed graph of migration (pollina- tion) paths between islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Emigration policy: How to select emigrants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', random or best) and whether to remove them from the source population (actual migration) or not (pollination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Immigration policy: How to insert immigrants into the target population, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', either add them (migration) or replace existing individuals (pollination, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', random or worst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate’s functional principle is outlined in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We consider multiple PEs (or workers) partitioned into islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Each worker processes one individual at a time and maintains a population to track evaluated and migrated individuals on its island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To mitigate the computational overhead of synchronized OF evalu- ations, Propulate leverages asynchronous propagation of continuous populations with interwoven, worker-specific generations (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In each iteration, each worker breeds and evaluates an individual which is added to its population list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It then sends the individual with its evaluation result to all workers on the Breed Evaluate Breed Evaluate Breed Evaluate Append indg Append indg Append Propagator Propagator Propagator indl Worker 1 Population list Population list Population list ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2 indel indel indel indel indga indfe Send Send synchronization Intra-island indgs Receive Receive Receive Receive Breed Evaluate Send Breed Evaluate Worker 2 Append inde?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Append Propagator Propagator inde Populationlist Population list indei indel indfz indge .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Optimizationprogress6 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Taubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Algorithm 2: Propulate with pollination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Input: Search-space limits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' hyperparameters n islands, island sizes Pi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' , n islands), number of iterations generations, evolutionary operators (including selection policy, crossover probability, mutation probability etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' ), pollination probability, pollination topology, emigration policy, immigration policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 1 Configure n islands islands with Pi workers each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Each worker evaluates one individual at a time and maintains its own population list pop of evaluated and migrated individuals on the island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 2 /* START OPTIMIZATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' / 3 for each worker do in parallel 4 while generation ≤ generations do // Loop over generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 5 Breed and evaluate individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Append it to pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send it to other workers on island to synchronize their populations lists: evaluate individual() // BREED AND EVALUATE 6 Check for and possibly receive individuals bred and evaluated by other workers on island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Append them to pop: receive intra isle individuals() // SYNCHRONIZE 7 if random ≤ pollination probability then // EMIGRATE 8 Choose pollinators from currently active individuals on island according to emigration policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send copies of pollinator(s) to workers of target islands according to pollination topology: send emigrants() 9 Check for and possibly receive pollinators sent by workers from other islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Add them to pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Determine individuals to be replaced by incoming pollinators according to immigration policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send individuals to be replaced to other workers on island for deactivation: receive immigrants() // IMMIGRATE 10 Check for and possibly receive individuals replaced by pollinators on other workers on island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Try to deactivate them in pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' If an individual to be deactivated is not yet in pop, append it to history list replaced and try again in the next generation: deactivate replaced individuals() // SYNCHRONIZE 11 Go to next generation: generation += 1 12 /* OPTIMIZATION DONE: FINAL SYNCHRONIZATION / 13 Wait for all other workers to finish: MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='COMM WORLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='barrier() 14 Final check for incoming messages so all workers hold complete population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 15 Probe individuals evaluated by other workers on island: receive intra isle individuals() 16 Probe for incoming pollinators immigrating from other islands: receive immigrants() 17 Probe for individuals replaced by other workers on island to be deactivated: deactivate replaced individuals() Result: n individuals with smallest OF values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate 7 same island and, in return, receives evaluated individuals dispatched by them for a mutual update of their population lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To avoid explicit synchronization points, the independently operating workers use asynchronous point-to-point communication via MPI to share their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Each one dispatches its result immediately after finishing an evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Directly afterwards, it non-blockingly checks for incoming messages from workers of its own island awaiting to be received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In the next iteration, it breeds a new individual by applying the evo- lutionary operators to its continuous population list of all evaluated individuals from any generation on the island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The workers thus proceed asynchronously without idle times despite the individuals’ varying computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' After the mutual update, asynchronous migration or pollination between is- lands happens on a per-worker basis with a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Each worker selects a number of emigrants from its current population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For actual migra- tion1, an individual can only exist actively on one island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' A worker thus may only choose eligible emigrants from an exclusive subset of the island’s popu- lation to avoid overlapping selections by other workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It then dispatches the emigrants to the target islands’ workers as specified in the migration topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Finally, it sends them to all workers on its island for island-wide deactivation of emigrated individuals before deactivating them in its own population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In the next step, the worker probes for and, if applicable, receives immigrants from other islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It then checks for individuals emigrated by other workers of its island and tries to deactivate them in its population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Due to the asyn- chronicity, individuals might be designated to be deactivated before arriving in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate continuously corrects these synchronization artefacts during the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For pollination (Figure 2), identical copies of individuals can exist on multiple islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Workers thus can choose emigrating pollinators from any active individ- uals in their current populations and do not deactivate them upon emigration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To control the population growth, pollinators replace active individuals in the target population according to the immigration policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For proper accounting of the population, one random worker of the target island selects the individual to be replaced and informs the other workers accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Individuals to be deac- tivated that are not yet in the population are cached to be replaced in the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This process is repeated until each worker has evaluated a set number of generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Finally, the population is synchronized among workers and the best individuals are returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate uses so-called propagators to breed child individuals from an ex- isting collection of parent individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It implements various standard genetic operators, including uniform, best, and worst selection, random initialization, stochastic and conditional propagators, point and interval mutation, and several forms of crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In addition, Propulate provides a default propagator: Having selected two random parents from the breeding pool consisting of a set num- ber of the currently most fit individuals, uniform crossover and point mutation 1 See github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='com/Helmholtz-AI-Energy/propulate/tree/master/supplementary for pseudocode with migration and explanatory figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 8 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Taubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Asynchronous pollination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Consider two islands with N (blue) and M (red) workers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We illustrate pollination (dark colors) by tracing worker N on island 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' After evaluation and mutual intra-island updates (light blue, see Figure 1), this worker performs pollination: it sends copies of the chosen pollinators to all workers of each target island, here island 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The target island’s workers receive the pollinators asynchronously (dark blue arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For proper accounting of the populations, worker 1 on island 2, selects the individual to be replaced and informs all workers on its island accordingly (middle red arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Afterwards, worker N receives incoming pollinators from island 2 to be included into its population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It then probes for individuals that have been replaced by other workers on its island, here worker 1, in the meantime and need to be deactivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' After these pollination-related intra-island population updates, it breeds the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' As pollination does not occur in this generation, it directly receives pollinators from island 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This time, worker N chooses the individual to be replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' are performed each with a specified probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Afterwards, interval mutation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To prevent premature trapping in a local optimum, a randomly initialized individual is added with a specified probability instead of one bred from the current population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4 Experimental Evaluation We evaluate Propulate on various benchmark functions (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='4) and an HPO use case in remote sensing classification (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5) which provides a real world application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We compare our results against Optuna, since it is the most widely used HPO software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='1 Experimental Environment We ran the experiments on the distributed-memory, parallel hybrid supercom- puter Hochleistungsrechner Karlsruhe (HoreKa) at the Steinbuch Centre for Com- puting, Karlsruhe Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Each of its 769 compute nodes Worker 1 Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send Receive Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propagate Evaluate Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send Receive Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Island 1 with Evaluate Nworkers Y pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='N pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='Y N Y pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='N pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='Y Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Worker N Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send Receive Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send Receive Propagate Evaluate Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Evaluate Y pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='Y pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='Y Y N pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='N pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='y N Pollination Worker1 Island 2 with Send Receive Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propagate Evaluate Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send Receive Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' M workers pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='Y pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='Y pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='y N Propagate pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='N N Evaluate WorkerM Evaluate Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send Receive Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propagate Evaluate Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Send Receive Sync?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' N pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='N pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='N Y Y pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='Y pollinators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='Y OptimizationprogressPropulate 9 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Benchmark functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Name Function Limits Global minimum Sphere f1 = x2 1 + x2 2 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='12 f (0, 0) = 0 Rosenbrock f2 = 100 � x2 1 − x2 �2 + (1 − x1)2 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='048 f (1, 1) = 0 Step f3 = �5 i=1 int (xi) ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='12 f (xi ≤ −5) = −25 Quartic f4 = �30 i=1 � ix4 i + Ni (0, 1) � ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='28 f (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', 0) = � i Ni Rastrigin f5 = 200 + �20 i=1 x2 i − 10 cos (2πxi) ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='12 f (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', 0) = 0 Griewank f6 = 1 + 1 4000 �10 i=1 x2 i − �10 i=1 cos xi √ i ±600 f (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', 0) = 0 Schwefel f7 = 10V − �10 i=1 xi sin � |xi| ±500 f (x∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', x∗ 10) = 0, with V = 418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='982887 x∗ i = 420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='968746 Bi-sphere f8 = min ��30 i=1 (xi − µ1)2 , ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='12 f (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', µ1) = 0 30 + s · �30 i=1 (xi − µ2)2� with µ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5, µ2 = − � s−1 � µ2 1 − 1 ��1/2 , s = 1 − � 2 √ 50 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2 �−1/2 Bi-Rastrigin f9 = f8 + 10 �30 i=1 1 − cos 2π (xi − µ1) ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='12 f (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', µ1) = 0 is equipped with two 38-core Intel Xeon Platinum 8368 processors at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='4 GHz base and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='4 GHz maximum turbo frequency, 256 GB (standard) or 512 GB (high-memory and accelerator) local memory, a local 960 GB NVMe SSD disk, and two network adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 167 of the nodes are accelerator nodes each equipped with four NVIDIA A100-40 GPUs with 40 GB memory connected via NVLink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Inter-node communication uses a low-latency, non-blocking NVIDIA Mellanox InfiniBand 4X HDR interconnect with 200 Gbit/s per port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' A Lenovo Xclar- ity controller measures full node energy consumption, excluding file systems, networking, and cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The operating system is Red Hat Enterprise Linux 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2 Benchmark Functions Benchmark functions are used to evaluate optimizers in terms of convergence, accuracy, and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The informative value of such studies is limited by how well we understand the characteristics making real-life optimization problems difficult and our ability to embed these features into benchmark functions [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We use Propulate to optimize a variety of traditional and recent benchmark functions emulating situations optimizers have to cope with in different kinds of problems (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Sphere is smooth, unimodal, strongly convex, symmetric, and thus simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Rosenbrock has a narrow minimum inside a parabola-shaped valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Step represents the problem of flat surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Plateaus pose obstacles to op- timizers as they lack information about which direction is favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Quartic is a unimodal function padded with Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' As it never returns the same value on the same point, algorithms that do not perform well on this test function will do poorly on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 10 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Taubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Grid search parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' All experiments use 144 CPUs equally distributed between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Random-initialization probability refers to the chance that a new individual is generated entirely randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Number of islands 2 4 8 16 32 Island population size 72 36 18 9 4 Migration (pollination) probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='9 Pollination True False Crossover probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='775 Point-mutation probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='775 Random-initialization probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='775 – Rastrigin is non-linear and highly multimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Its surface is determined by two external variables, controlling the modulation’s amplitude and fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The local minima are located at a rectangular grid with size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Their functional values increase with the distance to the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Griewank’s product creates sub-populations strongly codependent to par- allel GAs, while the summation produces a parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Its local optima lie above parabola level but decrease with increasing dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', the larger the search range, the flatter the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Schwefel has a second-best minimum far away from the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Lunacek’s bi-sphere’s [28] landscape structure is the minimum of two quadratic functions, each creating a single funnel in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The spheres are placed along the positive search-space diagonal, with the op- timal and sub-optimal sphere in the middle of the positive and negative quadrant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Their distance and the barrier’s height increase with dimensionality, creating a globally non-separable underlying surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' – Lunacek’s bi-Rastrigin [28] is a double-funnel version of Rastrigin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This function isolates global structure as the main difference impacting problem difficulty on a well understood test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='3 Meta-Optimizing the Optimizer Propulate itself has HPs influencing its optimization behavior, accuracy, and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To explore their effect systematically and give transparent recom- mendations for default values, we conducted a grid search across the six most prominent HPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The search space is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We ran the grid search five times for the quartic, Rastrigin, and bi-Rastrigin benchmark functions (see Table 1 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='4), each with a different seed consistently used over all points within a search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' All three functions have their global minimum at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' They were chosen for their high-dimensional parameter spaces (30, 20, and 30, respectively) and different levels of difficulty to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For quartic, Propulate found a minimum below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='005 for 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='12 % of all points across the five grid searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This increases to 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='94 % for minima found within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='05 of the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In comparison, the tolerances have to be relaxed considerably Propulate 11 for the more complex Rastrigin and bi-Rastrigin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' While only 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='57 % of all grid points had a function value less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5 for Rastrigin, only a single point resulted in an average value of less than 10 for bi-Rastrigin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Although the av- erage value of bi-Rastrigin was only less than 10 once, we found the minimum across each of the five searches to be less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='0 for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='31 % of the grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Considering grid points with at least one result smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='0, 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='61 % used either 16 or 36 islands, while the remainder used eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' As Propulate initializes different islands at different positions in the search space, the chance that one of them is at a very beneficial position increases with the number of islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This is further confirmed by a migration probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='7 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='9 for 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='41 % of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' If one of the islands is well-initialized, it thus will quickly notify others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' With every best grid point using pollination, we clearly find pollination to be favorable over real migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' To determine the other HPs, we compute the aver- ages of the results for the top ten grid points across all three functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The top ten were determined by grouping over the lowest average and standard deviation of the function values, sorting by the averages, and sorting by the standard de- viations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This method reduces the chances of a single run simply benefiting from an advantageous starting seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Average crossover, point-mutation, and random- initialization probabilities are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='655 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='056, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='363 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='133, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='423 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='135, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The average number of islands was 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='800 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='009 which equates to an island population of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The average migration probability was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='527±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' These values provide a reasonable starting point towards choosing default HPs for Propulate (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' As the grid searches only considered functions with independent parameters, we assume a relatively high random-ini- tialization probability to be useful due to the benefits of random search [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' On this account, we chose to reduce the default random-initialization probability to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' As the migration probability might also be lowered artificially by this phenomenon, we set its default to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The default probabilities for crossover and point-mutation were chosen as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='7 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The island size was set at four individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This is a practical choice as our test system has four accelerators per node and the number of CPUs per node is a multiple of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='4 Benchmark Function Optimization For each function, we ran each ten equivalent Propulate and Optuna optimiza- tions, using the same compute resources, degree of parallelization, and number of evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Figure 3 shows the optimization accuracy over walltime comparing Propulate with default parameters determined from our grid search (see Table 3) to Optuna’s default optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In terms of accuracy, Propulate and Optuna are comparable in most experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For many functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Schwefel, bi-Rastrigin, and Rastrigin, Propulate even achieves a better OF value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In terms of walltime, Propulate is consistently at least one order of magnitude faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This is due to Propulate’s MPI-based communication over the fast network, whereas Optuna uses relational databases with SQL and is limited by the slow file system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Since the functions are cheap to evaluate, optimization and communication dominate the walltime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In particular for problems where evaluations are cheap compared to 12 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Taubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate HPs for benchmark function minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Number of islands 38 Island population size 4 Pollination probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='7 Crossover probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='7 Point-mutation probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='4 Sigma factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='05 Random-initialization probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2 Generations per worker 256 Selection policy Best Pollination topology Fully connected Number of migrants 1 Emigration policy Best Immigration policy Worst the search itself, we find that Optuna’s computational efficiency suffers massively from the frequent file locking inherent to its parallelization strategy, reducing its usability for large-scale HPC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In addition, we inspected the evolution of the population over walltime for both Propulate and Optuna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' An example for minimizing the Rastrigin function is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate is roughly three orders of magnitude faster and makes significantly greater progress in terms of both OF values and distance to the global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Due to this drastic difference in runtime, we measure only 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='27 Wh for Propulate compared to Optuna’s 2646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='29 Wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5 HP Optimization for Remote Sensing Classification BigEarthNet [35] is a Sentinel-2 multispectral image dataset in remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It comprises 590 326 image patches each of which is assigned one or more of the 19 available CORINE Land Cover map labels [10,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Multiple computer vision networks for BigEarthNet classification have been trained [35], with ResNet- 50 [20] being the most accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' While a previous Propulate version was used to optimize a set of HPs and the architecture for this use case [13], a more versatile and efficient parallelization strategy in the current version makes it worthwhile to revisit this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Analogously to [13], we consider different optimizers, learning rate (LR) schedulers, activation functions, loss functions, number of filters in each convolutional block, and activation order [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The search space is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Optimizer parameters, LR functions, and LR warmup are included as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We only consider SGD-based optimizers as they share common parameters and thus exclude Adam-like optimizers from the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We theorize that including Adam led to the difficulties seen previously [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The training is exited if the validation loss has not been increasing for ten epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We prepared the data analogously to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' The network is implemented in TensorFlow [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For both Propulate and Optuna, we ran each three searches over 24 h on 32 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We use 1 − F val 1 with the validation F1 score as the OF to be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate 13 10 1 100 101 102 103 104 Walltime / s 10 7 10 5 10 3 10 1 101 103 Function value Sphere Rosenbrock Quartic Rastrigin Griewank Schwefel Bi-sphere Bi-Rastrigin Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Benchmark function minimization accuracy over walltime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Lowest function values found by Propulate (red) and Optuna (blue) versus walltimes to reach them, each averaged over ten runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Step is not shown since both optimizers achieve a perfect value of −25 within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='6 s and 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2 s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' On average, Optuna achieves its best OF value of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='01) h within (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='05 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='14) h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate beats Optuna’s average best after (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='30 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='41) h and achieves its best OF value of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='00) within (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='89 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='15) h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='6 Scaling Finally, we explore Propulate’s scaling behavior for the use case presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Figure 5 shows our results for weak and strong linear scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Our baseline configuration used two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Since each node has four GPUs, we cal- culate speedup and efficiency with respect to eight workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' For strong scaling, we fix the total number of evaluations at 512 and increase the number of work- ers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=', GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' We average over three runs with different seeds and keep four workers per island while increasing the number of islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Speedup increases up to 128 workers, where we reach approximately half the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This is an expected decline since each worker only processes few individuals, so the vari- ance in evaluation times leads to larger idle times of the faster workers before the final population synchronization at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Additionally, as the number of workers approaches the total number of evaluations, the randomly initial- ized evolutionary search in turn approaches a random search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This means that the search performance is likely to be worse than what the pure compute per- formance might suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' It is still possible to apply Propulate on these scales, but the other search parameters have to be adjusted accordingly as shown in the weak scaling plot Figure 5 top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Weak efficiency only drops to 95 % at our largest configuration of 128 workers The early super-scalar behavior is likely due to the non-sequential baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 14 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Taubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 0 5 10 Time / s 101 102 Function value Propulate 0 5000 10000 Time / s Optuna 100 101 Distance to optimum Maximum value Maximum distance Median value Median distance Minimum value Minimum distance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Evolution of the population over walltime for the Rastrigin func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate (left) versus Optuna (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' OF values (blue) use the left-hand scale, distances to the global optimum (purple) use the right-hand scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Pastel dots show each individual’s OF value/distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Solid (dashed) lines show the minimum (median) value and distance achieved so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Maximum value (distance) are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Both optimizers perform 38 912 evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Note the difference on the time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 5 Conclusion We presented Propulate, our HPC-adapted, asynchronous genetic optimization algorithm and software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Our experimental evaluation shows that the fully asyn- chronous evaluation, propagation, and migration enable a highly efficient and parallelizable genetic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Harder to quantify than performance but very important is ease of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Especially for HPC applications at scale, some parallelization and distribution models are more suited than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' A purely MPI-based implementation as in Propulate is not only extremely efficient for highly parallel and communication-intensive algorithms but also easy to set up and maintain, since the required infrastructure is commonly available on HPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This is not the case for any of the other tools investigated, except for the not publicly available MENNDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' This also facilitates a tighter coupling of in- dividuals during the optimization, which enables a more efficient evaluation of candidates and in particular early stopping informed by previously evaluated individuals in the NAS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Propulate was already successfully applied to HPO for various ML models on different HPC machines [13,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Another avenue for future work is including variable-length gene descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Mutually exclusive genes of different lengths, such as the parameter sets for Adam- and SGD-like optimizers in our NAS use case, can thus be explored efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' While this is already possible, it requires an inconvenient workaround of including inactive Propulate 15 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' HP search space of ResNet-50 for BigEarthNet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Optimizers Optimizer parameters LR warmup parameters Adagrad Initial accum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' value � 10−4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5 � LR warmup steps � 100, 104� SGD Clipnorm [−1, −1000] Initial LR � 10−5, 10−1� Adadelta Clipvalue [−1, 1000] Decay steps � 102, 105� RMSprop Use EMA Boolean LR warmup power � 10−1, 101� EMA momentum [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='0] EMA overwrite � 1, 103� Momentum [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='0] Nesterov Boolean Rho [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='99999] Epsilon � 10−9, 10−4� Loss functions LR parameters Binary CE Categorical CE Categorical hinge Decay rate [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='9999] Hinge KL divergence Squared hinge Staircase inverse Boolean time decay Activation functions Decay rate [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='9] ELU ReLU Softplus Staircase poly- Boolean Exponential SELU Softsign nomial decay Hard sigmoid Sigmoid Swish End LR � 10−4, 10−2� Linear Softmax Tanh Power [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='5] genes and adapting the propagators to manually prevent the evaluation of many individuals differing only in inactive genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Acknowledgments This work is supported by the Helmholtz AI platform grant and the Helmholtz Association Initiative and Networking Fund on the HAICORE@KIT partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Abadi, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2 Efficiency Weak 101 102 Number of workers 100 101 Speedup Linear Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Scaling with respect to a baseline of eight workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Weak efficiency (top) and strong linear speedup (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' Use case 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Patton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' : Optimizing Deep Learning Hyper-parameters through an Evolutionary Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' In: Pro- ceedings of the Workshop on Machine Learning in High-performance Computing Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' 1–5 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='1145/2834892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} +page_content='2834896' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFAT4oBgHgl3EQf2R4K/content/2301.08713v1.pdf'} diff --git a/U9E3T4oBgHgl3EQfawpi/content/tmp_files/load_file.txt b/U9E3T4oBgHgl3EQfawpi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a91d7d6541e626199fdff52030acf7991f78f287 --- /dev/null +++ b/U9E3T4oBgHgl3EQfawpi/content/tmp_files/load_file.txt @@ -0,0 +1,512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf,len=511 +page_content='Received 26 April 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Revised 6 June 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Accepted 6 June 2016 DOI: xxx/xxxx ORIGINAL ARTICLE Alpha tensor and dynamo excitation in turbulent fluids with anisotropic conductivity fluctuations Oliver Gressel1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='2 | Günther Rüdiger *1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='3 | Detlef Elstner1 1MHD & Turbulence Section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Leibniz Institute for Astrophysics Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' An der Sternwarte 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 14482 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Germany 2Niels Bohr International Academy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The Niels Bohr Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Blegdamsvej 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' DK-2100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Copenhagen Ø,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Denmark 3Institute of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' University of Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 24-25, 14476 Potsdam, Germany Correspondence Email: gruediger@aip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='de A mean-field theory of the electrodynamics of a turbulent fluid is formulated under the assumption that the molecular electric conductivity is correlated with the tur- bulent velocity fluctuation in the (radial) direction, 품.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' It is shown that for such homogeneous fluids a strong turbulence-induced field advection anti-parallel to 품 arises almost independently of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For rotating fluids, an extra 훼 effect appears with the known symmetries and with the expected maximum at the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Fast rotation, however, with Coriolis number exceeding unity suppresses this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Numerical simulations of forced turbulence using the NIRVANA code demonstrate that the radial advection velocity, 훾, always dominates the 훼 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We show finally with simplified models that 훼2 dynamos are strongly influenced by the radial pump- ing: for 훾 < 훼 the solutions become oscillatory, while for 훾 > 훼 they become highly exotic if they exist at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In conclusion, dynamo models for slow and fast solid- body rotation on the basis of finite conductivity–velocity correlations are unlikely to work, at least for 훼2훺 dynamos without strong shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' KEYWORDS: astrophysical plasma – dynamo theory 1 INTRODUCTION If apart from the velocity and magnetic field, by any reason also the electric conductivity in a turbulent fluid fluctuates around a certain value then also the local magnetic diffusiv- ity fluctuates around its average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Krause & Roberts (1973) started to consider the consequences of this constellation with the result that the effective decay time of a large-scale nonuni- form magnetic field is changed by reducing the effective eddy diffusivity of the turbulence field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Moreover, in convection-driven turbulent fields the always existing temperature fluctuations should produce magnetic resistivity fluctuations which are correlated with one of the velocity components, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=', the vertical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In this case, even a turbulent diffusivity-flux vector ⟨휂′풖′⟩ —with 휂 = 1∕휇0휎 denoting the magnetic resistivity and 풖′ the velocity fluctuations— occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This, in connection with the magnetic background field or electric current, may form new terms in the mean-field induction equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Pétrélis, Alexakis, & Gissinger (2016) suggested a new sort of 훼 effect arising in such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' They derived an expression for the diffusivity-current cor- relation, in which the diffusivity-flux vector, multiplied with the mean magnetic field, ̄푩, appears so that a new 훼 effect could be possible in spite of the assumed homogeneity of the turbulence field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' However, there are two possibilities for the relation between the electromotive force and the mean mag- netic field: the latter can be i) parallel to the electromotive force or ii) perpendicular to the electromotive force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Only in the first case, one formally speaks of an 훼 effect, which may lead to self-excitation of large-scale magnetic fields, while in the second case the expression describes a turbulent diamag- netism (also called “topological pumping”) which is known arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='04509v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='flu-dyn] 11 Jan 2023 2 Gressel, Rüdiger & Elstner to hamper dynamo instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' If the correlation ⟨휂′풖′⟩ exclu- sively defines a preferred direction 품 the resulting turbulent electromotive force is perpendicular to the mean magnetic field and an alpha-effect is not obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Later on, quasi-linear SOCA calculations applicable to rotating forced turbulence and/or magneto-convection indeed confirmed the existence of an 훼 effect in the presence of global rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Without rotation, the conductivity fluctuations lead to a reduction of the eddy diffusivity and —if correlated with one of the velocity components— to a new but rather strong dia- magnetic pumping effect (Rüdiger, Küker, & Käpylä 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In that work, rotating magneto-convection was numerically used to derive the radial turbulent electric current flux ⟨푢′ 푟curl푩′⟩ — where 푟 is the radial coordinate— which serves as a proxy of the turbulent diffusivity-current vector ⟨휂′curl푩′⟩ if 휂′ and 푢′ 푟 are correlated or anti-correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The flux vector always exists for rotating convection under the influence of an azimuthal magnetic background field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The result is a well-defined dia- magnetic pumping and, with rotation, an 훼 effect which is anti-symmetric with respect to the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' However, convection only exists if the fluid is stratified in the radial direction, 품.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The main difference caused by the fluctuating-conductivity concept is the occurrence of an 훼 effect in fully uniform fluids in which an anisotropy exists rather than any form of stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This makes the idea a promising one for a dynamo theory of planetary magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In the present paper, therefore, the existence of the 훼 effect in absolutely homogeneous fluids is shown by numerical sim- ulations of forced rotating turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We shall demonstrate that the 훼 effect indeed occurs, if the global rotation is not too slow or too fast but that it is, however, always accompanied by a dominating diamagnetic pumping term, 훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Even without rotation (and only slightly suppressed in its presence) a strong radial advection term occurs by which the horizontal field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' perpendicular to 품) is lifted to either of the radial boundary layers, depending on the sign of the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We note that a large-scale 훼2 dynamo can in principle oper- ate for very weak 훼 effect if only the region is big enough, or —with other words— if it hosts a sufficiently large number of eddies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In our final Section, the consequences of this puz- zling situation are shown by the presentation of a sequence of mean-field 훼2 dynamo models with stronger and stronger mag- netic pumping term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' turbulence-induced diamagnetism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We shall show that such dynamos can only operate as long as the 훼 term (in form of a pattern velocity) exceeds the pump- ing velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This condition is unfortunately not met – at least, according to the results of the derived electrodynamics, which is based on the correlations with conductivity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 2 THE EQUATIONS The basic equation of the problem is the induction equation 휕푩 휕푡 = curl ( 풖 × 푩 − 휂 curl푩 ) , (1) with the continuity condition div 푩 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Moreover, we assume div 풖 = 0 as the condition for an incompressible fluid for the analytic derivations, while for the numerical experi- ments, this constraint is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Here, 풖 is the fluid velocity, 푩 is the magnetic field vector and 휂 the (molecular) magnetic dif- fusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We consider a turbulent fluid with 풖 = ̄풖+풖′ and with a fluctuating magnetic diffusivity 휂 = ̄휂 + 휂′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For the expec- tation values of the perturbations we shall use the notations 푢rms = ⟨풖′2⟩1∕2 and 휂rms = ⟨휂′2⟩1∕2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Large-scale observables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=', mean values) are marked with overbars, while brack- ets are used for the correlations of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Low or high values of the magnetic Reynolds number Rm = 푢rms퓁∕̄휂 (2) (for Strouhal number ≃ 1, and with 퓁 the correlation length) distinguish between the regimes of low / high conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Within the realm of the electrodynamics with finite fluctua- tions, the high-conductivity limit ̄휂 →0 may not be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' If the fluctuations 풖′ and 휂′ exist and are correlated, then the turbulence-originated diffusivity flux 푼 = ⟨휂′풖′⟩ (3) forms a vector, which is polar by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The existence of the radial component of this vector is obvious for thermal convection, where both the radial velocity and the electric con- ductivity are due to temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The correlation (3) can be understood as transport of magnetic diffusivity in a certain direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' If, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=', the correlation between 휂′ and 푢′ 푟 is positive then resistivity is transported upwards – balanced by a downward radial velocity ∇(−휂) which “pumps” the horizon- tal field downwards in the direction where the magnetic decay is maximum (the “diamagnetic effect” of turbulent origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Also the magnetic field will fluctuate, hence 푩 = ̄푩+푩′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The magnetic fluctuation 푩′ fulfils a nonlinear induction equation which follows from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The turbulence-originated electromo- tive force \ue24b = ⟨풖′ ×푩′⟩ and the diffusivity-current correlation \ue250 = −⟨휂′curl푩′⟩ enter the induction equation for large-scale magnetic field via 휕 ̄푩 휕푡 = curl ( \ue24b + \ue250 − ̄휂 curl ̄푩 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (4) Under the assumption that the large-scale field, ̄푩, varies suf- ficiently slowly in space and time, the electromotive force can be written as \ue24b = 훼◦ ̄푩 − 휂tcurl ̄푩 , (5) where the tensor 훼 and the coefficient 휂t represent the 훼 effect and the turbulent magnetic diffusivity (Krause & Rädler Gressel, Rüdiger & Elstner 3 1980), respectively, and where ‘◦’ denotes a tensor multi- plication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The tensorial structure of 휂t under the presence of magnetic field and rotation has been discussed later by Kitchatinov, Pipin, & Rüdiger (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As in Rüdiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (2020), the spectral vector of the correlation (3) may be written as ̂푈푖 = 푢1(푘, 휔) ( 푔푖 − (품⋅풌) 푘푖 푘2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (6) The vector 품 gives the unit vector of the direction in which the correlation between velocity and diffusivity is non-vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The expression (6) must be odd in 품 and its real part must be even in the wave number 풌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The quantity 푢1 reflects the correlation of the velocity component 품⋅풖′ with 휂′ where 휔 is the Fourier frequency of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As it should, the transformation 품 → −품 only changes the sign of 푼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 3 THE DIFFUSIVITY-CURRENT CORRELATION It has been shown earlier that a relation \ue250 = −훾 품 × ̄푩 (7) between the the diffusivity-current correlation, \ue250 , and the large-scale magnetic field, ̄푩, results with 훾 = 1 3 ∫∫ ̄휂푘4푢1 휔2 + ̄휂2푘4 d풌 d휔 , (8) representing a turbulent advection of the magnetic background field where 풖adv = −훾품 is the advection velocity (Rüdiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We find a coefficient 훾 of the same sign as the dif- fusivity flux (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For positive 푢1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=', for positive correlation of 휂′ and 푢′ 푟), the advection velocity, 풖adv, points downward if 품 is the radial unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Anti-correlated 휂′ and 푢′ 푟 lead to an upward turbulent transport of the mean magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This means that the field is always attracted by the islands of lower resistivity – or, equivalently, of higher electric conduc- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As a consequence, the large-scale magnetic field favours the direction towards longer diffusive decay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The advec- tion velocity is opposite to the diffusivity flux (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The integral expression for 훾 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (8) scales linearly with Rm until it saturates for large magnetic Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Let ̂푉 be the spectral function of the two-point autocor- relation function 푉 (흃, 휏) = ⟨휂′(풙, 푡) 휂′(풙 + 흃, 푡 + 휏)⟩ of the diffusivity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For the diffusivity-current correlation \ue250 the term with ̂푉 leads to \ue250 = ⋯ + 2 3 ∫∫ 푘2 ̂푉 −i휔 + ̄휂푘2 d풌 d휔 curl ̄푩 , (9) which provides an extra contribution to the magnetic field dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The question is whether this term reduces or enhances the eddy diffusivity 휂t representing turbulence with- out 휂-fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The small-scale diffusivity fluctuations obviously lead to a reduction of the large-scale eddy diffusiv- ity 휂t which, however, is only weak as it runs with the small value (휂rms∕̄휂) in second order (Krause & Roberts 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Rüdi- ger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The actual value of the turbulence dissipation will not have relevance for the results of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Our assumed background turbulence is homogeneous but anisotropic, where the anisotropy is only implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' If the turbu- lence rotates, an additional pseudo-scalar 품 ⋅ 휴 appears with which a relation \ue250 = −훾 품 × ̄푩 − − 훼1 [ (품⋅ ̄푩) 휴 + (품⋅휴) ̄푩 ] − 훼2( ̄푩⋅휴) 품 (10) can be formulated – with yet unknown coefficients 훼1 and 훼2 for the diffusivity-current correlation, \ue250 , in presence of a large-scale magnetic field and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For the above expres- sion, 훾 is again given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Relation (10) formally describes the existence of an 훼 tensor which connects the correlation \ue250 with the large-scale magnetic field ̄푩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This con- nection exists despite the turbulence model being assumed as strictly homogeneous (so that the standard 훼 tensor cannot appear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The 훼 effect according to (10) is highly anisotropic, the middle term with the coefficient 훼1 provides the rotation- induced standard 훼 expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' While the diamagnetic term with 훾 also exists for 훺 = 0, the 훼 terms need global rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We shall show below that, independently of the sign of the cor- relations ⟨휂′푢′ 푟⟩, the values of 훼1 and 훾 are always of opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The dimensionless ratio ̂훾 = 훾 훼1훺 (11) of the pumping velocity 훾 and the rotation-induced 훼 effect indicates the ratio of anti-symmetric and symmetric elements in the complete 훼 tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Simulating electromotive forces for models of rotating magnetoconvection, Ossendrijver, Stix, & Brandenburg (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Ossendrijver, Stix, Brandenburg, & Rüdiger (2002) found ̂훾 ≃ 1 where both 훼 and 훾 were about 10% of the rms value of the convective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Also Käpylä, Korpi, & Brandenburg (2009) reached typical val- ues of order unity in their numerical models of turbulent magnetoconvection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Additionally, with their extensive numer- ical simulations, Gressel, Ziegler, Elstner, & Rüdiger (2008) derived ̂훾 = 푂(1) for interstellar turbulence driven by col- lective supernova explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' All these examples summarise the results of 훼 effect calculations from the relation between the electromotive force \ue24b and the mean magnetic field ̄푩, which only appears if the turbulence is nonuniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' On the other hand, we shall demonstrate in the following that for homogeneous models with fluctuating conductivities, the cor- responding ratio (11) reaches values even exceeding unity – with severe consequences for associated dynamo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 4 Gressel, Rüdiger & Elstner 4 NUMERICAL METHODS To probe the theoretical predictions we run artificially forced, fully nonlinear numerical simulations with the NIRVANA MHD code (Ziegler 2004), which solves the equations of com- pressible magnetohydrodynamics by means of a second-order Godunov approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In the simulations, the fluctuating compo- nent of the magnetic diffusivity is prescribed by 휂′ = 푐푢푢푧, where the coefficient 푐푢 is used to control the strength of the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We furthermore use 휂rms = 푐푢푢푧,rms to quantify the amplitude of the fluctuating part of the magnetic diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The simulation domain is a fully periodic cube with volume 퐿3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The units of length and time are [푥] = 푘−1 1 , [푡] = (푐s푘1)−1 where 푘1 is the wave number corresponding to the system size and 푐s is the constant speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The simulations employ standard non-helical forcing according to eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (7) of Hau- gen, Brandenburg, & Dobler (2004) and are characterised by the magnetic Reynolds number (2) with 푢rms volume averaged and 퓁 = (푘f)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The flows under consideration are weakly compressible with Mach number Ma = 푢rms∕푐s ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' All simulations have 푘f ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='5 (using isotropically sampled dis- crete wave vectors obeying 4 ≤ 푘f ≤ 5) and employ a grid resolution of 803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In code units, the molecular diffusivity is fixed at ̄휂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 5 THE TURBULENT FLUX OF ELECTRIC CURRENT Consider a homogeneous and isotropic turbulence that is influ- enced by uniform magnetic fields and global rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Let us write its correlation tensor, ⟨푢′ 푖 curl푗푩′⟩, as ⟨푢′ 푖 curl푗푩′⟩ = = 휅′휖푗푖푘 ̄퐵푘 + 휅1훺푖 ̄퐵푗 + 휅2훺푗 ̄퐵푖 + 휅3(휴 ⋅ ̄푩)훿푖푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (12) The tensor is not a pseudo-tensor and there is no reason that the dimensionless coefficients 휅 identically vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' It does not play a known role in the mean-field electrodynamics but it is exploited here as a proxy of the desired diffusivity-current correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The correlation vector ⟨푢′ 푟 curl푩′⟩ describes an upward or downward radial flux of electric current in a rotating magnetised turbulence which we shall use below to estimate the diffusion-current correlation \ue250 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We note that for 훺 = 0 it is ⟨(품 ⋅ 풖′) curl푩′⟩ = 휅′품 × ̄푩 for all directions 품.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' With 품 as the radial direction, one finds ⟨푢′ 푟curl휃푩′⟩ = −⟨푢′ 휃curl푟푩′⟩ = −휅′ ̄퐵휙 , (13) if the magnetic background field only has an azimuthal com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Based on SOCA calculations, the coefficient 휅′ is 휅′ = 1 15 ∞ ∫ 0 ∞ ∫ 0 휂푘4퐸(푘, 휔) 휔2 + 휂2푘4 d푘 d휔 , (14) with the positive spectral function 퐸 of the turbulence inten- sity, 푢2 rms = ∞ ∫ 0 ∞ ∫ 0 퐸(푘, 휔) d푘 d휔 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (15) As the spectrum 퐸(푘, 휔) is positive-definite, the tensor coeffi- cient 휅′ is positive-definite, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Figure 1 gives a numerical representation of the complete tensor (12) in Cartesian coordinates (푟, 휃, 휙) → (푥, 푦, 푧) where the rotation vector is 휴 = 훺0(cos 휃, − sin 휃, 0) and the mag- netic field ̄푩 = (0, 0, 퐵0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The details of the simulations were given in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Obviously, the 휅3 coefficient in (12) cannot be determined for this geometry as always 휴 ⟂ 푩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' It is clear from the uppermost and the lowermost curves in the left and the right panel that after (13) the sim- ulation gives 휅′ > 0 in accordance to the result (14) of the quasi-linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Only the 푥푦-component is anti-symmetric in its indices but the cross correlations 푥푧 and 푦푧 are sym- metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The diagonal components 푥푥, 푦푦 and 푧푧 vanish (not shown) in accordance to the relation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For the remaining off-diagonal tensor components, one finds 휅1 = 휅2 = 휅 with 휅 < 0 as ⟨푢′ 푟curl휙푩′⟩ = ⟨푢′ 휙curl푟푩′⟩ = 휅훺 ̄퐵0 cos 휃 < 0 (16) and ⟨푢′ 휃curl휙푩′⟩ = ⟨푢′ 휙curl휃푩′⟩ = −휅훺 ̄퐵0 sin 휃 > 0 , (17) hence for rotating and magnetised (but otherwise isotropic) turbulence, the tensor expression (12) becomes ⟨푢′ 푖 curl푗푩′⟩ = 휅′휖푗푖푘 ̄퐵푘 + + 휅(훺푖 ̄퐵푗 + 훺푗 ̄퐵푖) + 휅3(휴 ⋅ ̄푩)훿푖푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (18) In a rotating but otherwise isotropic turbulence with an azimuthal background field, the meridional flow fluctuations will always be correlated with the azimuthal electric-current fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We note that the simulations show that the anti- symmetric (푥푦)-component of the tensor is always much larger than the symmetric (푥푧)-component – which, in fact, will have important consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Replace now in the relations (13) and (16) 푢′ 푟 by 휂′ and the existence of correlations such as ⟨휂′curl휃푩′⟩ and ⟨휂′curl휙푩′⟩ becomes obvious in (rotating) homogeneous turbulence fields magnetised with an azimuthal background field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Just this find- ing is formulated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For positive correlation of the 휂-fluctuation and the radial velocities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=', positive 푈푟), the 훼1 in (10) becomes negative and for negative correlations it becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Note the negative sign in the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In the same relation, the 훾 results as positive – hence the pumping Gressel, Rüdiger & Elstner 5 FIGURE 1 The off-diagonal components (expectation value plus temporal variations) of the turbulence-induced electric- current flux tensor ⟨푢′ 푖curl푗푩′⟩ normalised with 푢rms푘f퐵0 for various co-latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The plot reflects the symmetry of the tensor except the 푥푦-component which is anti-symmetric in accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Rm = 100, 퐵0 = 2 × 10−8, Pm = 1, 훺 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' is downward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=', opposite to 푈푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We always obtain 훾훼1 ≤ 0 for both signs of 푈푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Another basic finding is that the term with 휅′ always exceeds those with 휅, which – in other words – means that, for rotating turbulence, the pumping term (a velocity) will always be larger than the 훼 term (also a velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As a consequence, in rotating conducting fluids, the diamagnetic effect may by far exceed the inducting action of the 훼 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The remainder of this paper will confirm this suggestion and will show that a dominating turbulent pumping precludes dynamo instability of the 훼2-type, that is, in the absence of large-scale shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Figure 2 numerically shows the influence of the mag- netic Prandtl number on the off-diagonal components of tensor (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The values are taken for mid-latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The Pm varies by more than one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The numerical values basically grow for growing Prandtl number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Nevertheless, the ratio of the negative quantities ⟨푢′ 푥curl푦푩′⟩ and ⟨푢′ 푥curl푧푩′⟩ remains numerically always much larger than one, also for the important case of Pm < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The following numerical simulations in a Cartesian box with the vertical (radial) vector 품 = (1, 0, 0) have been done with a negative correlation between diffusivity fluctuation and vertical velocity, hence 푈푥 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Again, the applied magnetic field is azimuthally directed, ̄푩 = (0, 0, 퐵0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We find 훾 = − ⟨휂′curl푦푩′⟩ 퐵0 , 훼1훺 = ⟨휂′curl푧푩′⟩ 퐵0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (19) Figure 3 displays the three components of the diffusivity- current vector as function of the co-latitude 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As it should, its radial component vanishes (left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' It is also understand- able that the advection term ⟨휂′curl푦푩′⟩ is positive and does hardly depend on the latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' According to the first relation in (19), 훾 < 0 – so that the advection velocity 풖adv is directed upwards (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=', opposite to 푈푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Contrary to this, the 푧-component of the correlation vector vanishes at the equator – as it is expected for a rotation- induced 훼-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Its maximum is obtained at the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Accord- ing to the second definition (19), one finds a positive 훼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Note that the negative sign of the product 훾훼1 is independent of the sign of the correlation of 휂′ and 푢′ 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The simulated components of the correlation vector ⟨휂′curl푩⟩ for fixed rotation rate have been given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For a characteristic value 휂rms∕̄휂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='1 of the diffusivity fluc- tuations, the rotation frequency is varied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 4 to obtain the characteristic numbers at the northern pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Obviously, the maximal correlation appears for rotation 훺 ≃ 1 and will be suppressed by faster rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For the ratio (11) we generally obtain a value of about five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The normalised 훼 effect is 퐶훼 = 훼1훺퐿 ̄휂 + 휂t ≃ 3훼1훺 푢rms 퐿 퓁 , (20) with 퐿 as the box length in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The characteristic turnover time of the turbulence is 휏corr ≃ 퓁∕푢rms ≃ 2 in the simulation (also in code units), where 푢rms ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='11 is set by the amplitude of the forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' It is 휂t∕̄휂 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='3푢2 rms휏corr∕̄휂 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 3 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (19), we have 훼1훺∕푢rms ≃ 5 ⋅ 10−3 so that 퐶훼 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='5 ⋅ 10−2 퐿 퓁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=" (21) ×10-1 ×10-1 ×10-1 4 2 0 2 4 6 0°15° 30° 45° 60° 75° 90° 0°15° 30° 45° 60° 75° 90° 0°15° 30° 45° 60° 75° 90 co-latitude e co-latitude e co-latitude e6 Gressel, Rüdiger & Elstner FIGURE 2 Similar to Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 1 but for 휃 = 45◦ and for various magnetic Prandtl numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The blue line in the middle panel (⟨푢′ 푥curl푦푩′⟩, leading to the advection term) and the orange line in the right panel (⟨푢′ 푥curl푧푩′⟩, leading to the 훼 effect) are of particular relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The ratio ⟨푢′ 푥curl푦푩′⟩∕⟨푢′ 푥curl푧푩′⟩ for all Pm always exceeds unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Rm = 11, 훺 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' FIGURE 3 The three components of the diffusivity-current vector ⟨휂′curl푩′⟩∕(푢rms퐵0) versus co-latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Rm = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 휂rms∕̄휂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='1, 퐵0 = 2 × 10−8, Pm = 1, 훺 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The ratio 퐿∕퓁 gives the number of cells along the vertical direction, which obviously must exceed 70 to reach 퐶훼 of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This is one of the arguments that it would not be easy to simulate such a dynamo in a box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The dependencies of the diffusivity-current vector compo- nents on the rotation rate 훺0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 4 , where for 훺0 = 1, the rotation period is 2휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As usual, the Strouhal num- ber St = 푢rms휏corr∕퓁 results of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We also note that 훺0 = 1 describes a rapid rotation with a Coriolis number of 2휏corr훺 ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='4 beyond which the 훼 effect is strongly quenched by the rotation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 4 , right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The maximum correla- tion exists for 훺 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' one cannot increase this value by faster rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For 훺 = 1 it is ̂훾 ≃ 5, and this ratio even grows for slower and/or faster rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' A weak rotational quenching can also be observed in the middle panel, where the advection term is reduced (only) by a factor of three when 훺0 grows by two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=" ×10-1 x10-1 x10-1 2 0 2 4 6 2 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='5 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='5 1 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='5 1 2 4 magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Prandtl number magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Prandtl number magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=" Prandtl number×10-2 ×10-2 ×10-2 3 2 1 0 0°15° 30° 45°60°75° 90° 0°15° 30° 45° 60° 75° 90' 0°15° 30° 45° 60°75° 90 co-latitude A co-latitude e co-latitude eGressel, Rüdiger & Elstner 7 FIGURE 4 Similar to Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 3 but at the northern pole and for increasing rotation rate 훺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' FIGURE 5 The three components of the vector ⟨휂′curl푩′⟩∕(푢rms퐵0) for non-rotating turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' (10) only the 푦-component (representing the topological pumping) remains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Rm = 11, Pm = 1, 훺 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Figure 5 refers to non-rotating turbulence with growing ratio of the diffusivity fluctuations, 휂rms∕̄휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As expected, the curve in the middle panel linearly runs with the normalised diffusivity fluctuation in accordance with the 훾 expression (8), and it vanishes for 휂′ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For non-rotating turbulence, of course, the two remaining components (including the 훼 effect) are identical zero – as shown in the left and the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As it should, the advection term plotted in the middle panel also exists for non-rotating turbulent fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We still have to find out how the calculated large values of ̂훾 influence the operation of a global dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 6 KINEMATIC 훂ퟐ DYNAMO MODELS WITH FIELD ADVECTION It has been shown that for rotating turbulence, the above for- mulated 훼 effect is always accompanied by a pumping effect in the direction of the component of the flow vector which is correlated with conductivity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For all rotation rates, the ratio ̂훾 exceeds unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We now turn our inquiry to the influence of the turbulent field advection on the operation of an 훼2 dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=" In earlier papers, we already found for disk ×10-2 ×10-2 ×10-2 3 2 1 0 1 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='1 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='1 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content="1 1 10 angular velocity Qo angular velocity 20 angular velocity Qo×10-2 ×10-2 x10-2 3 2 1 0 n'curlxB'> I." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content="B'> 1 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content="1 1 n'/n (nominal) n'/n (nominal) n'/n (nominal)8 Gressel, Rüdiger & Elstner dynamo models that a too strong field advection suppresses the field excitation even under the presence of differential rotation (Schultz, Elstner, & Rüdiger 1994)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The geometrically simplest model is posed by uniform quantities 훼 and 훾 existing in a gap between two parallel plates embedded in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The vertical distance between the boundaries is 퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The eddy diffusivity 휂0 between the plates is assumed as a free parameter, whose actual value is not important for the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' All quantities are assumed as uniform in the two horizontal directions 푦 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Then the condition div 푩 = 0 requires that the vertical field 퐵푥 does not depend on 푥 hence 퐵푥 = 0 without lost of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The equations for this kinematic 1D slab model may be formulated with the normalised quantities 퐶훼 = 훼퐻 휂0 , 퐶훾 = 훾퐻 휂0 (22) (let 훺 = 1 for simplicity) so that i휔퐵푦 − d2퐵푦 d푥2 = −퐶훾 d퐵푦 d푥 − 퐶훼 d퐵푧 d푥 (23) and i휔퐵푧 − d2퐵푧 d푥2 = −퐶훾 d퐵푧 d푥 + 퐶훼 d퐵푦 d푥 , (24) — see Moss, Shukurov, & Sokoloff (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Parker (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Rüdiger & Kitchatinov (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The real part of the complex frequency 휔 determines the oscillation frequency (in units of the diffusion rate) of a possible dynamo wave along the verti- cal direction, while the growth rate of the dynamo is given by the negative value of its imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We are mainly inter- ested to know the critical 퐶훼,0 for neutral instability, ℑ(휔) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Let us define the ratio ̂훾0 = 퐶훾 퐶훼,0 (25) as describing the influence of the pumping effect on the excitation of kinematic 훼2 dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The vacuum boundary conditions 퐵푦(0) = 퐵푧(0) = 퐵푦(1) = 퐵푧(1) = 0 are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For 훾 = 0 the lowest nontrivial eigen- value of a stationary solution is 퐶훼,0 = 2휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The solutions do not depend on the sign of 퐶훾 as they do also not depend on the sign of 퐶훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 6 gives the dynamo’s growth rates for three values of 퐶훾 as function of 퐶훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As usual, for sub-critical (super-critical) 훼 the modes are decaying (grow- ing), and we find that the 퐶훼,0 for neutral instability grows with growing 퐶훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' All the critical dynamo solutions for non- vanishing 훾 are oscillating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The lower panel of this figure demonstrates that for these eigensolutions the ratio ̂훾0 does never exceed unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The 1D 훼2 dynamo, therefore, has no neu- tral dynamo solution for 퐶훾 > 퐶훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' A too strong radial advec- tion effect is not compatible with the operation of 훼2 dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The reason for the suppression of the dynamo instability by FIGURE 6 Upper panel: Growth rates multiplied with the diffusion time vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 퐶훼 for three plane dynamos with 퐶훾 = 8, 16, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' All solutions describe waves travelling in vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Lower panel: The dimensionless ratio ̂훾0 versus 퐶훾 for neutral excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Kinematic dynamos only exist as long ̂훾 ≤ 1, the pumping term 퐶훾 suppresses the dynamo action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' dominating radial advection is that the field components per- pendicular to the advection vector are concentrated inwards (or outwards) so that the dynamo domain is reduced and the critical 퐶훼 must grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This destructive action proves to be even more drastic for 훼2 dynamos than for those of 훼훺-type (Brandenburg, Moss, & Tuominen 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Moss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Results for a very special spherical model with 훼 effect and pumping term are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='10 in Krause & Rädler (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The 훼 effect only exists in an outer hemisphere while the diamagnetic pumping only exists in its inner part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Simi- lar to the above slab model, for growing 퐶훾 also the critical 퐶훼,0 grows linearly so that the ̂훾0 never exceeds unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The mode with the lowest dynamo number is a nonaxisymmetric quadrupole with an azimuthally drifting magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Because of its relevance for the concept of conductivity fluctuations, we have designed a simple shell-type dynamo model with an outer turbulence domain filled with 훼 effect Gressel, Rüdiger & Elstner 9 independent of the radius, and with uniform radial 훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The 훼 term is anti-symmetric with respect to the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The def- initions (22) have been used with the replacement 퐻 → 퐷 with 퐷 = (1 − 푟in)푅 and 푅 the radius of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The inner boundary is a perfect-conducting one while the outer boundary mimics vacuum, so that the Poynting flux is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' To illus- trate the performance of the advection term, examples for the excited magnetic fields are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 7 for a turbulence with outward pumping (top) and inward pumping (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The inner part (or the outer part, in dependence on the sign of 훾) of the shell are field-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Eigensolutions with dipolar symmetry have the same eigenvalue as those with quadrupolar symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The sign of 퐶훾 differs in both models but with- out consequences for the excitation condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For both cases |̂훾| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='8 is prescribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The radial advection produces nonax- isymmetric solutions drifting in the azimuthal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For 훾 = 0 the critical eigenvalue for neutral excitation is 퐶훼,0 ≃ 5, independent of the value of 푟in (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 7 , middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For increasing ̂훾, the horizontal field will be more and more concentrated at the inner or the outer boundary (in dependence on the sign of 훾) while the bulk of the shell becomes field-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The values of 퐶훼 necessary for dynamo excitation grow to unrealistic high values (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 8 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' A fluid with values of ̂훾 > 1 and without shear cannot maintain large-scale fields via the 훼2 mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For the above calculated high value of ̂훾 ≃ 5, therefore, kinematic 훼2 dynamos are not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' With other words, the dynamo only works for 퐶훼 >∼ Max(5, 퐶훾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In case that 훼 ≃ ̂훼훺 (which is true for slow rotation), the dynamo only operates as long as the rotation rate exceeds the critical value of 훺 ≃ 훾∕̂훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The dynamo decays for 훺 < 훺1 where 훺1 denotes the rotation rate where ̂훾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The above men- tioned simulations for solar magneto-convection suggest that indeed 훺1 ≃ 훺⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Figure 8 also contains eigenvalues for an 훼2훺 dynamo with the rather flat rotation law 훺 = 훺0∕푟0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For the normalised rotation rate 퐶훺 = 훺0퐷∕휂0 the value 퐶훺 = 460 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' One only finds small deviations from the curves for the 훼2 dynamo with 퐶훺 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For weak field advec- tion the solutions with the lowest 훼0 are axisymmetric and oscillating while for stronger pumping the nonaxisymmetric modes prevail which are drifting in the azimuthal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' We note that we only considered the kinematic approxima- tion where any nonlinear feedback of the induced fields onto the turbulence is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' In any case, if dynamos ever existed for large values of ̂훾, they must be rather exotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 7 CONCLUSIONS If an anisotropy in a conducting turbulent fluid is defined as one that is (only) in the direction of the conductivity fluc- tuations, and the velocity fluctuation is correlated, then a FIGURE 7 Influence of the advection term on 훼2 dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The nonaxisymmetric dipolar mode A1 (top) and the quadru- polar mode S1 (bottom) for |̂훾0| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='8 are excited by the same value of 퐶훼,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The kinematic axisymmetric 훼2 dynamo for 훾 = 0 (middle) is shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The bottom of the turbulence domain is at a 푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='5, with a perfect-conducting boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The models are embedded in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='010 Gressel, Rüdiger & Elstner 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='0 ˆγ Cγ FIGURE 8 The values ̂훾0 critical for excitation versus 퐶훾 of spherical shell dynamo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The nonaxisymmetric (dashed lines) solutions possess (slightly) smaller 퐶훼 than the axisym- metric solutions (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Dynamo solutions for 훾 > 훼0 do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 퐶훺 = 0 (dark), 퐶훺 = 460 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The smallest eigenvalue is 퐶훼,0 = 5 for 훾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' 푟in = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' turbulent field-advection exists in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' It lifts large- scale fields oriented perpendicular to this direction downward or upward, depending on the sign of the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Our simulations provide the amplitude of this advection term in units of the turbulence velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' They are on the order of about 10% of the normalised resistivity fluctuation 휂rms∕̄휂, while the 훼 effect is generally smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Its amplitude grows for growing rotation rate until 훺 ≃ 1 – declining, however, for faster rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' On the other hand, the advection term 훾 is numerically almost uninfluenced by the rotation, in accor- dance with general expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As we have also shown that the Pm-dependence of the results is only weak, one can be sure that in rotating fluids with velocity-correlated conductiv- ity fluctuations, the resulting pumping term 훾 always exceeds the alpha term velocity 훼1훺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' As demonstrated in Section 6, this constellation has severe consequences for associated dynamo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' There we have considered two dynamo models with different geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' First, a simplified slab dynamo model with two insulat- ing plates and with a uniform 훼 effect, including a vertical turbulence-induced field advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This model only yields solutions with neutral stability if the 훼 velocity exceeds the advection velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The solution for 훾 = 0 is stationary while otherwise it forms a vertical dynamo wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For 훾 ≥ 훼, dynamo solutions no longer exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The results are very simi- lar for spherical shell dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' For growing advection effect the most unstable modes become oscillatory but always the dynamos need 퐶훼 > 퐶훾, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' the ratio ̂훾 never exceeds unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Pure 훼2 dynamos on the basis of resistivity fluctuations can thus not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' The same holds for shell dynamos with rather flat rotation laws while the behaviour of 훼훺 dynamos with large shear is still unknown for the case of strong pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' ACKNOWLEDGEMENTS OG thanks Petri Käpylä for a helpful correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' This work used the NIRVANA code version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='3, developed by Udo Ziegler at the Leibniz-Institut für Astrophysik Potsdam (AIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Elstner (2021), Alpha tensor and dynamo excitation in turbulent flu- ids with anisotropic conductivity fluctuations, Astronomical Notes, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='00:x–y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' How cite this article: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Gressel G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Rüdiger, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content=' Elst- ner (2021), Alpha tensor and dynamo excitation in turbulent fluids with anisotropic conductivity fluctuations, Astronomical Notes, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} +page_content='00:x–y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQfawpi/content/2301.04509v1.pdf'} diff --git a/V9AzT4oBgHgl3EQfJ_vP/content/tmp_files/load_file.txt b/V9AzT4oBgHgl3EQfJ_vP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2f3f7ee5a884ea415953a5a26693f18cd427b19 --- /dev/null +++ b/V9AzT4oBgHgl3EQfJ_vP/content/tmp_files/load_file.txt @@ -0,0 +1,618 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf,len=617 +page_content='Fitting mixed logit random regret minimization models using maximum simulated likelihood Ziyue Zhu ID Faculty of Sciences KU Leuven Leuven, Belgium ziyue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='zhu16@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='com ´Alvaro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas ID Faculty of Economics and Business KU Leuven Leuven, Belgium alvaro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='gutierrezvargas@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='be Martina Vandebroek ID Faculty of Economics and Business KU Leuven Leuven, Belgium martina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='vandebroek@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='be Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' This article describes the mixrandregret command, which extends the randregret command introduced in Guti´errez-Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2021, The Stata Journal 21: 626–658) incorporating random coefficients for Random Regret Min- imization models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The newly developed command mixrandregret allows the in- clusion of random coefficients in the regret function of the classical RRM model introduced in Chorus (2010, European Journal of Transport and Infrastructure Re- search 10: 181-196).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The command allows the user to specify a combination of fixed and random coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' In addition, the user can specify normal and log- normal distributions for the random coefficients using the commands’ options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The models are fitted using simulated maximum likelihood using numerical integration to approximate the choice probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Keywords: notag1, mixrandregret, mixrpred, mixrbeta, discrete choice models, mixed random regret minimization model 1 Introduction McFadden (1974) introduced conditional logit models to explain the choice behavior of individuals and to predict market shares of products and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The conditional logit models form the basis for the majority of discrete choice models, which assume that individuals use a decision rule based on Random Utility Maximization (RUM) when choosing between various alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' In contrast, Chorus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2008) proposed an alternative decision rule known as Random Regret Minimization (RRM), assuming that decision-makers aim to minimize regret when making their choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' McFadden and Train (2000) extended the random utility model by allowing the parameters to vary across individuals, leading to the so-called mixed logit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Similarly, Hensher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2016) modified the RRM models to include random effects, which account for preference heterogeneity and allow for correlation among choices made by the same individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='01091v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='EM] 3 Jan 2023 2 Fitting mixed logit random regret minimization models In this article, we extend the command randregret (Guti´errez-Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2021) into a mixed version called mixrandregret which allows the inclusion of random param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The new command allows the user to specify normal and log-normally distributed taste parameters inside the regret function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The parameters of the distribution of the co- efficients are estimated using Simulated Maximum Likelihood (SML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Specifically, given that there is no closed-form solution for the choice probabilities, we approximate them using simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We also developed the mixrpred post-estimation command that can predict the choice probabilities for each alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Additionally, the mixrbeta post-estimation command allows estimating the individual-level parameters for each in- dividual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We will illustrate the command’s usage in examples from van Cranenburgh and Chorus (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2 Classical Random Regret Models In contrast to the decision-making process of RUM models, which measure the benefits of selecting a particular alternative in terms of utility, RRM models focus on the regret resulting from not-chosen alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Regret occurs when, compared to other avail- able alternatives, the selected alternative is outperformed by the other alternatives in some of the attributes (Loomes and Sugden 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Accordingly, RRM models assume that the individuals intend to minimize regret when choosing among alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' For- mally, Chorus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2008) presented an initial model for random regret minimization models, and Chorus (2010) revised the regret function in order to obtain a smooth like- lihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Accordingly, he proposed (1) to denote the regret of individual n when choosing alternative i among the J possible alternatives Rin = J � j̸=i M � m=1 ln[1 + exp{βm · (xjn,m − xin,m)}] + αi, (1) Equation (1) represents the regret that an individual (referred to by n) experiences when choosing alternative i among J alternatives (referred to by j or i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Additionally, each alternative is described in terms of the value of M attributes (referred to by m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Consequently, xin,m represents the values of attribute m of alternative i for individual n, and βm is the taste parameter of attribute m for individual n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The parameter βm indicates that for each unit change of attribute m in a non-selected alternative, regret would either increase (if βm is positive) or decrease (if βm is negative) relative to the level of the same attribute in the selected alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Besides, the inclusion of Alternative Specific Constants (ASC) in the stated models is possible by simply adding them to the systematic part of the regret as αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The inclusion of the ASC serves the same purpose as in RUM models, which is to account for omitted attributes for a particular alternative i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' As usual, for identification purposes, we need to exclude one of the ASC from the model specification, so we define α = (αi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' , αJ−1) as the vector of J − 1 ASC included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A detailed discussion of the ASC in the context of RRM models see Van Cranenburgh and Prato (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Consequently, Rin describes the total systematic regret for an individual n choosing alternative i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Zhu, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Vandebroek 3 Similarly to RUM models, we can obtain the random regret function, RRin, by adding an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='d extreme value type I error term to the systematic regret function, Rin, that will account for the pure random noise and the impact of omitted attributes in the regret function: RRin = Rin + εin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Mathematically, the minimization of the random regret function is equivalent to maximizing the negative function, which results in the conventional closed-form logit formula for the choice probabilities given in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Pin = exp(−Rin) �J j=1 exp(−Rjn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2) The log-likelihood function of the regret model for N individuals is given by equation (3), where β = (β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' , βm) is the vector of taste parameters and yin is the dummy variable that takes the value of 1 when alternative i is chosen by individual n, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' LL(α, β) = N � n=1 S � s=1 J � i=1 yin × ln (Pin) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (3) In the literature, there exist several extensions to the classical RRM models (Chorus 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' van Cranenburgh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Chorus (2014) proposed the generalized RRM, which replaces the “1” in the regret function with a new parameter γm denoting the regret-weight for attribute m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' van Cranenburgh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2015) incorporated a scale pa- rameter into the RRM, which is now referred to as µRRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The Pure-RRM was proposed in the same article (van Cranenburgh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2015), as a special case of µRRM when µ arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' For a review that compares the different types of RRM models and RUM models, see Guti´errez-Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' In what follows, we will focus on the classical regret function of Chorus (2010), but we will allow for the inclusion of random taste parameters as introduced by Hensher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' This model will be referred to as the Mixed Random Regret Minimization (Mixed RRM) model and takes preference heterogeneity into consideration by assuming a parametric distribution for the taste parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 3 Mixed Random Regret Minimization Models In this section, we describe the Mixed RRM where we (i) allow that the taste pa- rameters follow a parametric distribution, and (ii) we are able to model data with panel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Consequently, (i) triggers a new sub-index to the taste parameters, βn = (βn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' , βn,m), which now follow a parametric distribution f(β|ϕ), where ϕ are the parameters that describe the distribution1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Hence, βn,m is now an individual- specific taste parameter that represents the regret sensitivity of individual n to changes in attribute m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Additionally, (ii) implies that multiple choice situations (referred to by s) are answered by the same individual, which triggers the inclusion of a new sub-index for the choice situations in our formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Hence, xins,m will now represent the value of attribute m for alternative i for individual n in choice situation s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Similarly, yins is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' For instance, if we assume a normal distribution, ϕ would contain its mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 4 Fitting mixed logit random regret minimization models now a binary variable that takes the value of 1 when individual n choose alternative i in choice situation s and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' That being said, we will define a new regret function that considers points (i) and (ii) in equation (4) where Rins describes the systematic regret for individual n choosing alternative i in choice situation s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Rins = J � j̸=i M � m=1 ln[1 + exp{βn,m · (xjns,m − xins,m)}] + αi, (4) Similarly, we add the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='d extreme value type I error term to the systematic regret function, and the choice probability is given by equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Pins = exp(−Rins) �J j=1 exp(−Rjns) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (5) Additionally, the probability of the observed sequence of choices of individual n (condi- tional on knowing βn) is given by equation (6), which differs from equation (2) in the sense that equation (6) consider responses from the same individual might be correlated, but responses from different individuals are treated as independent from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Pn(α, β) = S � s=1 J � j=1 {Pins}yins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (6) The unconditional choice probabilities of the observed sequence of choices are the con- ditional choice probabilities (see equation 6) integrated over the entire domain of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Consequently, the log-likelihood function of the Mixed RRM Model in equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' LL(α, ϕ) = N � n=1 ln �� β Pn(α, β)f(β|ϕ)dβ � (7) Given that the integral described in equation (7) does not have a closed-form solution, it is approximated using simulation (Train 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Accordingly, we estimate the model by SML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Hence, we maximize the simulated log-likelihood function of equation (8) where R is the number of draws and βr is the rth drawn from f(β|ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Finally, we use Halton draws to create the draws used to approximate the choice probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' SLL(α, ϕ) = N � n=1 ln � 1 R R � r=1 Pn(α, βr) � (8) 4 Individual-level Parameters After maximizing the simulated log-likelihood function to obtain estimates for ˆϕ and ˆα, we can also obtain estimates for the individual-level parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' That is to say, we can estimate the taste parameters for every individual conditional on their sequences Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Zhu, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Vandebroek 5 of choices (denoted by yn) and the attribute levels for every alternative and choice set, denoted by xn, that the individual faced when making the choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' For instance, we can compute the individual-level parameter ¯βn for every individual n which corresponds to the mean of the distribution of βn conditional on yn, xn, and our estimated ˆϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The expression for ¯βn is given in equation (9), and its derivation can be found in Train (2009): ¯βn = � β β × Pn(yn|xn, ˆα, β)f(β|ˆϕ)dβ � β Pn(yn|xn, ˆα, β)f(β|ˆϕ)dβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (9) Again, since there is no closed-form solution for the integrals in equation (9), we approximate them using simulations yielding to equation (10): � βn = R � r=1 � βr × Pn(yn|xn, ˆα, βr) �R r=1 Pn(yn|xn, ˆα, βr) � , (10) where R is the number of draws and βr is the rth drawn from f(β|ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 5 Commands 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1 mixrandregret Syntax mixrandregret depvar � indepvars � � if � � in � � weight � , id(varname) group(varname) rand(varlist) alernatives(varname) � basealternatives(#) noconstant cluster(varname) robust ln(#) nrep(#) burn(#) level(#) maximize options � depvar equal to 1 identifies the chosen alternative, whereas a 0 indicates that the al- ternative was not selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' There is only one chosen alternative for each choice set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' fweights, iweights, and pweights are allowed (see [U] weight), but they are applied to decision-makers, not to individual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Description mixrandregret estimates the mixed random regret minimization model described in Hensher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2016), which is a mixed version of the classic random regret minimization model introduced in Chorus (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' mixrandregret extends the randregret command 6 Fitting mixed logit random regret minimization models (Guti´errez-Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2021) and allows the user to specify normally and log-normally distributed taste parameters inside the regret function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The command uses simulated maximum likelihood for estimation (Train 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Options id(varname) is required and specifies a numeric identifier variable for the decision- makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' group(varname) is required and specifies a numeric identifier variable for the choice occasions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' rand(varlist) is required and specifies the independent variables whose coefficients are random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The random coefficients can be specified to be normally or log-normally dis- tributed (see the ln() option).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The variables immediately following the dependent variable in the syntax are specified to have fixed coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' alternatives(varname) is required to identify the alternatives available for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' basealternatives(#) sets base Alternative Specific Constants (ASC) if ASC is not suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' noconstant suppress the ASC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' cluster(varname), robust see [U] estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The cluster variable must be numeric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' ln(#) specifies that the last # variables in rand() have log-normally rather than nor- mally distributed coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The default is ln(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' nrep(#) specifies the number of Halton draws used for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The default is nrep(50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' burn(#) specifies the number of initial elements to be dropped when creating the Halton sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The default is burn(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Specifying this option helps reduce the correlation between the sequences in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' level(#) set the confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The default is level(95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' maximize options difficult, technique(algorithm spec), iterate(#), trace, gradient, showstep, hessian, tolerance(#), ltolerance(#), gtolerance(#), nrtolerance(#), from(init specs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' see [U] maximize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='2 mixrpred Syntax mixrpred newvar � if � � in � � , proba nrep(#) burn(#) � Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Zhu, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Vandebroek 7 Description Following mixrandregret, mixrpred can be used to obtain the predicted probabilities by specifying the option proba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Options proba calculate the choice probability for each alternative for each choice situation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' the default option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' nrep(#) specifies the number of Halton draws used for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The default is nrep(50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' burn(#) specifies the number of initial elements to be dropped when creating the Halton sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The default is burn(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Specifying this option helps reduce the correlation between the sequences in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3 mixrbeta Syntax mixrbeta varlist � if � � in � , saving(filename) � , plot nrep(#) burn(#) replace � Description mixrbeta can be used after mixrandregret to calculate individual-level parameters cor- responding to the variables in the specified varname using equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The individual- level parameters are stored in a user-specified data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Options saving(filename) saves individual-level parameters to filename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' plot create the plots of the distribution of individual-level parameters conditional on the estimates of mixrandregret for individual-level parameters for each individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' nrep(#) specifies the number of Halton draws used for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The default is nrep(50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' burn(#) specifies the number of initial sequence elements to be dropped when creating the Halton sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The default is burn(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Specifying this option helps reduce the correlation between the sequences in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' replace overwrites filename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 8 Fitting mixed logit random regret minimization models 6 Examples To show how we can fit Mixed RRM Models using mixrandregret, we use data from van Cranenburgh and Chorus (2018) on a Stated Choice (SC) experiment2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' These data are collected to analyse the impact of the different decision rules on the statistical efficiency of the design (Van Cranenburgh and Prato 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The participants answered 10 choice situations where they chose from three unlabelled route alternatives with two generic attributes: travel cost and travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The following variables are used in our illustration: altern: identify the alternative faced by the user (sub-index i or j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' choice: whether the alternative was chosen by the individual (dummy, 1 if cho- sen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' id: ID of the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' cs: ID of the choice situation faced by the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' tt: total travel time of the alternative in minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' tc: total travel cost of the alternative in euros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We follow the data setup in randregret (see [U] randregret), and the setup for mixrandregret is identical to that required by mixlogit (see [U] mixlogit), which is the panel representation in terms of individual-alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The data set is loaded from the server to Stata directly as illustrated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We keep the variables of interest and list the first 3 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The data loaded are in wide format as each row corresponds to a choice situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' scalar server = "https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='nl/ndownloader/" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' scalar doi = "files/24015353" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' import delimited "`=server + doi´",clear (encoding automatically selected: ISO-8859-1) (29 vars, 1,060 obs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' keep obs id tt1 tc1 tt2 tc2 tt3 tc3 choice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' list obs id tt1 tc1 tt2 tc2 tt3 tc3 choice in 1/3,sepby(obs) obs id tt1 tc1 tt2 tc2 tt3 tc3 choice 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 1 1 23 6 27 4 35 3 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2 1 27 5 35 4 23 6 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 3 1 35 3 23 5 31 4 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' You can download the dataset from 4TU ResearchData: https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='nl/articles/dataset/Small value- of-time experiment Netherlands/12681650 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Zhu, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Vandebroek 9 Following the data manipulation in Guti´errez-Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' (2021), we transform the data set using the reshape command and present the data in long format below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We list the first 12 rows, and each row now corresponds to an alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The dependent variable choice is 1 for the chosen alternative in each choice situation, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' altern identifies the alternatives in a choice situation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' cs identifies the choice situation faced by the individual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' and id identifies the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Furthermore, total time and total cost are obtained from the tt and tc variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' rename (choice) (choice_w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' qui reshape long tt tc, i(obs) j(altern) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' generate choice = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' replace choice = 1 if choice_w==altern .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' label define alt_label 1 "First" 2 "Second" 3 "Third" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' label values altern alt_label .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' gen cs = obs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' gen total_time = tt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' gen total_cost = tc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' list id cs altern total_time total_cost choice in 1/12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' sepby(cs) ab(10) noo id cs altern total_time total_cost choice 1 1 First 23 6 0 1 1 Second 27 4 0 1 1 Third 35 3 1 1 2 First 27 5 0 1 2 Second 35 4 1 1 2 Third 23 6 0 1 3 First 35 3 1 1 3 Second 23 5 0 1 3 Third 31 4 0 1 4 First 27 4 0 1 4 Second 23 5 0 1 4 Third 35 3 1 We begin by fitting a classical RRM Model using the randregret command to obtain reasonable starting values for mixrandregret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We also declare noncons suppressing the ASC given that alternatives are non-labeled in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' If we have labeled data, we can specify the base alternative by declaring base() option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' As we have repeated choices from a given individual, the standard errors are corrected by specifying cluster(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' As expected, both parameter estimates are negative and highly significant, suggesting that regret decreases as the level of travel time or travel cost increases in a non-chosen alternative compared with the same attribute level in the chosen one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The coefficients are saved in init mix rrm for later use as initial values for mixrandregret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' randregret choice total_time total_cost, group(cs) alternatives(altern) /// > rrmfn(classic) nocons cluster(id) Fitting Classic RRM Model initial: log likelihood = 1164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='529 10 Fitting mixed logit random regret minimization models alternative: log likelihood = -1156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5784 rescale: log likelihood = 1121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='29 Iteration 0: log likelihood = 1121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='29 Iteration 1: log likelihood = -1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4843 Iteration 2: log likelihood = -1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4784 Iteration 3: log likelihood = -1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4784 RRM: Classic Random Regret Minimization Model Case ID variable: cs Number of cases = 1060 Alternative variable: altern Number of obs = 3180 Wald chi2(2) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='41 Log likelihood = -1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4784 Prob > chi2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0000 (Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' adjusted for 106 clusters in id) Robust choice Coefficient std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' z P>|z| [95% conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' interval] RRM total_time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='102813 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0182526 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1385874 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0670386 total_cost .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='417101 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='068059 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5504943 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='2837078 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' matrix b_rrm = e(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' matrix zero = J(1,1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='01) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' matrix init_mix_rrm = b_rrm, zero .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' matrix li init_mix_rrm init_mix_rrm[1,3] RRM: RRM: total_time total_cost c1 y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='102813 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='41710104 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='01 We then fit a Mixed RRM Model in which the coefficient for total cost is fixed, but the coefficient for total time is normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We use the option from() in mixrandregret to initialize the optimization routine using the values saved in init mix rrm as the starting point for the mean for the total time parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We estimated the model using 500 Halton draws to approximate the choice probabilities of equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Additionally, we clustered our standard errors at the individual level using cluster(id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' mixrandregret choice total_cost, group(cs) alter(altern) rand(total_time) /// > id(id) nocons cluster(id) nrep(500) from(init_mix_rrm) tech(bhhh) Iteration 0: log likelihood = -2850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0956 Iteration 1: log likelihood = 2169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='409 Iteration 2: log likelihood = -861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='11253 Iteration 3: log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='96998 Iteration 4: log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='20333 Iteration 5: log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='09059 Iteration 6: log likelihood = 771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0649 Iteration 7: log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='05912 Iteration 8: log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='05774 Iteration 9: log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='05741 Iteration 10: log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='05733 Iteration 11: log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='05731 Case ID variable: cs Number of cases = 1060 Alternative variable: altern Random variable(s): total_time (Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' adjusted for 106 clusters in id) Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Zhu, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Vandebroek 11 Mixed random regret model Number of obs = 3,180 Wald chi2(2) = 606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='11 Log likelihood = -771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='05731 Prob > chi2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0000 OPG choice Coefficient std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' z P>|z| [95% conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' interval] Mean total_cost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='102136 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0449727 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='190281 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='013991 total_time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3580736 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0581449 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4720355 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='2441117 SD total_time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5068268 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='041366 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='425751 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5879027 The sign of the estimated standard deviations is irrelevant: interpret them as being positive .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' matrix b_mixrrm = e(b) On average, the regret decreases as the total travel time increases in a non-chosen alternative, compared to the same level of travel time in the chosen alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The interpretation is similar for the total travel cost attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Additionally, we observe that there is significant regret heterogeneity for total travel time, given that the standard deviation parameter for total travel time is statistically different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Further- more, after the estimation of the Mixed RRM Model, we can compute individual-level parameters using mixrbeta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' In the code below, we use equation (10) to approximate the value for the regret coefficient for each individual using 500 Halton draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Addi- tionally, mixrbeta creates a new data set with one observation per individual (id) and its corresponding parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Subsequently, we also display the estimates for the first five individuals in the sample, where we observe that some of them have a positive coefficient for the total time attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Besides, we plot the individual level parameters for total time in Figure 1 for all the individuals in the sample and observe that there are individuals with positive estimates for the total time coefficient, which is counter-intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' mixrbeta total_time, nrep(500) replace saving("${graphs_route}\\mixRRM_normal_idl") .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' use "${graphs_route}\\mixRRM_normal_idl", replace .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' list id total_time in 1/5 id total_time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='37640482 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='05517462 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='37672848 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='38495822 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='37607978 One solution to obtain non-positive estimates for the total time coefficient is to use a bounded distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' When using mixrandregret, we can specify that a coefficient is log-normally distributed for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' In our case, since we want a non-positive distribution for the total time coefficient, we have to multiply the total time attribute 12 Fitting mixed logit random regret minimization models 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5 Density kdensity total_time Distribution of Total Time Coefficient Figure 1: Distribution of Total Time Coefficient (Normal) for -1 to ensure that it is non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' To this end we create the new variable ntt, which corresponds to the negative of total time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' gen ntt = -1 * total_time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' mixrandregret choice total_cost, group(cs) alt(altern) rand(ntt) ln(1) id(id) /// > nocons cluster(id) nrep(500) tech(bhhh) from(b_mixrrm) Iteration 0: log likelihood = -994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='35461 Iteration 1: log likelihood = -858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='23241 Iteration 2: log likelihood = 798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4694 Iteration 3: log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='66872 Iteration 4: log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='30817 Iteration 5: log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='27945 Iteration 6: log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='27728 Iteration 7: log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='27686 Iteration 8: log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='27675 Iteration 9: log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='27672 Iteration 10: log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='27671 Case ID variable: cs Number of cases = 1060 Alternative variable: altern Random variable(s): ntt (Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' adjusted for 106 clusters in id) Mixed random regret model Number of obs = 3,180 Wald chi2(2) = 1230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='55 Log likelihood = -785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='27671 Prob > chi2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0000 OPG choice Coefficient std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' z P>|z| [95% conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' interval] Mean total_cost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='217682 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0442047 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='304321 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='131042 ntt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='312285 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1562202 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='618471 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='006099 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Zhu, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Vandebroek 13 SD ntt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='363632 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1185994 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='131181 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='596082 The sign of the estimated standard deviations is irrelevant: interpret them as being positive The estimated ntt parameters are the mean and standard deviation of the natural logarithm of the coefficient, and we can transform them back to the estimates of the coefficients themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The median of the coefficient is given by exp(bntt), the mean is given by exp(bntt + s2 ntt/2), and the standard deviation is given by exp(bntt + s2 ntt/2) × � exp(s2 ntt) − 1 (Train 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The sign change prior to the estimation is reversed by multiplying the estimates by -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' nlcom (mean_time: -1*exp([Mean]_b[ntt]+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5*[SD]_b[ntt]^2)) > (med_time: -1*exp([Mean]_b[ntt])) > (sd_time : exp([Mean]_b[ntt]+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5*[SD]_b[ntt]^2) > sqrt(exp([SD]_b[ntt]^2)-1)) mean_time: -1*exp([Mean]_b[ntt]+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5*[SD]_b[ntt]^2) med_time: -1*exp([Mean]_b[ntt]) sd_time: exp([Mean]_b[ntt]+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5*[SD]_b[ntt]^2)*sqrt(exp([SD]_b[ntt]^2)-1) choice Coefficient Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' z P>|z| [95% conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' interval] mean_time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='682127 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1587961 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='9933616 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3708923 med_time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='2692041 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0420551 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3516307 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1867776 sd_time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='588122 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='6295756 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='012 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3541763 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='822067 Again, we calculate individual-level parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' As we can observe in the listed data and distribution presented in Figure 2, all individual-level parameters are now negative as we expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' mixrbeta ntt, nrep(500) replace saving("${graphs_route}\\mixRRM_ln_idl") .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' use "${graphs_route}\\mixRRM_ln_idl" , replace .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' replace ntt = -1 * ntt /*reverse sign for graph*/ (106 real changes made) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' list id ntt in 1/5 id ntt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='04032598 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='08142616 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='04047817 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='04110615 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='04025335 We can also generate predictions after running mixrandregret using mixrpred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' To illustrate this command, we rerun the models using mixrandregret with normally distributed random coefficients, suppressing the output using the quietly command (see [U] quietly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Then, using the option proba, we generate the pred p variable containing the predicted probability for each alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The code and output are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 14 Fitting mixed logit random regret minimization models 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5 4 3 2 1 0 Density kdensity ntt Distribution of Total Time Coefficient Figure 2: Distribution of Total Time Coefficient (Log-normal) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' qui mixrandregret choice total_cost, group(cs) alter(altern) rand(total_time) /// > id(id) nocons cluster(id) nrep(500) from(init_mix_rrm) tech(bhhh) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' mixrpred pred_p, proba nrep(500) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' list id cs altern total_time total_cost choice pred_p in 151/162, sepby(cs) ab(10) noo id cs altern total_time total_cost choice pred_p 6 51 First 23 6 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1516009 6 51 Second 27 4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5547292 6 51 Third 35 3 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='2936699 6 52 First 27 5 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3153724 6 52 Second 35 4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='291449 6 52 Third 23 6 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3931786 6 53 First 35 3 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3134595 6 53 Second 23 5 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5523607 6 53 Third 31 4 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1341798 6 54 First 27 4 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3153724 6 54 Second 23 5 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3931786 6 54 Third 35 3 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='291449 Additionally, mixrandregret also allows for the inclusion of ASC if users have la- beled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Although the data set is unlabeled in this example, we treat it as a labeled one in that each alternative represents a distinct category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We run the model including basealternative(1) option, which specify that the first alternative is the reference group for ASC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' mixrandregret choice total_cost, group(cs) alt(altern) rand(total_time) id(id) /// Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Zhu, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Vandebroek 15 > basealternative(1) cluster(id) nrep(500) tech(bhhh) Iteration 0: log likelihood = 1164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='529 Iteration 1: log likelihood = -812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='87881 Iteration 2: log likelihood = -773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='05839 Iteration 3: log likelihood = 769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='1873 Iteration 4: log likelihood = -768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='22193 Iteration 5: log likelihood = -767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='97262 Iteration 6: log likelihood = -767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='90237 Iteration 7: log likelihood = 767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='8867 Iteration 8: log likelihood = -767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='88268 Iteration 9: log likelihood = -767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='88165 Iteration 10: log likelihood = -767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='88138 Iteration 11: log likelihood = -767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='88131 Iteration 12: log likelihood = -767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='88129 Case ID variable: cs Number of cases = 1060 Alternative variable: altern Random variable(s): total_time (Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' adjusted for 106 clusters in id) Mixed random regret model Number of obs = 3,180 Wald chi2(2) = 465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='50 Log likelihood = -767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='88129 Prob > chi2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0000 OPG choice Coefficient std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' z P>|z| [95% conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' interval] Mean total_cost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='06784 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0498243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='165494 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='9701866 total_time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='3455217 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0594409 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4620237 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='2290197 SD total_time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5095087 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0420965 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='5920163 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='4270012 ASC ASC_2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0064798 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0510223 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='024 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='0177131 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content='2551768 The sign of the estimated standard deviations is irrelevant: interpret them as being positive 7 Conclusions This article presents the command mixrandrgret to fit Random Regret Minimization models with random parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' We also developed the post-estimation command mixrpred for predicting the estimated probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Additionally, the mixrbeta post- estimation command allows the user to estimate individual-level parameters for the random coefficients included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' The commands’ usage and options are illustrated using discrete choice data from van Cranenburgh and Chorus (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 16 Fitting mixed logit random regret minimization models 8 Acknowledgments We thank Michel Meulders, Jan De Spiegeleer, and the participants from the 2022 London Stata Conference for their helpful comments and constructive suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Ad- ditionally, substantial portions of our programs were inspired by the book Maximum Likelihood Estimation with Stata, Fourth Edition by Willian Gould, Jeffrey Pitblado, and Brian Poi (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Finally, many of the previous checks to the data and the construc- tion of the log-likelihood functions were greatly inspired by the randregret (Guti´errez- Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2021) and mixlogit (Hole 2007) commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 9 Funding This work was produced while ´Alvaro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas was a PhD student at the Re- search Centre for Operations Research and Statistics (ORSTAT) at KU Leuven funded by Bijzonder Onderzoeksfonds KU Leuven (Special Research Fund KU Leuven).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 10 Conflict of interest Ziyue Zhu, ´Alvaro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas, and Martina Vandebroek declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 11 Contribution Ziyue Zhu and ´Alvaro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas contributed equally to the article by devel- oping the command and drafting the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Martina Vandebroek critically commented on both the article and the command’s functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 12 References Chorus, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' A new model of random regret minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' European Journal of Transport and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Train, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Discrete choice methods with simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Cambridge university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Van Cranenburgh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Prato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' On the robustness of random regret mini- mization modelling outcomes towards omitted attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Journal of choice modelling 18: 51–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' About the authors Ziyue Zhu is a master student of statistics and data science at KU Leuven in Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' She earned a Bachelor of Economics from Wuhan University and a Master of Economics from Barcelona School of Economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' ´Alvaro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Guti´errez-Vargas is a PhD student at the Research Centre of Operation Research and Statistics (ORSTAT) at KU Leuven in Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' He earned a Bachelor of Science in economics from the University of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' His research interests are mainly methodological and focused on computational statistics, machine learning, and discrete choice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' He has been published in The Stata Journal and Journal of Choice Modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' Martina Vandebroek is a full professor at the Faculty of Economics and Business at KU Leuven in Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' She earned a PhD in actuarial sciences from KU Leuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' She is interested in the design of experiments, discrete choice experiments, and multivariate statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} +page_content=' She has been published in Transportation Research B, Journal of Choice Modelling, Marketing Science, and Journal of Statistical Software, among other journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfJ_vP/content/2301.01091v1.pdf'} diff --git a/W9FQT4oBgHgl3EQfcjYS/content/tmp_files/2301.13327v1.pdf.txt b/W9FQT4oBgHgl3EQfcjYS/content/tmp_files/2301.13327v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c47eceb18cfb71da8daaaa6dde5077c7ed48f95 --- /dev/null +++ b/W9FQT4oBgHgl3EQfcjYS/content/tmp_files/2301.13327v1.pdf.txt @@ -0,0 +1,1839 @@ +arXiv:2301.13327v1 [math.OC] 30 Jan 2023 +Optimization Over the Pareto Front of Nonconvex +Multi-objective Optimal Control Problems +C. Yal¸cın Kaya∗ +Helmut Maurer† +February 1, 2023 +Abstract +Simultaneous optimization of multiple objective functions results in a set of trade-off, or +Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging +task: In the case of three or more objectives the Pareto front is usually difficult to view, if not +impossible, and even in the case of just two objectives constructing the whole Pareto front +so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or +efficient) set has been an active area of research. Although there is a wealth of literature +involving finite dimensional optimization problems in this area, there is a lack of problem +formulation and numerical methods for optimal control problems, except for the convex case. +In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex +constrained and time-delayed optimal control problems as a bi-level optimization problem. +Motivated by existing solution differentiability results, we propose an algorithm incorporating +(i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the +simple but effective bisection method, for optimal control problems with two objectives. We +illustrate the working of the algorithm on two example problems involving an electric circuit +and treatment of tuberculosis and discuss future lines of research for new computational +methods. +Key words: Multi-objective optimization, Optimal control, Optimization over Pareto +front, Optimization over efficient set, Numerical methods, Rayleigh problem, Tu- +berculosis, Time-delay problems. +1 +Introduction +We continue our study of optimal control problems where one wishes to minimize simul- +taneously a number of conflicting objective functionals. These problems are referred to as +multi-objective optimal control problems and can be expressed in the following concise form: +(P) +min +(x,u,tf)∈X (ϕ1(x(tf), tf), . . . , ϕr(x(tf), tf)) . +The constraint or the feasible set X in Problem (P) involves a system of differential equations +(DEs) in the state and control variables x(·) and u(·), respectively, over a time horizon [0, tf]. +The set X also typically involves point and path equality and inequality constraints. The +DEs and constraints in X might even include time delays in the variables x(·) and u(·). It is +∗Mathematics, UniSA STEM, University of South Australia, Mawson Lakes, S.A. 5095, Australia. E-mail: +yalcin.kaya@unisa.edu.au . +†Institut f¨ur Numerische und Angewandte Mathematik, Westf¨alische Wilhelms-Universit¨at M¨unster, +M¨unster, Germany. E-mail: helmut.maurer@uni-muenster.de . + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +2 +worth noting that although each of the objective functionals ϕi(x(tf), tf), i = 1, . . . , r, in (P) +above constitutes the so-called Mayer form, other forms (Bolza and Lagrange) can easily be +converted into this form conveniently. Therefore, the general model in (P) caters for a wide +range of conflicting objectives; for instance, minimization of the energy, the terminal time, +the deviations from a reference state trajectory, or the uncertainty in measurements, to name +just a few. +Broadly speaking, the simultaneous or Pareto minimization in Problem (P) is the process of +finding a compromise solution, referred to as a Pareto minimum, where the value of some cost +cannot be improved (i.e., reduced) further, without making the value of some other cost worse +(i.e., higher). One typical example is the case when one wants to minimize simultaneously +the fuel expenditure of an airplane travelling from one given city to another and the time the +airplane takes for this travel: A shorter travel time often requires a higher fuel consumption. +The set of all such compromise or trade-off solutions form the Pareto set in the optimization +space, or the Pareto front in the value space. Pareto set and Pareto front are also commonly +referred to as the efficient set and the efficient front, respectively1. +The authors of this paper have studied in [27] the problem of constructing the Pareto +front of Problem (P) involving ODEs and constraints of general form. They discussed and +demonstrated that for the nonconvex optimal control problems like the one in Problem (P), +it is better to use the so-called weighted Chebyshev-norm scalarization (or just Chebyshev +scalarization) to guarantee that the whole Pareto front can be constructed, instead of using +the traditional weighted-sum scalarization, i.e., a convex combination of the objective func- +tionals. They discretized the scalarized problem directly and utilized large-scale optimization +software (the AMPL–Ipopt suite [23,46]) to find the Pareto fronts of two constrained optimal +control problems as examples, one involving tumour anti-angiogenesis and the other a fed- +batch bioreactor, by means of what they called a scalarize–discretize–then–optimize approach. +This approach is in contrast with the other existing discretize–scalarize–then–optimize ap- +proach (see e.g. [28–30,39]) which scalarizes the discretized problem rather than the original +(continuous-time) problem. +An additional benefit of the Chebyshev scalarization is also reported and illustrated in [27]: +One can compute the whole Pareto front by using only those weights of the objective func- +tionals within what they name as the essential subinterval of weights, instead of the whole +interval. Having to compute fewer Pareto solutions over a smaller number of grid points in a +subinterval is obviously a computational advantage. For further details and an extensive list +of references on multi-objective optimal control the reader is referred to [27]. Other relevant +studies on the topic in more recent years have appeared in [13,16]. +Apart from certain trivial or special cases, the Pareto front consists of infinitely many +solutions to choose from. When a discrete approximation of the front is found the number +of solutions to choose from is still relatively large since the approximate front is required to +be accurate enough. Making a decision as to which Pareto solution in the front is the most +suitable (to the needs of a practitioner) is often very hard for the following reasons. +• In the case of three or more objectives, the Pareto front might be difficult (if not +impossible) to view and to carry out a visual inspection (or “eyeballing”) for a decision. +• Even with two objectives, a visual inspection alone may not be enough to choose a +desirable solution. +• Constructing the whole Pareto front might just be too costly a thing to do numerically. +1These and other definitions will be given in more precise terms in Section 2. + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +3 +Motivated by these drawbacks, minimization of an additional (single) objective function over +the Pareto front has been of great interest to many researchers over the past decades—see, +for example, [2, 3, 5, 6, 14, 15, 25, 26, 31, 41, 47]. Despite this rich collection of works, to the +knowledge of the authors, it was not before the reference [7] that optimization over the Pareto +front was studied and a numerical method proposed for convex multi-objective optimal control +problems. In the current paper, we extend the works in [7,27] to nonconvex multi-objective +optimal control problems and propose a numerical method for carrying out optimization over +the Pareto front. +We set the optimal control problem as a bi-level optimization problem as in [7]: One has +to minimize a master objective functional subject to the minimization of a scalarization of +Problem (P). The lower level problem uses the Chebyshev scalarization as in [27], as opposed +to the weighted-sum scalarization in [7]. The problems we consider is in much more general +form in this paper: We consider nonconvex instead of convex problems compared to [7] and +we consider problems with time-delay instead of those without time delay compared to [27]. +Just to re-iterate, [27] only proposes a technique to construct the Pareto front, otherwise it +does not carry out optimization over the Pareto front. +As the optimization technique over the Pareto front, we propose the simplest possible +technique, namely the bisection method, over the set of weights for the bi-objective problem, +which are the parameters of the lower level optimal control problem. Even in this simplest +case, it is necessary to obtain derivatives with respect to the weight, for which we employ +difference approximations. +However, is it guaranteed that these derivatives exist? +This +question is answered by [32, 33, 36, 37] which studied the differentiability of a solution of a +parametric optimal control problem with respect to the parameters. We add a discussion +concerning these studies in the paper. +The main algorithm first finds the essential interval of weights over which the first step of +the bisection method is taken to find a new subinterval. Then the subsequent steps of the +bisection method are carried out until the stopping criterion is met. +The algorithm is illustrated on two challenging numerical examples: the Rayleigh problem, +which comes from an electric circuit, and a compartmental optimal control model for tuber- +culosis. In the first problem there are constraints on the control variables, and the second +problem not only has constraints on the two control variables but also time delays on both +the control and state variables. +The paper is organized as follows. In Section 2, we introduce the multi-objective optimal +control problem, discuss scalarization, introduce the problem of optimization over the Pareto +front, and elaborate on solution differentiability. In Section 3, we first define and explain +the essential interval of weights, and then introduce the bisection method for our problem +and provide the detailed algorithm. In Section 4, we illustrate the algorithm on two example +optimal control problems. Finally, in Section 5, we provide concluding remarks. +2 +Problem Statement and Preliminaries +2.1 +Multi-objective optimal control problem +We consider the following general multi-objective optimal control problem (similar to that +in [27] but made look slightly more general here) to underlie our study on minimization over +its Pareto front. The ensuing notation and definitions can also be found in [27] but given + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +4 +here for completeness as well as convenience. +(OCP) + + + + + + + + + + + + + + + + + + + + + + + + + +min +(ϕ1(x(tf), tf), . . . , ϕr(x(tf), tf)) +subject to +˙x(t) = f(x(t), u(t), t) , +for a.e. t ∈ [0, tf] , +θ(x(0), x(tf), tf) = 0 , +�θ(x(0), x(tf ), tf) ≤ 0 , +C(x(t), u(t), t) ≤ 0 , +for a.e. t ∈ [0, tf] , +S(x(t), t) ≤ 0 , +for all t ∈ [0, tf] , +where r ∈ {2, 3, 4, . . .} is fixed, the state variable x ∈ W 1,∞(0, tf; IRn), ˙x := dx/dt, and +the control variable u ∈ L∞(0, tf; IRm), with x(t) := (x1(t), . . . , xn(t)) ∈ IRn and u(t) := +(u1(t), . . . , um(t)) ∈ IRm. The functions ϕi : IRn × IR+ → IR, f : IRn × IRm × IR+ → IRn, +θ : IRn × IRn × IR+ → IRp1, �θ : IRn × IRn × IR+ → IRp2, C : IRn × IRm × IR+ → IRp3, and +S : IRn × IR+ → IRp4, are continuous in their arguments. In this problem, tf is either fixed +or free. Here, L∞(0, tf; IRm) corresponds to the space of essentially bounded, measurable +functions equipped with the essential supremum norm. Furthermore, W 1,∞(0, tf; Rn) is the +Sobolev space consisting of functions x : [0, tf] → Rn whose first derivatives lie in L∞. +Assume that ϕi(x(tf), tf) ≥ 0, for all i = 1, . . . , r. Note that this assumption can easily be +met by adding a large enough positive number to each objective functional. +Note that Problem (OCP) is in general a nonsmooth problem, because it does not require +differentiability of the objective functionals or the constraints. Moreover, although we have +stated Problem (OCP) in very broad terms, it can further be generalized, for example by +adding multi-point constraints, partial differential equations, time delays, etc. In other words, +although Problem (OCP) is already in a more general form than what one usually encounters +in applications, it can be further made look more general. +Of the possible extensions mentioned above, time delays in the state and control vari- +ables, for instance, can be incorporated into Problem (OCP) by replacing the ODEs in +Problem (OCP) with +˙x(t) = f(x(t), x(t − dx), u(t), u(t − du), t) , +for a.e. t ∈ [0, tf] , +(1a) +x(t) = x0(t) , +for all t ∈ [−dx, 0) , +(1b) +u(t) = u0(t) , +for all t ∈ [−du, 0) , +(1c) +where dx, du > 0 are the time delays in the state and control variables, respectively. +For technical convenience, let tf ≤ tmax +f +, where tmax +f +> 0 is some constant. Next, we define +the feasible set, X ⊂ W 1,∞(0, tf; IRn) × L∞(0, tf; IRm) × IR+, such that +X := {(x, u, tf) : ˙x(t) = f(x(t), x(t − dx), u(t), u(t − du), t) , +for a.e. t ∈ [0, tf] ; +x(t) = x0(t) , +for all t ∈ [−dx, 0]; +u(t) = u0(t) , +for all t ∈ [−du, 0) ; +θ(x(0), x(tf), tf) = 0 ; �θ(x(0), x(tf ), tf) ≤ 0 ; +C(x(t), u(t), t) ≤ 0 , for a.e. t ∈ [0, tf]; S(x(t), t) ≤ 0, for all t ∈ [0, tf]} . +Note that, for the case of time delays in the state and control variables, we have included +Equations (1a)–(1c) instead of the ODEs +˙x(t) = f(x(t), u(t), t) in the set X. +Define the vector of objective functionals, ϕ(x(tf), tf) := (ϕ1(x(tf), tf), . . . , ϕr(x(tf), tf)). +The triplet (x∗, u∗, t∗ +f) ∈ X is said to be a Pareto minimum if there exists no (x, u, tf) ∈ X + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +5 +such that ϕ(x(tf), tf) ̸= ϕ(x∗(t∗ +f), t∗ +f) and +ϕi(x(tf), tf)) ≤ ϕi(x∗(t∗ +f), t∗ +f) , +for all i = 1, . . . , r . +On the other hand, (x∗, u∗, t∗ +f) ∈ X is said to be a weak Pareto minimum if there exists no +(x, u, tf) ∈ X such that +ϕi(x(tf), tf)) < ϕi(x∗(t∗ +f), t∗ +f) , +for all i = 1, . . . , r . +The set of all the Pareto and weak Pareto minima is said to be the Pareto set. On the other +hand, the set of all vectors of objective functional values at the Pareto and weak Pareto min- +ima is said to be the Pareto front (or the efficient set) of Problem (OCP) in the r-dimensional +objective value, or outcome, space. Note that the coordinates of a point in the Pareto front +are simply ϕi(x∗(t∗ +f), t∗ +f), i = 1, . . . , r. Obviously, when r = 2 the Pareto front is in general a +curve; and when r = 3 the Pareto front is in general a surface. +2.2 +Scalarization +In [27], to compute a solution of Problem (OCP), the following single-objective problem (Pw), +i.e., scalarization, was employed. +(Pw) +min +(x,u,tf)∈X max{w1 ϕ1(x(tf), tf), . . . , wr ϕr(x(tf), tf)} , +where wi, i = 1, . . . , r, are referred to as weights, with the vector of weights w defined +as w := (w1, . . . , wr) ∈ IRr, such that �r +i=1 wi = 1. +Problem (Pw) is referred to as the +weighted Chebyshev problem (or Chebyshev scalarization) because of the weighted Chebyshev +norm, maxi |wi ϕi(x(tf), tf)| = maxi wi ϕi(x(tf), tf), appearing in the objective. This type of +scalarization is typically used for nonconvex multi-objective finite-dimensional optimization +problems, as opposed to the weighted sum scalarization which is effective for convex problems +but not the nonconvex ones—see, for example, [38]. +Define the set of weights +Y := +� +w ∈ IRr | +r +� +i=1 +wi = 1 +� +. +The following theorem was originally presented in [27, Theorem 1] for the case when there +was no delay in the state and control variables. It still holds with the set X modified with +the delayed state equations. +Theorem 1 (Bijection between sets of weights and Pareto minima [27]) The triplet +(x∗, u∗, t∗ +f) is a weak Pareto minimum of (OCP) if, and only if, (x∗, u∗, t∗ +f) is a solution of +(Pw) for some w1, . . . , wr > 0. +Remark 1 Suppose that Z ⊂ X denotes the Pareto set, namely the set of all Pareto minima +of (OCP). Then Theorem 1 establishes that there is a bijection between the set of weights +Y and the Pareto set Z. This implies that by solving (Pw) for all w ∈ Y , one can obtain +the whole Pareto set Z and in turn get the Pareto front. With numerical computations on +the other hand, one would of course carry out some discretization of the weight space Y and +typically get a discrete approximation of the Pareto front. The bijection between Y and Z +will also help us devise our algorithm for optimization over the Pareto front. +✷ + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +6 +An ideal cost ϕ∗ +i , i = 1, . . . , r, associated with Problem (Pw) is the optimal value of the +optimal control problem, +min +(x,u,tf )∈X ϕi(x(tf), tf) . +(2) +Let (x, u, tf) be a minimizer of the single-objective problem in (2). Then ϕ∗ +i := ϕi(x(tf), tf) +and we also define ϕj := ϕj(x(tf), tf), for j ̸= i and j = 1, . . . , r. +In the case when ϕ∗ +i is negative, one can simply add a large enough positive number to the +ith objective, to make the objective positive. In general, it is useful to add a positive number +to each objective in order to obtain an even spread of the Pareto points approximating the +Pareto front – see for example [21] for further discussion and geometric illustration. To serve +this purpose, it is common practice to define the so-called utopian objective values. +A utopian objective vector associated with Problem (OCP) is given as β∗ := (β∗ +1, . . . , β∗ +r), +with β∗ +i := ϕ∗ +i − ηi and ηi > 0 for all i = 1, . . . , r. Problem (Pw) can then be equivalently +written as +min +(x,u,tf)∈X max{w1 (ϕ1(x(tf), tf) − β∗ +1), . . . , wr (ϕr(x(tf), tf) − β∗ +r)} . +In the case when the objective functionals and the constraints in Problem (OCP) are +differentiable in their arguments, it is worth reformulating Problem (Pw) using a standard +technique from mathematical programming in the following (smooth) form. +(OCPw) + + + + + + + + + + + + + +min +α≥0 +(x,u,tf )∈X +α +subject to +w1 (ϕ1(x(tf), tf) − β∗ +1) ≤ α , +... +wr (ϕr(x(tf), tf) − β∗ +r) ≤ α . +Problem (OCPw) is referred to as goal attainment method [38], as well as Pascoletti-Serafini +scalarization [22]. We will solve Problem (OCPw) in an algorithm we present in the next +section, for the two examples we want to study. +We re-iterate that the “popular” weighted-sum scalarization, given below, fails to generate +the “nonconvex parts” of a Pareto front. +(Pws) +min +(x,u,tf)∈X +r +� +i=1 +wi ϕi(x(tf), tf) . +This deficiency is illustrated with a multi-objective optimal control problem, for example, in +the fed-batch bioreactor problem in [27]. +2.3 +Optimization over the Pareto front +The main task in this paper is to devise a numerical algorithm for solving the problem of +decision making as to which Pareto point should be chosen. +This obviously depends on +the criterion a decision maker uses in making his/her choice. As pointed in Remark 1, the +whole Pareto front can be parameterized in terms of the vector of weights w. Therefore, +Problem (Pw), or equivalently (OCPw), can be regarded as a parametric optimal control +problem, and it also makes sense to express the decision maker’s objective as the minimization +of a function of w. + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +7 +Before going ahead with the statement of this problem, we re-write the variables of the +optimal control problem, with a slight abuse of notation, as xw(t) := x(t, w), uw(t) := u(t, w), +and tw +f := tf(w) to emphasize their dependence on the vector of weights w. +We call the decision maker’s objective function the master objective function, expressed by +ϕ0(xw, uw, tw +f ). With the weight vector w of the scalarization treated now as a variable, the +problem of optimization over the Pareto front reduces to the problem of finding an optimal +weight w∗. Then the corresponding Pareto minimum is a solution of Problem (OCPw∗). +The problem of optimizing a master objective function over the Pareto front of (OCP) +with r ≥ 2 objectives is nothing but a bilevel programming problem and can be written as +(OPF) + + + + + + + + + + + + + + + + + + + +min +w∈Y +ϕ0(xw, uw, tw +f ) +subject to +min +α≥0 +(x,u,tf )∈X +α +subject to w1 (ϕ1(x(tf, w), tf) − β∗ +1) ≤ α , +... +wr (ϕr(x(tf, w), tf) − β∗ +r) ≤ α . +Remark 2 The lower-level problem in (OPF) for some given w is simply Problem (OCPw). +A solution of (OCPw) is nothing but a point in the Pareto set Z of (OCP) and is described +by the triplet Zw := (x∗(t, w), u∗(t, w), t∗ +f(w)). Then the (whole) Pareto set can be expressed +as Z = ∪w∈Y Zw. Now Problem (OPF) can equivalently be written as +� +min +w∈Y +ϕ0(xw, uw, tw +f ) +subject to +(xw, uw, tw +f ) ∈ Zw . +We note that the optimization variable of the upper-level problem is the “unknown” param- +eter w. If the solution (x∗(t, w), u∗(t, w), t∗ +f(w)) of Problem (OCPw) is differentiable in the +parameter w, then powerful differentiable optimization techniques can be employed in solving +Problem (OPF) (or in a more concise form the above problem). This is what was done in [7] +for convex multi-objective optimal control problems. In this paper, we are extending the +work in [7] to the nonconvex setting by also incorporating the Chebyshev scalarization and +the concept of essential interval of weights given in [27]. +✷ +2.4 +Solution differentiability +We briefly review results on solution differentiability or C1-sensitivity of solutions to the +following parametric optimal control problems depending on a parameter p ∈ P, where P is +a Banach space: +(OCP(p)) + + + + + + + + + + + + + + + + + +min x,u,p g(x(tf), tf, p) +subject to ˙x(t) = �f(x(t), u(t), p) , +for a.e. t ∈ [0, tf] , +ψ(x(0), x(tf), tf, p) = 0 , +˜ψ(x(0), x(tf), tf, p) ≤ 0 , +˜C(x(t), u(t), p) ≤ 0 , +for a.e. t ∈ [0, tf] , +˜S(x(t), p) ≤ 0 , +for a.e. t ∈ [0, tf] . +We note that problem (OCPw) is a special case of the parametric problem (OCP(p)) by +simply taking the parameter as the weight, p = w, which then appears only in the terminal +inequality constraints. The problem (OCP(p0)) corresponding to a reference parameter p0 +is considered as the nominal or unperturbed problem. It is assumed that a local solution + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +8 +(x0, u0) of the reference solution exists. Let p be a parameter in a neighbourhood of the +nominal parameter p0 and denote the solution to (OCP(p)) by (x(t, p), u(t, p)). Dontchev +and Hager [17] gave conditions under which the mapping p �→ (x(·, p), u(·, p)) is Lipschitz. +Malanowski and Maurer [32, 33] and Maurer and Pesch [36, 37] investigated the solution +differentiability or C1-sensitivity of the optimal solution. +The authors derived conditions +such that an optimal solution (x(·, p), u(·, p)) of the perturbed control problem OCP(p) exists +for all parameters p in a neighborhood of p0 and, moreover, the solution (x(t, p), u(t, p)) is +a C1 function with respect to both arguments (t, p). In broad descriptions, these conditions +include certain smoothness of the functions in Problem (OCP1), satisfaction of the strict +Legendre–Clebsch condition, uniqueness of the optimal control minimizing the Hamiltonian, +nonsingularity of the Jacobian of an associated boundary-value problem, and boundedness +of the symmetric solution of an associated Riccati ODE. +Fixing an increment d ∈ P, the differentials +zd(t, p0) = ∂x +∂p(t, p0)d, +vd(t, p0) = ∂u +∂p(t, p0)d, +satisfy a linear boundary value problem that contains only information obtained in the process +of computing the unperturbed solution. The computations of these sensitivity differentials +can also be performed by discretization methods applied to the parametric optimal control +problem; see B¨uskens [11] and B¨uskens and Maurer [12]. The sensitivity differentials can be +conveniently used in the minimization of a master function defined on the Pareto front; see +Section 2.3. +The above mentioned conditions for showing solution differentiability exclude optimal con- +trol problems with control appearing linearly, since for this class of problems the strict +Legendre-Clebsch condition does not hold. Here, optimal controls are combinations of bang- +bang and singular arcs. In case of finitely many switching times and junction times with +the boundary of a mixed control-state constraint or a pure state constraint, one can set up +a finite-dimensional optimization problem, the Induced Optimization Problem, where the +switching and junction times are optimized directly; see Maurer et al. [34] and Osmolovskii +and Maurer [40]. +If second-order sufficient conditions hold for the Induced Optimization +Problem (see [40]), one immediately obtains the result that the switching and junction times +locally are differentiable functions of the parameter p. +To our knowledge extensions of these results on solution differentiability to optimal control +problems with control and state delays can not be found in the literature. +3 +An Algorithm For Optimization Over the Pareto Front +As discussed in Section 2.4, the results [36, Theorem 3.1] and [37, Theorem 5.1] lay the ground +for devising and implementing numerical methods for solving Problem (OPF). Bonnel and +Kaya propose in [7] a barrier method for convex bi-objective optimal control problems with +pure control constraints. +Their method relies on twice continuous differentiability of the +solution (class C2) in the weight w, using the result in [36, Theorem 3.1]. +In this paper, we propose a bisection method also for the case of two objectives, which +relies on the solution of Problem (OCPw) being of class C1 w.r.t. the weight w, and thus +taking the result in [37, Theorem 5.1] as a basis. +Although a mathematical justification +of the applicability of our proposed method, i.e., solution differentiability, is given only for +Problem (OCPw), the working of the method will also be illustrated on problems of more +general class as in Problem (OCPw). + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +9 +PSfrag replacements +Pareto front +ϕ1 +ϕ2 +(ϕ∗ +1, ϕ2) +(ϕ1, ϕ∗ +2) +wf (ϕ1 − β∗ +1) = (1 − wf) (ϕ2 − β∗ +2) +w0 (ϕ1 − β∗ +1) = (1 − w0) (ϕ2 − β∗ +2) +(β∗ +1, β∗ +2) +Figure 1: Determination of the essential subinterval of weights [w0, wf] [27]. +In the scalarized problem (OCPw) with two objectives (r = 2), by choosing w1 = w, and +w2 = 1 − w, where w ∈ [0, 1], one can simply consider the single parameter w. +3.1 +Essential interval of weights +With the Chebyshev scalarization, it would usually be enough for the weight w to take values +over a (smaller) subinterval [w0, wf] ⊂ [0, 1], with w0 > 0 and wf < 1, for the generation of +the whole front. Figure 3.1 illustrates the geometry to compute the subinterval end-points, +w0 and wf. In the illustration, the points (ϕ∗ +1, ϕ2) and (ϕ1, ϕ∗ +2) represent the boundary of +the Pareto front. The equations of the “rays” which emanate from the utopia point (β∗ +1, β∗ +2) +and pass through the boundary points are also shown. By substituting the boundary values +of the Pareto curve into the respective equations, and solving each equation for w0 and wf +one simply gets +w0 = +(ϕ∗ +2 − β∗ +2) +(ϕ1 − β∗ +1) + (ϕ∗ +2 − β∗ +2) +and +wf = +(ϕ2 − β∗ +2) +(ϕ∗ +1 − β∗ +1) + (ϕ2 − β∗ +2) . +(3) +From the geometry depicted in Figure 3.1, as also discussed in [27], one can deduce that +with every w ∈ [0, w0] the solution of (OCPw) will yield the same boundary point (ϕ1, ϕ∗ +2) +on the Pareto front. Likewise with every w ∈ [wf, 1] the same boundary point (ϕ∗ +1, ϕ2) is +generated. This observation justifies the avoidance of the weights w ∈ [0, w0) ∪ (wf, 1] in +order not to keep getting the boundary points of the Pareto front, as otherwise one would +end up wasting valuable computational effort and time. +As a result of the above argument, the bisection method, implemented in the algorithm +described in the next section, starts with the essential interval [w0, wf] rather than [0, 1]. It +is worth re-iterating that our main concern here, unlike in [27], is not really to construct the +Pareto front, but rather do a search (in this case using the bisection method) over the Pareto +front, at the same time avoiding the task of constructing the front, so as to find in some sense +the best solution point in the Pareto front. + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +10 +3.2 +Bisection method for solving Problem (OPF) +The problem of finding a best point in the Pareto front/set has now been transformed into +a problem of finding best w, by virtue of the surjection from the set of weights to the set of +Pareto minima furnished by Theorem 1. This has resulted in Problem (OPF) and its concise +form: Find some weight w ∈ [w0, wf] such that the master objective function ϕ0(xw, uw, tw +f ) is +minimized, where (xw, uw, tw +f ) is found by solving (OCPw) for that w. For a simpler setting, it +is helpful to define a function F : [0, 1] → IR+ representing the function we want to minimize +over the Pareto front: +F(w) := ϕ0(xw, uw, tw +f ) , +(4) +such that (xw, uw, tw +f ) solves (OCPw). In other words, an evaluation of the function F(·) at +w requires the solution of Problem (OCPw) with that w. +Problem (OPF) can now be re-written in an even more concise form as +min +w∈[w0,wf] F(w) , +(5) +where F(·) is evaluated as in (4). In [7], a log-barrier method is proposed and implemented +to solve (5), with an underlying convex and smooth optimal control problem with no state +constraints for which the solution can be assumed to be of class C2, and so Newton-like +methods are used with heuristic barrier parameter updates. For the general form we have in +Problem (OCP), which is nonconvex and has state constraints, we assume that the solution +is of class C1. As elaborated in Section 2.4, under certain regularity conditions which can +in many cases be checked, this assumption is guaranteed to hold. Therefore we apply the +bisection method [10] as an effective and simple approach to solving (5) in the case of this +paper. +Albeit elementary and standard, a statement of the optimality conditions in the fact below +will be useful in formulating a computational algorithm later in this section. +Fact 1 Consider the minimization problem in (5) with F(·) of class C1. +(a) The interior point w∗ ∈ (w0, wf) is a strict local minimizer of F(·) if, and only if, +F ′(w∗) = 0 , +(6) +and, for arbitrarily small ε > 0, +F ′(w∗ − ε) < 0 +and +F ′(w∗ + ε) > 0 . +(7) +(b) The end point w0 (resp. wf) is a strict local minimizer of F(·) if, and only if, either +(i) F ′(w0) > 0 (resp. F ′(wf) < 0) or +(ii) F ′(w0) = 0 (resp. F ′(wf) = 0) and, for arbitrarily small ε > 0, F ′(w0 + ε) > 0 +(resp. F ′(wf − ε) < 0). +Remark 3 (Three Cases for the End Points of [w0, wf]) We will apply the bisection +method starting with the essential interval [w0, wf]. Before introducing the pertaining algo- +rithm, we consider below the cases for the end points of this interval. +Case I. F ′(w0)F ′(wf) < 0 : Since F ′(·) is assumed to be continuous the bisection method +is guaranteed to find a numerical solution to (6) by the intermediate value theorem. +Condition (7) needs to be check to see if w∗ is a strict local minimizer. + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +11 +Case II. F ′(w0)F ′(wf) > 0 : By the conditions in Fact 1(b)(i), at least one of w0 and wf is +a strict local minimizer. +Case III. F ′(w0)F ′(wf) = 0 : If one of the inequalities in Fact 1(b)(ii) is satisfied, then w0 +or wf is a strict local minimizer. It is possible that both w0 and wf are, or only one or +neither is, a local minimizer. +In Case I, the bisection method starts with the interval [w0, wf] and terminates with an +approximate solution in the interior of the interval. In Case II, a local minimum is found +immediately, and so in principle there is no need to do a further search. In Case III, however, +the conclusion might be that neither w0 nor wf is a strict local minimizer, in which case it +would be necessary to start the bisection method with a subinterval of [w0, wf], and consider +Cases I–III again. +✷ +Remark 4 In any of the scenarios elaborated in Remark 3, consideration of another subin- +terval of [w0, wf] might as well yield a better (lower-value) solution, since the problem is +nonconvex and we can only hope to get a locally optimal solution. In our approach here, +however, we do not endeavour to obtain a global minimum. As a result of our discussion +in Remark 3, we will consider only Case I, which clearly prompts us to use the bisection +method directly. As suggested above, in the event of Case III not yielding a solution, the +new subinterval could be chosen in such a way that one would fall into Case I. +✷ +The derivative of F(·) is defined at the end points of the interval [w0, wf] as one-sided +limits, +F ′(w0) := lim +δ→0+ +F(w0 + δ) − F(w0) +δ +and +F ′(wf) := lim +δ→0− +F(wf + δ) − F(wf) +δ +, +and in the interior, i.e., for w ∈ (w0, wf), as +F ′(w) := lim +δ→0 +F(w + δ) − F(w) +δ +, +where F(·) is evaluated as in (4). In computations, we will use the forward, and backward, +finite difference approximations of F ′(·). Namely, for some small δ > 0, we will set +F ′(w) ≈ + + + + + + + +F(w + δ) − F(w) +δ +, +if w ∈ [w0, wf − δ) , +F(w) − F(w − δ) +δ +, +if w ∈ [wf − δ, wf] . +(8) +The step δ in the difference approximation formula (8) is small for an accurate estimation of +the derivative but not too small in order not to divide one very small number by another and +cause numerical instabilities. +In what follows we provide an algorithm to solve Problem (OPF). The algorithm first finds +the essential interval [w0, wf], computes the signs of F ′(w0) and F ′(wf) and checks the cases +I–III in Remark 3, and then if F ′(w0)F ′(wf) < 0 it uses the bisection method, to find a +numerical solution to Problem (OPF). +Algorithm 1 +Step 0.0 (Initialization) +Choose utopia parameters, η1, η2 > 0, a small numerical differ- +entiation step δ > 0, a stopping tolerance ǫ > 0, and a maximum number of iterations +kmax . Set k := 1. + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +12 +Step 0.1 (Boundary points of the front) Solve (2) to get (xi, ui, ti +f), i = 1, 2. Set +(ϕ∗ +1, ϕ2) := (ϕ1(x1(t1 +f), t1 +f), ϕ2(x1(t1 +f), t1 +f)), (ϕ1, ϕ∗ +2) := (ϕ1(x2(t2 +f), t2 +f), ϕ2(x2(t2 +f), t2 +f)) . +Step 0.2 (Utopia point) Set β∗ := (β∗ +1, β∗ +2) with β∗ +i := ϕ∗ +i − ηi, i = 1, 2. +Step 0.3 (Essential interval) Determine the subinterval [w0, wf] ⊂ [0, 1] using (3). +Step 0.4 (Signs at end points) Compute F ′(w0) and F ′(wf) using (8), with F(·) evaluated +as in (4). +• If Fact 1(b)(i) or (ii) is satisfied then w∗ = w0 or w∗ = wf appropriately; STOP. +• If F ′(w0)F ′(wf) = 0 and neither of the inequalities in Fact 1(b)(ii) is satisfied then +declare “Algorithm failed. Change the interval [w0, wf].” and STOP. +Let a := w0 and b := wf. +Step k.1 (Bisection) Find the midpoint c := a + (b − a)/2 of the interval [a, b]. +Step k.2 (Stopping criterion) Compute F ′(c) using (8), with F(·) evaluated as in (4). +• If F ′(c) = 0 or (b − a)/2 < ǫ then set w∗ = c and STOP. +• If k = kmax then declare “Maximum number of iterations exceeded.” and STOP. +Step k.3 (New subinterval) +Set k := k + 1 . If F ′(a)F ′(c) > 0 then update the subinterval +as [a, b] := [c, b]; otherwise, set [a, b] := [a, c]. GO TO Step k.1. +4 +Numerical Examples +In this section, we illustrate the working of Algorithm 1 on two optimal control problems, +one involving an electric circuit in Section 4.1 and the other a tuberculosis (TB) epidemic in +Section 4.2. +In computations, we use direct discretization of optimal control problems for which con- +vergence theory has been an active topic of research in the literature (see for example +[1,4,18–20,42], and see [27] for additional references and discussion). +We employ the scalarize–discretize–then–optimize approach that was previously used in [27]. +Under this approach, one first scalarizes the multi-objective problem in the infinite-dimensional +space, and then discretizes the scalarized problem directly and applies a usually large-scale +finite-dimensional optimization method to find a discrete approximate solution of the scalar- +ized problem. By the existing theory of discretization mentioned above, under certain as- +sumptions, the discrete approximate solution converges to a solution of the continuous-time +scalarization of the original problem, yielding a Pareto minimum of the original problem. +When possible, we will also check a posteriori to see if the necessary optimality conditions +are satisfied by an accurate-enough numerical solution. +In Step 0.4 of Algorithm 1, a direct discretization of Problem (OCPw), for example em- +ploying a Runge–Kutta scheme, such as Euler’s method or the Trapezoidal rule, is solved by +using Ipopt, version 3.12.13, four times. In Step k.2, Problem (OCPw) is solved in a similar +way two times. Ipopt is a popular optimization software based on an interior point method; +see [46]. We use AMPL [23] as an optimization modelling language, which employs Ipopt as +a solver. + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +13 +4.1 +Example: Tunnel-diode oscillator (Rayleigh problem) +The tunnel-diode oscillator problem, also referred to as the Rayleigh problem in the literature, +involves dynamics represented by the following differential equations. +˙x1(t) = x2(t) , +˙x2(t) = −x1(t) + x2(t) (1.4 − 0.14 x2 +2(t)) + 4 u(t) , +for a.e. t ∈ [0, tf] , +where the state variable x1(t) denotes electric current, and the control variable u(t) stands +for a suitable transformation of the voltage at a generator, both at time t ∈ [0, tf]—see [35] +for a detailed exposition of the problem. In this particular instance of the problem, the initial +and terminal values of the state variables are specified as +(x1(0), x2(0)) = (−5, −5) +and +(x1(tf), x2(tf)) = (0, 0) , +and the dynamics are subject to constraints on the control variable such that +−1 ≤ u(t) ≤ 1 , +for a.e. t ∈ [0, tf] . +The optimal control problem is posed as a bi-objective problem with +min +� +tf , +� tf +0 +� +x2 +1(t) + u2(t) +� +dt +� +, +where the competing objectives are the minimization of the final time tf and the minimization +of the sum of the square L2-norms, or in some sense the magnitudes, of the current and the +generator voltage. Define a new state variable x3 such that +˙x3(t) = x2 +1(t) + u2(t) , +for a.e. t ∈ [0, tf] , +x3(0) = 0 . +Then the two objective functionals as in Problem (OCP), or Problem (OCPw), can be ex- +pressed as +ϕ1(x(tf), tf) = tf +and +ϕ2(x(tf), tf) = x3(tf) . +As we have stated above, the bi-objective Rayleigh problem is in the same form as Prob- +lem (OCP) and, in particular, Problem (OCPw). The decision maker’s objective for this +problem will be to minimize a weighted distance to the origin of the value space. We choose +ϕ0(xw, uw, tw +f ) := 100 ϕ2 +1(xw(tf), tw +f ) + ϕ2 +2(xw(tf), tw +f ) , +where the scaling multiplier 100 is used to make the orders of magnitudes of ϕ1 and ϕ2 the +same. We aim to solve Problem (OPF), to determine a scalar w ∈ (0, 1) with w1 := w and +w2 := 1 − w that results in the best Pareto solution in the sense that ϕ0(·, ·, ·) is minimized, +subject to the solution of Problem (OCPw). +In [35], Maurer and Oberle numerically illustrate that an optimal solution does not exist +for the single objective problem minimizing the quadratic functional ϕ2(x(tf), tf), in that tf +tends to infinity. They carry out a numerical test for checking the second-order sufficient +conditions (SSC) of optimality and show that the test fails to confirm the SSC. Therefore, +we will impose a bound on the terminal time, namely set tf ≤ 5. On the other hand, they +illustrate also in [35] that for certain instances of the weighted-sum problem, the SSC of +optimality are satisfied. + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +14 +Problem (OCPw) can now explicitly be written for the Rayleigh problem as + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +min +α≥0 +x(·),u(·),tf +α +subject to +˙x1(t) = x2(t) , +x1(0) = −5 , x1(tf) = 0 , +˙x2(t) = −x1(t) + x2(t) (1.4 − 0.14 x2 +2(t)) + 4 u(t) , x2(0) = −5 , x2(tf) = 0 , +˙x3(t) = x2 +1(t) + u2(t) , +x3(0) = 0 , +−1 ≤ u(t) ≤ 1 , +for a.e. t ∈ [0, tf] , +tf ≤ 5 , +w (tf − β∗ +1) ≤ α , +(1 − w) (x3(tf) − β∗ +2) ≤ α . +The Hamiltonian H : IR3 × IR × IR3 → IR for this problem simply is +H(x, u, λ) := λ1x2 + λ2 +� +(−x1 + x2 (1.4 − 0.14 x2 +2) + 4u +� ++ λ3(x2 +1 + u2) , +where λ(t) := (λ1(t), λ2(t), λ3(t)) ∈ IR3 is referred to as the adjoint variable vector. Using +the convenient notation H[t] := H(x(t), u(t), λ(t)), suppose that +˙λ1(t) := −Hx1[t] = λ2(t) − 2λ3(t)x1(t) , +(9a) +˙λ2(t) := −Hx2[t] = −λ1(t) − λ2(t)(1.4 − 0.42x2 +2(t)) , +(9b) +˙λ3(t) := −Hx3[t] = 0 , +(9c) +for all t ∈ [0, tf], with certain transversality conditions as required by the maximum principle. +In (9a)–(9c), Hxi := ∂H/∂xi, i = 1, 2, 3. We will not go into the details of these (boundary) +conditions here. However we note that λ3(t) = λ3, a constant, for all t ∈ [0, tf]. Then the +maximum principle states that if (x, u, tf) is an optimal solution triplet then there exists +a continuous function λ(·) satisfying (9a)–(9c), along with certain transversality conditions, +such that λ(t) ̸= 0, for all t ∈ [0, tf], and +u(t) = argmin +v∈[−1,1] +H(x(t), v, λ(t)) = argmin +v∈[−1,1] +� +4λ2(t)v + λ3(t)v2� +. +(10) +for a.e. t ∈ [0, tf]. If w = 1, then the problem is a single-objective one, referred to as a +time-optimal control problem, and the condition (10) reduces to +u(t) = argmin +v∈[−1,1] +λ2(t)v , +resulting in +uw(t) = + + + + + +1 , +if λw +2 (t) < 0 , +−1 , +if λw +2 (t) > 0 , +undetermined , +if λw +2 (t) = 0 , +(11) +for a.e. +t ∈ [0, tf]. By the discussion given in Section 3.1 (also see [27]), uw(t) given in +(11) is the same for all w ∈ [wf, 1]. +Recall that if one does not have λw +2 (t) = 0 for all +[t′, t′′] ⊂ [0, tf], where t′ < t′′, then uw(t) in (11) is referred to as optimal control of bang–bang +type. We assume (and therefore will numerically double-check) that the optimal control for +the particular instance of the problem is of bang–bang type. +The optimality condition (10) can be shown to yield, for any given w ∈ [w0, wf), +uw(t) = + + + + + + + +1 , +if 2λw +2 (t) < −λ +w +3 , +−2λw +2 (t)/λ +w +3 , +if +− λ +w +3 ≤ 2λw +2 (t) ≤ λ +w +3 , +−1 , +if 2λw +2 (t) > λ +w +3 , +(12) + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +15 +for all t ∈ [0, tf], provided λ +w +3 ̸= 0. Again by virtue of the discussion in Section 3.1, uw(t) in +(12) is the same for all w ∈ [0, w0]. We define the switching function as +σw(t) := +� +2 λw +2 (t)/λ +w +3 , +if 0 ≤ w < wf , +16 λw +2 (t) , +if wf ≤ w ≤ 1 . +(13) +The constant coefficients 2 and 16 above are used for scaling purposes, so that the graphs in +Figure 2(b) can be viewed more easily. Now, using (13), we can summarize and combine the +expressions for the optimal control in (11) and (12) as follows. +uw(t) = + + + + + + + + + + + + + + + + + + + + + + +1 , +if σw(t) < −1 +−2λw +2 (t)/λ +w +3 , +if +− 1 ≤ σw(t) ≤ 1 , +−1 , +if σw(t) > 1 . + + + + + +, +if 0 ≤ w < wf , +� +1 , +if σw(t) < 0 , +−1 , +if σw(t) > 0 . +� +, +if wf ≤ w ≤ 1 . +(14) +As to why σw(·) is referred to as the switching function should now be more clear from (14): +the value of σw(·) determines when to switch from one case of the control function uw(·) to +another. +For Problem (OCPw) written for the Rayleigh problem above, we have chosen the utopia +vector as (β∗ +1, β∗ +2) = (0, 0), since ϕi(x(tf), tf) > 0, for i = 1, 2. +Figure 2(a) depicts the +Pareto front for the instance of the multi-objective Rayleigh problem we consider here. It +also displays the iterations of Algorithm 1. The Rayleigh problem is discretized using the +trapezoidal rule, the number of grid points is set to be N = 5000, and the Ipopt’s tolerance +to 10−10, so as to get solutions for w accurate at least up to four decimal places (dp). +The essential interval is found to be [w0, wf] = [0.8994, 0.9269], with +(ϕw0 +1 , ϕw0 +2 ) = (5.000, 44.71) +and +(ϕwf +1 , ϕwf +2 ) = (3.668, 46.50) , +correct to four significant figures, where ϕw +i := ϕi(xw(tf), tw +f ), i = 1, 2, with w = w0 or wf, or +as will be the case below, w = w∗. Optimization over the Pareto front results in w∗ = 0.9247, +after 14 iterations of Algorithm 1, yielding +ϕw∗ +0 += 58.71 +and +(ϕw∗ +1 , ϕw∗ +2 ) = (3.709, 45.51) . +If there is a need to save the computational resources further, the algorithm can be asked to +yield a less accurate result, say correct to three dp, which then yields w∗ = 0.925 in eight +iterations with (ϕw∗ +1 , ϕw∗ +2 ) = (3.71, 45.5). In Figure 2(a) only five iterations are displayed +(labels 1–5 appearing to the right of each iteration) for clarity in viewing. +The Pareto +(master) solution with w = w∗ is represented by a square. +The numerical Pareto-optimal state and control variable solutions are presented in Fig- +ures 2(c)–(d) for w = w0, w∗, wf. One of the boundary Pareto-optimal solutions is shown +using solid (blue) curves for w = w0, which is the same solution for all w ∈ [0, w0], as pre- +viously discussed in Section 3.1. +On the other hand, the other boundary Pareto-optimal +solution for w = wf, which holds for all w ∈ [wf, 1], is shown using dashed (green) curves. +The latter is nothing but a time-optimal control solution for the Rayleigh problem (a solu- +tion with the smallest tf), resulting in a bang–bang type function with the sequence of values +{1, −1, 1}, namely with two switchings. The master Pareto solution is given for w = w∗ using +dashed-and-dotted (red) curves. +The switching function σw(·) plotted in Figure 2(b) by using (13) (recall that discrete +approximations of λw +2 (t) and λw +3 (t) can readily be obtained from AMPL) furnishes the means + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +16 +3.6 +3.8 +4 +4.2 +4.4 +4.6 +4.8 +5 +45 +45.5 +46 +46.5 + 1 + 2 + 3 + 4 + 5 +(a) Pareto front, and iterations of Algorithm 1: +Master solution is depicted by a (red) square and +iterates by (light blue) circles. +0 +1 +2 +3 +4 +5 +-3 +-2 +-1 +0 +1 +2 +3 +(b) Switching function as defined in (13). +-8 +-6 +-4 +-2 +0 +2 +-6 +-4 +-2 +0 +2 +4 +6 +PSfrag replacements +singular control switching curve +(c) Phase plane trajectories. +0 +1 +2 +3 +4 +5 +-1 +-0.5 +0 +0.5 +1 +(d) Control variable. +Figure 2: Rayleigh problem—Boundary Pareto solutions, corresponding to w0 = 0.8994 and +wf = 0.9269, are shown with (blue) solid curves and (green) dashed curves, respectively. Master +Pareto solution, corresponding to w∗ = 0.9247, is shown with dashed-and-dotted (red) curves. +to verify the optimality condition for uw(·) expressed in (14). It is evident from the dashed +(green) plot of the switching function that, for w ∈ [wf, 1], when σw(·) crosses the time +axis there is a jump (from 1 to −1 or vice versa) in the value of the corresponding uw(·) +plot. Likewise, for w ∈ [0, w0] and for w = w∗ ∈ [w0, wf), whenever σw(·) crosses one of the +lines σw(t) = 1 and σw(t) = −1 (shown by two black lines in Figure 2(b) for convenience) +the expression for the control function uw(·) switches from one case in (14) to another, as +required. +4.2 +Example: Compartmental model for tuberculosis +In 2020 and 2021, tuberculosis (TB) was the second leading cause of death from an infectious +disease worldwide after COVID-19 [44]. Active TB refers to disease that occurs in someone +infected with Mycobacterium tuberculosis. It is characterized by signs or symptoms of active +disease, or both, and is distinct from latent tuberculosis infection, which occurs without signs +or symptoms of active disease. Only individuals with active TB can transmit the infection. +Many people with active TB do not experience typical TB symptoms in the early stages of the + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +17 +disease. These individuals are unlikely to seek care early, and may not be properly diagnosed +when seeking care. Delays to diagnosis of active TB present a major obstacle to the control of +a TB epidemic, it may worsen the disease, increase the risk of death and enhance tuberculosis +transmission to the community. Both patient and the health system may be responsible for +the diagnosis delay. +We study the control model with control and state delays presented in Silva et al. [43]. In +this model, reinfection and post-exposure interventions for tuberculosis are considered. The +population is divided into five categories (compartments) (i.e., the control system has five +state variables): +S +: +susceptible individuals, +L1 +: +early latent individuals, recently infected (less than two years), +I +: +infectious individuals, who have active TB, +L2 +: +persistent latent individuals, +R +: +recovered individuals, +N +: +total population N = S + L1 + I + L2 + R , assumed constant. +The model has two control variables and three delays: +u1 +: +effort on early detection and treatment of recently infected individuals L1, +du1 +: +delay on the diagnosis of latent TB, and commencement of latent TB treatment, +u2 +: +chemotherapy or post-exposure vaccine to persistent latent individuals L2, +du2 +: +delay in the prophylactic treatment of persistent latent L2, +dI +: +delay in I, i.e., delay in diagnosis. +The dynamical system is given by + + + + + + + + + + + + + + + + + + + + + +˙S(t) = µN − β +N I(t)S(t) − µS(t), +˙L1(t) = β +N I(t) (S(t) + σL2(t) + σRR(t)) − (δ + τ1 + ǫ1u1(t − du1) + µ) L1(t), +˙I(t) = φ δ L1(t) + ωL2(t) + ωRR(t) − τ0I(t − dI) + µI(t), +˙L2(t) = (1 − φ)δL1(t) − σ β +N I(t)L2(t) − (ω + ǫ2u2(t − du2) + τ2 + µ)L2(t). +(15) +The recovered population is defined by +R(t) := N − S(t) − L1(t) − I(t) − L2(t) , +(16) +with N = 30000. The system and delay parameters in the model (15) along with their values +are listed in Table 1. In view of the delays the initial conditions and functions are: +S(0) = 76 N/120, L1(0) = 36 N/120, L2(0) = 2 N/120, R(0) = N/120, +I(t) = 5 N/120 +for −dI ≤ t ≤ 0, +uk(t) = 0 +for −duk ≤ t < 0, +(k = 1, 2). +(17) +The control constraints are given by +0 ≤ uk(t) ≤ 1 , +∀t ∈ [0, tf] , +(k = 1, 2). +(18) +We consider the following parametric objective functional with control weights a1, a2 ≥ 0: +tf +� +0 +(I(t) + L2(t) + a1u1(t) + a2u2(t)) dt . +(19) + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +18 +Symbol +Description +Value +β +Transmission coefficient +variable +µ +Death and birth rate +1/70 yr−1 +δ +Rate at which individuals leave L1 +12 yr−1 +φ +Proportion of individuals going to I +0.05 +ω +Endogenous reactivation rate for persistent latent infections +0.0002 yr−1 +ωR +Endogenous reactivation rate for treated individuals +0.00002 yr−1 +σ +Factor reducing the risk of infection as a result of acquired +immunity to a previous infection for L2 +0.25 +σR +Rate of exogenous reinfection of treated patients +0.25 +τ0 +Rate of recovery under treatment of active TB +2 yr−1 +τ1 +Rate of recovery under treatment of early latent individuals L1 +2 yr−1 +τ2 +Rate of recovery under treatment of persistent latent individuals L2 +1 yr−1 +N +Total population +30, 000 +ǫ1 +Efficacy of treatment of early latent L1 +0.5 +ǫ2 +Efficacy of treatment of persistent latent TB L2 +0.5 +tf +Total simulation duration +5 years +dI +delay in the diagnosis of I +0.1 years +du1 +delay in the diagnosis of early latent individuals L1 +0.2 years +du2 +delay in the prophylactic treatment of persistent latent individuals L2 +0.2 years +Table 1: Parameter values for the TB control model. +Depending on the priorities, the weights a1, a2 can be chosen in different ways (for example, +both can be chosen to be very small or very large) giving rise to competing objectives. Namely, +x5(tf) := +tf +� +0 +� +I(t) + L2(t) + a11 u1(t) + a12 u2(t) +� +dt , +x6(tf) := +tf +� +0 +� +I(t) + L2(t) + a21 u1(t) + a22 u2(t) +� +dt . +(20) +with control weights a11, a12, a21, a22 ≥ 0, constitute two competing objective functionals. +Both functionals are given in Lagrange form. The standard method to obtain an optimal +control problem of Bolza type is to introduce additional state variables x5 and x6 defined by +˙x5(t) = I(t) + L2(t) + a11 u1(t) + a12 u2(t) , +x5(0) = 0 , +˙x6(t) = I(t) + L2(t) + a21 u1(t) + a22 u2(t) , +x6(0) = 0 . +(21) +Denoting the (augmented) state vector by x(t) = (S(t), L1(t), I(t), L2(t), x5(t), x6(t)) ∈ R6 +and the control vector u(t) := (u1(t), u2(t)) ∈ R2, the two competing objectives in the general +problem (P) are given by +ϕ1(x(tf), tf) = x5(tf) =: F1(x, u) +and +ϕ2(x(tf), tf) = x6(tf) =: F2(x, u) , +where F1(x, u) and F2(x, u) denote the two functionals in Lagrange form. +The bi-objective TB problem is now in the same form as Problem (OCP) and, in particular, +Problem (OCPsd). The decision maker’s objective for this problem will be to minimize the +distance to the origin of the value space. We therefore choose +ϕ0(xw, uw, tw +f ) := ϕ2 +1(xw(tf), tw +f ) + ϕ2 +2(xw(tf), tw +f ) , + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +19 +Our aim is to solve Problem (OPF), to determine a scalar w ∈ (0, 1) with w1 := w and +w2 := 1 − w that results in the best Pareto solution in the sense that ϕ0(·, ·, ·) is minimized, +subject to the solution of Problem (OCPw). +Next we focus on the solution of Problem (OCPw): We aim to find a pair of functions +(x, u) ∈ W 1,∞([0, tf], R6) × L∞([0, tf], R2) that minimizes the parameter α subject to the +time-delayed dynamics (15) and the auxiliary dynamics (21), initial conditions (17), control +constraints (18) and auxiliary weighted inequalities involving ϕ1 and ϕ2. +We consider the necessary optimality conditions for the time-delayed optimal control +problem (OCPw); see G¨ollmann and Maurer [24], Vinter [45]. +For this purpose we in- +troduce the delayed state variable y3(t) = x3(t − dI) = I(t − dI) and delayed control +variables vk(t) = uk(t − duk), k = 1, 2. +Denoting the adjoint variable vector by λ(t) := +(λS(t), λL1(t), λI(t), λL2(t), λ5(t), λ6(t)) ∈ R6 the Hamiltonian or Pontryagin function is given +by +H(x, y3, λ, u1, v1, u2, v2) = λs (µN − β +N IS − µS) ++ λL1 ( β +N I (S + σL2 + σRR) − (δ + τ1 + ǫ1v1 + µ) L1) ++ λI ( φ δL1 + ωL2 + ωR R − τ0y3 + µI) ++ λL2 ((1 − φ)δL1 − σ β +N IL2 − (ω + ǫ2v2 + τ2 + µ)L2) ++ λ5 (I + L2 + a11u1 + a12u2) ++ λ6 (I + L2 + a21u1 + a22u2) , +(22) +where R is given as in (16). The Minimum Principle [24,45] yields the adjoint equations +˙λS(t) = −∂H +∂S [t], +˙λL1(t) = − ∂H +∂L1 +[t], +˙λL2(t) = − ∂H +∂L2 +[t], +˙λx5(t) = − ∂H +∂x5 +[t] = 0 , +˙λx6(t) = − ∂H +∂x6 +[t] = 0 , +and the advanced adjoint equation +˙λI(t) = −∂H +∂I [t] − χ[0,tf −dI](t)∂H +∂I [t + dI] , +where the argument [t] stands for evaluating all arguments at time t. We note that λw +5 (t) = λ +w +5 +and λw +6 (t) = λ +w +5 , constants, for any fixed w ∈ [0, 1]. In the last equation, the term χ[0,tf −dI](t) +denotes the characteristic function of the interval [0, tf − dI] at time t. The minimization of +the Hamiltonian with respect to the controls u1, u2 and delayed controls v1, v2 involves the +switching functions σk(t) for k = 1, 2: +σw +k (t) = ∂H +∂uk +[t] + χ[0,tf −duk](t)∂H +∂vk +[t + duk] += +� +a1kλ +w +5 + a2kλ +w +6 − ǫkλw +Lk(t + duk)Lw +k (t + duk) , if 0 ≤ t ≤ tf − duk , +a1kλ +w +5 + a2kλ +w +6 , +if tf − duk ≤ t ≤ tf . +(23) +As in the Rayleigh problem, the superscript “w” above denotes dependence on the scalariza- +tion parameter/weight w. Then the controls minimizing the Hamiltonian are characterized +by the switching conditions (control law) +uw +k (t) = +� 0 , +if σw +k (t) > 0 , +1 , +if σw +k (t) < 0 , +k = 1, 2. +(24) + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +20 +Figure 3: TB problem—Pareto front, and iterations of Algorithm 1: Master solution is depicted +by a (red) square and iterates by (light blue) circles.. +for all w ∈ [0, 1]. In particular, for positive weights a1 > 0, a2 > 0, the switching functions +(23) and the control law (24) imply +uw +k (t) = 0 +∀ tf − duk ≤ t ≤ tf , +for all w ∈ [0, 1]. +In what follows we choose the control weights as a11 = a12 = 10 (small) and a21 = a22 = +1000 (large) in the objective functionals ϕ1 and ϕ2. +For Problem (OCPw) written for the TB problem, we have chosen the utopia vector as +(β∗ +1, β∗ +2) = (0, 0). Figure 3 depicts the Pareto front for the TB problem we consider here. +The plot also displays the iterations of Algorithm 1. The TB problem is discretized using the +trapezoidal rule, the number of grid points is set to be N = 5000, and the Ipopt’s tolerance +to 10−10, so as to get solutions for w accurate at least up to four decimal places (dp). +The essential interval in this case is found to be [w0, wf] = [0.5251, 0.5709], with +(ϕw0 +1 , ϕw0 +2 ) = (28155, 31133) +and +(ϕwf +1 , ϕwf +2 ) = (26459, 35205) , +where ϕw +i := ϕi(xw(tf), tw +f ), i = 1, 2, with w = w0 or wf, or as will be the case below, w = w∗. +Optimization over the Pareto front results in w∗ = 0.5358, after 10 iterations of Algorithm 1, +yielding +ϕw∗ +0 += 41621 +and +(ϕw∗ +1 , ϕw∗ +2 ) = (27255, 31455) . +In Figure 3 only five iterations are displayed (labelled 1–5) for clarity in viewing. The Pareto +(master) solution with w = w∗ is represented by a square. +The numerical Pareto-optimal control variable solutions uw +1 (·) and uw +2 (·) are presented +in Figures 4(a)–(b) for w = w0, w∗, wf. +As with Rayleigh, one of the boundary Pareto- +optimal solutions is shown using solid (blue) curves for w = w0, the same solution for all +w ∈ [0, w0]. The other boundary Pareto-optimal solution for w = wf, which holds for all +w ∈ [wf, 1], is shown using dashed (green) curves. Both of the control solutions are of bang– +bang type (as required by (24)), with one switching (the number of switchings not dictated + +X10 +0 +3.5 +3.4 +P2 +3.35 +3 +0 +4 +2.75 +2.8 +P13.2 +1 +2 +3.1 +2.65 +2.7Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +21 +0 +1 +2 +3 +4 +5 +-0.5 +0 +0.5 +1 +(a) Control variable uw +1 (24) and scaled +switching function σw +1 (23) superposed. +0 +1 +2 +3 +4 +5 +-0.5 +0 +0.5 +1 +(b) Control variable uw +2 (24) and scaled +switching function σw +2 (23) superposed. +Figure 4: TB problem—Boundary Pareto solutions, corresponding to w0 = 0.5251 and wf = +0.5709, are shown with (blue) solid curves and (green) dashed curves, respectively. Master Pareto +solution, corresponding to w∗ = 0.5358, is shown with dashed-and-dotted (red) curves. +Scalarization +Functional values +Switching times +Terminal state values +weight w +xw +5 (tf) +xw +6 (tf) +tw +s1 +tw +s2 +Sw(tf) +Lw +1 (tf) +Iw(tf) +Lw +2 (tf) +Rw(tf) +w0 = 0.5251 : +28155 +31133 +0.145 +2.864 +1193.1 +28.2 +13.3 +864.0 +27901.4 +w∗ = 0.5358: +27255 +31455 +0.809 +3.439 +1205.8 +27.5 +13.0 +747.6 +28006.1 +wf = 0.5709 : +26459 +35205 +4.083 +4.752 +1238.2 +23.8 +11.2 +419.3 +28307.5 +Table 2: TB problem. +by (24) alone). The master Pareto solution is given for w = w∗ using dashed-and-dotted +(red) curves, in which the controls are also of bang–bang type with one switching. +The switching functions for each control and case, σw +k (·), k = 1, 2, scaled as indicated, are +plotted with (black) dotted curves and superposed with the control plots in Figures 4(a)–(b). +We remind that, by using (23) (recall that discrete approximations of λw +Lk(t), k = 1, 2, λw +5 (t) +and λw +6 (t) can readily be obtained as constraint multipliers from AMPL), one verifies the +optimality condition in (24). +In each strategy, the two control efforts are “on” until the times tw +sk, k = 1, 2, at which +the respective uw +k (·) is switched “off” (down to zero). These types of bang–bang controls +are also referred to as on–off controls. In Table 2 the switching times for the boundary as +well as the optimal weights are listed. Under these controls, the resulting terminal values +of the state variables are also listed in Table 2. The plots of these variables are not pro- +vided as they are difficult to distinguish at earlier times (as expected) and that they become +distinguishable/comparable only near the terminal time. +Under the controls minimizing x5(tf) (with w = wf = 0.5709 and minimum xwf +5 (tf) = +26459) the number of persistent latent individuals L2(tf) turns out to be about 419 (in a +population of 30000). This number is more than doubled to 864 if x6(tf) is minimized (with +w = w0 = 0.5709 and minimum xw0 +6 (tf) = 31133). The optimal Pareto solution minimizing +the distance in value space to the origin yields with w = w∗ = 0.5358 the optimal L2(tf) as +748. + +Optimization Over the Pareto Front of Multi-objective Optimal Control Problems by C. Y. Kaya and H. Maurer +22 +5 +Conclusion +We have proposed an algorithm to solve the problem of optimization over the Pareto front. +The algorithm employs bisection method which starts with an essential interval of weights +of the Chebyshev scalarization. It is applicable to a wide range of optimal control problems, +including state- and control-constrained problems with time delay. Numerical solution of two +challenging optimal control problems has demonstrated the effectiveness of the algorithm. +The main motive behind the algorithm we have proposed is that one can find the optimal +solution minimizing a master objective functional without having to construct the Pareto +front. The algorithm solves the challenging optimal control problem (OCPw) a relatively +smaller number of times than the case of constructing the Pareto front. In the examples +we have studied the algorithm had to solve (OCPw) 20 to 30 times. On the other hand, +without the algorithm we propose, it is necessary to construct the Pareto front by solving +(OCPw) thousands of times in order to obtain the same solution with the same computational +accuracy. +The proposed algorithm can be improved/modified in various ways. For example, scalar- +ization techniques other than Chebyshev might be employed; see for example [8, 9] and the +references therein. Bisection method might be replaced by methods with higher convergence +rates, for example regula falsi and secant methods (see [10]), at the expense of approximating +higher order derivatives of course, although the latter would make the algorithm applicable +to problems with more than just two objective functionals. +References +[1] Alt, W., Baier, R. , Lempio, F., Gerdts, M.: Approximations of linear control problems +with bang–bang solutions. Optimization, 62, 9–32 (2013) +[2] Benson, H.P.: Optimization over the efficient set, J. Math. Anal. 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Global Optim., 22(1-4), +285–317 (2002). + diff --git a/X9FRT4oBgHgl3EQfODd9/content/tmp_files/2301.13512v1.pdf.txt b/X9FRT4oBgHgl3EQfODd9/content/tmp_files/2301.13512v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..71657433b39a3bbef9305ce248ccb7395af97373 --- /dev/null +++ b/X9FRT4oBgHgl3EQfODd9/content/tmp_files/2301.13512v1.pdf.txt @@ -0,0 +1,934 @@ +OpTaS: An Optimization-based Task Specification Library for +Trajectory Optimization and Model Predictive Control +Christopher E. Mower, Jo˜ao Moura, Nazanin Zamani Behabadi, +Sethu Vijayakumar, Tom Vercauteren∗, Christos Bergeles∗ +Abstract— This paper presents OpTaS, a task specification +Python library for Trajectory Optimization (TO) and Model +Predictive Control (MPC) in robotics. Both TO and MPC +are increasingly receiving interest in optimal control and in +particular handling dynamic environments. While a flurry +of software libraries exists to handle such problems, they +either provide interfaces that are limited to a specific problem +formulation (e.g. TracIK, CHOMP), or are large and stati- +cally specify the problem in configuration files (e.g. EXOTica, +eTaSL). OpTaS, on the other hand, allows a user to specify +custom nonlinear constrained problem formulations in a single +Python script allowing the controller parameters to be modified +during execution. The library provides interface to several open +source and commercial solvers (e.g. IPOPT, SNOPT, KNITRO, +SciPy) to facilitate integration with established workflows in +robotics. Further benefits of OpTaS are highlighted through +a thorough comparison with common libraries. An additional +key advantage of OpTaS is the ability to define optimal control +tasks in the joint space, task space, or indeed simultaneously. +The code for OpTaS is easily installed via pip, and the source +code with examples can be found at github.com/cmower/optas. +I. INTRODUCTION +High-dimensional motion planners and controllers are +integrated in many of the approaches for solving complex +manipulation tasks. Consider, for example, a robot operating +in an unstructured and dynamic environment that, e.g. places +an object onto a shelf, or drilling during pedicle screw +fixation in surgery (see Fig. 1). In such cases, a planner +and controller must account for objectives/constraints like +bi-manual coordination, contact constraints between robot- +object and object-environment, and be robust to disturbances. +Efficient motion planning and fast controllers are an effective +way of enabling robots to perform these tasks subject to +C. E. Mower, C. Bergeles and T. Vercauteren are with the School of +Biomedical Engineering & Imaging Sciences, King’s College London, UK. +J. Moura and S. Vijaykumar are with School of Informatics, University of +Edinburgh, UK. Correspondence: christopher.mower@kcl.ac.uk. +This research received funding from the European Union’s Horizon 2020 +research and innovation program under grant agreement No. 101016985 +(FAROS). Further, this work was supported by core funding from the +Wellcome/EPSRC [WT203148/Z/16/Z; NS/A000049/1]. T. Vercauteren is +supported by a Medtronic / RAEng Research Chair [RCSRF1819\7\34], +and C. Bergeles by an ERC Starting Grant [714562]. This work has +received funding from the European Union’s Horizon 2020 research and +innovation programme under grant agreement No 101017008, Enhancing +Healthcare with Assistive Robotic Mobile Manipulation (HARMONY). +This work was supported by core funding from the Wellcome/EPSRC +[WT203148/Z/16/Z; NS/A000049/1]. This research is supported by Kawada +Robotics Corporation, Japan and the Alan Turing Institute, UK. +∗C. Bergeles and T. Vercauteren equally contributed to the work. +For the purpose of open access, the authors have applied a CC BY public +copyright license to any Author Accepted Manuscript version arising from +this submission. +(a) +(b) +Fig. 1: Examples of contact-rich manipulation showing (a) +a robot placing an item on a shelf, (b) a human interacting +with a robot performing a drilling task during pedicle screw +fixation. Image credit: University Hospital Balgrist, Daniel +Hager Photography & Film GmbH. +motion constraints, system dynamics, and changing task +objectives. +Sampling-based planners [1] are effective, however, they +typically require considerable post-processing (e.g. trajectory +smoothing). Optimal planners (i.e. that are provably asymp- +totically optimal, e.g. RRT∗) are promising but inefficient (in +terms of computation duration) for solving high-dimensional +problems [2]. +Gradient-based trajectory optimization (TO) is a key ap- +proach in optimal control, and has also been utilized for mo- +tion planning. This approach underpins many recent works +in robotics for planning and control, e.g. [3], [4], [5], [6], +[7], [8], [9], [10]. Given an initialization, optimization finds a +locally optimal trajectory, comprised of a stream of state and +control commands subject to motion constraints and system +dynamics (i.e. equations of motion). +Several reliable open-source and commercial optimization +solvers exist for solving TO problems, e.g. IPOPT [11], KNI- +TRO [12], and SNOPT [13]. However, despite the success +of the optimization approaches proposed in the literature +and motion planning frameworks such as MoveIt [14], there +is a lack of libraries enabling fast development/prototyping +of optimization-based approaches for multi-robot setups that +easily interfaces with these efficient solvers. +To fill this gap, this paper proposes OpTaS, a user-friendly +task-specification library for rapid development and deploy- +ment of nonlinear optimization-based planning and control +approaches such as Model Predictive Control (MPC). The +library leverages the symbolic framework of CasADi [15], +enabling function derivatives to arbitrary order via automatic +differentiation. This is important since some solvers (e.g. +SNOPT) utilize the Jacobian and Hessian. +arXiv:2301.13512v1 [cs.RO] 31 Jan 2023 + +ACEFig. 2: System overview for the proposed OpTaS library. Red highlights the main features of the proposed library. Green +shows configuration parameter input. Grey shows third-party frameworks/libraries. Finally, the image in the top-right corner +shows integration with the ROS-PyBullet Interface [16]. +A. Related work +In this section, we review popular optimization solvers +and their interfaces. Next, we describe works similar (in +formulation) to our proposed library. Finally, we summarize +the key differences and highlight our contributions. Table I +summarizes alternatives and how they compare to OpTaS. +There are several capable open-source and commercial +optimization solvers. First considering quadratic program- +ming, the OSQP method provides a general purpose solver +based on the alternating direction method of multipliers [17]. +Alternatively, CVXOPT implements a custom interior-point +solver [18]. IPOPT implements an interior-point solver for +constrained nonlinear optimization. SNOPT provides an in- +terface to an SQP algorithm [13]. KNITRO also solves gen- +eral mixed-integer programs [12]. Please note that SNOPT +and KNITRO are proprietary. +These solvers are often implemented in low-level program- +ming languages such as C, C++, or FORTRAN. However, +there are also many interfaces to these methods via higher +level languages, such as Python, to make implementation and +adoption easier. The SciPy library contains the optimize +module [19] to interface with low-level routines, e.g. conju- +gate gradient and BFGS algorithm [20], the Simplex method +[21], COBYLA [22], and SLSQP [23]. A requirement when +using optimization-based methods is the need for function +gradients. Several popular software packages implement +automatic differentiation [24], [15], [25]. We leverage the +CasADi framework [15] for deriving gradients. Our choice +for CasADI is based on the fact that it comes readily +integrated with common solvers for optimal control. To the +best of our knowledge, JAX and PyTorch are not currently +integrated with constrained nonlinear optimization solvers. +Similar to our proposed library are the following pack- +ages. The MoveIt package provides the user with specific +TABLE I: Comparison between OpTaS and common alter- +natives in literature. +Languages +End-pose Traj. MPC Solver +AutoDiff ROS Re-form +OpTaS +Python + + + +QP/NLP  + + +EXOTica +Python/C++  + + +QP/NLP  + + +MoveIt +Python/C++  + + +QP + + + +TracIK +Python/C++  + + +QP + + + +RBDL +Python/C++  + + +QP + + + +eTaSL +C++ + + + +QP + +1 + +OpenRAVE Python + + + +QP + + + +IK/planning formulations and provides interfaces to solvers +for the particular problem [14]. The eTaSL library [26] +allows the user to specify custom tasks specifications, but +only supports problems formulated as quadratic programs. +The CASCLIK library uses CasADi and provides support +for constraint-based inverse kinematic controllers [27], to the +best of our knowledge they allow optimization in the joint +space. We provide joint space, task space optimization and +also the ability to simultaneously optimize in the joint/task +space. Furthermore, our framework supports optimization of +several robots in a single formulation. The EXOTica library +allows the user to specify a problem formulation from an +XML file [28]. The package, however, requires the user to +supply analytic gradients for additional sub-task models. +B. Contributions +This paper makes the following contributions: +• A task-specification library, in Python, for rapid devel- +opment/deployment of TO approaches for multi-robot +setups. +• Modeling of the robot kinematics (forward kinematics, +geometric Jacobian, etc.), to arbitrary derivative order, +given a URDF specification. +1Enabled with external pluggins. + +Task specification +Goals +Obstacles +Regularization +Optimization builder +URDF +ROS-PyBullet +Robot model 1 +Interface + Model options +Decision +Linear (In)equality +.. +variables +constraints +Cost +URDF +terms +":ROS +Robot model N +(In)equality +Parameters +Model options +constraints +Solver +Solver options +Optimization problem type identifie +Deployment +Feedback +Joint + configuration +Solver interface +Optimization problem +Obstacle +tracking +Teleoperator +Controller/Planner +input +Controller options• An interface that allows a user to easily reformulate +an optimal control problem, and define parameterized +constraints for online modification of the optimization +problem. +• Analysis comparing the performance of the library (i.e. +solver convergence, solution quality) versus existing +software packages. Further demonstrations highlight the +ease in which nonlinear constrained optimization prob- +lems can be set up and deployed in realistic settings. +II. PROBLEM FORMULATION +We can write an optimal control formulation of a TO or +planning problems as +min +x,u cost(x, u; T) +subject to +� +� +� +� +� +˙x = f(x, u) +x ∈ X +u ∈ U +(1) +where t denotes time, and x = x(t) ∈ Rnx and u = +u(t) ∈ Rnu denote the states and controls, with T being the +time-horizon for the planned trajectory. The scalar function +cost : Rnx ×Rnu → R represents the cost function (typically +a weighted sum of terms each modeling a certain sub-task), +the dot notation denotes a derivative with respect to time +(i.e. ˙x ≡ dx +dt ), f represents the system dynamics (equations +of motion), and X ⊆ Rnx and U ⊆ Rnu are feasible +regions for the states and controls respectively (modeled by +a set of equality and inequality constraints). Direct optimal +control, optimizes for the controls u for a discrete set of time +instances, using numerical methods (e.g. Euler or Runge- +Kutta), to integrate the system dynamics over the time +horizon T [29]. Given an initialization xinit, uinit, a locally +optimal trajectory x∗, u∗ is found by solving (1). +As discussed in Sec. I, many works propose optimization- +based approaches for planning and control. These can all be +formulated under the same framework, i.e. a TO problem as +in (1). The goal of our work is to deliver a library that allows +a user to quickly develop and prototype constrained nonlinear +TO for multi-robot problems, and deploy them for motion +generation. The library includes two types of problems, IK +and task-sace TO, and indeed both simultaneously. Common +steps, such as transcription that transforms the problem’s +task-level description into a form accepted by numerical +optimization solver routines, should be automated and thus +not burden the user. Furthermore, many works in practice +require the ability to adapt constraints dynamically to handle +changes in the environment (e.g. MPC). This motivates a +constraint parameterization feature. +III. PROPOSED FRAMEWORK +In this section, we describe the main features of the +proposed library shown in Fig. 2. The library is completely +implemented in the Python programming language. We chose +Python because it is simple for beginners but also versatile +with many well-developed libraries, and it easily facilitates +fast prototyping. +A. Robot model +The robot model (RobotModel) provides the kinematic +modeling and specifies the time derivative orders required for +the optimization problem. The only requirement is a URDF +to instantiate the object2. A key feature is that we can include +several robots in the TO, which is useful for dual arm and +whole-body optimization. Additional base frames and end- +effector links can be added programatically (for example, +when several robots are included the optimization their +base frames should be registered within a global coordinate +frame). +The RobotModel class allows access to data such as: +the number of degrees of freedom, the names of the actuated +joints, the upper and lower actuated joint limits, and the kine- +matics model. Furthermore, we provide methods to compute +the forward kinematics and geometric Jacobian in any given +reference frame. Several methods modeling the kinematics +are supplied, given a specification from the user for the base +frame and end-effector frame. These methods include: the 4× +4 homogeneous transformation matrix, translation position, +rotational representations (e.g. Euler angles, quaternions), +the geometric and analytical Jacobian. Each of the methods +above depend on a joint state (supplied as either a Python +list, NumPy array, or CasADi symbolic array). +B. Task model +Several works optimize robot motion in the task space +and then compute the IK as a secondary step, e.g. [8], [9]. +The task model (TaskModel) provides a representation for +any arbitrary trajectory. For example, the three dimensional +position trajectory of an end-effector. In the same way as +the robot model, the time derivatives can be specified in the +interface an arbitrary order. +C. Optimization builder +This section introduces and describes the optimization +builder class (OptimizationBuilder). The purpose of +this class is to aid the user to easily setup a TO problem, +and then automatically build an optimization problem model +(Sec. III-D) that interfaces with a solver interface (Sec. +III-E). The development cycle consists in specifying the +task (i.e. decision variables, parameters, cost function, and +constraints) using intuitive syntax and symbolic variables. +Then, the builder creates an optimization problem class, +which interfaces with several solvers. +D. Optimization problem model +The standard TO is stated in (1). This task/problem is +specified by the optimization builder class in intuitive syntax +for the user. Transcribing the problem to a form that can be +solved by off-the-shelf solvers is non-trivial. The output of +the optimization builder method build is an optimization +problem model that allows us to interface with several +solvers. +2http://wiki.ros.org/urdf + +The most general optimization problem that is modeled +by OpTaS is given by +X∗ = arg min +X +f(X; P) +(2a) +subject to +k(X; P) = M(P)X + c(P) ≥ 0 +(2b) +a(X; P) = A(P)X + b(P) = 0 +(2c) +g(X; P) ≥ 0 +(2d) +h(X; P) = 0 +(2e) +where X = [vec(x)T , vec(u)T ]T ∈ RnX is the decision +variable array such that x, u are as defined in (1) and vec(·) +is a function that returns its input as a 1-dimensional vector, +P ∈ RnP is the vectorized parameters, f : RnX → R +denotes the objective function, k : RnX → Rnk denotes +the linear inequality constraints, a : RnX → Rna denotes +the linear equality constraints, g : RnX → Rng denotes +the nonlinear inequality constraints, and h : RnX → Rnh +denotes the nonlinear equality constraints. The decision vari- +ables X are all the joint states and other variables specified +by the user stacked into a single vector. Similarly for the +parameters, cost terms, and constraints. Vectorization is made +possible by the SXContainer data structure implemented +in the sx container module. This data structure enables +automatic transcription of the TO problem specified in (1) +into the form (2). +Of course, not all task specifications will require defini- +tions for each of the functions in (2). Depending on the struc- +ture of the objective function and constraints, the required +time budget, and accuracy, some solvers will be more appro- +priate for solving (2). For example, a quadratic programming +solver that only handles linear constraints (e.g. OSQP [17]) +is unsuitable for solving a problem with nonlinear objective +function and nonlinear constraints. The build process auto- +matically identifies the optimization problem type, exposing +only the relevant solvers. Several problem types are available +to the user: unconstrained quadratic cost, linearly constrained +with quadratic cost, nonlinear constrained with quadratic +cost, unconstrained with nonlinear cost, linearly constrained +with nonlinear cost, nonlinear cost and constraints. +1) Initialization: Upon initialization of the optimization +builder class we can specify (i) the number of time steps +in the trajectory, (ii) several robot and task models (given a +unique name for each), (iii) the joint states (positions and +required time-derivatives) that integrate the decision variable +array, (iv) task space labels, dimensions, and derivatives +to also integrate the decision variable array, (v) a Boolean +describing the alignment of the derivatives (Fig. 3), and (vi) +a Boolean indicating whether to optimize time steps. +The alignment of time-derivatives can be specified in +two ways. Each derivative is aligned with its corresponding +state (alignement), or otherwise. This is specified by the +derivs align flag in the optimization builder interface +and shown diagramatically in Fig. 3. +In addition, the user can also optimize the time-steps +between each state. The time derivatives can be integrated +Fig. 3: Joint state alignment with time. User supplies +derivs align that specifies how joint state time deriva- +tives should be aligned. +over time, e.g. qt+1 = qt + δτt ˙qt, where δτt is an increment +in time. When optimize time=True, then each δτt is +included as decision variables in the optimal control problem. +2) Decision variables and parameters: Decision variables +are specified in the optimization builder class interface for +the joint space, task space, and time steps. Each group +of variables is given a unique label and can be retrieved +using the get model state method. States are retrieved +by specifying a robot name or task name, the required time +index, and the time derivative order required. Additional +decision variables can be included in the problem by using +the add decision variables method given a unique +name and dimension. +Parameters for the problem (e.g. safe distances) can be +specified using the add parameter method. To specify a +new parameter, a unique name and dimension is required. +3) Cost and constraint functions: The cost function in (1) +is assumed to be made up of several cost terms, i.e. +cost(x, u; T) = +� +i +ci(x, u; T) +(3) +where ci : Rnx × Rnu → R is an individual cost term +modeling a specific sub-task. For example, let us define the +cost terms c0 = ∥ψ(xT ) − ψ∗∥2 and c1 = λ +� T +0 +∥u∥2 dt +(note, discretization is implicit in this formulation) where +ψ : Rnx → R3 is a function for the forward kinematics +position (note, this can be provided by the robot model +class as described in Sec. III-A), ψ∗ ∈ R3 is a goal task +space position, and 0 < λ ∈ R is a scaling term used +to weight the relative importance of one constraint against +the other. Thus, c0 describes an ideal state for the final +state, and c1 encourages trajectories with minimal control +signals (e.g. minimize joint velocities). Each cost term is +added to the problem using the add cost term method; +the build sequence ensures each term is added to the +objective function. +Several constraints can be added to the optimization +problem by using the add equality constraint and +add leq inequality constraint methods that add +equality and inequality constraints respectively. When the +constraints are added to the problem, they are first checked to +see if they are linear constraints with respect to the decision +variables. This functionality allows the library to differentiate +between linear and nonlinear constraints. +Additionally, OpTaS offers several methods that provide +an implementation for common constraints, as, for example, + +derivs_align=True +1- +-1- +—... +q0 +q1 +q2 +q3 +q4 +QT-1 +qT +:b +:pb +q0 +q1 +q2 +q3 +q4 +QT-1 +qT +derivs_align=False +1 +..· +-1 +qo +q1 +q2 +q3 +q4 +: b +qT-1 +qT +qo +q1 +q2 +q3 +: pb +qT-1joint position/velocity limits and time-integration for the +system dynamics f (e.g joint velocities can be integrated +to positions). +E. Solver interface +OpTaS provides interfaces to solvers (open-source and +commercial) that interface with CasADi [15] (such as +IPOPT [11]), SNOPT [13], KNITRO [12], and Gurobi [30]), +the Scipy minimize method [19], OSQP [17], and CVX- +OPT [18]. +1) Initialization of solver: When the solver is initialized, +several variables are setup and the optimization problem +object is set as a class attribute. The user must then call +the setup method - that itself is an interface to the solver +initialization that the user has chosen. The requirement of +this method is to setup the interface for the specific solver; +relevant solver parameters are passed to the interface at this +stage. +2) Resetting the interface: When using the solver as a +controller, it is expected that the solver should be called more +than once. In the case for feedback controllers or controllers +with parameterized constraints (e.g. obstacles), this requires +a way to reset the problem parameters. Furthermore, the +initial seed for the optimizer is often required to be reset +at each control loop cycle. To reset the initial seed and +problem parameters the user calls reset initial seed, +and reset parameters, respectively. Both the initial +seed and parameters are initialized by giving the name of the +variables. The required vectorization is internally performed +by the solver utilizing features of the SXContainer data +structure. Note, if any decision variables or parameters are +not specified in the reset methods then they automatically +default to zero. This enables warm-starting the optimization +routine, e.g. with the solution of the previous time-step +problem. +3) Solving an optimization problem: The optimization +problem is solved by calling the solve method. This +method passes the optimization problem to the desired +solver. The resulting data from the solver is collected and +transformed back into the state trajectory for each robot. A +method is provided, named interpolate, is used to in- +terpolate the computed trajectories across time. Additionally, +the method stats retrieves available optimization statistics +(e.g. number of iterations). +4) Extensible solver interface: The solver interface has +been implemented to allow for extensibility, i.e. additional +optimization solvers can be easily integrated into the frame- +work. When a user would like to include a new solver +interface, they must create a new class that inherits from +the Solver class. In their sub-class definition they must +implement three methods: (i) setup which (as described +above) initializes the solver interface, (ii) solve that calls +the solver and returns the optimized variable X∗, and (iii) +stats that returns any statistics from the solver. +F. Additional features +Support for integration with ROS [31] is provided out- +of-the-box. The ROS node provided is integrated with the +import optas +# Setup robot and optimization builder +T = 100 # number of time steps in trajectory +urdf = ’/path/to/robot.urdf’ +r = optas.RobotModel(urdf, time_deriv=[0, 1]) +n = r.get_name() +b = optas.OptimizationBuilder(T=T, robots=[r]) +# Retrieve variables and setup parameters +q0 = b.get_model_state(n, t=0) +qT = b.get_model_state(n, t=-1) # final state +pg = b.add_parameter(’pg’, 3) # goal pos. +qc = b.add_parameter(’qc’, r.ndof) # init q +o = b.add_parameter(’o’, 3) # obstacle pos. +r = b.add_parameter(’r’) +# obstacle radius +dt = b.add_parameter(’dt’) # time step +# Forward kinematics +p = r.get_global_link_position(tip, qT) +# Cost and constraints +b.add_cost_term(’c’, optas.sumsqr(p - pg)) +b.integrate_model_states( +n, time_deriv=1, dt=dt) +b.add_equality_constraint(’init’, q0, qc) +for t in range(T): +b.add_leq_inequality_constraint( +optas.sumsqr(p - o), r**2) +# Build optimization problem and setup solver +solver = optas.CasADiSolver( +b.build()).setup(’ipopt’) +Fig. 4: Example code for TO described in Section IV. +ROS-PyBullet Interface [16] so the publishers/subscribers +can connect a robot in the optimization problem with a robot +simulated in PyBullet. +In addition, we provide a port of the spatialmath +library by Corke [32] that supports CasADi variables. This +library defines methods for manipulating homogeneous trans- +formation matrices, quaternions, Euler angles, etc. using +CasADi symbolic variables. +IV. CODE EXAMPLE +In this section, we describe a common TO problem and +give the code that models the problem. We aim to highlight +how straightforward it is to setup a problem. +Consider a serial link manipulator, and goal to find a +collision-free plan over time horizon T to a goal end- +effector position pg given a starting configuration qc. A single +spherical collision is represented by a position o and radius r. +The robot configuration qt represent states, and the velocities +˙qt are controls. +The cost function is given by ∥p(qT ) − pg∥2 where p +is the position of the end-effector given by the forward +kinematics. We solve the problem by minimizing the cost +function subject to the constraints: (i) initial configuration, +q0 = qc, (ii) joint limits q− ≤ qt ≤ q+, and (iii) obstacle +avoidance, ∥p(qt) − o∥2 ≥ r2. The system dynamics is +represented by several equality constraints qt+1 = qt + δt ˙qt +that can be specified by methods already in-built into OpTaS. +The code for the TO problem above, is shown in Fig. 4. + +(a) +(b) +Fig. 5: Comparison of end-effector task space trajectories +computed using two different formulations. (a) Shows the +start (left), and final configurations (right) for the robot under +each approach. (b) Plots the end-effector position trajectory +two dimensions. +V. EXPERIMENTS +A. Optimization along custom dimensions +Popular solvers, such as TracIK [33], require the user to +provide a 6D pose as the task space goal. Whilst this is ap- +plicable to several robotics problems (e.g. pick-and-place) it +may not be necessary to optimize each task space dimension +(e.g. spraying applications does not require optimization in +the roll angular direction). Furthermore, optimizing in more +dimensions than necessary may be disadvantageous. +OpTaS can optimize or neglect any desired task space +dimension. This can have certain advantages, for example +increasing the robot workspace. Consider a non-prehensile +pushing task along the plane, optimizing the full 6D pose +may not be ideal since the task is two dimensional. By +optimizing in the two dimensional plane and specifying +boundary constraints on the third linear spatial dimension, +increases the robots workspace. +We setup a tracking experiment in OpTaS using a sim- +ulated Kuka LWR robot arm to compare the two cases: +(i) optimize the full 6D pose, and (ii) optimize 2D linear +position. The robot is given an initial configuration (Fig. 5a +left) and the task is to move the end-effector with velocity of +constant magnitude and direction in the 2D plane. The end +configuration for each approach is shown in Fig. 5a right and +the end-effector trajectories are shown in Fig. 5b. We see +that the 2D optimization problem is able to reach a greater +distance, highlighting that the robot workspace is increased. +B. Performance comparison +In this section, we demonstrate that OpTaS can formulate +similar problems and compare its performance to alterna- +tives. First, we model, with OpTaS, the same problem as +used in TracIK [33] and in addition we also model the +problem using EXOTica [28]. The Scipy SLSQP solver [23] +was used for OpTaS and EXOTica. With same Kuka LWR +Fig. 6: Figure-of-eight trajectory tracked by the Kuka LWR. +(a) +(b) +Fig. 7: Solver duration comparisons for figure of eight +motion. (a) Compares an IK tracking approach described +in Section V, (b) is a similar comparison that includes a +maximization term for manipulability. Green is OpTaS, red +is TracIK, and blue is EXOTica. +robot arm in the previous experiment, we setup a task where +the robot must track a figure-of-eight motion in task space +(Fig. 6) and record the CPU time for the solver duration at +each control loop cycle. The results are shown in Fig. 7a. +TracIK is the fastest (0.049 ± 0.035ms), which is expected +since it is optimized for a specific problem formulation. We +see that OpTaS (2.608 ± 0.239ms) is faster than EXOTica +(3.694 ± 0.300ms) +A second experiment, using the same setup as be- +fore, was performed comparing the performance of OpTaS +against EXOTica with an additional cost term to maxi- +mize manipulability [34]. The results are shown in Fig. +7b. Despite using the same formulation and solver, OpTaS +(2.650 ± 0.270ms) achieved better performance than EXOT- +ica (7.640±1.404ms). Without extensive profiling it is diffi- +cult to precisely explain this difference. However, EXOTica +requires the user to supply analytical gradients for sub-tasks +(called task maps in the EXOTica documentation). EXOTica +does not provide the gradients for the manipulability task, +and thus falls-back to using the finite difference method to +estimate the gradient - this can can be slow to compute. +VI. CONCLUSIONS +In this paper, we have proposed OpTaS: an optimization- +based task tpecification Python library for TO and MPC. +OpTaS allows a user to setup a constrained nonlinear pro- +grams for custom problem formulations and has been shown +to perform well against alternatives. Parameterization enables +programs to act as feedback controllers, motion planners, and +benchmark problem formulations and solvers. +We hope OpTaS will be used by researchers, students, and +industry to facilitate the development of control and motion +planning algorithms. The code base is easily installed via +pip and has been made open-source under the Apache 2 +license: https://github.com/cmower/optas. + +0.1 +Optimize6D +Optimize2D +Ideal path +Start +(m) +0.0 +-0.1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +X (m)CPU Time (ms) +OpTaS +TraclK +EXOTica +7 +0 +0.0 +2.5 +5.0 +7.5 +5 10.0 12.5 15.0 17.5 20.0 +Time (s)CPU Time (ms) +10 +OpTaS +EXOTica +8 +6 +4 +2 +0.0 +2.5 +5.0 +7.5 +10.0 12.5 15.0 17.5 20.0 +Time (s)REFERENCES +[1] S. M. LaValle, Planning algorithms. +Cambridge university press, +2006. +[2] S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimal +motion planning,” The international journal of robotics research, +vol. 30, no. 7, pp. 846–894, 2011. +[3] N. Ratliff, M. Zucker, J. A. Bagnell, and S. Srinivasa, “Chomp: +Gradient optimization techniques for efficient motion planning,” in +2009 IEEE International Conference on Robotics and Automation, +2009, pp. 489–494. +[4] J. Schulman, Y. Duan, J. Ho, A. Lee, I. Awwal, H. 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Available: https://doi.org/10.1177/027836498500400201 + diff --git a/X9FRT4oBgHgl3EQfODd9/content/tmp_files/load_file.txt b/X9FRT4oBgHgl3EQfODd9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3444e40dfe25650ce4416e117dfcb6c95e2b9323 --- /dev/null +++ b/X9FRT4oBgHgl3EQfODd9/content/tmp_files/load_file.txt @@ -0,0 +1,773 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf,len=772 +page_content='OpTaS: An Optimization-based Task Specification Library for Trajectory Optimization and Model Predictive Control Christopher E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Mower, Jo˜ao Moura, Nazanin Zamani Behabadi, Sethu Vijayakumar, Tom Vercauteren∗, Christos Bergeles∗ Abstract— This paper presents OpTaS, a task specification Python library for Trajectory Optimization (TO) and Model Predictive Control (MPC) in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Both TO and MPC are increasingly receiving interest in optimal control and in particular handling dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' While a flurry of software libraries exists to handle such problems, they either provide interfaces that are limited to a specific problem formulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' TracIK, CHOMP), or are large and stati- cally specify the problem in configuration files (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' EXOTica, eTaSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' OpTaS, on the other hand, allows a user to specify custom nonlinear constrained problem formulations in a single Python script allowing the controller parameters to be modified during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The library provides interface to several open source and commercial solvers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' IPOPT, SNOPT, KNITRO, SciPy) to facilitate integration with established workflows in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Further benefits of OpTaS are highlighted through a thorough comparison with common libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' An additional key advantage of OpTaS is the ability to define optimal control tasks in the joint space, task space, or indeed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The code for OpTaS is easily installed via pip, and the source code with examples can be found at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='com/cmower/optas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' INTRODUCTION High-dimensional motion planners and controllers are integrated in many of the approaches for solving complex manipulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Consider, for example, a robot operating in an unstructured and dynamic environment that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' places an object onto a shelf, or drilling during pedicle screw fixation in surgery (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' In such cases, a planner and controller must account for objectives/constraints like bi-manual coordination, contact constraints between robot- object and object-environment, and be robust to disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Efficient motion planning and fast controllers are an effective way of enabling robots to perform these tasks subject to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Mower, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Bergeles and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Vercauteren are with the School of Biomedical Engineering & Imaging Sciences, King’s College London, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Moura and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Vijaykumar are with School of Informatics, University of Edinburgh, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Correspondence: christopher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='mower@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This research received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 101016985 (FAROS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Further, this work was supported by core funding from the Wellcome/EPSRC [WT203148/Z/16/Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' NS/A000049/1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Vercauteren is supported by a Medtronic / RAEng Research Chair [RCSRF1819\\7\\34], and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Bergeles by an ERC Starting Grant [714562].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 101017008, Enhancing Healthcare with Assistive Robotic Mobile Manipulation (HARMONY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This work was supported by core funding from the Wellcome/EPSRC [WT203148/Z/16/Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' NS/A000049/1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This research is supported by Kawada Robotics Corporation, Japan and the Alan Turing Institute, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' ∗C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Bergeles and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Vercauteren equally contributed to the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' For the purpose of open access, the authors have applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 1: Examples of contact-rich manipulation showing (a) a robot placing an item on a shelf, (b) a human interacting with a robot performing a drilling task during pedicle screw fixation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Image credit: University Hospital Balgrist, Daniel Hager Photography & Film GmbH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' motion constraints, system dynamics, and changing task objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Sampling-based planners [1] are effective, however, they typically require considerable post-processing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' trajectory smoothing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Optimal planners (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' that are provably asymp- totically optimal, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' RRT∗) are promising but inefficient (in terms of computation duration) for solving high-dimensional problems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Gradient-based trajectory optimization (TO) is a key ap- proach in optimal control, and has also been utilized for mo- tion planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This approach underpins many recent works in robotics for planning and control, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' [3], [4], [5], [6], [7], [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Given an initialization, optimization finds a locally optimal trajectory, comprised of a stream of state and control commands subject to motion constraints and system dynamics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' equations of motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Several reliable open-source and commercial optimization solvers exist for solving TO problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' IPOPT [11], KNI- TRO [12], and SNOPT [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' However, despite the success of the optimization approaches proposed in the literature and motion planning frameworks such as MoveIt [14], there is a lack of libraries enabling fast development/prototyping of optimization-based approaches for multi-robot setups that easily interfaces with these efficient solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' To fill this gap, this paper proposes OpTaS, a user-friendly task-specification library for rapid development and deploy- ment of nonlinear optimization-based planning and control approaches such as Model Predictive Control (MPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The library leverages the symbolic framework of CasADi [15], enabling function derivatives to arbitrary order via automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This is important since some solvers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' SNOPT) utilize the Jacobian and Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='13512v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='RO] 31 Jan 2023 ACEFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 2: System overview for the proposed OpTaS library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Red highlights the main features of the proposed library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Green shows configuration parameter input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Grey shows third-party frameworks/libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Finally, the image in the top-right corner shows integration with the ROS-PyBullet Interface [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Related work In this section, we review popular optimization solvers and their interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Next, we describe works similar (in formulation) to our proposed library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Finally, we summarize the key differences and highlight our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Table I summarizes alternatives and how they compare to OpTaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' There are several capable open-source and commercial optimization solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' First considering quadratic program- ming, the OSQP method provides a general purpose solver based on the alternating direction method of multipliers [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Alternatively, CVXOPT implements a custom interior-point solver [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' IPOPT implements an interior-point solver for constrained nonlinear optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' SNOPT provides an in- terface to an SQP algorithm [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' KNITRO also solves gen- eral mixed-integer programs [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Please note that SNOPT and KNITRO are proprietary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' These solvers are often implemented in low-level program- ming languages such as C, C++, or FORTRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' However, there are also many interfaces to these methods via higher level languages, such as Python, to make implementation and adoption easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The SciPy library contains the optimize module [19] to interface with low-level routines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' conju- gate gradient and BFGS algorithm [20], the Simplex method [21], COBYLA [22], and SLSQP [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' A requirement when using optimization-based methods is the need for function gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Several popular software packages implement automatic differentiation [24], [15], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We leverage the CasADi framework [15] for deriving gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Our choice for CasADI is based on the fact that it comes readily integrated with common solvers for optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' To the best of our knowledge, JAX and PyTorch are not currently integrated with constrained nonlinear optimization solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Similar to our proposed library are the following pack- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The MoveIt package provides the user with specific TABLE I: Comparison between OpTaS and common alter- natives in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Languages End-pose Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' MPC Solver AutoDiff ROS Re-form OpTaS Python \x13 \x13 \x13 QP/NLP \x13 \x13 \x13 EXOTica Python/C++ \x13 \x13 \x17 QP/NLP \x17 \x13 \x13 MoveIt Python/C++ \x13 \x13 \x17 QP \x17 \x13 \x17 TracIK Python/C++ \x13 \x17 \x17 QP \x17 \x13 \x17 RBDL Python/C++ \x13 \x17 \x17 QP \x17 \x17 \x17 eTaSL C++ \x13 \x17 \x17 QP \x13 \x171 \x13 OpenRAVE Python \x17 \x13 \x17 QP \x17 \x13 \x17 IK/planning formulations and provides interfaces to solvers for the particular problem [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The eTaSL library [26] allows the user to specify custom tasks specifications, but only supports problems formulated as quadratic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The CASCLIK library uses CasADi and provides support for constraint-based inverse kinematic controllers [27], to the best of our knowledge they allow optimization in the joint space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We provide joint space, task space optimization and also the ability to simultaneously optimize in the joint/task space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Furthermore, our framework supports optimization of several robots in a single formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The EXOTica library allows the user to specify a problem formulation from an XML file [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The package, however, requires the user to supply analytic gradients for additional sub-task models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Contributions This paper makes the following contributions: A task-specification library, in Python, for rapid devel- opment/deployment of TO approaches for multi-robot setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Modeling of the robot kinematics (forward kinematics, geometric Jacobian, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' ), to arbitrary derivative order, given a URDF specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 1Enabled with external pluggins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Task specification Goals Obstacles Regularization Optimization builder URDF ROS-PyBullet Robot model 1 Interface Model options Decision Linear (In)equality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' variables constraints Cost URDF terms ":ROS Robot model N (In)equality Parameters Model options constraints Solver Solver options Optimization problem type identifie Deployment Feedback Joint configuration Solver interface Optimization problem Obstacle tracking Teleoperator Controller/Planner input Controller options• An interface that allows a user to easily reformulate an optimal control problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' and define parameterized constraints for online modification of the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Analysis comparing the performance of the library (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' solver convergence, solution quality) versus existing software packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Further demonstrations highlight the ease in which nonlinear constrained optimization prob- lems can be set up and deployed in realistic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' PROBLEM FORMULATION We can write an optimal control formulation of a TO or planning problems as min x,u cost(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' T) subject to � � � � � ˙x = f(x, u) x ∈ X u ∈ U (1) where t denotes time, and x = x(t) ∈ Rnx and u = u(t) ∈ Rnu denote the states and controls, with T being the time-horizon for the planned trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The scalar function cost : Rnx ×Rnu → R represents the cost function (typically a weighted sum of terms each modeling a certain sub-task), the dot notation denotes a derivative with respect to time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' ˙x ≡ dx dt ), f represents the system dynamics (equations of motion), and X ⊆ Rnx and U ⊆ Rnu are feasible regions for the states and controls respectively (modeled by a set of equality and inequality constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Direct optimal control, optimizes for the controls u for a discrete set of time instances, using numerical methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Euler or Runge- Kutta), to integrate the system dynamics over the time horizon T [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Given an initialization xinit, uinit, a locally optimal trajectory x∗, u∗ is found by solving (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' I, many works propose optimization- based approaches for planning and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' These can all be formulated under the same framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' a TO problem as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The goal of our work is to deliver a library that allows a user to quickly develop and prototype constrained nonlinear TO for multi-robot problems, and deploy them for motion generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The library includes two types of problems, IK and task-sace TO, and indeed both simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Common steps, such as transcription that transforms the problem’s task-level description into a form accepted by numerical optimization solver routines, should be automated and thus not burden the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Furthermore, many works in practice require the ability to adapt constraints dynamically to handle changes in the environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' MPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This motivates a constraint parameterization feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' PROPOSED FRAMEWORK In this section, we describe the main features of the proposed library shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The library is completely implemented in the Python programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We chose Python because it is simple for beginners but also versatile with many well-developed libraries, and it easily facilitates fast prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Robot model The robot model (RobotModel) provides the kinematic modeling and specifies the time derivative orders required for the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The only requirement is a URDF to instantiate the object2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' A key feature is that we can include several robots in the TO, which is useful for dual arm and whole-body optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Additional base frames and end- effector links can be added programatically (for example, when several robots are included the optimization their base frames should be registered within a global coordinate frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The RobotModel class allows access to data such as: the number of degrees of freedom, the names of the actuated joints, the upper and lower actuated joint limits, and the kine- matics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Furthermore, we provide methods to compute the forward kinematics and geometric Jacobian in any given reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Several methods modeling the kinematics are supplied, given a specification from the user for the base frame and end-effector frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' These methods include: the 4× 4 homogeneous transformation matrix, translation position, rotational representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Euler angles, quaternions), the geometric and analytical Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Each of the methods above depend on a joint state (supplied as either a Python list, NumPy array, or CasADi symbolic array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Task model Several works optimize robot motion in the task space and then compute the IK as a secondary step, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The task model (TaskModel) provides a representation for any arbitrary trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' For example, the three dimensional position trajectory of an end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' In the same way as the robot model, the time derivatives can be specified in the interface an arbitrary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Optimization builder This section introduces and describes the optimization builder class (OptimizationBuilder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The purpose of this class is to aid the user to easily setup a TO problem, and then automatically build an optimization problem model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' III-D) that interfaces with a solver interface (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' III-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The development cycle consists in specifying the task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' decision variables, parameters, cost function, and constraints) using intuitive syntax and symbolic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Then, the builder creates an optimization problem class, which interfaces with several solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Optimization problem model The standard TO is stated in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This task/problem is specified by the optimization builder class in intuitive syntax for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Transcribing the problem to a form that can be solved by off-the-shelf solvers is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The output of the optimization builder method build is an optimization problem model that allows us to interface with several solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 2http://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='org/urdf The most general optimization problem that is modeled by OpTaS is given by X∗ = arg min X f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' P) (2a) subject to k(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' P) = M(P)X + c(P) ≥ 0 (2b) a(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' P) = A(P)X + b(P) = 0 (2c) g(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' P) ≥ 0 (2d) h(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' P) = 0 (2e) where X = [vec(x)T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' vec(u)T ]T ∈ RnX is the decision variable array such that x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' u are as defined in (1) and vec(·) is a function that returns its input as a 1-dimensional vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' P ∈ RnP is the vectorized parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' f : RnX → R denotes the objective function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' k : RnX → Rnk denotes the linear inequality constraints,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' a : RnX → Rna denotes the linear equality constraints,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' g : RnX → Rng denotes the nonlinear inequality constraints,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' and h : RnX → Rnh denotes the nonlinear equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The decision vari- ables X are all the joint states and other variables specified by the user stacked into a single vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Similarly for the parameters, cost terms, and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Vectorization is made possible by the SXContainer data structure implemented in the sx container module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This data structure enables automatic transcription of the TO problem specified in (1) into the form (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Of course, not all task specifications will require defini- tions for each of the functions in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Depending on the struc- ture of the objective function and constraints, the required time budget, and accuracy, some solvers will be more appro- priate for solving (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' For example, a quadratic programming solver that only handles linear constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' OSQP [17]) is unsuitable for solving a problem with nonlinear objective function and nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The build process auto- matically identifies the optimization problem type, exposing only the relevant solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Several problem types are available to the user: unconstrained quadratic cost, linearly constrained with quadratic cost, nonlinear constrained with quadratic cost, unconstrained with nonlinear cost, linearly constrained with nonlinear cost, nonlinear cost and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 1) Initialization: Upon initialization of the optimization builder class we can specify (i) the number of time steps in the trajectory, (ii) several robot and task models (given a unique name for each), (iii) the joint states (positions and required time-derivatives) that integrate the decision variable array, (iv) task space labels, dimensions, and derivatives to also integrate the decision variable array, (v) a Boolean describing the alignment of the derivatives (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 3), and (vi) a Boolean indicating whether to optimize time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The alignment of time-derivatives can be specified in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Each derivative is aligned with its corresponding state (alignement), or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This is specified by the derivs align flag in the optimization builder interface and shown diagramatically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' In addition, the user can also optimize the time-steps between each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The time derivatives can be integrated Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 3: Joint state alignment with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' User supplies derivs align that specifies how joint state time deriva- tives should be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' over time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' qt+1 = qt + δτt ˙qt, where δτt is an increment in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' When optimize time=True, then each δτt is included as decision variables in the optimal control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 2) Decision variables and parameters: Decision variables are specified in the optimization builder class interface for the joint space, task space, and time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Each group of variables is given a unique label and can be retrieved using the get model state method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' States are retrieved by specifying a robot name or task name, the required time index, and the time derivative order required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Additional decision variables can be included in the problem by using the add decision variables method given a unique name and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Parameters for the problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' safe distances) can be specified using the add parameter method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' To specify a new parameter, a unique name and dimension is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 3) Cost and constraint functions: The cost function in (1) is assumed to be made up of several cost terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' cost(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' T) = � i ci(x, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' T) (3) where ci : Rnx × Rnu → R is an individual cost term modeling a specific sub-task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' For example, let us define the cost terms c0 = ∥ψ(xT ) − ψ∗∥2 and c1 = λ � T 0 ∥u∥2 dt (note, discretization is implicit in this formulation) where ψ : Rnx → R3 is a function for the forward kinematics position (note, this can be provided by the robot model class as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' III-A), ψ∗ ∈ R3 is a goal task space position, and 0 < λ ∈ R is a scaling term used to weight the relative importance of one constraint against the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Thus, c0 describes an ideal state for the final state, and c1 encourages trajectories with minimal control signals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' minimize joint velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Each cost term is added to the problem using the add cost term method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' the build sequence ensures each term is added to the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Several constraints can be added to the optimization problem by using the add equality constraint and add leq inequality constraint methods that add equality and inequality constraints respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' When the constraints are added to the problem, they are first checked to see if they are linear constraints with respect to the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This functionality allows the library to differentiate between linear and nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Additionally, OpTaS offers several methods that provide an implementation for common constraints, as, for example, derivs_align=True 1- 1- —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' q0 q1 q2 q3 q4 QT-1 qT :b :pb q0 q1 q2 q3 q4 QT-1 qT derivs_align=False 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='.· 1 qo q1 q2 q3 q4 : b qT-1 qT qo q1 q2 q3 : pb qT-1joint position/velocity limits and time-integration for the system dynamics f (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g joint velocities can be integrated to positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Solver interface OpTaS provides interfaces to solvers (open-source and commercial) that interface with CasADi [15] (such as IPOPT [11]), SNOPT [13], KNITRO [12], and Gurobi [30]), the Scipy minimize method [19], OSQP [17], and CVX- OPT [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 1) Initialization of solver: When the solver is initialized, several variables are setup and the optimization problem object is set as a class attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The user must then call the setup method - that itself is an interface to the solver initialization that the user has chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The requirement of this method is to setup the interface for the specific solver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' relevant solver parameters are passed to the interface at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 2) Resetting the interface: When using the solver as a controller, it is expected that the solver should be called more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' In the case for feedback controllers or controllers with parameterized constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' obstacles), this requires a way to reset the problem parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Furthermore, the initial seed for the optimizer is often required to be reset at each control loop cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' To reset the initial seed and problem parameters the user calls reset initial seed, and reset parameters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Both the initial seed and parameters are initialized by giving the name of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The required vectorization is internally performed by the solver utilizing features of the SXContainer data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Note, if any decision variables or parameters are not specified in the reset methods then they automatically default to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This enables warm-starting the optimization routine, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' with the solution of the previous time-step problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 3) Solving an optimization problem: The optimization problem is solved by calling the solve method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This method passes the optimization problem to the desired solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The resulting data from the solver is collected and transformed back into the state trajectory for each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' A method is provided, named interpolate, is used to in- terpolate the computed trajectories across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Additionally, the method stats retrieves available optimization statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' number of iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 4) Extensible solver interface: The solver interface has been implemented to allow for extensibility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' additional optimization solvers can be easily integrated into the frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' When a user would like to include a new solver interface, they must create a new class that inherits from the Solver class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' In their sub-class definition they must implement three methods: (i) setup which (as described above) initializes the solver interface, (ii) solve that calls the solver and returns the optimized variable X∗, and (iii) stats that returns any statistics from the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Additional features Support for integration with ROS [31] is provided out- of-the-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The ROS node provided is integrated with the import optas # Setup robot and optimization builder T = 100 # number of time steps in trajectory urdf = ’/path/to/robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='urdf’ r = optas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='RobotModel(urdf, time_deriv=[0, 1]) n = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='get_name() b = optas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='OptimizationBuilder(T=T, robots=[r]) # Retrieve variables and setup parameters q0 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='get_model_state(n, t=0) qT = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='get_model_state(n, t=-1) # final state pg = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='add_parameter(’pg’, 3) # goal pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' qc = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='add_parameter(’qc’, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='ndof) # init q o = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='add_parameter(’o’, 3) # obstacle pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' r = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='add_parameter(’r’) # obstacle radius dt = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='add_parameter(’dt’) # time step # Forward kinematics p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='get_global_link_position(tip, qT) # Cost and constraints b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='add_cost_term(’c’, optas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='sumsqr(p - pg)) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='integrate_model_states( n, time_deriv=1, dt=dt) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='add_equality_constraint(’init’, q0, qc) for t in range(T): b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='add_leq_inequality_constraint( optas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='sumsqr(p - o), r**2) # Build optimization problem and setup solver solver = optas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='CasADiSolver( b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='build()).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='setup(’ipopt’) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 4: Example code for TO described in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' ROS-PyBullet Interface [16] so the publishers/subscribers can connect a robot in the optimization problem with a robot simulated in PyBullet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' In addition, we provide a port of the spatialmath library by Corke [32] that supports CasADi variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This library defines methods for manipulating homogeneous trans- formation matrices, quaternions, Euler angles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' using CasADi symbolic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' CODE EXAMPLE In this section, we describe a common TO problem and give the code that models the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We aim to highlight how straightforward it is to setup a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Consider a serial link manipulator, and goal to find a collision-free plan over time horizon T to a goal end- effector position pg given a starting configuration qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' A single spherical collision is represented by a position o and radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The robot configuration qt represent states, and the velocities ˙qt are controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The cost function is given by ∥p(qT ) − pg∥2 where p is the position of the end-effector given by the forward kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We solve the problem by minimizing the cost function subject to the constraints: (i) initial configuration, q0 = qc, (ii) joint limits q− ≤ qt ≤ q+, and (iii) obstacle avoidance, ∥p(qt) − o∥2 ≥ r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The system dynamics is represented by several equality constraints qt+1 = qt + δt ˙qt that can be specified by methods already in-built into OpTaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The code for the TO problem above, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 5: Comparison of end-effector task space trajectories computed using two different formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' (a) Shows the start (left), and final configurations (right) for the robot under each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' (b) Plots the end-effector position trajectory two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Optimization along custom dimensions Popular solvers, such as TracIK [33], require the user to provide a 6D pose as the task space goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Whilst this is ap- plicable to several robotics problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' pick-and-place) it may not be necessary to optimize each task space dimension (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' spraying applications does not require optimization in the roll angular direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Furthermore, optimizing in more dimensions than necessary may be disadvantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' OpTaS can optimize or neglect any desired task space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' This can have certain advantages, for example increasing the robot workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Consider a non-prehensile pushing task along the plane, optimizing the full 6D pose may not be ideal since the task is two dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' By optimizing in the two dimensional plane and specifying boundary constraints on the third linear spatial dimension, increases the robots workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We setup a tracking experiment in OpTaS using a sim- ulated Kuka LWR robot arm to compare the two cases: (i) optimize the full 6D pose, and (ii) optimize 2D linear position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The robot is given an initial configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 5a left) and the task is to move the end-effector with velocity of constant magnitude and direction in the 2D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The end configuration for each approach is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 5a right and the end-effector trajectories are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We see that the 2D optimization problem is able to reach a greater distance, highlighting that the robot workspace is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Performance comparison In this section, we demonstrate that OpTaS can formulate similar problems and compare its performance to alterna- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' First, we model, with OpTaS, the same problem as used in TracIK [33] and in addition we also model the problem using EXOTica [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The Scipy SLSQP solver [23] was used for OpTaS and EXOTica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' With same Kuka LWR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 6: Figure-of-eight trajectory tracked by the Kuka LWR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 7: Solver duration comparisons for figure of eight motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' (a) Compares an IK tracking approach described in Section V, (b) is a similar comparison that includes a maximization term for manipulability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Green is OpTaS, red is TracIK, and blue is EXOTica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' robot arm in the previous experiment, we setup a task where the robot must track a figure-of-eight motion in task space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 6) and record the CPU time for the solver duration at each control loop cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' TracIK is the fastest (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='035ms), which is expected since it is optimized for a specific problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We see that OpTaS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='608 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='239ms) is faster than EXOTica (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='694 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='300ms) A second experiment, using the same setup as be- fore, was performed comparing the performance of OpTaS against EXOTica with an additional cost term to maxi- mize manipulability [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Despite using the same formulation and solver, OpTaS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='650 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='270ms) achieved better performance than EXOT- ica (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='640±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='404ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Without extensive profiling it is diffi- cult to precisely explain this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' However, EXOTica requires the user to supply analytical gradients for sub-tasks (called task maps in the EXOTica documentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' EXOTica does not provide the gradients for the manipulability task, and thus falls-back to using the finite difference method to estimate the gradient - this can can be slow to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' CONCLUSIONS In this paper, we have proposed OpTaS: an optimization- based task tpecification Python library for TO and MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' OpTaS allows a user to setup a constrained nonlinear pro- grams for custom problem formulations and has been shown to perform well against alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' Parameterization enables programs to act as feedback controllers, motion planners, and benchmark problem formulations and solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' We hope OpTaS will be used by researchers, students, and industry to facilitate the development of control and motion planning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' The code base is easily installed via pip and has been made open-source under the Apache 2 license: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='com/cmower/optas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='1 Optimize6D Optimize2D Ideal path Start (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='4 X (m)CPU Time (ms) OpTaS TraclK EXOTica 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='0 2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='0 Time (s)CPU Time (ms) 10 OpTaS EXOTica 8 6 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FRT4oBgHgl3EQfODd9/content/2301.13512v1.pdf'} +page_content='5 15.' 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index 0000000000000000000000000000000000000000..1d7a94d7d5daf86c5c0026627dc22c23376c156d --- /dev/null +++ b/XNAyT4oBgHgl3EQf9PpX/content/tmp_files/2301.00870v1.pdf.txt @@ -0,0 +1,1356 @@ +TUM-HEP 1447/22, IPMU22-0070 +Enhanced prospects for direct detection of +inelastic dark matter from a non-galactic diffuse +component +Gonzalo Herrera1,2, Alejandro Ibarra1, and Satoshi Shirai3 +1Physik-Department, Technische Universität München, James-Franck-Straße, 85748 +Garching, Germany +2Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6,80805 +München, Germany +3Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of +Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa 277-8583, Japan +Abstract +In some scenarios, the dark matter particle predominantly scatters inelastically with +the target, producing a heavier neutral particle in the final state. In this class of scenarios, +the reach in parameter space of direct detection experiments is limited by the velocity of +the dark matter particle, usually taken as the escape velocity from the Milky Way. On the +other hand, it has been argued that a fraction of the dark matter particles in the Solar +System could be bound to the envelope of the Local Group or to the Virgo Supercluster, +and not to our Galaxy, and therefore could carry velocities larger than the escape velocity +from the Milky Way. In this paper we estimate the enhancement in sensitivity of current +direct detection experiments to inelastic dark matter scatterings with nucleons or electrons +due to the non-galactic diffuse components, and we discuss the implications for some well +motivated models. +1 +Introduction +The existence of dark matter in galaxies, clusters of galaxies and the Universe at large scale is +by now established by their gravitational effects on ordinary matter (for reviews, see e.g. [1– +4]). If the dark matter is constituted by new particles, it is plausible that they could interact +with the ordinary matter through other interactions aside from gravity. A promising avenue to +probe these putative interactions consists in the search for nuclear or electron recoils induced +by dark matter particles entering a dedicated detector at the Earth [5, 6] (for reviews, see e.g. +[7–9]). This search strategy, denominated direct detection, has seen an impressive increase in +sensitivity since it was first proposed more than three decades ago. Yet, no conclusive dark +matter signal has been found to date. +Assuming that the dark matter scatters elastically with the nucleus, current direct detection +experiments restrict the spin-independent interaction cross-section to be smaller than ∼ 1 +1 +arXiv:2301.00870v1 [hep-ph] 2 Jan 2023 + +zeptobarn in the mass range ∼ 10 GeV - 1 TeV [10]. These stringent constraints put pressure on +several well motivated dark matter scenarios, especially those for which the dark matter particle +couples at tree level with the valence quarks in models addressing the electroweak hierarchy +problem [1]. On the other hand, there are many other dark matter scenarios, arguably also +well motivated theoretically, which are largely unconstrained by current searches. +In this paper we will focus on scenarios where the dark matter cannot scatter elastically with +a nucleus (or an electron), so that the stringent limits on the elastic scattering cross-section do +not necessarily hold. This seemingly strong assumption naturally arises in some models. For +instance, the elastic scattering mediated by vector current is forbidden for Majorana dark matter +χ, due to the Majorana nature of fermion: ¯χγµχ = 0 [11]. However, Majorana dark matter +particles may leave an imprint in direct search experiments if they could scatter inelastically +producing a heavier Majorana fermion χ′ in the final state, since there is an off-diagonal fermion +current ¯χ′γµχ ̸= 0. This scenario is approximately realized in the Minimal Supersymmetric +Standard Model, when the lightest supersymmetric particle is almost a pure Higgsino state, +and the other supersymmetric particles are very heavy. In this case, the elastic scattering of +the Higgsino dark matter is suppressed by the large sfermion and gaugino masses, while it has +a large inelastic scattering cross section by the electroweak gauge interactions [12]. Scenarios +of inelastic dark matter have also been motivated phenomenologically, e.g. in [12–22]. +The kinematics of the inelastic scattering differs from the one in the elastic scenario. In +order to allow the production of a heavier neutral particle in the final state, the velocity of the +incoming dark matter particle must be larger than a certain threshold. Therefore, as the mass +difference between the initial and final neutral particles increases, faster and faster dark matter +particles are necessary in order to open kinematically the inelastic process. For dark matter +particles bound to our galaxy, and which have speeds smaller than the escape velocity from +the Milky Way, vesc = 544 km/s [23, 24], the inelastic scattering off a nucleus is kinematically +allowed when the mass difference between the two states is δm < 1/2µv2 +esc, with µ the reduced +mass of the DM-nucleus system; for the scattering off an electron, the inelastic channel is open +when δm < 1/2µev2 +esc − |Enl|, where µe is the reduced mass of the DM-electron system, and +|Enl| is the binding energy of an electron in the (n, l) shell of the target nucleus. In practice, +experiments can only detect recoiling nuclei/ionized electrons within a given energy range, +therefore the mass difference that can be probed in direct searches is smaller than this value. +In this letter we argue that the parameter space of inelastic dark matter scenarios that +can be probed in direct search experiments is larger than the one previously considered in +the literature, that implicitly assumes that the Milky Way is an isolated galaxy. Instead, the +Milky Way is one among the various members of the Local Group, which include M31, M33 +and several dwarf galaxies. It has been argued that the Local Group contains a diffuse dark +matter component, which is not bound to any individual galaxy, and which is distributed +roughly homogeneously over the Local Group [25–27]. Notably, a non-negligible fraction of +the dark matter particles in the Solar System is expected to be associated to this non-galactic +diffuse component, rather than to the Milky Way halo, and could have velocities larger than +the escape velocity from the Milky Way. Consequently, the mass splitting that could be probed +in experiments correspondingly increases. Likewise, the Local Group is one among the many +groups of galaxies embedded in the Virgo Supercluster, which could also contain a diffuse +component [28]. Although the fraction of dark matter particles in the Solar System associated +to the Virgo Supercluster is fairly small, they have very large velocities, and could be pivotal in +generating a signal in direct search experiments when the inelastic scattering is kinematically +2 + +inaccessible for the dark matter bound to the Milky Way and to the Local Group. +The paper is organized as follows. In section 2, we present the non-galactic dark matter +flux at Earth. In section 3, we derive constraints on inelastic dark matter from nuclear recoil +searches, and in section 4, we derive constraints from electron recoil searches. Finally, in section +5, we present our conclusions. +2 +Dark matter flux at Earth +A correct description of the dark matter flux at Earth is crucial for assessing the prospects for +detection of a given dark matter model. The largest contribution to the flux is expected to arise +from dark matter particles in the Milky Way halo. The local density of dark matter particles +and their velocity distribution is unknown. However, it is common in the literature to adopt +the Standard Halo Model (SHM), characterized by a local density ρloc +SHM = 0.3 GeV/cm3 and +an isotropic velocity distribution described by a Maxwell-Boltzmann distribution truncated at +the escape velocity of the Milky Way [29, 30]. In the galactic frame, the velocity distribution +reads: +fSHM(⃗v) = +1 +(2πσ2 +v)3/2Nesc +exp +� +− v2 +2σ2 +v +� +for v ≤ vesc , +(1) +where v = |⃗v|, σv ≈ 156 km/s is the velocity dispersion [30, 31], and vesc = 544 km/s is the +escape velocity from our Galaxy [23, 24]. Further, Nesc is a normalization constant, given by: +Nesc = erf +� vesc +√ +2σv +� +− +� +2 +π +vesc +σv +exp +� +−v2 +esc +2σ2 +v +� +. +(2) +For our chosen parameters, Nesc ≃ 0.993. The contribution to the local dark matter flux from +the Milky Way halo then reads: +FSHM(⃗v) = ρloc +SHM +mDM +vfSHM(⃗v) . +(3) +It is plausible that the dark matter flux at Earth also contains a contribution from dark mat- +ter particles not bound to the Milky Way. Astronomical observations indicate the presence of +diffuse dark matter components homogeneously distributed between clusters and Superclusters +of galaxies [32]. Since these dark matter particles are not gravitationally bound to the Milky +Way, they carry larger velocities than the escape velocity of the Milky Way. In this work, we +consider the contribution to the dark matter flux from the Local Group and from the Virgo +Supercluster. The dark matter particles from the Local Group contribute at the Solar System +with a local density of ρLG ∼ 10−2 GeV/cm3, and are expected to move isotropically with a +narrow velocity distribution, σv.LG ∼ 20 km/s, and with mean velocity vLG ∼ 600 km/s [33]. +The contribution from the Local Group to the dark matter flux at the location of the Solar +System then reads: +FLG(⃗v) = ρloc +LG +mDM +δ(v − vLG) +4πv +. +(4) +Dark matter particles bound to the Virgo Supercluster give a small contribution to the local +dark matter density. Observations indicate that the average density in the diffuse component +3 + +of the Virgo Supercluster is close to the cosmological value ∼ 10−6 GeV/cm3 [28]. However, the +gravitational focusing due to the Local Group leads to an increase in the density at the location +of the Sun by a factor ∼ 1 + v2 +esc/v2 +σVS, where vσVS is the velocity dispersion of the dark matter +particles from the Virgo Supercluster [33]. This value is highly uncertain, but it is expected +to be comparable to that of the observable members of the Supercluster, which ranges from +vσVS ∼ 50 km/s to vσVS ∼ 500 km/s [28, 34]. We consider for concreteness an enhancement on +the local density of dark matter particles from the Virgo Supercluster of ∼ 10, consistent with +the value of the velocity dispersion of the observable members of the Supercluster, which leads +to ρloc +VG ∼ 10−5 GeV/cm3. Current knowledge on the dark matter velocity distribution in the +Virgo Supercluster is much poorer. Following [33], we assume that the dark matter particles +have the typical velocities of the members of the Virgo Supercluster, corresponding to (at least) +vVS ∼ 1000 km/s. The contribution to the dark matter flux at the location of the Solar System +from the Virgo Supercluster can then be written as: +FVS(⃗v) = ρloc +VS +mDM +δ(v − vVS) +4πv +. +(5) +The total (galactic plus non-galactic) dark matter flux at the Solar System is therefore +approximately given by: +F(⃗v) = FSHM(⃗v) + FLG(⃗v) + FVS(⃗v). +(6) +Following [33], we adopt values for the local density of each component such that the total sum +yields the canonical value of the local density used by direct detection experiments ρloc = 0.3 +GeV/cm3, namely ρloc +SHM = 0.26 GeV/cm3 (∼ 88%), ρloc +LG = 0.037 GeV/cm3 (∼ 12%), and +ρloc +VS = 10−5 GeV/cm3 (∼ 0.00003%). +3 +Impact on nuclear recoils +The differential rate of nuclear recoils induced by inelastic up-scatterings of dark matter parti- +cles traversing a detector at the Earth is given by: +dR +dER += +� +i +ξi +mAi +� +v≥vi +min(ER) +d3vF(⃗v + ⃗v⊙) dσi +dER +(v, ER) . +(7) +Here, ⃗v is the dark matter velocity in the rest frame of the detector, F(⃗v + ⃗v⊙) is the dark +matter flux in the detector frame, and ⃗v⊙ is the velocity of the Sun with respect to the Galactic +frame with |⃗v⊙| ≈ 232 km/s [35]. For the inelastic scattering with mass splitting between two +dark matter states, δDM, the minimum velocity necessary to induce a recoil with energy ER of +the nucleus i with mass mAi and mass fraction ξi in the detector reads +vi +min(ER) = +1 +� +2ERmAi +�ERmAi +µAi ++ δDM +� +. +(8) +Further, for spin-independent interactions, the differential dark matter-nucleus cross section +reads, +dσSI +i +dER +(v, ER) = +mAi +2µ2 +Aiv2σSI +0,iF 2 +i (ER) . +(9) +4 + +Here mAi is mass of the nucleus i, µAi is the reduced mass of the dark matter-nucleus i system +and F 2 +i (ER) is the nuclear form-factor, for which we adopt the Helm prescription. Besides, +σSI +0,i is the spin-independent dark matter-nucleus scattering cross section at zero momentum +transfer, which depends on the details of the dark matter model and the target nucleus. From +the differential rate, one can calculate the total recoil rate using: +R = +� ∞ +0 +dER ϵi(ER) dR +dER +, +(10) +where ϵi(ER) is the efficiency of that experiment. Finally, the total number of expected recoil +events is N = R · E, with E the exposure (i.e. mass multiplied by live-time). +In our analysis, we will consider two scenarios for the coupling of dark matter to nucleons. +First, we will consider a Majorana dark matter candidate. In this case +σSI +0,i = 4µ2 +Ai +π +� +Zif p +S + (Ai − Zi)f n +S +�2 +, +(11) +where f p +S and f n +S parametrize the strength of the scalar interactions to the proton and the +neutron (see e.g. [7, 36]). It is common to write Eq. (11) as +σSI +0,i = µ2 +Ai +µ2 +p +� +Zi + (Ai − Zi)f n +S +f p +S +�2 +σDM,p , +(12) +with µp the reduced mass of the DM-proton system and σDM,p an effective DM-proton inter- +action cross-section. Within the Majorana dark matter scenario, we will consider in particular +the widely adopted benchmark case where the interaction is “isoscalar”, i.e. when the dark +matter couples with equal strength to protons and neutrons, for which +σSI +0,i = µ2 +Ai +µ2 +p +A2 +i σDM,p . +(13) +We will also consider a scenario where the dark matter has hypercharge Y , and interacts +with the quarks via the exchange of a Z boson. +In this case, σSI +0,i has the same form as +Eq. (11), replacing the scalar couplings by the corresponding vector couplings, f p,n +S +→ f p,n +V . +For interactions with the Z boson, f p +V and f n +V are explicitly given by: +f p +V = GFζY +2 +√ +2 (1 − 4 sin2 θW) , +f n +V = −GFζY +2 +√ +2 , +(14) +with ζ = 1 (ζ = 2) for fermionic (bosonic) dark matter [5, 21, 37]. In this scenario, the dark +matter-nucleus cross section can be related to the dark matter-proton cross-section through: +σSI +0,i = µ2 +Ai +µ2 +p +� +Zi − +(Ai − Zi) +(1 − 4 sin2 θW) +�2 +σDM,p , +(15) +which is independent of the dark matter hypercharge and spin. +To assess the impact of the non-galactic diffuse components for direct detection experiments, +we plot in Figure. 1 the differential rate of inelastic scatterings in the LUX-ZEPLIN experiment +5 + +10 +20 +30 +40 +50 +60 +70 +80 +ER [keV] +10−6 +10−4 +10−2 +100 +102 +104 +106 +dR +dER [keV−1] +mDM = 1 TeV +σDM−p = 10−38cm2 +LUX-ZEPLIN (SHM, δDM = 100 keV) +LUX-ZEPLIN (SHM, δDM = 200 keV) +LUX-ZEPLIN (SHM+Non-galactic, δDM = 100 keV) +LUX-ZEPLIN (SHM+Non-galactic, δDM = 200 keV) +CEνNS (Solar neutrinos) +Figure 1: Differential rate for the inelastic scattering of a Majorana dark matter candidate in +the “isoscalar” scenario with mass mDM = 1 TeV, for δDM = 100 keV (light blue) and 200 keV +(dark blue), for a dark matter flux at Earth as modelled by the Standard Halo Model (dotted +line) or including also the contribution from the non-galactic diffuse dark matter component +(solid line). For the plots it was assumed σDM,p = 10−38 cm2. +for the “isoscalar” scenario, assuming mDM = 1 TeV and σDM,p = 10−38 cm2, for δDM = 100 +keV (light blue) and 200 keV (dark blue), including in the flux only the contribution from dark +matter bound to the Milky Way (dotted lines), as commonly assumed in the literature, and +including the contribution from the non-galactic diffuse component (solid lines). The impact +of the non-galactic component in the differential rate is apparent from the figure, and increases +the number of events at all recoil energies, especially in the region with low ER which is not +kinematically accessible to the galactic dark matter. The non-galactic dark matter, therefore, +has implications not only for enhancing the sensitivity of the experiment, but also for the +interpretation of a putative dark matter signal. +Current direct search experiments have not observed a significant excess of nuclear recoils, +which allows to derive upper limits on the dark matter nucleon cross section for given com- +binations of the dark matter mass and mass splitting between the dark matter particle and +the neutral particle in the final state. In Figure 2, we show upper limits on the dark matter- +proton spin-independent scattering cross section versus mass splitting for mDM = 1 TeV from +LUX-ZEPLIN (blue) [10], PICO60 (green) [38], CRESST-II (red) [39], and from a radiopurity +measurement in a CaWO4 crystal (orange) [40, 41]. The dotted lines represent the limits ob- +tained considering the galactic dark matter (described by the SHM) as the only contribution +to the dark matter flux, while the solid lines were obtained including also the contributions to +the flux from the non-galactic diffuse component in the Solar System. In the upper left plot, +we show the limits for a Majorana dark matter candidate in the “isoscalar” scenario, and in the +upper right plot, the most conservative limit for the Majorana dark matter, without making +assumptions on the coupling strengths, derived following the approach of [42]. Lastly, in the +lower plot we show the limits for a scenario where the dark matter interacts with the nucleus via +the exchange of a Z-boson. In the latter plot we also show the dark matter-proton scattering +cross-section for scenarios of a fermionic dark matter, and Y = 1/2 (corresponding to the well +motivated scenario of the Higgsino dark matter in the limit of high scale supersymmetry [12]), +6 + +0 +200 +400 +600 +800 +1000 +1200 +δDM [keV] +10−48 +10−46 +10−44 +10−42 +10−40 +10−38 +10−36 +10−34 +10−32 +10−30 +σSI +DM−p[cm2] +mDM = 1 TeV +Majorana DM, f n = f p +LUX-ZEPLIN (SHM) +LUX-ZEPLIN (SHM + Non-galactic) +PICO60 (SHM) +PICO60 (SHM + Non-galactic) +CRESST II (SHM) +CRESST II (SHM + Non-galactic) +CaWO4 (SHM) +CaWO4 (SHM + Non-galactic) +0 +200 +400 +600 +800 +1000 +1200 +δDM [keV] +10−48 +10−46 +10−44 +10−42 +10−40 +10−38 +10−36 +10−34 +10−32 +10−30 +σSI +DM−p[cm2] +mDM = 1 TeV +Majorana DM, f n, f p free +LUX-ZEPLIN (SHM) +LUX-ZEPLIN (SHM + Non-galactic) +PICO60 (SHM) +PICO60 (SHM + Non-galactic) +CRESST II (SHM) +CRESST II (SHM + Non-galactic) +CaWO4 (SHM) +CaWO4 (SHM + Non-galactic) +0 +200 +400 +600 +800 +1000 +1200 +δDM [keV] +10−48 +10−46 +10−44 +10−42 +10−40 +10−38 +10−36 +10−34 +10−32 +σSI +DM−p[cm2] +mDM = 1 TeV +Y=1/2 +Y=1 +Y=3/2 +Z-boson mediation +LUX-ZEPLIN (SHM) +LUX-ZEPLIN (SHM + Non-galactic) +PICO60 (SHM) +PICO60 (SHM + Non-galactic) +CRESST II (SHM) +CRESST II (SHM + Non-galactic) +CaWO4 (SHM) +CaWO4 (SHM + Non-galactic) +Figure 2: 90% C.L upper limits on the spin-independent dark matter-proton inelastic cross +section for a dark matter mass of 1 TeV as a function of the mass splitting, from LUX-ZEPLIN +(blue), PICO60 (green), CRESST-II (red and orange) and from a CaWO4 detector radiopurity +measurement (orange). We show the limits for three different scenarios: Majorana dark matter +with scalar interactions f p = f n (upper left plot), arbitrary f p and f n (upper right plot), +and dark matter interacting via the Z-boson (lower plot). In the lower plot, we also show for +reference the predicted value of the cross-section with a xenon target for scenarios of fermionic +dark matter with hypercharge Y = 1/2, 1, 3/2. +Y = 1 and Y = 3/2 (which correspond to different scenarios of minimal dark matter [37]), for a +xenon target. For other targets, the expected cross section for mDM = 1 TeV scales as ∼ Ai/Zi, +being indistinguishable in the Figure. +As seen in the plots, for all the scenarios the non-galactic diffuse component enhances the +sensitivity of experiments to inelastic dark matter, allowing to probe larger mass splittings. +For instance, for our representative dark matter mass of 1 TeV, the LUX-ZEPLIN experiment +is insensitive to dark matter particles of the Milky Way scattering inelastically if the mass +difference with the neutral particle in the final state is δDM ≳ 300 keV. However, the presence +of dark matter in the Solar System from the envelope of the Local Group extends the reach +up to δDM ≃ 330 keV and allows to probe uncharted parameter space for large mass splittings. +7 + +Concretely, the LUX-ZEPLIN experiment sets for the isoscalar scenario the limit σSI +DM−p ≲ +10−44 cm2 for δDM = 250 keV, which is about three orders of magnitude stronger than the limit +obtained assuming that all dark matter is bound to the Milky Way, and only a factor of 100 +weaker than the limit on the elastic scattering cross-section i.e. for δDM = 0. For the interaction +mediated by the Z-boson the upper limit is σSI +DM−p ≲ 10−44 cm2, and the most conservative limit +without making assumptions on the form of the interaction is σSI +DM−p ≲ 10−40 cm2, obviously +much weaker than for concrete scenarios. The dark matter particles from the Virgo Supercluster +extend the reach to even larger mass differences, up to δDM ≃ 450 keV and sets for the isoscalar +scenario the limit σSI +DM−p ≲ 5 × 10−40 cm2 for δDM = 450 keV; for the interaction mediated +by the Z-boson the upper limit is σSI +DM−p ≲ 10−41 cm2, while the model independent limit is +σSI +DM−p ≲ 5 × 10−36 cm2. Similar conclusions apply for the PICO and CRESST experiments, +and from the radiopurity measurements on a CaWO4 target. +It is interesting to note the complementarity of the different experiments in probing the +parameter space of inelastic dark matter scenarios. Both in the scenario of a Majorana dark +matter with f n = f p and for the scenario with Z-boson mediation, LUX-ZEPLIN is the most +sensitive probe for small δDM, whereas the radiopurity measurements on a CaWO4 is the most +sensitive probe for large δDM. +PICO-60 is relevant for intermediate values of δDM, and is +in fact the most sensitive current probe of some well motivated dark matter scenarios, as +suggested by the gray lines in the Figure, which correspond to the expected cross-section for +different scenarios of electroweakly interacting fermionic dark matter. The complementarity +of experiments in probing these scenarios is investigated in Figure 3. The dotted lines show +the upper limit on the mass splitting as a function of the dark matter mass assuming the +Standard Halo Model. Under this common assumption, LUX-ZEPLIN is the most constraining +experiment over the whole parameter space considered. However, when including the non- +galactic components, different experiments contribute to set the upper limit, as reflected by the +breaks in the solid lines in the Figure: LUX-ZEPLIN remains as the most sensitive experiment +for small dark matter masses, while PICO-60 is the best experiment for larger masses. Further, +the dark matter mass at which PICO-60 becomes the leading experiment becomes larger and +larger as the dark matter hypercharge increases. As seen in the Figure, for this class of scenarios +the non-galactic components in the dark matter flux enhance the sensitivity of experiments to +the mass splitting by a factor ∼ 2 for mDM = 100 GeV - 1 TeV. +It is noteworthy the pivotal role of the radiopurity measurements on a CaWO4 target to +probe large mass splittings in inelastic dark matter scenarios. This can be understood from the +expression for the minimum DM velocity required to induced a recoil with energy ER, Eq. (8). +Let us consider a velocity distribution where the maximum speed is v∗. Then, for an experiment +capable of detecting a recoil of a nucleus Ai with energy ER, the maximum mass splitting that +can be probed is: +δDM ≤ +� +2ERmAiv∗ − ERmAi +µAi +≤ 1 +2µAiv2 +∗ , +(16) +where the absolute maximum is reached when ER = µ2 +Aiv2 +∗/(2mAi). This is shown in Figure 4, +for v∗ = 764 km/s, v∗ = 820 km/s, v∗ = 1220 km/s (solid lines), corresponding respectively to +the maximal velocity at the Earth of dark matter particles bound to the Milky Way (described +by the Standard Halo Model), from the Local Group envelope and from the Virgo Supercluster. +The plot also shows the range of recoil energies that can be detected by the CRESST-II ex- +periment and by the radiopurity measurements in CaWO4 crystals. As seen in the plot, while +8 + +102 +103 +104 +mDM [GeV] +100 +200 +300 +400 +500 +δDM [keV] +Upper limits at 90% CL from LZ+PICO60+CaWO4, Dirac dark matter +Y = 1/2, SHM +Y = 1/2, SHM + Non-galactic +Y = 1, SHM +Y = 1, SHM + Non-galactic +Y = 3/2, SHM +Y = 3/2, SHM + Non-galactic +Figure 3: Upper limits on the mass splitting for electroweakly charged (pseudo-)dirac dark +matter as a function of the dark matter mass, for different choices of the hypercharge, and +including in the flux only the Standard Halo Model component (dotted lines) or also the non- +galactic diffuse components (solid lines). +CRESST-II can only probe up to δDM ∼ 700 keV, the radiopurity measurements allow to probe +up to δDM ∼ 1200 keV, when including the flux component from the dark matter bound to the +Virgo Supercluster (however with a lower sensitivity due to the smaller exposure). From this +plot it follows that the CRESST experiment would have an enhanced sensitivity to inelastic +dark matter scenarios if the window of recoil energies used in the analysis were extended to +larger values. Let us note that for low dark matter masses, extending the search window to +higher recoil energies would not help in probing larger values of the mass splitting. This is +illustrated in the Figure for mDM = 100 GeV, from where it is apparent that to increase the +reach in mass splittings it is necessary to extend the search to lower recoil energies. +Finally, we show in Figure 5 the isocontours with the 90% C.L. upper limits on the cross- +section for different dark matter masses and mass splittings, from LUX-ZEPLIN (top panels), +PICO60 (middle panels) and from radiopurity measurements on a CaWO4 target (bottom +panels), considering that all dark matter in the Solar System is bound to the Milky Way, as +commonly assumed (left panels), and including the non-galactic components (right panels). +The enhancement in sensitivity is clear from the plots. +4 +Impact on electron recoils +The differential ionization rate induced by dark matter-electron inelastic scattering in liquid +xenon, with mass splitting between the two dark matter states given by δDM, reads: +dRion +dlnEer += NT +� +n,l +� +v≥vnl +min(Eer) +d3vF(⃗v + ⃗v⊙) dσnl +ion +dlnEer +(v, Eer) , +(17) +where NT is the number of target nuclei and +vnl +min(Eer) = +� +2 +mDM +(Eer + |Enl| + δDM) +(18) +9 + +100 +101 +102 +103 +104 +ER [keV] +200 +400 +600 +800 +1000 +1200 +1400 +δDM [keV] +mDM = 100 GeV +CaWO4 +CRESST-II +SHM +SHM+LG +SHM+LG+VS +100 +101 +102 +103 +104 +ER [keV] +200 +400 +600 +800 +1000 +1200 +1400 +δDM [keV] +mDM = 1 TeV +CaWO4 +CRESST-II +SHM +SHM+LG +SHM+LG+VS +Figure 4: Values of the mass splitting δDM that can produce a recoil energy in a 184W target +for mDM = 100 GeV (left plot) and mDM = 1 TeV (right plot) when the maximal velocity of +the dark matter particles at Earth is v∗ = 764 km/s (dotted lines), v∗ = 820 km/s (dashed +lines) and v∗ = 1220 km/s (solid lines), corresponding respectively to dark matter bound to +the Milky Way (described by the Standard Halo Model), bound to the Local Group and bound +to the Virgo Supercluster. +For comparison, we also show the range of recoil energies that +can be detected by the CRESST-II experiment (red band) and by the CaWO4 radiopurity +measurement (yellow band). +is the minimum dark matter velocity necessary to ionize a bound electron in the (n, l) shell of +a xenon atom (with energy Enl), giving a free electron with energy Eer. Further, dσnl +ion/dlnEer +is the differential ionization cross section, given by: +dσnl +ion +dlnEer +(v, Eer) = +¯σDM−e +8µ2 +DM,ev2 +� qnl +max +qnl +min +dqq +��f nl +ion(k′, q) +��2 |FDM(q)|2 . +(19) +Here, µDM,e is the reduced mass of the dark matter-electron system, ¯σDM−e is the dark matter- +free electron scattering cross section at fixed momentum transfer q = αme, +��f nl +ion(k′, q) +��2 is the +ionization form factor of an electron in the (n, l) shell with final momentum k′ = √2meEer +and momentum transfer q, and FDM(q) is a form factor that encodes the q-dependence of the +squared matrix element for dark matter-electron scattering and depends on the mediator under +consideration. The maximum and minimum values of the momentum transfer needed to ionize +a bound electron in the (n, l) shell recoil with energy Eer from the interaction of a dark matter +particle with speed v are: +qnl +max +min(Eer) = mDMv +� +�1 ± +� +1 − +�vnl +min(Eer) +v +�2� +� , +(20) +with vnl +min(Eer) defined in Eq. (18). Finally, the total number of expected ionization events reads +N = Rion · E, with Rion the total ionization rate, calculated from integrating Eq.(17) over the +experimentally measured recoil energies, and E the exposure (i.e. mass multiplied by live-time) +of the experiment. +10 + +102 +103 +104 +mDM [GeV] +100 +200 +300 +400 +500 +600 +δDM [keV] +Upper limits at 90% C.L from LUX-ZEPLIN, SHM, Isoscalar +10−47 +10−45 +10−43 +10−41 +10−39 +10−37 +σDM−p +102 +103 +104 +mDM [GeV] +100 +200 +300 +400 +500 +600 +δDM [keV] +Upper limits at 90% C.L from LUX-ZEPLIN, Non-galactic, Isoscalar +10−47 +10−45 +10−43 +10−41 +10−39 +10−37 +σDM−p +102 +103 +104 +mDM [GeV] +100 +200 +300 +400 +500 +600 +δDM [keV] +Upper limits at 90% C.L from PICO60, SHM, Isoscalar +10−45 +10−43 +10−41 +10−39 +10−37 +10−35 +σDM−p +102 +103 +104 +mDM [GeV] +100 +200 +300 +400 +500 +600 +δDM [keV] +Upper limits at 90% C.L from PICO60, Non-galactic, Isoscalar +10−45 +10−43 +10−41 +10−39 +10−37 +10−35 +σDM−p +102 +103 +104 +mDM [GeV] +200 +400 +600 +800 +1000 +1200 +1400 +δDM [keV] +Upper limits at 90% C.L from CaWO4, SHM, Isoscalar +10−41 +10−39 +10−37 +10−35 +10−33 +10−31 +10−29 +10−27 +σDM−p +102 +103 +104 +mDM [GeV] +200 +400 +600 +800 +1000 +1200 +1400 +δDM [keV] +Upper limits at 90% C.L from CaWO4, Non-galactic, Isoscalar +10−41 +10−39 +10−37 +10−35 +10−33 +10−31 +10−29 +10−27 +σDM−p +Figure 5: Isocontours of the 90% C.L. upper limits on the spin-independent dark matter-proton +inelastic cross-section for the isoscalar scenario (f p = f n) in the parameter space spanned by +the dark matter mass and mass splitting, from LUX-ZEPLIN (top panels), PICO60 (middle +panels) and radiopurity measurements in a CaWO4 target (lower panels), assuming that all +dark matter in the Solar System is bound to the Milky Way (left panels) or including the +non-galactic diffuse component (right panels). +11 + +In semiconductor detectors, the electron excitation rate induced by dark matter-electron +inelastic scatterings, with a mass splitting δDM, reads [43, 44] +R = 1 +ρT +¯σDM−e +µ2 +DM,e +π +α +� +d3vF(⃗v + ⃗v⊙) +v +� +d3q +(2π)3q2 |FDM(q)|2 +� dω +2π +1 +1 − e−βω Im +� +−1 +ϵ(ω, ⃗q) +� +δ +� +ω + δDM + +q2 +2mχ +− ⃗q · ⃗v +� +, +(21) +where w is the energy deposited in the material, ⃗q is the momentum transfer of the process, +and ρT is the target density. The rate involves an integration of the Electronic Loss Function +(ELF) of the target material, which we calculate with DarkELF [44]. For the dielectric function +ϵ(ω, q), we use the Lindhard method, which treats the target as a non-interacting Fermi liquid. +Finally, the total number of events reads N = R · E, with E the exposure (i.e. mass multiplied +by live-time) of the experiment. +The non-observation of a significant excess of electron recoils in a given experiment allows +to set upper limits on the dark matter-electron scattering cross section, for a given dark matter +mass and a given mass splitting between the dark matter particle and the heavier neutral state. +We show in Figure 6, upper limits on the inelastic dark matter-electron cross section versus mass +splitting for a fixed dark matter mass of mDM = 1 GeV from XENON1T [45](blue lines), and +from the semiconductor experiment SENSEI [46](purple lines), both when considering the SHM +flux only (solid lines), and when including the non-galactic components to the dark matter flux +(dotted lines). In the upper plots, we take the form factor FDM = α2m2 +e/q2, corresponding to +an ultralight or massless mediator. In the middle plots, we take the form factor FDM = αme/q, +corresponding to an electric dipole interaction, and in the lower plots we take the form factor +FDM = 1, corresponding to a heavy mediator [47, 48]. +As can be seen in the Figure, the non-galactic components enhance the sensitivity to the +mass splitting of both XENON1T and SENSEI by a factor of ∼ 2, compared to the sensitivity +estimated from considering just the galactic component. This conclusion holds independently +of the choice of the dark matter form factor. Further, the reach in cross-section is enhanced due +to the non-galactic components, especially at low mass splittings, being the effect stronger for +XENON1T than for SENSEI. For comparison, we also show as a grey band the cross section +for which the observed dark matter abundance is reproduced via freeze-in in the case of an +ultralight mediator [49], or via freeze-out in the case of a heavy mediator [50]. Clearly, the +non-galactic dark matter components allow to probe larger values of the mass splitting. +5 +Conclusions +We have investigated the impact of a non-galactic diffuse dark matter component inside the +Solar System for the detection of the inelastic scattering of a dark matter particle in direct +search experiments. Concretely, we have considered the contribution to the dark matter flux +from dark matter particles in the envelope of the Local Group and from the Virgo Supercluster. +Their speeds in the galactic frame are ∼ 600 km/s and ∼ 1000 km/s, respectively, which are +larger than the maximal speed of dark matter particles bound to the Milky Way, ∼ 540 km/s. +As a result, the region of parameter space that can be probed with current experiments is larger +than reported in previous works, that implicitly assumed that the Milky Way is an isolated +galaxy in the Universe. +12 + +100 +101 +102 +δDM [eV] +10−47 +10−44 +10−41 +10−38 +10−35 +10−32 +10−29 +10−26 +¯σe[cm2] +FDM = α2m2 +e/q2 +mDM = 1 GeV +Freeze-in +Ultralight mediator +SENSEI (SHM) +SENSEI (SHM+Non-galactic) +XENON1T (SHM) +XENON1T (SHM+Non-galactic) +100 +101 +102 +δDM [eV] +10−47 +10−44 +10−41 +10−38 +10−35 +10−32 +10−29 +10−26 +¯σe[cm2] +FDM = αme/q +mDM = 1 GeV +Dipole interaction +SENSEI (SHM) +SENSEI (SHM+Non-galactic) +XENON1T (SHM) +XENON1T (SHM+Non-galactic) +100 +101 +102 +δDM [eV] +10−47 +10−44 +10−41 +10−38 +10−35 +10−32 +10−29 +10−26 +¯σe[cm2] +FDM = 1 +mDM = 1 GeV +Freeze-out (Pseudo-Dirac fermion) +Massive mediator +SENSEI (SHM) +SENSEI (SHM+Non-galactic) +XENON1T (SHM) +XENON1T (SHM+Non-galactic) +Figure 6: 90% C.L upper limits on the spin-independent dark matter-electron inelastic cross +section for a dark matter mass of 1 GeV, as a function of the mass splitting, from XENON1T +(blue) and SENSEI (purple), when the dark matter-electron interaction is mediated by an +ultralight dark photon (upper left plot), by a dipole operator (upper right plot), or by a heavy +mediator (lower plot). +For nuclear recoils, the non-galactic component expands the reach in mass splitting at +the LUX-ZEPLIN, PICO60, and CRESST-II experiments by a factor ∼ 2 in the mass range +mDM = 10 GeV- 10 TeV, and enhances significantly the reach in cross-section, especially close +to the kinematic threshold for the galactic dark matter. For instance, for mDM = 1 TeV and +δDM = 250 keV, the sensitivity to the cross-section improves by about three orders of magnitude. +We have also stressed the relevance of experiments capable of detecting high recoil energies +for probing the parameter space of inelastic dark matter scenarios. We have illustrated this +capability with the radiopurity measurements in CaWO4 crystals performed by the CRESST +collaboration, and which allows to probe up to δDM ∼ 1.2 MeV (1.4 MeV) for mDM = 1 TeV +(10 TeV). For electron recoils, the conclusions are analogous, allowing to increase reach in mass +splitting of the XENON1T and SENSEI experiments also by a factor ∼ 2 for dark matter +13 + +10−2 +10−1 +100 +101 +mDM [GeV] +5 +10 +15 +20 +25 +30 +δDM [eV] +Upper limits at 90% C.L from SENSEI, SHM, Massive mediator +10−37 +10−35 +10−33 +10−31 +10−29 +10−27 +10−25 +¯σDM−e +10−2 +10−1 +100 +101 +mDM [GeV] +5 +10 +15 +20 +25 +30 +δDM [eV] +Upper limits at 90% C.L from SENSEI, Non-galactic, Massive mediator +10−37 +10−35 +10−33 +10−31 +10−29 +10−27 +10−25 +¯σDM−e +10−2 +10−1 +100 +101 +mDM [GeV] +100 +200 +300 +400 +500 +600 +δDM [eV] +Upper limits at 90% C.L from XENON1T, SHM, Massive mediator +10−41 +10−40 +10−39 +10−38 +10−37 +10−36 +10−35 +10−34 +¯σDM−e +10−2 +10−1 +100 +101 +mDM [GeV] +100 +200 +300 +400 +500 +600 +δDM [eV] +Upper limits at 90% C.L from XENON1T, Non-galactic, Massive mediator +10−41 +10−40 +10−39 +10−38 +10−37 +10−36 +10−35 +10−34 +¯σDM−e +Figure 7: Isocontours of the 90% C.L. upper limits on the dark matter-electron inelastic scat- +tering cross-section for the heavy mediator scenario (FDM = 1) in the parameter space spanned +by the dark matter mass and mass splitting, from SENSEI (top panels), and XENON1T (lower +panels), assuming that all dark matter in the Solar System is bound to the Milky Way (left +panels) or including the non-galactic component diffuse (right panels). +masses in the range mDM = 0.01 GeV-10 GeV, +Acknowledgments +The work of GH and AI was supported by the Collaborative Research Center SFB1258 and by +the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s +Excellence Strategy - EXC-2094 - 390783311. The work of SS is supported by Grant-in-Aid +for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technol- +ogy (MEXT), Japan, 18K13535, 20H01895, 20H05860 and 21H00067, and by World Premier +International Research Center Initiative (WPI), MEXT, Japan. +14 + +A +Derivation of upper limits from direct detection exper- +iments +To derive upper limits on the inelastic dark matter-nucleon scattering cross section, as a function +of the dark matter mass and/or the dark matter mass splitting, we follow a poissonian-likelihood +approach, and we calculate the rates for the different experiments/detectors independently. For +the LUX-ZEPLIN experiment, we use the data from [10], with an exposure of 0.904 tonne×year, +a region of interest extending from 2 keV to 70 keV, and the efficiency function reported by +the collaboration. Given the agreement of the number of signal events with the background +prediction reported by the collaboration, we take a 90% C.L. upper limit on the number of +signal events of 2.71. For the PICO-60 experiment, we use the results from [38], corresponding +to an exposure of 9.356 kg×year, a region of interest extending from 13.5 keV to 100 keV, and +the efficiency function reported by the collaboration. Since PICO-60 observed no signal events, +we take a 90% C.L. upper limit on the number of signal events of 2.71. For CRESST-II, we use +the published data [39], corresponding to an exposure of 52 kg×days. We do not consider as +signal events those belonging to the acceptance region of the experiment at low recoil energies, +but instead, we consider the recoil energy region extending from 30 keV to 120 keV, which gives +an upper limit of 4 signal events. Finally, for the CaWO4 radiopurity measurement from [40], +we take an exposure of 90.10 kg×days, with a recoil energy region extending from 300 keV to +2000 keV, and a number of 3 signal events. +For the inelastic dark matter-electron scattering cross-section, we derive upper limits at 90% +C.L at fixed momentum transfer q = αme using data from XENON1T [45] and SENSEI [46]. +We consider the observed event rate XENON1T between 150-3000 photoelectrons (PE), which +corresponds to the range 0.18 keVee to 3.5 keVee (kiloelectronvolt electron equivalent). We take +the efficiency function from [45], an exposure of 22 ± 3 tonne-days and an upper limit on the +number of events of 39.2. For SENSEI, we sum-up the observed events in the energy bins +ranging from 4.91 eV to 16.31 eV, resulting in an upper limit of 4.957 events per gram day of +exposure. 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In: JHEP 04 (2022), p. 060. doi: 10.1007/JHEP04(2022)060. arXiv: +2108.13422 [hep-ph]. +19 + diff --git a/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf b/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b30e8bc983df793bb945baf2070e74b25c0f3cf5 --- /dev/null +++ b/XtAzT4oBgHgl3EQfKfuP/content/2301.01098v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:03fe3e629cce619c8f4b150c0e0a20cb51a14f979eca4f6abcc6f143dd88a1e0 +size 4501676 diff --git a/YNFQT4oBgHgl3EQfdzaO/content/tmp_files/2301.13332v1.pdf.txt b/YNFQT4oBgHgl3EQfdzaO/content/tmp_files/2301.13332v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..77931ae1638d197b4e7373cab9d546365215ee20 --- /dev/null +++ b/YNFQT4oBgHgl3EQfdzaO/content/tmp_files/2301.13332v1.pdf.txt @@ -0,0 +1,1307 @@ +2CIM: Area-Efficient 2-Cycle Integer Multipliers +Ahmad Houraniah +Department of Computer Science +¨Ozye˘gin University +Istanbul, Turkey +ahmad.houraniah@ozu.edu.tr +H. Fatih Ugurdag +Department of Electrical +and Electronics Engineering +¨Ozye˘gin University +Istanbul, Turkey +fatih.ugurdag@ozyegin.edu.tr +Cengiz Emre Dedeagac +Department of Computer Science +¨Ozye˘gin University +Istanbul, Turkey +emre.dedeagac@ozu.edu.tr +Abstract—Fast multipliers with large bit widths can occupy +significant silicon area, which, in turn, can be minimized by +employing multi-cycle multipliers. This paper introduces archi- +tectures and parameterized Verilog circuit generators for 2- +cycle integer multipliers. When implementing an algorithm in +hardware, it is common that less than 1 multiplication needs +to be performed per clock cycle. It is also possible that the +multiplications per cycle is a fractional number, e.g., 3.5. In such +case, we can surely use 4 multipliers, each with a throughput of 1 +result per cycle. However, we can instead use 3 such multipliers +plus a multiplier with a throughput of 1/2. Resource sharing +allows a multiplier with a lower throughput to be smaller, hence +area savings. These multipliers offer customization in regards +to the latency and clock frequency. All proposed designs were +automatically synthesized and tested for various bit widths. Two +main architectures are presented in this work, and each has +several variants. Our 2-cycle multipliers offer up to 21%, 42%, +32%, 41%, and 48% of area savings for bit widths of 8, 16, 32, +64, and 128, with respect to synthesizing the “*” operator with +throughput of 1. Furthermore, some of the proposed designs also +offer power savings under certain conditions. +Index Terms—computer arithmetic, multi-cycle multiplier, +resource-sharing, pipelining +I. INTRODUCTION +An integer multiplier is an essential building block for +various ASICs and CPUs. Integer multipliers can get quite +expensive in terms of area as bit width increases. Large +integers are used in a wide range of applications that require a +high degree of precision. Floating-point operations are unsuit- +able for some applications because of so-called “catastrophic +cancellations” [1]. The addition and subtraction of floating- +point numbers that greatly vary in magnitude can produce +rounding errors, resulting in significant data loss. Floating- +point operations are significantly more expensive than fixed- +point operations regarding area complexity and latency. For +these reasons, fixed point representation is preferred for many +applications. The CUDA [2] programming model and software +environment, which NVIDIA develops, recently introduced a +128-bit integer data type, which is intended for applications +that require a higher degree of precision within a predeter- +mined range. This signifies the importance of large fixed- +point integers. Multiplying such large integers can require +significant hardware resources. For this reason, decreasing the +area complexity for integer multipliers can be very beneficial. +There exists a large number of applications containing data +flow paths that require less than one multiplication every +two or more clock cycles. Such a case is found in RSIC-V +softcore implementations, where area efficiency is vital, even +if it comes at the expense of increasing the latency of ALU +operations such as multiplication. +Applications can contain numerous multiplications in their +data flow paths. Due to the area complexity of these multipli- +ers, the same multiplication circuits are used multiple times to +minimize the area complexity of the system. These multiplica- +tions typically need to be computed within a predefined period. +Since a conventional multiplier can only be used once in every +clock cycle, using a single multiplier can severely limit the +system’s throughput. The throughput can be maintained by +using several multiplication circuits operating in parallel. The +number of multipliers necessary can be calculated by dividing +the number of multiplications by the period they need to be +computed within (in clock cycles). Using this formula, the +number of multipliers required is not always an integer. The +conventional approach is to round up this value, which causes +one of these multipliers to be underutilized. A fully-pipelined +multiplier that can accept new inputs in consecutive clock +cycles would remain underutilized in such applications. +In this work, various 2-cycle unsigned integer multiplier +(2CIM) designs are proposed to decrease the area complexity +for these underutilized multipliers. 2CIM architectures can +be extended for signed integer multiplication as well. 2CIM +designs offer partial multipliers that can offer area-efficient +designs with a throughput of 1/2. The conventional approach +of multiplying unsigned integers comprises two steps: partial +product generation (PPG) and partial production summation. +The PPG stage is typically done using AND gates, where one +integer is shifted and multiplied with each bit of the other +integer. The PPG stage produces multiple variables that must +be summed to produce the multiplication result. This is done +using a tree of full adder (FA) and half adder (HA) cells. +Using a standard ripple carry adder structure to handle the +partial product summation can result in a long critical path +and an increased area complexity. The optimization of integer +multiplications has been a thoroughly studied topic, and the +conventional approach is to use a carry-save adder for the +summation stage. +Multiplication can be solved using a divide and conquer +arXiv:2301.13332v1 [cs.AR] 30 Jan 2023 + +strategy, where any multiplication can be divided into multiple +smaller ones, this is often seen as an algorithmic problem, and +a great deal of research was done to reduce the algorithmic +complexity of multiplications. The same concepts used for +reducing the algorithmic complexity can be applied to hard- +ware for area reductions since multiplication can be spread +across multiple clock cycles. The conventional schoolbook +approach is to divide multiplication using the distributive +property. One multiplication can be split into various smaller +multiplications. Equation 1 shows how a single multiplication +can be divided into two smaller multiplications using the +schoolbook approach, where N is the bit width of the second +multiplicand B. +Y = A ∗ B = A ∗ {B1, B0} = A ∗ B0 + A ∗ B1 ∗ 2N/2 (1) +Such an approach can be applied for area reductions, where a +smaller multiplication circuit can be used repetitively to com- +pute several smaller multiplications. Although multiplication +can be implemented using a recursive approach, implementing +several division levels requires more control logic. Since the +smaller multiplier needed for a single division level is expected +to be used twice for each multiplication, new inputs can only +arrive once every two clock cycles (CCs), which will be called +the initiation interval (II). +The rest of the paper is organized as follows: Section +II presents the previous work. Section III presents the ar- +chitectures proposed in this work. Section IV describes the +methodology used for the design generation, synthesis, and +verification. Section V presents the implementation results of +all the proposed architectures and explains the implications of +these results. Section VI concludes the paper. +II. PREVIOUS WORK +A great degree of research has been done to improve the +efficiency of integer multipliers. This is due to the large area +complexities that can require. In [3], Wallace et al. proposed +an efficient approach to dealing with integer multiplications +by using carry-save trees for row and column compression. +Their approach significantly decreased the critical path for the +summation of more than two numbers. A carry-save adder +structure uses FA and HA cells in a parallel fashion, reducing +the number of rows to two before the ripple carry adder, +thus significantly reducing the critical path. The architecture +proposed by Wallace [3] heavily impacted the industry and +continues to be applied in modern research. In [4], Dadda +presented an improvement over the previously proposed carry +save adder tree structure proposed by [3]. This reduced the +number of FA and HA cells required for the column and row +compression. Ugurdag et al. in [5] proposed a new and faster +carry-save tree structure called row and column compression +trees (RoCoCo). Their structure allowed for smaller and faster +final adders for integer multiplication circuits. Using these +structures, they presented faster multiplication circuits than +much of the literature on FPGAs, outperforming Dadda mul- +tipliers [4] and the built-in multiplication circuits by Xilinx. +Considering how RoCoCo trees can offer an improvement +over Wallace [3] and Dadda [4] trees, their architectures were +utilized in this work. +Multi-cycle (MC) multipliers have also been studied in the +past, both for FPGA and ASIC applications. The focus of +these studies was to decrease the area complexity for integer +multiplication (similar to ours). However, recent work only +target FPGA applications due to the limited resources avail- +able. The authors in [6] extensively studied the topic of large +integer multiplication, with a focus on FPGA implementation. +They proposed several MC architectures involving Schoolbook +multiplication, Comba multiplication, Karatsuba multiplica- +tion, and Number Theoretic Transforms. These architectures +are each suited for different applications, each having different +characteristics. In their architectures, the latency depends on +the bit widths of the multiplicands, allowing for a higher +degree of resource sharing depending on the multiplication +size, minimizing the number of required DSP slices. DSP +slices are an FPGA-specific type of resource. Thus, more op- +timized architectures can be presented for ASIC applications. +In [7], a design implementing the Karatsuba algorithm was +implemented. Due to the recursive nature of the Karatsuba +algorithm, they proposed using a “Coprocessor,” which was +responsible for the sub-functions of the Karatsuba algorithm, +such as multiplication, addition, and shifting operations. The +“Coprocessor” has a fixed size, and a Block RAM was used +to store intermediate results. They were able to produce a +circuit that could handle different bit widths using the same +area resources. The only varying part is the latency required. +This approach means the latency can increase rapidly as the +multiplication size increases, requiring 120 clock cycles to im- +plement a 128×128 multiplication. This approach also implies +a long II since the “Coprocessor” is expected to be reused sev- +eral times for a single computation. This architecture provides +an area-efficient approach to computing large multiplications +on FPGAs, yet, it is only suitable for very low-bandwidth +applications. This architecture made further improvements to +other FPGA-based MC multiplication circuits proposed in +[8], [9], and [10]. Such architectures heavily rely on the +usage of DSP slices and Block RAMs, which are expected to +increase the speed of an FPGA application while decreasing +the slice usage of the design. Since ASICs do not have these +built-in hardware resources, more optimized solutions can be +proposed. The authors in [11] proposed a design based on an +MC Karatsuba multiplication. Their design achieves significant +area reduction for FPGAs. Their design required 1/9th of the +DSP resources for a conventional 2048×2048 multiplication. +For a 2048×2048 multiplication, they achieve a latency of +118 cycles and an II of 9 cycles. Although the design has +a relatively low throughput of 1/9, they achieve significant +area savings for large multiplications, which would otherwise +be too costly in terms of area requirements. This design is +only feasible for very large multiplications with a more limited +number of applications. +A few work have evaluated MC multiplications for ASIC, +offering significant area reductions at the cost of speed/latency. +Li et al. presented an area-efficient MC multiplier in [12]. + +Their architecture heavily relies on resource sharing. Their +designs require an II of N for N×N multiplication and have +a latency of N+1. Such a high II and latency can severely +limit the design’s applications and scalability. In [13], an +iterative multiplication circuit for 64×64 was proposed, having +an II of 4 CCs and a latency of 10 CCs. They used an +internally generated clock using NOT gates. Therefore, the +clock generation circuitry requires manual modification to +work with different system clocks. Such an approach can limit +the design’s capability of being pipelined. Another design was +proposed in [14], implementing an MC multiplication circuit +based on a modified-Booth encoding, similar to [13]. They use +self-timed clocks that do not require manual modification to +change the operating frequency, allowing them to work with +any system clock. Their architecture offered an area reduction +of 86.6% in comparison with an array implementation, coming +at the cost of an 18.8% reduction in speed. Although this +architecture presents significant area savings, a speed reduction +can be a major limitation for high-speed applications. Further- +more, using a self-timed clock can limit the design’s ability to +be pipelined. This architecture is considered a combinational +circuit; hence, the design can accept inputs in consecutive +clock cycles. The system clock, however, would be limited +by the maximum frequency of the design. +III. PROPOSED ARCHITECTURES +Integer multiplication can be represented by three stages, +partial product generation (PPG), partial product reduction +(PPR), and the final addition. Since multiplication can be +described as the sum of several smaller multiplications as +shown in equation 1. These smaller multiplications can be +computed using the same hardware resource when applying +resource-sharing. Thus, the circuit responsible for the smaller +multiplications will be used multiple times for each multipli- +cation, reducing the throughput to less than one. This approach +reduces the area requirements and the critical path of the +circuit. Both the critical path and the area requirements for the +PPG stage for a conventional integer multiplier can be reduced +by 50% when implementing a 2-cycle approach (implementing +equation 1). For any multiplication greater than 4×4, more +than two rows are generated after the PPG stage. This means +that the PPR stage is expected to handle large reductions. +Furthermore, all the intermediate values must be stored in +registers until they reach the PPR stage. Such an approach +can limit the area reduction that can be achieved. A more +optimized approach is to separate the PPR stage into two parts. +The first part would be connected to the outputs of the PPG +stage. This allows for applying resource sharing on a large +part of the PPR stage since this circuit is expected to be used +twice for each multiplication. This combination of a PPG and a +smaller PPR is called a partial product multiplier (PPM). PPMs +have a shorter critical path than regular multipliers since they +do not require the final addition stage. The PPM will be used +twice for each multiplication, producing four results. These +results are then reduced using the second part of the PPR, +which will be called a compressor. +A. Sub-module Architectures +Our architectures consist of three core sub-modules: PPM, +compressor, and final adder. Several architectures for these +sub-modules were used in this work, each having certain +advantages. DW02 multp is a synthesis-based PPM offered by +Synopsys [15] which can produce fast and efficient PPMs. The +outputs’ sizes for an M × N multiplication is M+N+2 due to +the nature of the design. DW02 multp produces signed results +that require sign extension. Since the sign of both outputs can +vary, the sign extension was implemented using the following +three steps. +1) Applying a NOT gate to the most significant bits of all +PPMs’ outputs (bit position M+N+2, where M and N +are the sizes of the multiplicands). +2) Pad the outputs with 1’s after the most significant bit +(bit positions greater than M+N+2). +3) Sum all constants to reduce the compression size (before +synthesis). +Steps 1 and 2 allow DW02 multp to be used for unsigned +multiplications, even though DW02 multp produces signed +results. Step 3 sums up all the constants, including the sign +extension padding bits, reducing the size of the numbers that +should be compressed in the next stage, and providing further +area reductions. Like DW02 mult, DW02 tree is a synthesis- +based compression tree offered by Synopsys [15], which can +provide fast and efficient compressors. +RoCoCo, proposed in [5], presents a row and column com- +pression tree that can be used to create fast and efficient integer +multipliers and compressors. RoCoCo aims to maximize the +reduction in the row and column compression tree, allowing +for a smaller final addition. RoCoCo multipliers can be used +as PPMs by omitting the final addition stage. RoCoCo also +presents RTL generators that can produce compressors. Their +architecture can reduce the area complexity and the critical +path of the overall design. The compressor used for the Ro- +CoCo multiplier maximizes the reduction made, where several +of the least significant bits of the second output are reduced to +0. This reduces required computations in the following stages, +reducing the overall area complexity and the critical path. +A custom area-efficient compression tree is proposed in this +work as well. This compression tree aims at reducing the num- +ber of rows into two while minimizing the resources required. +This compressor is tailored for a feed-forward architecture +(which will be discussed in III-C) utilizing a DW02 multp +PPM. Since this compressor is designed to minimize the +required hardware resources, it does not achieve identical bit +reductions as RoCoCo. This approach can be helpful since the +final addition can be spread into multiple cycles using resource +sharing or pipelining. +The final addition stage, which comes after the partial +product reduction stage, typically consists of ripple carry +adders, which can often be the critical path of a design. +This can increase the area complexity since the synthesis +tool is forced to use larger library cells with greater driving +strength and a shorter propagation delay. Since the designs + +target MC multiplications, the final adder will remain idle +50% of the time for 2-cycle multipliers. This presents another +opportunity to implement resource sharing. The size of the +adder can be reduced by 50% by applying resource-sharing +and using it in two consecutive cycles. This approach creates +a loop around the adders, which makes pipelining the design +complex, requiring additional control logic. Such a 2-cycle +resource shared adder (2CA) architecture is unsuitable for +strict timing targets. The feedback loop presents a limitation +since the designs cannot always meet the timing target. For +more relaxed timing targets, creating a feedback loop around +a smaller adder reduces the area complexity because it reduces +the size of the final adder. Pipelining the 2CA architecture can +allow resource sharing to be implemented without limiting the +maximum frequency. A pipelined 2-cycle adder (2CPA) can +be achieved by placing registers in the path of the final adders. +Such an approach would not work since variables would start +to overlap. This was solved by adding a delay register in the +path, making the total latency for the final adder five clock +cycles. However, when inputs arrive at odd intervals with +respect to the previous inputs, this again causes an overlap +of variables. Thus, an internal state machine is required for +such cases, which requires more complicated control logic and +memory elements. +MC multiplication circuits offer various opportunities for +resource sharing since any multiplication stage can be reused +for II times. However, resource sharing can also create feed- +back loops in the design. Feedback loops limit a design’s +ability to be pipelined. And thus, there exists a trade-off +between the design’s ability to be pipelined (which determines +the maximum frequency) and the degree of resource sharing +to be implemented. A design containing no feedback loops +can easily be pipelined by placing registers in the path. This +allows a design to meet very strict timing targets at the +cost of increased latency. Designs containing feedback loops +have a fixed critical path since the feedback loop cannot +be pipelined. This means that the maximum frequency can +become a limitation for high-speed applications. Furthermore, +since the area of a standard library cell depends on its driving +strength and speed, pipelining allows for the reduction of area +usage since the synthesis tool can use smaller cells that have +longer propagation delays. Due to this, a feed-forward design +can significantly outperform a design with a feedback loop +in its data flow path when the clock target is strict enough, +even if the designs implementing a feedback loop can meet +timing. Nevertheless, feedback loops enable a greater degree of +resource sharing, maximizing the area reductions when dealing +with more relaxed timing targets. As a result, both approaches +have their benefits. Depending on the target frequency, either +approach could outperform the other. +Two designs are proposed in this work using the previously +discussed concepts, each having different variations in the +sub-modules used. These designs are optimized for specific +applications in terms of operating frequency, maximizing the +area reductions that can be achieved. +B. Feedback Design +Feedback loops allow for a greater degree of resource +sharing. As a result, we present an architecture that uses feed- +back loops to reduce the area complexity. In this architecture, +all three multiplier stages are resource-shared: the PPM, the +compressor, and the final adder. These stages are fully utilized +for 100% of clock cycles under regular operation (when +inputs arrive back to back). This is achieved by creating a +feedback loop around the compressor and final adder, reducing +the size of both. The feedback loop in this design contains +a 3:2 compressor and a ripple carry adder. For an M×N +multiplication, this architecture uses an M × (⌈N/2⌉) PPM, +a final adder, and a 3:2 compressor of width M+⌈N/2⌉. This +design is represented by figure 1. The number of FA cells in +the feedback loop, which determines the critical path of the +design, can be calculated using equation 2, where M and N +are the bit widths of the multiplicands. +CriticalPath = 1 + (⌈M/2⌉ + N), +(2) +This approach uses the least amount of resources out of our +designs. However, due to the loop, it can be outperformed +by feed-forward designs for strict timing targets, where the +feedback loop can be a limiting factor. This design is based on +the schoolbook approach and can be represented by figure 1 for +2C multiplication. This architecture can be extended for any +3:2 +Compressor +PPM ++ +REG +MUX +0 +Fig. 1. +2-cycle feedback design +II, decreasing the size of the core sub-modules. The II and the +area complexity of the three stages (PPM, compressor, and the +final adder) have an exponential decay relationship. However, +the number of registers storing intermediate results increases +linearly. Due to this relationship, diminishing returns will be +seen as the II increases. Furthermore, continuously increasing +the II after some point would also increase the area complexity +of the design. + +C. Feed-forward Design +Multiplication circuits are frequently used with high- +frequency applications; ergo, architectures with short critical +paths can be very beneficial. The critical path directly affects +both area and speed. The critical path limits the maximum +operating frequency, and the area complexity is indirectly +affected by the critical path. A circuit synthesized using its +maximum frequency consumes significantly more area than +one synthesized for a relaxed frequency. When operating under +strict timing conditions, the synthesis tool instantiates larger +library cells with shorter delays to meet timing. In contrast, +the synthesis tool can use small library cells with longer prop- +agation delays when dealing with relaxed timing targets. The +critical path is caused by the combinational circuit that has to +be executed within the same clock cycle. If this combinational +logic were to be divided across multiple clock cycles, the +critical path would be decreased, and thus, both speed and area +would improve. As previously discussed, having a feedback +loop in the design limits the design’s ability to be pipelined. +For this reason, we propose a design that contains no feedback +loops. This architecture has three main steps. Firstly, the partial +product multiplications are computed using one module, thus +requiring two clock cycles. In the final clock cycle, four results +need to be added, these are first sent to a compressor, and then +the result of the compression is sent to a ripple carry adder. +Only the PPM is used twice in this architecture which does not +create any feedback loops. This design can easily be pipelined +to meet very strict timing targets. The area savings of this +design come from the fact that it requires smaller PPG and +PPR stages, which contribute a large portion of the overall +complexity for an integer multiplier. All the stages in this +design can be efficiently pipelined, which allows the critical +path to be continuously decreased at the cost of longer latency. +The PPM’s size equals M + ⌈(N/2)⌉. The compressor’s size +depends on the bit widths of the inputs, as well as the type of +PPM used. The compression tree has at most four rows to be +reduced, and several bit-positions contain fewer rows due to +the shifting operations. DW02 multp produces signed outputs +that require sign extension, while RoCoCo produces unsigned +results; ergo, the compression trees needed for these two PPMs +are not identical. This architecture is ideal to be used with an +II of 2. In such a case, the stage can be reduced by around +50% while keeping the control logic simple and not requiring a +significant number of registers for storing intermediate results. +Since the PPM has two outputs and it is used twice, a 4:2 +compressor is required, where the inputs would be the first +2 PPM results and the shifted version of the second 2 PPM +results. This design is represented by figure 2. +This architecture can also be extended for different IIs; +however, maintaining a feed-forward approach becomes a +limitation. A feed-forward design with an II of 3 would +require a 6:2 compressor. This increase in the compressor’s +size undermines the area reductions gained by the smaller +PPM. Furthermore, such a design requires significantly more +registers to store the intermediate results until all PPMs are +REG +REG +4:2 +Compressor +Final +Adder +PPM +Fig. 2. 2 cycle feed-forward with a 4:2 comp +computed. Such an architecture is not expected to provide +any area savings. A better approach would be to add a loop +around the 4:2 compressor. Such a design allows the II to +be increased without significantly affecting the architecture. +However, The feedback loop limits the design’s ability to be +pipelined. This loop has a short critical path of only 2 FAs, +but it also requires more registers to store the intermediate +results. Moreover, this approach also requires a significantly +larger compressor, requiring 96% more FAs and HAs for the +case of 3C versus 2C II. Although a 4:2 compressor is used +in both cases, the 4:2 compressor required for a 2C design +contains more columns with only 2 bits to be reduced. All this +results in a higher area requirement when compared to the 2C +version. Therefore, a feed-forward design with this approach +is only viable for an II of 2. +IV. IMPLEMENTATION +The proposed designs were tested thoroughly, using differ- +ent bit widths, latencies, and timing targets. This required sev- +eral steps, including design generation, synthesis, pipelining, +simulation, and reporting power. These steps were automated +using a series of scripts that handle these tasks accordingly. +This automation allows for a greater degree of testing required +for a complete evaluation of all these designs. The designs +are generated using RTL generation scripts written in Python, +synthesized using the Synopsys Design Compiler and a TSMC +40 nm technology, and simulated using Icarus Verilog. +In this work, two main architectures are proposed, which +can have variations in the type of compressors, PPMs, and +final adders. Due to this, design generation scripts were used +to accommodate these variations easily. All designs are instan- +tiated with a wrapper, automatically setting the input/output +delays and loads. The wrapper applies a register to each of +the inputs and outputs of the design except the clock signal. + +There are two different generators for 2CIM designs, one for +each architecture. All the generators create both a wrapper +and a testbench, thus allowing for easy and accurate testing +for each design. The feedback architecture’s generator takes +the size of the multiplicands, the type of PPM (DW02 multp +or RoCoCo), and the added pipeline stages as inputs and +then generates the design. The feed-forward design has some +differences in the design generation parameters since it has +multiple options for the compressor and final adder to be +used. This generator takes the multiplicands’ size, the PPM +and compressor’s type, and the number of pipeline stages +as the input parameters. These variations in sub-modules are +required to achieve optimal performance since each offers +some advantages. +Retiming is a technique used to optimize digital circuits +by moving flip-flops. It can significantly reduce the critical +path and improve the area. Since strict timing targets require +larger library cells, reducing the critical path can also decrease +the area complexity. The Synopsys Design Compiler offers +the retiming feature, which was utilized to achieve optimized +pipelining for any design. To increase the depth of the pipeline, +registers are added at the end of the design. The synthesis tool +can freely move these registers as long as they do not affect +a feedback loop in the design. Increasing the pipeline’s depth +also increases the latency, but it can decrease both the area +complexity and the critical path while maintaining the same +throughput. +Synthesis is not a linear computation. It has to handle var- +ious constraints to meet timing, optimize area, and minimize +power consumption. Over-constraining is another technique to +meet strict timing targets when the synthesis tool fails. This +is done by further restricting the timing target, allowing the +synthesis tool to make decisions that expect a more strict +timing target. This can often allow our designs to meet the +required timing target even if the initial synthesis attempt +was unsuccessfully in meeting timing. The automation scripts +attempt over-constraining whenever a design does not meet +timing; this is done by reducing the timing target by 5% +and re-attempting synthesis. Over-constraining is attempted +three times before increasing the depth of the pipeline (adding +registers to the design and applying retiming). +In this work, area complexities of all 2CIM designs and the +standard multiplication circuit generated by Synopsys using +the “*” operator (which will be referred to as Star) are +compared. The same target frequency should be used for a fair +comparison of results between any two designs. The area com- +plexity, power consumption, and operating frequency are all +interdependent. However, in most applications, the operating +frequency is a design decision that needs to be preserved.The +initial timing target is set as the maximum frequency achieved +by the Star multiplier without adding any additional pipeline +stages. In addition, over-constraining is used to test if a higher +maximum frequency can be achieved. Timing is then increased +by 15% and 30% to see how the designs perform in more +relaxed timing targets. Furthermore, the target is set to 0.31, +representing the clock-to-q + setup + hold delay for 1 FA +placed between registers, using the smallest library cell for +a FA. However, such a strict timing target is not suitable +for multiplications larger than 32×32 since large multipliers +require many pipeline stages to meet the target timing. Larger +multipliers used less strict timing targets according to the +actual multiplication size. Using such strict timing targets is +useful to check how effectively the designs can be pipelined +to meet very strict timing targets. The designs are also +synthesized with additional pipeline stages until the latency +reaches the maximum latency of other designs. This is because +increasing the pipeline depth can decrease the area complexity +as well. All this was accomplished by using automation scripts. +The scripts first synthesize the Star multiplier to get the three +timing targets. Two more timing targets are used, representing +strict and relaxed timing conditions. For each timing target, it +follows a series of steps to produce the complete results table. +The first step is the design generation, which is done using +the RTL generators of each architecture. It then synthesizes +each design using retiming and over-constraining to meet the +timing target. After that, it simulates the generated netlist and +estimates the power consumption. This is repetitively done for +all designs, variations, and timing targets. Since the designs are +tested using a strict timing target, the target is not always met +from the initial synthesis attempt. The script will first attempt +to meet timing using over-constraining. If the design is still +unable to meet timing, a pipeline stage is added. This iteration +is repetitively done until either timing is met or the number of +pipeline stages exceeds 8. This usually means that the target +timing cannot be met even when pipelined. All architectures +are simulated using a set of 200 randomly generated inputs. +Self-checking test benches were used, where the output is +sampled depending on the latency of the design. Furthermore, +both design and netlist simulations were performed. +Power consumption can be an important aspect to consider +when designing a digital circuit. All the proposed designs were +analyzed for power consumption. Randomly generated inputs +were used, being set every 2 clock cycles. This represents a +worst-case scenario, where the power consumption is analyzed +under heavy loads. Post-synthesis simulation was performed +on all the designs, generating a file that contains the switching +activities of all nets and ports. This file is then used by +the synthesis tool (Synopsys Design Compiler) to estimate +the power consumption of a design, including both dynamic +and static power consumption in the estimation. Analyzing +power using the post-synthesis netlist simulation can generate +more accurate results when compared to the RTL simulation +since it contains more accurate switching activities. Power +consumption can be affected by several factors. However, there +are three main factors to consider, the critical path of the +circuit, the length of the MC path (which includes the parts to +be resource shared), and the area of the design. Under strict +timing conditions, the synthesis tool uses large library cells to +meet timing, which consumes more power than smaller library +cells having longer delays. Longer MC paths produce more +glitches in the design, and these glitches (part of the switching +activity) increase power consumption. And lastly, a larger area + +implies that there are more gates in the design, thus consuming +more power since even idle gates contribute to static power +consumption. +V. RESULTS +All the proposed 2CIM designs were tested thoroughly in +this work. Each design has advantages and disadvantages, +offering significant area saving under the right conditions. All +the proposed 2CIM designs and variants should be synthesized +under the same timing conditions and compared to perform a +thorough evaluation. All 2CIM designs were synthesized under +a wide range of multiplication sizes and various timing targets. +All the designs were tested using a wide range of operating +frequencies since each 2CIM architecture targets a specific +type of application with respect to the operating frequency. +Moreover, the scalability of these designs is an important +feature. The designs were tested for various multiplication +sizes to show that these area savings are consistent for various +multiplication sizes. This section will present two tables that +can represent strict and relaxed timing conditions. Since our +designs usually achieve a longer latency, they should be +compared with a Star multiplier design using the same number +of pipeline stages. However, the Star multiplier is relatively +faster, so it will reach its minimum area using a smaller +number of pipeline stages than that of 2CIM designs. The +area results that are reported for the Star multiplier are the +best area results that can be achieved using any number of +pipeline stages. +A. Synthesis Under Relaxed Timing Conditions +A relaxed timing target can show the actual area require- +ments of each design since a strict timing target would affect +the results depending on the critical path. The relaxed timing +conditions case is represented by a timing target of 10 ns. +Such a target allowed all the designs to meet timing without +requiring additional pipeline stages while being able to use +small library cells. Table I presents the synthesis results for a +16×16 multiplication. Since the timing target is very relaxed, +retiming and over-constraining were not required. Thus, the +latency presented in table I represents the minimum latency +(L) for these designs. +The feedback (FB) designs best suit applications that do +not require very strict timing targets. The area complexity of +these designs is always the lowest under such circumstances. +For 16×16 multiplications, the feedback designs can offer +around 30% area savings. But it comes at the cost of an +increase in power consumption. And so, a trade-off exists +between the area complexity and power consumption for such +multiplications with relaxed timing targets. It can be seen that +the 2CA final adder cannot offer any benefits in terms of +area savings compared to either the feed-forward (FF) design +implementing no spread (SCA) or the feedback design. This +can be explained by the fact that under very relaxed timing +conditions, the area complexity coming from the final adder is +lower than the required logic to implement resource sharing. +Thus, 2CA is not expected to offer additional area savings +TABLE I +SYNTHESIS RESULTS FOR 16×16 MULTIPLICATIONS UNDER RELAXED +TIMING CONDITIONS (TARGET = 10 ns) +Design +Final +PPM +Comp. +L +Area +Power +Adder +(uW) +Star +N/A +N/A +N/A +1 +1348 +79 +FB +N/A +DW02 +FAs +2 +942 +100 +N/A +RoCoCo +FAs +2 +960 +108 +FF +RCA +DW02 +DW02 +2 +1096 +124 +RCA +DW02 +RoCoCo +2 +1145 +127 +RCA +DW02 +Custom +2 +1105 +124 +RCA +RoCoCo +DW02 +2 +1051 +120 +RCA +RoCoCo +RoCoCo +2 +1122 +122 +2CA +DW02 +DW02 +3 +1352 +159 +2CA +DW02 +RoCoCo +3 +1181 +148 +2CA +DW02 +Custom +3 +1420 +168 +2CA +RoCoCo +DW02 +3 +1115 +138 +2CA +RoCoCo +RoCoCo +3 +1155 +140 +2CPA +DW02 +DW02 +6 +2102 +250 +2CPA +DW02 +RoCoCo +6 +2053 +251 +2CPA +DW02 +Custom +6 +2086 +248 +2CPA +RoCoCo +DW02 +6 +2120 +244 +2CPA +RoCoCo +RoCoCo +6 +2129 +243 +under these conditions. 2CPA is designed to provide area +savings for strict timing targets. Therefore, it cannot offer any +area savings under such relaxed timing conditions. This is due +to the added complexity in the control logic, which is required +for scheduling inputs based on their arrival time, and the +increased number of registers needed for storing intermediate +results. The proposed designs consume more power than +the Star multiplier since they have a longer MC path. A +longer MC path increases the circuit’s glitches, increasing the +dynamic power consumption. The Star multiplier employs no +resource sharing. Therefore, only a part of the circuit is active +simultaneously when a throughput of less than one is needed. +In our testing, inputs are received every two cycles, making +the Star multiplier remain idle 50% of the time. +B. Synthesis Under Strict Timing Conditions +An essential aspect of ASICs is their ability to operate at +high frequencies. For this reason, the ability of any multi- +plication circuit to operate at high frequencies is vital. All +2CIM designs were tested under high frequencies as well. +They were synthesized using a very strict timing target of +0.31 ns, equal to the clock-to-q + setup + hold delay for 1 +FA placed between registers. Such a strict target shows how +well these designs can be pipelined. Table II contains the +synthesis results of the proposed designs, both retiming and +over-constraining were used to meet timing. The feed-forward +designs that use a 2CA final adder have a longer critical path +than the feedback designs; ergo, they are unsuitable for high- +frequency applications. Designs using the 2CPA final adders +could achieve high frequencies as intended. However, the +added complexity from the more complicated control logic +resulted in a greater area complexity. Therefore, 2CPA is +consistently outperformed by a pipelined RCA using the same +latency. For such high-frequency applications, the only viable +options are fully feed-forward designs because of their ability + +to be efficiently pipelined without requiring additional control +logic. As seen in table II, the feed-forward (FF) designs can +TABLE II +SYNTHESIS RESULTS FOR 16×16 MULTIPLICATIONS UNDER STRICT +TIMING CONDITIONS (TARGET = 0.31 ns) +Design +FA +PPM +Comp. +L +Area +Timing +Power +(ns) +(mW) +Star +N/A +N/A +N/A +7 +5178 +0.31 +7.52 +FF +RCA +RoCoCo +DW02 +9 +3963 +0.31 +7.13 +RCA +DW02 +DW02 +9 +3984 +0.31 +7.92 +RCA +RoCoCo +RoCoCo +7 +4065 +0.31 +6.57 +RCA +DW02 +Custom +9 +4065 +0.31 +7.83 +RCA +DW02 +RoCoCo +9 +4200 +0.31 +7.98 +2CPA +DW02 +DW02 +11 +4971 +0.31 +10.17 +2CPA +DW02 +Custom +10 +5115 +0.31 +9.72 +2CPA +RoCoCo +RoCoCo +12 +5192 +0.31 +9.58 +2CPA +RoCoCo +DW02 +12 +5202 +0.31 +10.11 +2CPA +DW02 +RoCoCo +11 +5307 +0.31 +10.06 +2CA +DW02 +DW02 +5 +4394 +0.46 +4.43 +2CA +DW02 +RoCoCo +7 +4208 +0.49 +4.18 +2CA +DW02 +Custom +5 +4600 +0.48 +4.46 +2CA +RoCoCo +DW02 +5 +3255 +0.55 +2.96 +2CA +RoCoCo +RoCoCo +6 +3434 +0.58 +3.05 +FB +N/A +DW02 +FAs +4 +3712 +0.46 +4.47 +N/A +RoCoCo +FAs +6 +3554 +0.49 +4.34 +offer up to 23% area savings and 13% power reduction. The +Star multiplier is not able to meet timing without pipelining. +It requires a minimum pipeline depth of 6 for the designs to +meet timing and a depth of 7 to achieve the optimal area. +The Star multiplier achieves a similar latency to the proposed +designs while having a greater area complexity. Therefore, the +MC path for 2CIM designs is shorter or equivalent to that of +Star, explaining the power reduction 2CIM designs offer for +such strict timing targets. +C. Discussions +The proposed 2CIM designs can offer significant area +savings for various applications. Table III contains the area +savings provided by the best-performing design under different +bit widths and timing targets. This table presents the optimal +design under either very strict or semi-relaxed timing condi- +tions. The feed-forward (FF) design is best suited for strict +timing targets, and the feedback (FB) design is best suited for +more relaxed timing targets. The 2CA and 2CPA final adders +do not offer any area savings when compared to the other +designs. The 2CA creates a feedback loop in the feed-forward +design. This means that a feed-forward design implementing +a 2CA final adder will consistently be outperformed by either +the feedback designs or a feed-forward design that uses an +RCA final adder. +2CIM designs have a throughput of 1/2, i.e., they can +compute one multiplication every two clock cycles. 2CIM +designs can be used when i multiplications are required within +j clock cycles, and (i mod j)/j is less than or equal to 1/2. +Another case in which 2CIM designs can be used is when +there is a latency constraint. For 128×128 multipliers with +a clock target of 0.8, both the feed-forward design and the +Star multiplier achieve the same latency. This is because the +feed-forward architecture can be pipelined very efficiently. +TABLE III +AREA SAVINGS FOR DIFFERENT BIT WIDTHS +Design +PPM +Comp. +Timing +L +Area +Savings +(ns) +8×8 +Star +N/A +N/A +0.31 +4 +1377 +- +FF +RoCoCo +RoCoCo +0.31 +5 +1088 +21% +Star +N/A +N/A +0.5705 +2 +738 +- +FB +DW02 +FAs +0.5705 +4 +600 +19% +16×16 +Star +N/A +N/A +0.31 +7 +5179 +- +FF +RoCoCo +DW02 +0.31 +9 +3964 +23% +Star +N/A +N/A +1.001 +1 +2160 +- +FB +DW02 +FAs +1.001 +3 +1255 +42% +32×32 +Star +N/A +N/A +0.31 +10 +17790 +- +FF +DW02 +Custom +0.31 +9 +13653 +23% +Star +N/A +N/A +1.287 +2 +6057 +- +FB +DW02 +FAs +1.287 +3 +4093 +32% +64×64 +Star +N/A +N/A +0.4 +7 +51638 +- +FF +DW02 +Custom +0.4 +7 +47496 +8% +Star +N/A +N/A +1.3915 +2 +22841 +- +FB +DW02 +FAs +1.3915 +3 +13389 +41% +128×128 +Star +N/A +N/A +0.8 +4 +121634 +- +FF +DW02 +Custom +0.8 +4 +63777 +48% +Star +N/A +N/A +1.457 +2 +89165 +- +FB +DW02 +FAs +1.457 +3 +48911 +45% +Suppose there is a latency constraint of 4 cycles, and the +throughput of these multipliers does not exceed 1/2. In such +a case, multiple instances of the feed-forward design can be +used to calculate any number of multiplications, providing +a great degree of area savings. Such cases are only seen +for large multiplications operating under high frequencies. In +most cases, however, a 2CIM design can slightly increase the +latency. This is because conventional multipliers are somewhat +faster than the proposed 2CIM designs. Thus, 2CIM designs +might require more pipeline stages to meet a strict timing +target. This is usually around one or two additional clock +cycles. For large multipliers with very strict timing targets, +however, the feed-forward architecture can be pipelined more +efficiently, achieving identical latencies as those of the Star +multiplier. +VI. CONCLUSION +The number of multiplications required in a clock cycle is +not always an integer, which is the case when an odd number +of multiplications is required within two clock cycles, e.g., +three multiplications are required within two clock cycles, i.e., +1.5 multiplications per clock cycle. Such cases can be found +in a variety of applications across any domain since multipli- +cation circuits are essential building blocks. The conventional +way of dealing with such cases is to utilize complete multi- +pliers for these fractional multiplications. Or in other words, +using a multiplier for only 50% of the time. This approach +is not optimal since it requires more area than necessary +and does not take full advantage of the allocated resources. +This work presents a range of MC unsigned integer multi- + +pliers that offer fractional multipliers. These designs provide +significant area savings for a variety of applications. 2CIM +designs are designed to replace fully-pipelined multipliers, +i.e., multipliers that can accept inputs in a consecutive clock +cycle, for applications where they remain underutilized. 2CIM +designs have a throughput of 1/2, i.e., they can compute one +multiplication every two clock cycles. This work presents two +main architectures, each having multiple possible variations. +The feed-forward architecture has the advantage of speed. It +can run at very high frequencies due to the feed-forward design +aspect, which allows it to be continuously pipelined until the +timing is met. The feedback design implements a higher degree +of resource sharing, enabling it to require significantly fewer +hardware resources. However, it also requires a feedback loop. +Feedback loops can limit a design’s ability to operate at high +frequencies since feedback loops are not easily pipelined. Both +architectures have advantages, and depending on the target +application, either can be the optimal choice. 2CIM designs +can be used for any application that requires i multiplications +within j clock cycles, and (i mod j)/j is less than or equal +to 1/2. This can be especially useful for area-efficient low- +bandwidth applications. 2CIM designs can provide up to 21%, +42%, 32%, 41%, and 48% area savings for multiplications +bit widths of 8, 16, 32, 64, and 128, respectively. 2CIM +designs were tested thoroughly through automation scripts +using various multiplication sizes and timing targets. 2CIM +designs consistently offered significant area savings through- +out our testing. Moreover, 2CIM designs can provide power +savings when dealing with strict timing targets. All designs +were synthesized using the Synopsys Design Compiler and a +40 nm TSMC technology. +REFERENCES +[1] D. Goldberg, “What every computer scientist should know about +floating-point arithmetic,” ACM Computing Surveys, vol. 23, pp. 5–48, +1991. +[2] C. +Hoekstra, +K. +Shukla, +and +M. +Harris, +“Im- +plementing +high-precision +decimal +arithmetic +with +CUDA +int128,” +https://developer.nvidia.com/blog/ +implementing-high-precision-decimal-arithmetic-with-cuda-int128 +(accessed 14 July 2022). +[3] C. S. Wallace, “A suggestion for a fast multiplier,” IEEE Transactions +on Electronic Computers, vol. EC-13, pp. 14–17, 1964. +[4] L. Dadda, “Some schemes for parallel multipliers,” Alta Frequenza, +vol. 34, pp. 349–356, 1965. +[5] F. Ugurdag, O. Keskin, C. Tunc, F. Temizkan, G. Fici, and S. Dedeoglu, +“RoCoCo: Row and column compression for high-performance multipli- +cation on FPGAs,” in Proc. East-West Design & Test Symp. (EWDTS), +2011, pp. 98–101. +[6] C. Rafferty, M. O’Neill, and N. Hanley, “Evaluation of large integer +multiplication methods on hardware,” IEEE Transactions on Computers, +vol. 66, pp. 1369–1382, 2017. +[7] I. San and N. At, “On increasing the computational efficiency of long +integer multiplication on FPGA,” in Proc. IEEE Int. Conf. on Trust, +Security and Privacy in Computing and Communications (TrustCom), +2012, pp. 1149–1154. +[8] J. Von Zur Gathen and J. Shokrollahi, “Efficient FPGA-based karat- +suba multipliers for polynomials over F2,” in Proc. Selected Areas in +Cryptography (SAC). +Springer, 2005, pp. 359–369. +[9] S. Gao, D. Al-Khalili, and N. Chabini, “Efficient scheme for implement- +ing large size signed multipliers using multigranular embedded DSP +blocks in FPGAs,” Int. Journal of Reconfigurable Computing, 2009. +[10] F. de Dinechin and B. Pasca, “Large multipliers with fewer DSP blocks,” +in Proc. IEEE Int. Conf. on Field Programmable Logic and Applications +(FPL), 2009, pp. 250–255. +[11] M. Langhammer and B. Pasca, “Folded integer multiplication for +FPGAs,” in Proc. ACM/SIGDA Int. Symp. on Field-Programmable Gate +Arrays (FPGA), 2021, pp. 160–170. +[12] J. Li, Y. Du, and J. Wang, “Design a pocket multi-bit multiplier in +FPGA,” in Proc. IEEE Int. Conf. on ASIC (ASICON), 1996, pp. 275– +279. +[13] M. R. Santoro and M. A. Horowitz, “SPIM: a pipelined 64*64-bit +iterative multiplier,” IEEE Journal of Solid-state Circuits, vol. 24, pp. +487–493, 1989. +[14] M.-C. Shin, S.-H. Kang, and I.-C. Park, “An area-efficient iterative +modified-booth multiplier based on self-timed clocking,” in Proc. IEEE +Int. Conf. on Computer Design: VLSI in Computers and Processors +(ICCD), 2001, pp. 511–512. +[15] Synopsys, +“DesignWare +Library,” +https://www.synopsys.com/dw/ +buildingblock.php (accessed: 2021-03-01). + diff --git a/YNFQT4oBgHgl3EQfdzaO/content/tmp_files/load_file.txt b/YNFQT4oBgHgl3EQfdzaO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cee7f09fcad252a42c4488e95568f0d5d02d4885 --- /dev/null +++ b/YNFQT4oBgHgl3EQfdzaO/content/tmp_files/load_file.txt @@ -0,0 +1,729 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf,len=728 +page_content='2CIM: Area-Efficient 2-Cycle Integer Multipliers Ahmad Houraniah Department of Computer Science ¨Ozye˘gin University Istanbul, Turkey ahmad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='houraniah@ozu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='tr H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Fatih Ugurdag Department of Electrical and Electronics Engineering ¨Ozye˘gin University Istanbul, Turkey fatih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='ugurdag@ozyegin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='tr Cengiz Emre Dedeagac Department of Computer Science ¨Ozye˘gin University Istanbul, Turkey emre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='dedeagac@ozu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='tr Abstract—Fast multipliers with large bit widths can occupy significant silicon area, which, in turn, can be minimized by employing multi-cycle multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This paper introduces archi- tectures and parameterized Verilog circuit generators for 2- cycle integer multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' When implementing an algorithm in hardware, it is common that less than 1 multiplication needs to be performed per clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' It is also possible that the multiplications per cycle is a fractional number, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In such case, we can surely use 4 multipliers, each with a throughput of 1 result per cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, we can instead use 3 such multipliers plus a multiplier with a throughput of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Resource sharing allows a multiplier with a lower throughput to be smaller, hence area savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These multipliers offer customization in regards to the latency and clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All proposed designs were automatically synthesized and tested for various bit widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Two main architectures are presented in this work, and each has several variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Our 2-cycle multipliers offer up to 21%, 42%, 32%, 41%, and 48% of area savings for bit widths of 8, 16, 32, 64, and 128, with respect to synthesizing the “*” operator with throughput of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Furthermore, some of the proposed designs also offer power savings under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Index Terms—computer arithmetic, multi-cycle multiplier, resource-sharing, pipelining I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' INTRODUCTION An integer multiplier is an essential building block for various ASICs and CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Integer multipliers can get quite expensive in terms of area as bit width increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Large integers are used in a wide range of applications that require a high degree of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Floating-point operations are unsuit- able for some applications because of so-called “catastrophic cancellations” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The addition and subtraction of floating- point numbers that greatly vary in magnitude can produce rounding errors, resulting in significant data loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Floating- point operations are significantly more expensive than fixed- point operations regarding area complexity and latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For these reasons, fixed point representation is preferred for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The CUDA [2] programming model and software environment, which NVIDIA develops, recently introduced a 128-bit integer data type, which is intended for applications that require a higher degree of precision within a predeter- mined range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This signifies the importance of large fixed- point integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Multiplying such large integers can require significant hardware resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For this reason, decreasing the area complexity for integer multipliers can be very beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' There exists a large number of applications containing data flow paths that require less than one multiplication every two or more clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such a case is found in RSIC-V softcore implementations, where area efficiency is vital, even if it comes at the expense of increasing the latency of ALU operations such as multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Applications can contain numerous multiplications in their data flow paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Due to the area complexity of these multipli- ers, the same multiplication circuits are used multiple times to minimize the area complexity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These multiplica- tions typically need to be computed within a predefined period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since a conventional multiplier can only be used once in every clock cycle, using a single multiplier can severely limit the system’s throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The throughput can be maintained by using several multiplication circuits operating in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The number of multipliers necessary can be calculated by dividing the number of multiplications by the period they need to be computed within (in clock cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Using this formula, the number of multipliers required is not always an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The conventional approach is to round up this value, which causes one of these multipliers to be underutilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A fully-pipelined multiplier that can accept new inputs in consecutive clock cycles would remain underutilized in such applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In this work, various 2-cycle unsigned integer multiplier (2CIM) designs are proposed to decrease the area complexity for these underutilized multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM architectures can be extended for signed integer multiplication as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs offer partial multipliers that can offer area-efficient designs with a throughput of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The conventional approach of multiplying unsigned integers comprises two steps: partial product generation (PPG) and partial production summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The PPG stage is typically done using AND gates, where one integer is shifted and multiplied with each bit of the other integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The PPG stage produces multiple variables that must be summed to produce the multiplication result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is done using a tree of full adder (FA) and half adder (HA) cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Using a standard ripple carry adder structure to handle the partial product summation can result in a long critical path and an increased area complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The optimization of integer multiplications has been a thoroughly studied topic, and the conventional approach is to use a carry-save adder for the summation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Multiplication can be solved using a divide and conquer arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='13332v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='AR] 30 Jan 2023 strategy, where any multiplication can be divided into multiple smaller ones, this is often seen as an algorithmic problem, and a great deal of research was done to reduce the algorithmic complexity of multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The same concepts used for reducing the algorithmic complexity can be applied to hard- ware for area reductions since multiplication can be spread across multiple clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The conventional schoolbook approach is to divide multiplication using the distributive property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' One multiplication can be split into various smaller multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Equation 1 shows how a single multiplication can be divided into two smaller multiplications using the schoolbook approach, where N is the bit width of the second multiplicand B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Y = A ∗ B = A ∗ {B1, B0} = A ∗ B0 + A ∗ B1 ∗ 2N/2 (1) Such an approach can be applied for area reductions, where a smaller multiplication circuit can be used repetitively to com- pute several smaller multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Although multiplication can be implemented using a recursive approach, implementing several division levels requires more control logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since the smaller multiplier needed for a single division level is expected to be used twice for each multiplication, new inputs can only arrive once every two clock cycles (CCs), which will be called the initiation interval (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The rest of the paper is organized as follows: Section II presents the previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Section III presents the ar- chitectures proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Section IV describes the methodology used for the design generation, synthesis, and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Section V presents the implementation results of all the proposed architectures and explains the implications of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' PREVIOUS WORK A great degree of research has been done to improve the efficiency of integer multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is due to the large area complexities that can require.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In [3], Wallace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' proposed an efficient approach to dealing with integer multiplications by using carry-save trees for row and column compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Their approach significantly decreased the critical path for the summation of more than two numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A carry-save adder structure uses FA and HA cells in a parallel fashion, reducing the number of rows to two before the ripple carry adder, thus significantly reducing the critical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The architecture proposed by Wallace [3] heavily impacted the industry and continues to be applied in modern research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In [4], Dadda presented an improvement over the previously proposed carry save adder tree structure proposed by [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This reduced the number of FA and HA cells required for the column and row compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Ugurdag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' in [5] proposed a new and faster carry-save tree structure called row and column compression trees (RoCoCo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Their structure allowed for smaller and faster final adders for integer multiplication circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Using these structures, they presented faster multiplication circuits than much of the literature on FPGAs, outperforming Dadda mul- tipliers [4] and the built-in multiplication circuits by Xilinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Considering how RoCoCo trees can offer an improvement over Wallace [3] and Dadda [4] trees, their architectures were utilized in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Multi-cycle (MC) multipliers have also been studied in the past, both for FPGA and ASIC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The focus of these studies was to decrease the area complexity for integer multiplication (similar to ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, recent work only target FPGA applications due to the limited resources avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The authors in [6] extensively studied the topic of large integer multiplication, with a focus on FPGA implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' They proposed several MC architectures involving Schoolbook multiplication, Comba multiplication, Karatsuba multiplica- tion, and Number Theoretic Transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These architectures are each suited for different applications, each having different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In their architectures, the latency depends on the bit widths of the multiplicands, allowing for a higher degree of resource sharing depending on the multiplication size, minimizing the number of required DSP slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' DSP slices are an FPGA-specific type of resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Thus, more op- timized architectures can be presented for ASIC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In [7], a design implementing the Karatsuba algorithm was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Due to the recursive nature of the Karatsuba algorithm, they proposed using a “Coprocessor,” which was responsible for the sub-functions of the Karatsuba algorithm, such as multiplication, addition, and shifting operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The “Coprocessor” has a fixed size, and a Block RAM was used to store intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' They were able to produce a circuit that could handle different bit widths using the same area resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The only varying part is the latency required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This approach means the latency can increase rapidly as the multiplication size increases, requiring 120 clock cycles to im- plement a 128×128 multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This approach also implies a long II since the “Coprocessor” is expected to be reused sev- eral times for a single computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This architecture provides an area-efficient approach to computing large multiplications on FPGAs, yet, it is only suitable for very low-bandwidth applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This architecture made further improvements to other FPGA-based MC multiplication circuits proposed in [8], [9], and [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such architectures heavily rely on the usage of DSP slices and Block RAMs, which are expected to increase the speed of an FPGA application while decreasing the slice usage of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since ASICs do not have these built-in hardware resources, more optimized solutions can be proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The authors in [11] proposed a design based on an MC Karatsuba multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Their design achieves significant area reduction for FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Their design required 1/9th of the DSP resources for a conventional 2048×2048 multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For a 2048×2048 multiplication, they achieve a latency of 118 cycles and an II of 9 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Although the design has a relatively low throughput of 1/9, they achieve significant area savings for large multiplications, which would otherwise be too costly in terms of area requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This design is only feasible for very large multiplications with a more limited number of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A few work have evaluated MC multiplications for ASIC, offering significant area reductions at the cost of speed/latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' presented an area-efficient MC multiplier in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Their architecture heavily relies on resource sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Their designs require an II of N for N×N multiplication and have a latency of N+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such a high II and latency can severely limit the design’s applications and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In [13], an iterative multiplication circuit for 64×64 was proposed, having an II of 4 CCs and a latency of 10 CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' They used an internally generated clock using NOT gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Therefore, the clock generation circuitry requires manual modification to work with different system clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such an approach can limit the design’s capability of being pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Another design was proposed in [14], implementing an MC multiplication circuit based on a modified-Booth encoding, similar to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' They use self-timed clocks that do not require manual modification to change the operating frequency, allowing them to work with any system clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Their architecture offered an area reduction of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='6% in comparison with an array implementation, coming at the cost of an 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='8% reduction in speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Although this architecture presents significant area savings, a speed reduction can be a major limitation for high-speed applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Further- more, using a self-timed clock can limit the design’s ability to be pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This architecture is considered a combinational circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' hence, the design can accept inputs in consecutive clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The system clock, however, would be limited by the maximum frequency of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' PROPOSED ARCHITECTURES Integer multiplication can be represented by three stages, partial product generation (PPG), partial product reduction (PPR), and the final addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since multiplication can be described as the sum of several smaller multiplications as shown in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These smaller multiplications can be computed using the same hardware resource when applying resource-sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Thus, the circuit responsible for the smaller multiplications will be used multiple times for each multipli- cation, reducing the throughput to less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This approach reduces the area requirements and the critical path of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Both the critical path and the area requirements for the PPG stage for a conventional integer multiplier can be reduced by 50% when implementing a 2-cycle approach (implementing equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For any multiplication greater than 4×4, more than two rows are generated after the PPG stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This means that the PPR stage is expected to handle large reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Furthermore, all the intermediate values must be stored in registers until they reach the PPR stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such an approach can limit the area reduction that can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A more optimized approach is to separate the PPR stage into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The first part would be connected to the outputs of the PPG stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This allows for applying resource sharing on a large part of the PPR stage since this circuit is expected to be used twice for each multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This combination of a PPG and a smaller PPR is called a partial product multiplier (PPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' PPMs have a shorter critical path than regular multipliers since they do not require the final addition stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The PPM will be used twice for each multiplication, producing four results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These results are then reduced using the second part of the PPR, which will be called a compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Sub-module Architectures Our architectures consist of three core sub-modules: PPM, compressor, and final adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Several architectures for these sub-modules were used in this work, each having certain advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' DW02 multp is a synthesis-based PPM offered by Synopsys [15] which can produce fast and efficient PPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The outputs’ sizes for an M × N multiplication is M+N+2 due to the nature of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' DW02 multp produces signed results that require sign extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since the sign of both outputs can vary, the sign extension was implemented using the following three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 1) Applying a NOT gate to the most significant bits of all PPMs’ outputs (bit position M+N+2, where M and N are the sizes of the multiplicands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2) Pad the outputs with 1’s after the most significant bit (bit positions greater than M+N+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 3) Sum all constants to reduce the compression size (before synthesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Steps 1 and 2 allow DW02 multp to be used for unsigned multiplications, even though DW02 multp produces signed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Step 3 sums up all the constants, including the sign extension padding bits, reducing the size of the numbers that should be compressed in the next stage, and providing further area reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Like DW02 mult, DW02 tree is a synthesis- based compression tree offered by Synopsys [15], which can provide fast and efficient compressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' RoCoCo, proposed in [5], presents a row and column com- pression tree that can be used to create fast and efficient integer multipliers and compressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' RoCoCo aims to maximize the reduction in the row and column compression tree, allowing for a smaller final addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' RoCoCo multipliers can be used as PPMs by omitting the final addition stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' RoCoCo also presents RTL generators that can produce compressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Their architecture can reduce the area complexity and the critical path of the overall design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The compressor used for the Ro- CoCo multiplier maximizes the reduction made, where several of the least significant bits of the second output are reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This reduces required computations in the following stages, reducing the overall area complexity and the critical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A custom area-efficient compression tree is proposed in this work as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This compression tree aims at reducing the num- ber of rows into two while minimizing the resources required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This compressor is tailored for a feed-forward architecture (which will be discussed in III-C) utilizing a DW02 multp PPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since this compressor is designed to minimize the required hardware resources, it does not achieve identical bit reductions as RoCoCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This approach can be helpful since the final addition can be spread into multiple cycles using resource sharing or pipelining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The final addition stage, which comes after the partial product reduction stage, typically consists of ripple carry adders, which can often be the critical path of a design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This can increase the area complexity since the synthesis tool is forced to use larger library cells with greater driving strength and a shorter propagation delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since the designs target MC multiplications, the final adder will remain idle 50% of the time for 2-cycle multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This presents another opportunity to implement resource sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The size of the adder can be reduced by 50% by applying resource-sharing and using it in two consecutive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This approach creates a loop around the adders, which makes pipelining the design complex, requiring additional control logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such a 2-cycle resource shared adder (2CA) architecture is unsuitable for strict timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feedback loop presents a limitation since the designs cannot always meet the timing target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For more relaxed timing targets, creating a feedback loop around a smaller adder reduces the area complexity because it reduces the size of the final adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Pipelining the 2CA architecture can allow resource sharing to be implemented without limiting the maximum frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A pipelined 2-cycle adder (2CPA) can be achieved by placing registers in the path of the final adders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such an approach would not work since variables would start to overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This was solved by adding a delay register in the path, making the total latency for the final adder five clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, when inputs arrive at odd intervals with respect to the previous inputs, this again causes an overlap of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Thus, an internal state machine is required for such cases, which requires more complicated control logic and memory elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' MC multiplication circuits offer various opportunities for resource sharing since any multiplication stage can be reused for II times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, resource sharing can also create feed- back loops in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Feedback loops limit a design’s ability to be pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' And thus, there exists a trade-off between the design’s ability to be pipelined (which determines the maximum frequency) and the degree of resource sharing to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A design containing no feedback loops can easily be pipelined by placing registers in the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This allows a design to meet very strict timing targets at the cost of increased latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Designs containing feedback loops have a fixed critical path since the feedback loop cannot be pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This means that the maximum frequency can become a limitation for high-speed applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Furthermore, since the area of a standard library cell depends on its driving strength and speed, pipelining allows for the reduction of area usage since the synthesis tool can use smaller cells that have longer propagation delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Due to this, a feed-forward design can significantly outperform a design with a feedback loop in its data flow path when the clock target is strict enough, even if the designs implementing a feedback loop can meet timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Nevertheless, feedback loops enable a greater degree of resource sharing, maximizing the area reductions when dealing with more relaxed timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' As a result, both approaches have their benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Depending on the target frequency, either approach could outperform the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Two designs are proposed in this work using the previously discussed concepts, each having different variations in the sub-modules used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These designs are optimized for specific applications in terms of operating frequency, maximizing the area reductions that can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Feedback Design Feedback loops allow for a greater degree of resource sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' As a result, we present an architecture that uses feed- back loops to reduce the area complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In this architecture, all three multiplier stages are resource-shared: the PPM, the compressor, and the final adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These stages are fully utilized for 100% of clock cycles under regular operation (when inputs arrive back to back).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is achieved by creating a feedback loop around the compressor and final adder, reducing the size of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feedback loop in this design contains a 3:2 compressor and a ripple carry adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For an M×N multiplication, this architecture uses an M × (⌈N/2⌉) PPM, a final adder, and a 3:2 compressor of width M+⌈N/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This design is represented by figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The number of FA cells in the feedback loop, which determines the critical path of the design, can be calculated using equation 2, where M and N are the bit widths of the multiplicands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' CriticalPath = 1 + (⌈M/2⌉ + N), (2) This approach uses the least amount of resources out of our designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, due to the loop, it can be outperformed by feed-forward designs for strict timing targets, where the feedback loop can be a limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This design is based on the schoolbook approach and can be represented by figure 1 for 2C multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This architecture can be extended for any 3:2 Compressor PPM + REG MUX 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2-cycle feedback design II, decreasing the size of the core sub-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The II and the area complexity of the three stages (PPM, compressor, and the final adder) have an exponential decay relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, the number of registers storing intermediate results increases linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Due to this relationship, diminishing returns will be seen as the II increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Furthermore, continuously increasing the II after some point would also increase the area complexity of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Feed-forward Design Multiplication circuits are frequently used with high- frequency applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' ergo, architectures with short critical paths can be very beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The critical path directly affects both area and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The critical path limits the maximum operating frequency, and the area complexity is indirectly affected by the critical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A circuit synthesized using its maximum frequency consumes significantly more area than one synthesized for a relaxed frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' When operating under strict timing conditions, the synthesis tool instantiates larger library cells with shorter delays to meet timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In contrast, the synthesis tool can use small library cells with longer prop- agation delays when dealing with relaxed timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The critical path is caused by the combinational circuit that has to be executed within the same clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' If this combinational logic were to be divided across multiple clock cycles, the critical path would be decreased, and thus, both speed and area would improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' As previously discussed, having a feedback loop in the design limits the design’s ability to be pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For this reason, we propose a design that contains no feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This architecture has three main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Firstly, the partial product multiplications are computed using one module, thus requiring two clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In the final clock cycle, four results need to be added, these are first sent to a compressor, and then the result of the compression is sent to a ripple carry adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Only the PPM is used twice in this architecture which does not create any feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This design can easily be pipelined to meet very strict timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The area savings of this design come from the fact that it requires smaller PPG and PPR stages, which contribute a large portion of the overall complexity for an integer multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All the stages in this design can be efficiently pipelined, which allows the critical path to be continuously decreased at the cost of longer latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The PPM’s size equals M + ⌈(N/2)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The compressor’s size depends on the bit widths of the inputs, as well as the type of PPM used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The compression tree has at most four rows to be reduced, and several bit-positions contain fewer rows due to the shifting operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' DW02 multp produces signed outputs that require sign extension, while RoCoCo produces unsigned results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' ergo, the compression trees needed for these two PPMs are not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This architecture is ideal to be used with an II of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In such a case, the stage can be reduced by around 50% while keeping the control logic simple and not requiring a significant number of registers for storing intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since the PPM has two outputs and it is used twice, a 4:2 compressor is required, where the inputs would be the first 2 PPM results and the shifted version of the second 2 PPM results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This design is represented by figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This architecture can also be extended for different IIs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' however, maintaining a feed-forward approach becomes a limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A feed-forward design with an II of 3 would require a 6:2 compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This increase in the compressor’s size undermines the area reductions gained by the smaller PPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Furthermore, such a design requires significantly more registers to store the intermediate results until all PPMs are REG REG 4:2 Compressor Final Adder PPM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2 cycle feed-forward with a 4:2 comp computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such an architecture is not expected to provide any area savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A better approach would be to add a loop around the 4:2 compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such a design allows the II to be increased without significantly affecting the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, The feedback loop limits the design’s ability to be pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This loop has a short critical path of only 2 FAs, but it also requires more registers to store the intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Moreover, this approach also requires a significantly larger compressor, requiring 96% more FAs and HAs for the case of 3C versus 2C II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Although a 4:2 compressor is used in both cases, the 4:2 compressor required for a 2C design contains more columns with only 2 bits to be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All this results in a higher area requirement when compared to the 2C version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Therefore, a feed-forward design with this approach is only viable for an II of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' IMPLEMENTATION The proposed designs were tested thoroughly, using differ- ent bit widths, latencies, and timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This required sev- eral steps, including design generation, synthesis, pipelining, simulation, and reporting power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These steps were automated using a series of scripts that handle these tasks accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This automation allows for a greater degree of testing required for a complete evaluation of all these designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The designs are generated using RTL generation scripts written in Python, synthesized using the Synopsys Design Compiler and a TSMC 40 nm technology, and simulated using Icarus Verilog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In this work, two main architectures are proposed, which can have variations in the type of compressors, PPMs, and final adders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Due to this, design generation scripts were used to accommodate these variations easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All designs are instan- tiated with a wrapper, automatically setting the input/output delays and loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The wrapper applies a register to each of the inputs and outputs of the design except the clock signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' There are two different generators for 2CIM designs, one for each architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All the generators create both a wrapper and a testbench, thus allowing for easy and accurate testing for each design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feedback architecture’s generator takes the size of the multiplicands, the type of PPM (DW02 multp or RoCoCo), and the added pipeline stages as inputs and then generates the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feed-forward design has some differences in the design generation parameters since it has multiple options for the compressor and final adder to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This generator takes the multiplicands’ size, the PPM and compressor’s type, and the number of pipeline stages as the input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These variations in sub-modules are required to achieve optimal performance since each offers some advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Retiming is a technique used to optimize digital circuits by moving flip-flops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' It can significantly reduce the critical path and improve the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since strict timing targets require larger library cells, reducing the critical path can also decrease the area complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The Synopsys Design Compiler offers the retiming feature, which was utilized to achieve optimized pipelining for any design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' To increase the depth of the pipeline, registers are added at the end of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The synthesis tool can freely move these registers as long as they do not affect a feedback loop in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Increasing the pipeline’s depth also increases the latency, but it can decrease both the area complexity and the critical path while maintaining the same throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Synthesis is not a linear computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' It has to handle var- ious constraints to meet timing, optimize area, and minimize power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Over-constraining is another technique to meet strict timing targets when the synthesis tool fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is done by further restricting the timing target, allowing the synthesis tool to make decisions that expect a more strict timing target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This can often allow our designs to meet the required timing target even if the initial synthesis attempt was unsuccessfully in meeting timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The automation scripts attempt over-constraining whenever a design does not meet timing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' this is done by reducing the timing target by 5% and re-attempting synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Over-constraining is attempted three times before increasing the depth of the pipeline (adding registers to the design and applying retiming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In this work, area complexities of all 2CIM designs and the standard multiplication circuit generated by Synopsys using the “*” operator (which will be referred to as Star) are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The same target frequency should be used for a fair comparison of results between any two designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The area com- plexity, power consumption, and operating frequency are all interdependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, in most applications, the operating frequency is a design decision that needs to be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='The initial timing target is set as the maximum frequency achieved by the Star multiplier without adding any additional pipeline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In addition, over-constraining is used to test if a higher maximum frequency can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Timing is then increased by 15% and 30% to see how the designs perform in more relaxed timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Furthermore, the target is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31, representing the clock-to-q + setup + hold delay for 1 FA placed between registers, using the smallest library cell for a FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, such a strict timing target is not suitable for multiplications larger than 32×32 since large multipliers require many pipeline stages to meet the target timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Larger multipliers used less strict timing targets according to the actual multiplication size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Using such strict timing targets is useful to check how effectively the designs can be pipelined to meet very strict timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The designs are also synthesized with additional pipeline stages until the latency reaches the maximum latency of other designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is because increasing the pipeline depth can decrease the area complexity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All this was accomplished by using automation scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The scripts first synthesize the Star multiplier to get the three timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Two more timing targets are used, representing strict and relaxed timing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For each timing target, it follows a series of steps to produce the complete results table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The first step is the design generation, which is done using the RTL generators of each architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' It then synthesizes each design using retiming and over-constraining to meet the timing target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' After that, it simulates the generated netlist and estimates the power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is repetitively done for all designs, variations, and timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since the designs are tested using a strict timing target, the target is not always met from the initial synthesis attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The script will first attempt to meet timing using over-constraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' If the design is still unable to meet timing, a pipeline stage is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This iteration is repetitively done until either timing is met or the number of pipeline stages exceeds 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This usually means that the target timing cannot be met even when pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All architectures are simulated using a set of 200 randomly generated inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Self-checking test benches were used, where the output is sampled depending on the latency of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Furthermore, both design and netlist simulations were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Power consumption can be an important aspect to consider when designing a digital circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All the proposed designs were analyzed for power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Randomly generated inputs were used, being set every 2 clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This represents a worst-case scenario, where the power consumption is analyzed under heavy loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Post-synthesis simulation was performed on all the designs, generating a file that contains the switching activities of all nets and ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This file is then used by the synthesis tool (Synopsys Design Compiler) to estimate the power consumption of a design, including both dynamic and static power consumption in the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Analyzing power using the post-synthesis netlist simulation can generate more accurate results when compared to the RTL simulation since it contains more accurate switching activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Power consumption can be affected by several factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, there are three main factors to consider, the critical path of the circuit, the length of the MC path (which includes the parts to be resource shared), and the area of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Under strict timing conditions, the synthesis tool uses large library cells to meet timing, which consumes more power than smaller library cells having longer delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Longer MC paths produce more glitches in the design, and these glitches (part of the switching activity) increase power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' And lastly, a larger area implies that there are more gates in the design, thus consuming more power since even idle gates contribute to static power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' RESULTS All the proposed 2CIM designs were tested thoroughly in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Each design has advantages and disadvantages, offering significant area saving under the right conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All the proposed 2CIM designs and variants should be synthesized under the same timing conditions and compared to perform a thorough evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All 2CIM designs were synthesized under a wide range of multiplication sizes and various timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All the designs were tested using a wide range of operating frequencies since each 2CIM architecture targets a specific type of application with respect to the operating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Moreover, the scalability of these designs is an important feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The designs were tested for various multiplication sizes to show that these area savings are consistent for various multiplication sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This section will present two tables that can represent strict and relaxed timing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since our designs usually achieve a longer latency, they should be compared with a Star multiplier design using the same number of pipeline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, the Star multiplier is relatively faster, so it will reach its minimum area using a smaller number of pipeline stages than that of 2CIM designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The area results that are reported for the Star multiplier are the best area results that can be achieved using any number of pipeline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Synthesis Under Relaxed Timing Conditions A relaxed timing target can show the actual area require- ments of each design since a strict timing target would affect the results depending on the critical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The relaxed timing conditions case is represented by a timing target of 10 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such a target allowed all the designs to meet timing without requiring additional pipeline stages while being able to use small library cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Table I presents the synthesis results for a 16×16 multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Since the timing target is very relaxed, retiming and over-constraining were not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Thus, the latency presented in table I represents the minimum latency (L) for these designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feedback (FB) designs best suit applications that do not require very strict timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The area complexity of these designs is always the lowest under such circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For 16×16 multiplications, the feedback designs can offer around 30% area savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' But it comes at the cost of an increase in power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' And so, a trade-off exists between the area complexity and power consumption for such multiplications with relaxed timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' It can be seen that the 2CA final adder cannot offer any benefits in terms of area savings compared to either the feed-forward (FF) design implementing no spread (SCA) or the feedback design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This can be explained by the fact that under very relaxed timing conditions, the area complexity coming from the final adder is lower than the required logic to implement resource sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Thus, 2CA is not expected to offer additional area savings TABLE I SYNTHESIS RESULTS FOR 16×16 MULTIPLICATIONS UNDER RELAXED TIMING CONDITIONS (TARGET = 10 ns) Design Final PPM Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='Area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='Power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='Adder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='(uW) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='Star ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='251 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='2CPA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='DW02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='Custom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='2086 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='248 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='2CPA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='RoCoCo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='DW02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='2120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='244 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='2CPA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='RoCoCo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='RoCoCo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='2129 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='243 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='under these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CPA is designed to provide area savings for strict timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Therefore, it cannot offer any area savings under such relaxed timing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is due to the added complexity in the control logic, which is required for scheduling inputs based on their arrival time, and the increased number of registers needed for storing intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The proposed designs consume more power than the Star multiplier since they have a longer MC path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' A longer MC path increases the circuit’s glitches, increasing the dynamic power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The Star multiplier employs no resource sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Therefore, only a part of the circuit is active simultaneously when a throughput of less than one is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In our testing, inputs are received every two cycles, making the Star multiplier remain idle 50% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Synthesis Under Strict Timing Conditions An essential aspect of ASICs is their ability to operate at high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For this reason, the ability of any multi- plication circuit to operate at high frequencies is vital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All 2CIM designs were tested under high frequencies as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' They were synthesized using a very strict timing target of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 ns, equal to the clock-to-q + setup + hold delay for 1 FA placed between registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such a strict target shows how well these designs can be pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Table II contains the synthesis results of the proposed designs, both retiming and over-constraining were used to meet timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feed-forward designs that use a 2CA final adder have a longer critical path than the feedback designs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' ergo, they are unsuitable for high- frequency applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Designs using the 2CPA final adders could achieve high frequencies as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, the added complexity from the more complicated control logic resulted in a greater area complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Therefore, 2CPA is consistently outperformed by a pipelined RCA using the same latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For such high-frequency applications, the only viable options are fully feed-forward designs because of their ability to be efficiently pipelined without requiring additional control logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' As seen in table II, the feed-forward (FF) designs can TABLE II SYNTHESIS RESULTS FOR 16×16 MULTIPLICATIONS UNDER STRICT TIMING CONDITIONS (TARGET = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 ns) Design FA PPM Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' L Area Timing Power (ns) (mW) Star N/A N/A N/A 7 5178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='52 FF RCA RoCoCo DW02 9 3963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='13 RCA DW02 DW02 9 3984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='92 RCA RoCoCo RoCoCo 7 4065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='57 RCA DW02 Custom 9 4065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='83 RCA DW02 RoCoCo 9 4200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='98 2CPA DW02 DW02 11 4971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='17 2CPA DW02 Custom 10 5115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='72 2CPA RoCoCo RoCoCo 12 5192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='58 2CPA RoCoCo DW02 12 5202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='11 2CPA DW02 RoCoCo 11 5307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='06 2CA DW02 DW02 5 4394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='43 2CA DW02 RoCoCo 7 4208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='49 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='18 2CA DW02 Custom 5 4600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='46 2CA RoCoCo DW02 5 3255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='96 2CA RoCoCo RoCoCo 6 3434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='05 FB N/A DW02 FAs 4 3712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='47 N/A RoCoCo FAs 6 3554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='49 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='34 offer up to 23% area savings and 13% power reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The Star multiplier is not able to meet timing without pipelining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' It requires a minimum pipeline depth of 6 for the designs to meet timing and a depth of 7 to achieve the optimal area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The Star multiplier achieves a similar latency to the proposed designs while having a greater area complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Therefore, the MC path for 2CIM designs is shorter or equivalent to that of Star, explaining the power reduction 2CIM designs offer for such strict timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Discussions The proposed 2CIM designs can offer significant area savings for various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Table III contains the area savings provided by the best-performing design under different bit widths and timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This table presents the optimal design under either very strict or semi-relaxed timing condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feed-forward (FF) design is best suited for strict timing targets, and the feedback (FB) design is best suited for more relaxed timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The 2CA and 2CPA final adders do not offer any area savings when compared to the other designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The 2CA creates a feedback loop in the feed-forward design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This means that a feed-forward design implementing a 2CA final adder will consistently be outperformed by either the feedback designs or a feed-forward design that uses an RCA final adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs have a throughput of 1/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=', they can compute one multiplication every two clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs can be used when i multiplications are required within j clock cycles, and (i mod j)/j is less than or equal to 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Another case in which 2CIM designs can be used is when there is a latency constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For 128×128 multipliers with a clock target of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='8, both the feed-forward design and the Star multiplier achieve the same latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is because the feed-forward architecture can be pipelined very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' TABLE III AREA SAVINGS FOR DIFFERENT BIT WIDTHS Design PPM Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Timing L Area Savings (ns) 8×8 Star N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 4 1377 FF RoCoCo RoCoCo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 5 1088 21% Star N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='5705 2 738 FB DW02 FAs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='5705 4 600 19% 16×16 Star N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 7 5179 FF RoCoCo DW02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 9 3964 23% Star N/A N/A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='001 1 2160 FB DW02 FAs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='001 3 1255 42% 32×32 Star N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 10 17790 FF DW02 Custom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='31 9 13653 23% Star N/A N/A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='287 2 6057 FB DW02 FAs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='287 3 4093 32% 64×64 Star N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='4 7 51638 FF DW02 Custom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='4 7 47496 8% Star N/A N/A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='3915 2 22841 FB DW02 FAs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='3915 3 13389 41% 128×128 Star N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='8 4 121634 FF DW02 Custom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='8 4 63777 48% Star N/A N/A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='457 2 89165 FB DW02 FAs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='457 3 48911 45% Suppose there is a latency constraint of 4 cycles, and the throughput of these multipliers does not exceed 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In such a case, multiple instances of the feed-forward design can be used to calculate any number of multiplications, providing a great degree of area savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such cases are only seen for large multiplications operating under high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' In most cases, however, a 2CIM design can slightly increase the latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is because conventional multipliers are somewhat faster than the proposed 2CIM designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Thus, 2CIM designs might require more pipeline stages to meet a strict timing target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This is usually around one or two additional clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' For large multipliers with very strict timing targets, however, the feed-forward architecture can be pipelined more efficiently, achieving identical latencies as those of the Star multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' CONCLUSION The number of multiplications required in a clock cycle is not always an integer, which is the case when an odd number of multiplications is required within two clock cycles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=', three multiplications are required within two clock cycles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='5 multiplications per clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Such cases can be found in a variety of applications across any domain since multipli- cation circuits are essential building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The conventional way of dealing with such cases is to utilize complete multi- pliers for these fractional multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Or in other words, using a multiplier for only 50% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This approach is not optimal since it requires more area than necessary and does not take full advantage of the allocated resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This work presents a range of MC unsigned integer multi- pliers that offer fractional multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' These designs provide significant area savings for a variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs are designed to replace fully-pipelined multipliers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=', multipliers that can accept inputs in a consecutive clock cycle, for applications where they remain underutilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs have a throughput of 1/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=', they can compute one multiplication every two clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This work presents two main architectures, each having multiple possible variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feed-forward architecture has the advantage of speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' It can run at very high frequencies due to the feed-forward design aspect, which allows it to be continuously pipelined until the timing is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' The feedback design implements a higher degree of resource sharing, enabling it to require significantly fewer hardware resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' However, it also requires a feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Feedback loops can limit a design’s ability to operate at high frequencies since feedback loops are not easily pipelined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Both architectures have advantages, and depending on the target application, either can be the optimal choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs can be used for any application that requires i multiplications within j clock cycles, and (i mod j)/j is less than or equal to 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' This can be especially useful for area-efficient low- bandwidth applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs can provide up to 21%, 42%, 32%, 41%, and 48% area savings for multiplications bit widths of 8, 16, 32, 64, and 128, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs were tested thoroughly through automation scripts using various multiplication sizes and timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' 2CIM designs consistently offered significant area savings through- out our testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Moreover, 2CIM designs can provide power savings when dealing with strict timing targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' All designs were synthesized using the Synopsys Design Compiler and a 40 nm TSMC technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFQT4oBgHgl3EQfdzaO/content/2301.13332v1.pdf'} +page_content=' Goldberg, “What every computer 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b/_9E1T4oBgHgl3EQfDAK2/content/tmp_files/2301.02872v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8276bbbb7d5d1920bd7ee2966b587cc9dcf6ff78 --- /dev/null +++ b/_9E1T4oBgHgl3EQfDAK2/content/tmp_files/2301.02872v1.pdf.txt @@ -0,0 +1,375 @@ +Machine Learning to Estimate Gross Loss of +Jewelry for Wax Patterns +Mihir Jain1, Kashish Jain2, and Sandip Mane3 +1 Purdue School of Industrial Engineering, West Lafayette, IN 47906, USA +jain574@purdue.edu +2 Sardar Patel Institute of Technology, Mumbai, 400058, India +kashish.jain@spit.ac.in +3 D.J Sanghvi College of Engineering, Mumbai, 400056, India +sandip.mane@djsce.ac.in +Abstract. In mass manufacturing of jewellery, the gross loss is esti- +mated before manufacturing to calculate the wax weight of the pattern +that would be investment casted to make multiple identical pieces of +jewellery. Machine learning is a technology that is a part of AI which +helps create a model with decision-making capabilities based on a large +set of user-defined data. In this paper, the authors found a way to use +Machine Learning in the jewellery industry to estimate this crucial Gross +Loss. Choosing a small data set of manufactured rings and via regres- +sion analysis, it was found out that there is a potential of reducing the +error in estimation from ±2-3 to ±0.5- using ML Algorithms from his- +toric data and attributes collected from the CAD file during the design +phase itself. To evaluate the approach’s viability, additional study must +be undertaken with a larger data set. +Keywords: CAD · Gross Loss · Jewelry · Machine Learning · Wax +Model. +1 +Introduction +Loss is an inevitable component of manufacturing. In the manufacturing of jew- +ellery from precious metals, accounting and calculating the losses is a very cru- +cial. Gross Loss of jewellery is the total metal loss during its manufacturing. +Loss in metal happens during casting, filing, polishing, setting and at almost +every stage. Even though most of this lost metal is recovered and refined in the +refinery to get a recovery of 92%, on average, these losses are extremely crucial +not to be accounted for. +The loss on each piece of jewellery varies, based on various factors. Estimat- +ing this gross loss beforehand was very crucial for the manufacturing of that +jewellery. This estimated gross loss was used for while pulling wax patterns dur- +ing the process of injection moulding [1]. Jewelry made from the heavier wax +piece will have surplus metal that must be filed down and recovered later, which +arXiv:2301.02872v1 [cs.LG] 7 Jan 2023 + +2 +M. Jain et al. +is a waste of time and materials because only some of the metal will be recov- +ered. Therefore, estimating the total loss provides a general estimate of the wax +weight and can be used as a guide for how each procedure should be carried out. +In a production process, a step wise loss of each of step of manufacturing is +collected. This is done by weighing the jewelry after each step. Hence after the +jewelry has been manufactured it can assess the final data of gross loss that the +company bore. Total recovery that was done was also considered, and added to +the database. +This gross loss found out was further collected out of which a wide set of +databases is manufactured by an in-house engineer. Calculations based on cur- +rent trends are made where a few other variables are also taken into considera- +tion. +Variables like, weight of the final product, metal type (White Gold, Yellow +Gold, Pink Gold, Silver, Platinum and Palladium), cartage of metal (8k, 9k, 10k, +12k, 14k, 18k, 20k etc.), the customer for whom the jewelry is being manufac- +tured, the setting of diamond (whether the piece is handset or wax set) and of +course the type of jewelry it is (whether it is a ring, a pendant, an earring, a +bracelet or a bangle.) +Currently, the estimation comes with a variance of ±4-5%. Hence there is a +scope here by which, using the powerful tools of Machine Learning [2,3,4,5] we +can consider the variable constants to find out the gross loss in jewelry. These +variable constants can most often than not be fetched directly from the CAD +files which are made way before the actual manufacturing process even begins. +The aim of the paper is to estimate the gross loss of jewelry at the CAD level +with greater and repeatable accuracy using machine learning algorithms. This +paper will systematically narrow down the variables responsible for gross loss of +jewelry during its manufacturing, create a machine learning model that predicts +the final gross loss based on the data collected from the CAD file generated +before manufacturing and ensure greater accuracy of the model as compared to +the traditional methods of estimating loss. +2 +Methodology +As the project is a proof of concept, it only takes into account 26 rings as a +sample size. This project will only use information from the last several months +of production for all ring kinds for which CAD files were available (developed in +Rhino 3D [6]) and for which the company knew the associated gross losses. It is +important to highlight that only information that could be shown publicly has +been included in this report. There were notably three stages to the project’s +execution. +2.1 +Creating the Dataset +The first phase comprised of selecting all possible attribute of the rings from +the CAD file and listing them down with their corresponding values on an excel +sheet. This data was paired with its corresponding historic gross loss. + +Machine Learning to Estimate Gross Loss of Jewelry for Wax Patterns +3 +Table 1. Parameters of the Dataset +# Attribute +Datatype +1 +Volume +mm3 +2 +Surface Area +mm2 +3 +Metal +Karat-metal +4 +Weight/ Piece (Estimated) +gm +5 +Total Lot Quantity +integer +6 +Total Weight of Lot +gm +7 +Inner Diameter +mm +8 +Outer Diameter +mm +9 +Minimum Shank Thickness +mm +10 Maximum Shank Thickness +mm +11 Minimum Shank Width +mm +12 Maximum Shank Width +mm +13 Total Height +mm +14 Top Height +mm +15 Number of Components +integer +16 Number of Rings +integer +17 Tone +1/2/3 +18 True Miracle +binary +19 No. of True Miracle +integer +20 Diamond – Handset and Wax Set integer +21 Filigree +binary +22 J Back +binary +23 Gallery +binary +24 Fake Beads +integer +25 Plating +binary +2.2 +Preparation of Data +The compiled data was obtained from the CAD files. This data had irrelevant +parameters that are currently unknown but will be filtered out through process- +ing. The reason why all possible data was collected was to avoid any human +generated discrepancies in the very first stage of the project. Even though 26 +is a small number for a machine learning algorithm, its corresponding volume +would still suffice to give us the proof of concept required to carry on with the +project. But in an ideal situation, the number of rows should be 4x the number +of columns. So, it can safely consider that the results obtained will only improve +when the amount of data of rings increases. Before testing algorithms, the data +was checked for any missing values and such values were designated a weighted +average value. Feature scaling was done using Standard Scaling which standard- +ize features by removing the mean and scaling to unit variance [7]. The standard +score of a sample x is calculated as: +z = (x − u)/s +(1) +where u is the mean of the training samples, and s is the standard deviation of +the training samples. + +4 +M. Jain et al. +2.3 +Models +The above-mentioned data was split tested, and the mean value graph of each +feature was derived. The data set was split as 80-20, where 80 was the train +dataset and 20 was the test dataset. After the data set is split, feature scaling +was done to make the comparison easier. This data was the processed through +various algorithms. +1. Linear Regression Algorithm +Fig. 1. Architecture of Linear Regression Algorithm +(a) Linear Regression [8] is a supervised learning-based machine learning +technique. One of its functions is to carry out a regression analysis. +Through the use of independent variables, the regression model can pre- +dict a desired outcome. Its primary function is to investigate causal links +between factors and predicting. +(b) Predicting a value (y) of a dependent variable (x) from known values +(y) of independent variables (x) is the job of linear regression (x). Ac- +cordingly, this method of regression establishes a linear connection be- +tween the input variable (x) and the output variable (y) (output). Linear +Regression perfectly describes this method [9]. Hypothesis function for +Linear Regression : +y = θ1 + θ2.x +(2) +(c) When training the model – it fits the best line to predict the value of +y for a given value of x. The model gets the best regression fit line by +finding the best θ1 and θ2 values where θ1 is the intercept and θ2 is the +coefficient of x. + +dependent Variable +Datapoints +Line of +regression +independentVariables +XMachine Learning to Estimate Gross Loss of Jewelry for Wax Patterns +5 +(d) The best fit line is obtained by locating the optimal values of θ1 and θ2. +When our model is used for prediction, it will give us y as a function of +x. +2. Random Forest Regression +Fig. 2. Architecture of Random Forest Algorithm +(a) Random Forest Regression [10] algorithm is an example of a supervised +learning algorithm that use the ensemble learning approach of regression. +By combining the results of several different machine learning algorithms, +an ensemble learning method can produce a more precise forecast than +any one of them could on its own. +(b) Random Forest relies on the ”wisdom of the crowds” principle, which +states that a large number of independent models working in concert +can achieve better results than any of their parts working alone. +(c) This is owing to the fact that the trees buffer one another from their +particular errors. Since a random forest is completely random, there is no +communication between the trees that make up the forest. Random forest +is an estimator technique that takes the outputs of multiple decision +trees, compiles them, and then generates the ideal answer for the given +situation. [11] +3. Decision Tree Regression +(a) Decision tree [12] builds regression or classification models in the form of +a tree structure. It breaks down a dataset into smaller and smaller sub- +sets while at the same time an associated decision tree is incrementally +developed. The result is a tree with decision nodes and leaf nodes. +(b) There are three distinct sorts of nodes in this regression tree. The Root +Node is the primary node, representing the whole sample and potentially +being subdivided into further nodes. Features of a dataset are represented +by Interior Nodes, while decision rules are shown by Branches. In the end, +the result is shown by the Leaf Nodes. If you have an issue that requires +a choice, this algorithm is excellent. [13] + +6 +M. Jain et al. +Fig. 3. Architecture of ecision Tree Algorithm +(c) A single data point is processed all the way down to the leaf node by +asking and answering True/False queries. Ultimately, the dependent vari- +able value in each leaf node is averaged to arrive at a final forecast. The +Tree is able to provide an accurate prediction after going through several +rounds. +(d) The benefits of using decision trees include their simplicity, the fact that +they require less data cleaning, the fact that non-linearity has no effect +on the performance of the model, and the fact that the number of hyper- +parameters to be set is practically zero. +4. K-Nearest Neighbors Regression +Fig. 4. Example of KNN Algorithm + +Root +node +Interior +Interior +node +node +Leaf node +Leaf node +Interior +Interior +node +node +Leaf node +Leaf node +Leaf node +Leaf nodeMachine Learning to Estimate Gross Loss of Jewelry for Wax Patterns +7 +(a) K-Nearest Neighbors [14] is an easy-to-implement method that remem- +bers past examples and makes a numerical prediction based on a simi- +larity measure (e.g., distance functions). KNN is a non-parametric tech- +nique that has been utilised for statistical estimates and pattern recog- +nition since the early 1970s. +(b) We need to use cross-validation to choose K. Unlike classification we +cannot accuracy use as metric, since our predictions will almost never +exactly match the true response variable values. Therefore in the con- +text of KNN regression we will use root mean square prediction error +(RMSPE) instead. The mathematical formula for calculating RMSPE +is: +RMSPE = +��n +1 (yi − ˆyi)2 +n +(3) +Where n is the number of observations, yi is the observed value for the +ith observation, and ˆyi is the predicted value for the ith observation. +(c) To put it another way, for each observation in our test (or validation) +set, we calculate the squared difference between the anticipated and true +response value, average over observations, and then square root. Since +differences can be either positive or negative—that is, we can over- or +under-estimate the genuine response value—we utilise the squared dif- +ference rather than merely the difference. [15] +3 +Results +As we’ve seen, we have put our datasets through their paces with a wide range +of ML algorithms. The Mean Absolute Error (MAE) represents the error that +was introduced into our findings. The formula for the same is +MAE = +�n +i=1 |yi − xi|) +n +(4) +Where, yi is the prediction, xi is the true value and n specifies the total number +of data points. +Table 2. Mean Absolute Error Of Different Methods +Method +Mean Absolute Error +Linear Regression +0.56 +Random Forest Regressor +1.72 +Decision Tree Regressor +1.49 +K-Nearest Neighbour Regressor 2.02 +It was observed that all 4 algorithms performed well considering how small +the data set was. All algorithms gave promising results with Linear Regression + +8 +M. Jain et al. +lending the lowest MAE. Though with increasing data set, it would be wise to +consider all the remaining models as well. The scores will only improve as the +data set increases. +4 +Conclusion +The results show us that the gross loss can be predicted to an error margin of +±0.5. The proof of concept needed to be derived from the results was sufficient +to take to the company to act on it. Each of the 4 algorithms have potential, +Linear Regression being the most promising one so far. Further testing needs to +be done by increasing the number of data set and even expanding to different +category of jewelry. The implementation of this innovation in the field of jewelry +manufacturing would be a big undertaking and would be a time consuming and +labour-intensive processes but one which would bare fruitful results. +References +1. F.R. Sias. +Lost-wax Casting: Old, New, and Inexpensive Methods. +Woodsmere +Press, 2005. +2. S.J. Raudys and A.K. Jain. 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Curran Associates Inc., Red Hook, +NY, USA, 2019. + +Machine Learning to Estimate Gross Loss of Jewelry for Wax Patterns +9 +15. Kevin Beyer, Jonathan Goldstein, Raghu Ramakrishnan, and Uri Shaft. When +is “nearest neighbor” meaningful? In Catriel Beeri and Peter Buneman, editors, +Database Theory — ICDT’99, pages 217–235, Berlin, Heidelberg, 1999. Springer +Berlin Heidelberg. + diff --git a/_9E1T4oBgHgl3EQfDAK2/content/tmp_files/load_file.txt b/_9E1T4oBgHgl3EQfDAK2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a37c9e3a63a85d99c14a62f77b802e6ca3987535 --- /dev/null +++ b/_9E1T4oBgHgl3EQfDAK2/content/tmp_files/load_file.txt @@ -0,0 +1,281 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf,len=280 +page_content='Machine Learning to Estimate Gross Loss of Jewelry for Wax Patterns Mihir Jain1, Kashish Jain2, and Sandip Mane3 1 Purdue School of Industrial Engineering, West Lafayette, IN 47906, USA jain574@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='edu 2 Sardar Patel Institute of Technology, Mumbai, 400058, India kashish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='jain@spit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='in 3 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='J Sanghvi College of Engineering, Mumbai, 400056, India sandip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mane@djsce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='in Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' In mass manufacturing of jewellery, the gross loss is esti- mated before manufacturing to calculate the wax weight of the pattern that would be investment casted to make multiple identical pieces of jewellery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Machine learning is a technology that is a part of AI which helps create a model with decision-making capabilities based on a large set of user-defined data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' In this paper, the authors found a way to use Machine Learning in the jewellery industry to estimate this crucial Gross Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Choosing a small data set of manufactured rings and via regres- sion analysis, it was found out that there is a potential of reducing the error in estimation from ±2-3 to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='5- using ML Algorithms from his- toric data and attributes collected from the CAD file during the design phase itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' To evaluate the approach’s viability, additional study must be undertaken with a larger data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Keywords: CAD · Gross Loss · Jewelry · Machine Learning · Wax Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 1 Introduction Loss is an inevitable component of manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' In the manufacturing of jew- ellery from precious metals, accounting and calculating the losses is a very cru- cial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Gross Loss of jewellery is the total metal loss during its manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Loss in metal happens during casting, filing, polishing, setting and at almost every stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Even though most of this lost metal is recovered and refined in the refinery to get a recovery of 92%, on average, these losses are extremely crucial not to be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The loss on each piece of jewellery varies, based on various factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Estimat- ing this gross loss beforehand was very crucial for the manufacturing of that jewellery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' This estimated gross loss was used for while pulling wax patterns dur- ing the process of injection moulding [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Jewelry made from the heavier wax piece will have surplus metal that must be filed down and recovered later, which arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='02872v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='LG] 7 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' is a waste of time and materials because only some of the metal will be recov- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Therefore, estimating the total loss provides a general estimate of the wax weight and can be used as a guide for how each procedure should be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' In a production process, a step wise loss of each of step of manufacturing is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' This is done by weighing the jewelry after each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Hence after the jewelry has been manufactured it can assess the final data of gross loss that the company bore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Total recovery that was done was also considered, and added to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' This gross loss found out was further collected out of which a wide set of databases is manufactured by an in-house engineer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Calculations based on cur- rent trends are made where a few other variables are also taken into considera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Variables like, weight of the final product, metal type (White Gold, Yellow Gold, Pink Gold, Silver, Platinum and Palladium), cartage of metal (8k, 9k, 10k, 12k, 14k, 18k, 20k etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' ), the customer for whom the jewelry is being manufac- tured, the setting of diamond (whether the piece is handset or wax set) and of course the type of jewelry it is (whether it is a ring, a pendant, an earring, a bracelet or a bangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=') Currently, the estimation comes with a variance of ±4-5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Hence there is a scope here by which, using the powerful tools of Machine Learning [2,3,4,5] we can consider the variable constants to find out the gross loss in jewelry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' These variable constants can most often than not be fetched directly from the CAD files which are made way before the actual manufacturing process even begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The aim of the paper is to estimate the gross loss of jewelry at the CAD level with greater and repeatable accuracy using machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' This paper will systematically narrow down the variables responsible for gross loss of jewelry during its manufacturing, create a machine learning model that predicts the final gross loss based on the data collected from the CAD file generated before manufacturing and ensure greater accuracy of the model as compared to the traditional methods of estimating loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 2 Methodology As the project is a proof of concept, it only takes into account 26 rings as a sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' This project will only use information from the last several months of production for all ring kinds for which CAD files were available (developed in Rhino 3D [6]) and for which the company knew the associated gross losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' It is important to highlight that only information that could be shown publicly has been included in this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' There were notably three stages to the project’s execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='1 Creating the Dataset The first phase comprised of selecting all possible attribute of the rings from the CAD file and listing them down with their corresponding values on an excel sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' This data was paired with its corresponding historic gross loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Machine Learning to Estimate Gross Loss of Jewelry for Wax Patterns 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Parameters of the Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='# Attribute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Datatype ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Volume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Surface Area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Metal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Karat-metal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Weight/ Piece (Estimated) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='gm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Total Lot Quantity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Total Weight of Lot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='gm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Inner Diameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Outer Diameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='Minimum Shank Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='10 Maximum Shank Thickness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='11 Minimum Shank Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='12 Maximum Shank Width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='13 Total Height ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='14 Top Height ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='15 Number of Components ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='16 Number of Rings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='17 Tone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='1/2/3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='18 True Miracle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='binary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='19 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' of True Miracle integer 20 Diamond – Handset and Wax Set integer 21 Filigree binary 22 J Back binary 23 Gallery binary 24 Fake Beads integer 25 Plating binary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='2 Preparation of Data The compiled data was obtained from the CAD files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' This data had irrelevant parameters that are currently unknown but will be filtered out through process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The reason why all possible data was collected was to avoid any human generated discrepancies in the very first stage of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Even though 26 is a small number for a machine learning algorithm, its corresponding volume would still suffice to give us the proof of concept required to carry on with the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' But in an ideal situation, the number of rows should be 4x the number of columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' So, it can safely consider that the results obtained will only improve when the amount of data of rings increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Before testing algorithms, the data was checked for any missing values and such values were designated a weighted average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Feature scaling was done using Standard Scaling which standard- ize features by removing the mean and scaling to unit variance [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The standard score of a sample x is calculated as: z = (x − u)/s (1) where u is the mean of the training samples, and s is the standard deviation of the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='3 Models The above-mentioned data was split tested, and the mean value graph of each feature was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The data set was split as 80-20, where 80 was the train dataset and 20 was the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' After the data set is split, feature scaling was done to make the comparison easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' This data was the processed through various algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Linear Regression Algorithm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Architecture of Linear Regression Algorithm (a) Linear Regression [8] is a supervised learning-based machine learning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' One of its functions is to carry out a regression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Through the use of independent variables, the regression model can pre- dict a desired outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Its primary function is to investigate causal links between factors and predicting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' (b) Predicting a value (y) of a dependent variable (x) from known values (y) of independent variables (x) is the job of linear regression (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Ac- cordingly, this method of regression establishes a linear connection be- tween the input variable (x) and the output variable (y) (output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Linear Regression perfectly describes this method [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Hypothesis function for Linear Regression : y = θ1 + θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='x (2) (c) When training the model – it fits the best line to predict the value of y for a given value of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The model gets the best regression fit line by finding the best θ1 and θ2 values where θ1 is the intercept and θ2 is the coefficient of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' dependent Variable Datapoints Line of regression independentVariables XMachine Learning to Estimate Gross Loss of Jewelry for Wax Patterns 5 (d) The best fit line is obtained by locating the optimal values of θ1 and θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' When our model is used for prediction, it will give us y as a function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Random Forest Regression Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Architecture of Random Forest Algorithm (a) Random Forest Regression [10] algorithm is an example of a supervised learning algorithm that use the ensemble learning approach of regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' By combining the results of several different machine learning algorithms, an ensemble learning method can produce a more precise forecast than any one of them could on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' (b) Random Forest relies on the ”wisdom of the crowds” principle, which states that a large number of independent models working in concert can achieve better results than any of their parts working alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' (c) This is owing to the fact that the trees buffer one another from their particular errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Since a random forest is completely random, there is no communication between the trees that make up the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Random forest is an estimator technique that takes the outputs of multiple decision trees, compiles them, and then generates the ideal answer for the given situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' [11] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Decision Tree Regression (a) Decision tree [12] builds regression or classification models in the form of a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' It breaks down a dataset into smaller and smaller sub- sets while at the same time an associated decision tree is incrementally developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The result is a tree with decision nodes and leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' (b) There are three distinct sorts of nodes in this regression tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The Root Node is the primary node, representing the whole sample and potentially being subdivided into further nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Features of a dataset are represented by Interior Nodes, while decision rules are shown by Branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' In the end, the result is shown by the Leaf Nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' If you have an issue that requires a choice, this algorithm is excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' [13] 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Architecture of ecision Tree Algorithm (c) A single data point is processed all the way down to the leaf node by asking and answering True/False queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Ultimately, the dependent vari- able value in each leaf node is averaged to arrive at a final forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The Tree is able to provide an accurate prediction after going through several rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' (d) The benefits of using decision trees include their simplicity, the fact that they require less data cleaning, the fact that non-linearity has no effect on the performance of the model, and the fact that the number of hyper- parameters to be set is practically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' K-Nearest Neighbors Regression Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Example of KNN Algorithm Root node Interior Interior node node Leaf node Leaf node Interior Interior node node Leaf node Leaf node Leaf node Leaf nodeMachine Learning to Estimate Gross Loss of Jewelry for Wax Patterns 7 (a) K-Nearest Neighbors [14] is an easy-to-implement method that remem- bers past examples and makes a numerical prediction based on a simi- larity measure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=', distance functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' KNN is a non-parametric tech- nique that has been utilised for statistical estimates and pattern recog- nition since the early 1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' (b) We need to use cross-validation to choose K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Unlike classification we cannot accuracy use as metric, since our predictions will almost never exactly match the true response variable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Therefore in the con- text of KNN regression we will use root mean square prediction error (RMSPE) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The mathematical formula for calculating RMSPE is: RMSPE = ��n 1 (yi − ˆyi)2 n (3) Where n is the number of observations, yi is the observed value for the ith observation, and ˆyi is the predicted value for the ith observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' (c) To put it another way, for each observation in our test (or validation) set, we calculate the squared difference between the anticipated and true response value, average over observations, and then square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Since differences can be either positive or negative—that is, we can over- or under-estimate the genuine response value—we utilise the squared dif- ference rather than merely the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' [15] 3 Results As we’ve seen, we have put our datasets through their paces with a wide range of ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The Mean Absolute Error (MAE) represents the error that was introduced into our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The formula for the same is MAE = �n i=1 |yi − xi|) n (4) Where, yi is the prediction, xi is the true value and n specifies the total number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Mean Absolute Error Of Different Methods Method Mean Absolute Error Linear Regression 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='56 Random Forest Regressor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='72 Decision Tree Regressor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='49 K-Nearest Neighbour Regressor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='02 It was observed that all 4 algorithms performed well considering how small the data set was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' All algorithms gave promising results with Linear Regression 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' lending the lowest MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Though with increasing data set, it would be wise to consider all the remaining models as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The scores will only improve as the data set increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' 4 Conclusion The results show us that the gross loss can be predicted to an error margin of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' The proof of concept needed to be derived from the results was sufficient to take to the company to act on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Each of the 4 algorithms have potential, Linear Regression being the most promising one so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Further testing needs to be done by increasing 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} +page_content=' Springer Berlin Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfDAK2/content/2301.02872v1.pdf'} diff --git a/_NE1T4oBgHgl3EQfUwPr/content/tmp_files/2301.03095v1.pdf.txt b/_NE1T4oBgHgl3EQfUwPr/content/tmp_files/2301.03095v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f28a5c8fff83b857e98bc24018d1cf43bba93b4 --- /dev/null +++ b/_NE1T4oBgHgl3EQfUwPr/content/tmp_files/2301.03095v1.pdf.txt @@ -0,0 +1,974 @@ +MEGAnno: Exploratory Labeling for NLP in Computational Notebooks +Dan Zhang∗, Hannah Kim∗, Rafael Li Chen, Eser Kandogan, Estevam Hruschka +Megagon Labs +{dan_z,hannah,rafael,eser,estevam}@megagon.ai +Abstract +We present MEGAnno, a novel exploratory +annotation framework designed for NLP re- +searchers and practitioners. Unlike existing la- +beling tools that focus on data labeling only, +our framework aims to support a broader, iter- +ative ML workflow including data exploration +and model development. With MEGAnno’s +API, users can programmatically explore the +data through sophisticated search and auto- +mated suggestion functions and incrementally +update labeling schema as their project evolve. +Combined with our widget, the users can inter- +actively sort, filter, and assign labels to mul- +tiple items simultaneously in the same note- +book where the rest of the NLP project re- +sides. We demonstrate MEGAnno’s flexible, +exploratory, efficient, and seamless labeling +experience through a sentiment analysis use +case. +1 +Introduction +Data labeling is an important step in the machine +learning life cycle since the quality and quantity +of training data directly affect the model perfor- +mance (Geiger et al., 2021). Unfortunately, existing +annotation tools tend to consider the data labeling +step in isolation from the broader ML life cycle, +ignoring the iterative workflow of researchers and +practitioners. However, activities such as data se- +lection, exploratory data analysis, data labeling, +model training, and evaluation do not happen se- +quentially (Rahman and Kandogan, 2022). Instead, +continuous iterations are required to improve data, +annotation, and models (Hohman et al., 2020). +To further investigate this gap, we examine the +data annotation practices within the ML model de- +velopment life cycle. Based on a formative study +with six researchers from our organization, we char- +acterize their annotation practices as a “dual-loop” +model shown in Fig. 1. After data preprocess- +∗ Equal contribution. +Raw +Corpus +Unlabeled +Data +Data +Pre-processing +1 +Labeling +Model +Training +Model +Debugging +Annotation +Schema +Downstream +Model +Traditional +Labeling Tools +MEGAnno +Framework +2 +3 +5 +6 +Defining + Labeling Task +Data +Understanding +& Exploration +Loop +Training +Data +Model +Evaluation + Loop +Data +Debugging +4 +Figure 1: Dual-loop model for data annotation: (1) +data understanding/exploration loop (yellow): +itera- +tively update annotation schema while exploring and +annotating data and (2) model evaluation loop (green): +train and improve a downstream model over iterations +by debugging data. Most tools are focused on the la- +beling step only (red box). MEGAnno aims to cap- +ture both loops seamlessly within the framework (green +box). +ing (Fig. 1 +1⃝), researchers define their annota- +tion schema in terms of what labels to collect, how +many data points are needed, and so on (Fig. 1 2⃝). +As they explore and annotate the data (Fig. 1 3⃝), +they often go back and refine the annotation schema +due to their improved understanding of the data and +updated mental models for the tasks (Fig. 1 +4⃝). +For example, in a document classification task, a +user may start with loosely defined category labels +and add more choices as she discovers relevant +documents (Felix et al., 2018). Throughout this +paper, we refer to this cycle as data understand- +ing/exploration loop. Next, the labeled data is +exported from the annotation tool and is used to +train a model (Fig. 1 5⃝). However, in practice, ML +model training is rarely completed in one pass and +usually goes through many iterations of labeling, +training, data, and model debugging (Pustejovsky +and Stubbs, 2012). Fig. 1 +6⃝ refers to the cases +where researchers may need to collect more data +(e.g., for the less represented classes due to a sub- +optimal prediction performance of the downstream +model) or further refine the schema. We call this cy- +cle (i.e., model training, evaluating and debugging, +arXiv:2301.03095v1 [cs.HC] 8 Jan 2023 + +collecting more data, and training again) model +evaluation loop. +We find that the iterative dual-loop workflow +of NLP researchers and practitioners is rarely sup- +ported in most existing tools. More specifically, we +identified three main challenges towards supporting +the full annotation life cycle: +• Gaps between ML toolings. Most of the ex- +isting tools are standalone and designed for +a specific ML step, which results in frequent +context switching and data migration over- +head in the researchers’ daily workflow. +• Lack of customizable and granular con- +trol. Not all data points are equally impor- +tant. There are often cases where users might +want to prioritize a particular batch (e.g., to +achieve better class or domain coverage or +focus on the data points that the downstream +model cannot predict well). Although some +recent active learning based tools (Montani +and Honnibal, 2018; hua) can provide sugges- +tions for the next batch, most tools do not offer +customizable and fine-grained control with or +with a downstream model (i.e., covering both +loops). +• Lack of support for project evolution. Cur- +rent annotation tools usually work with the +assumption that the data collection task is +well-defined and immutable and ignore that +annotation projects can evolve as explorations +happen and insights are gathered. Thus they +lack the support to help users make evolution +decisions, and their immutable nature makes +it hard to apply these changes. +To +address +the +challenges, +we +present +MEGAnno, a flexible, exploratory, efficient, and +seamless labeling framework for NLP researchers +and practitioners. +It provides a seamless expe- +rience where data pre-processing, annotation, +analysis, model development and evaluation can +happen in the same notebook, a popular daily +working environment for data science practitioners. +MEGAnno +provides +customizable +interfaces +to help users drive their project to the desired +directions through rich heuristic-based search, +automatic suggestion, and active learning based +suggestions of the next data batch. With project +evolution in mind, MEGAnno is designed to work +with flexible task schema and provides a built-in +analysis dashboard to aid decision-making. To +our knowledge, MEGAnno is the first flexible, +exploratory labeling framework that can support +ML +workflow +seamlessly +in +computational +notebooks (Fig. 1: green box). +2 +MEGAnno +2.1 +Framework Overview +Database + server +Web +Server +Jupyter +Server +Jupyter Notebook +Python API +Direct REST requests +MEGAnno +service +MEGAnno + toolkit +In [1]: +MEGAnno + toolkit +Widget +Figure 2: +The MEGAnno framework provides ex- +ploratory annotation services through a toolkit (instal- +lable as Python libraries) providing programmatic inter- +faces, a web server providing language-agnostic REST +APIs, and an internal database to store data, annotation, +and related artifacts. Solid lines show programmatic +interactions through Python APIs calls and REST calls +delegated by our notebook widget or directly issued by +authenticated applications. Dotted lines show internal +interfaces where MEGAnno toolkit handles communi- +cation with the database and are hidden from the users. +MEGAnno provides service through 1) an inter- +nal database that stores the data, annotations, and +various artifacts produced in the annotation pro- +cess, 2) a MEGAnno toolkit that provides python +API for programmatic and visual data exploration +and labeling, and 3) a web server that provides +language-agnostic REST APIs (Fig 2). After in- +stalling the toolkit on a Jupyter server, users will +have access to our Python APIs and React-based +widget to manage their project, explore and anno- +tate their data from any connected notebook. +Data model +A Data record refers to an item +in the pre-processed data corpus for labeling. It +can be a sentence, a paragraph, a document, or a +flattened text from multiple texts such as a question- +answer pair. A Label is the smallest unit of user +labeling output. MEGAnno currently supports +record-level (e.g., topics for document or sentence) +and span-level (e.g., named entities) labels. An +Annotation is a set of labels given to a data +record by an annotator. Metadata refers to addi- +tional information related to the content of a data +record (e.g., externally generated part-of-speech +tags, embeddings) or of an annotation (e.g., time +taken to label, disagreement among annotators). + +File +Edit +View +Insert +Cell +Kernel +Widgets +Help ++ +% +心 +个 +Run +c +Codesubset = L.search(3o0, by='id order') +subset.show() +MEGAnno +Q Search... +Single +Table + Submit + Clear filter +1 +1 + data +sentiment_label + Bulk label +0 +@JetBlueI'vehad + no shampoo sad +no winter coat sad +no deodorant sad +no flatiron sad +and the rest if n +neg +@JetBlue right now completely pissed off sad +neg +2 +didn't go through. Was tol +neg +@SouthwestAir no self help way to put in tsa pre check number for +3 +existing reservation. Very annoying sad . 80o is not reachable. +neg +300 showingjupyterFigure 3: The table view to show multiple data records. +Hovering over a data record shows its full text in a pop- +up. This view allows exploration by searching, sorting, +and filtering over labels and single/bulk annotation. +Such information can be helpful in various steps of +the ML iterations. A Subset is a slice of the data +records in the database. Subsets can be of random +data records or can be generated through search +queries that match certain characteristics. +Task schema +We support a wide variety of tasks +through our customizable schema in JSON format. +To collect a label, users need to specify the level +(i.e., record or span) and provide a list of options +to choose from. For a task, MEGAnno supports +arbitrary numbers of both types of labels. We’ll +see concrete schema examples for a sentiment pre- +diction and extraction task in Section 3. At any +stage, users can always update the schema to re- +flect the evolution of the project. There are certain +constraints to schema updates to keep the consis- +tency of data. Adding new labels or new label +options will always be allowed. Removal of la- +bels and options will trigger a database query and +MEGAnno will warn the user if there exist such +labeled instances. 1 +2.2 +MEGAnno Jupyter Notebook Widget +MEGAnno’s interactive notebook widget features +1) our novel table view to facilitate exploratory +and batch labeling and 2) the single view, which is +similar to traditional labeling UIs. +Table view +The table view (Fig. 3) shows data +items in a Subset and their annotations if any. +Each record-level label is shown as a column, and +span-level labels are shown together with the high- +lighted textual span in the data column. Users can +hover over an item to see full text in a pop-up. The +search box supports three types of search (exact, +fuzzy, and regex-based) to filter the data subset fur- +ther. Users can sort and filter rows based on any +1MEGAnno provides an option to clean up legacy labels +and retry automatically. +record-level labels using the dropdown menu. To +assign a record-level label, the users can click on +the cell’s arrow button and select from the drop- +down options. Alternatively, the users can assign +the same label to multiple records simultaneously +by selecting those records and clicking the bulk +label button. +Single view +The table view is good for explo- +ration, but the limited space makes span-level an- +notation cumbersome. So we also provide a single +view where users can have a more zoomed-in ex- +perience. By clicking the “Single” button on the +top-right corner or double-clicking on a data record, +the widget switches to the single view (Fig. 4). In +this view, users can assign record-level labels on +the right side and span-level labels on the left side +by selecting/dragging target spans and choosing +the label from the options drop-down. Users can +loop through the subset using the prev/next button +based on the order specified in the table view. At +any time, users can switch between the two views +by clicking the top right buttons, and the widget +preserves all uncommitted annotations during view +changes. +Figure 4: The single view to annotate data one by one. +In this view, users can drag and label spans for extrac- +tion tasks. +Working with multiple annotators +Annotation +is rarely done by a single person. As an initial +step towards collaborative annotation, MEGAnno +provides virtually separated namespaces for each +annotator. Users identify themselves by a unique +authentication token while connecting to the ser- +vice and only update their own labels through the +widgets. MEGAnno provides a reconciliation view +(Fig. 5) to look at labels from different individuals +and resolve potential conflicts. +Dashboard +MEGAnno also provides a built- +in visual monitoring dashboard (Fig. 6). +As +projects evolve, users would need to understand +the project’s status to make decisions about the +next steps, like collecting more data points with +certain characteristics or adding a new class to the + +In[109]: +subset = L.search(3o0, by='id_order') +subset.show() +MEGAnno +Q Search... +Single +Table ++ Submit +Clear filter +Search box +Sorting & +filtering +::8: +:8: + data +V +1 sentiment_label +Bulk label +assignment +sentiment_label +0 +@JetBlue I've had no shampoo sad , no winter coat sad +no deodorant sad +no flatiron sadand the res +neg +12 Sort asc. +↑ Sort desc. +1 +@JetBlue right now completely pissed off sad +neg +Filter by +2 +didn't go through. Wa +neg +@SouthwestAir no self help way to put in tsa pre check number for +Positive +pos +3 +existing reservation. Very annoying sad . 8oo is not reachable. +neg +Negative +neg +4 +@SouthwestAir no self help way to put in tsa pre check number for existing reservation. Very annoying sad +neg +Neutral +neu +5 +@united is there a referral program for the Milageplus explorer card? +300 showingMEGAnno +Single +Table +← Prev. +← Xan +sentiment_label +@SouthwestAir no self help way to put in tsa pre check number for existing reservation. +O Positive (pos) +Very annoying sad . 800 is + not reachable +Negative (neg) +Label as +Neutral (neu) +Happy +hap +Sad +sad +300 showing +Remove labelFigure 5: Reconcilation view showing the existing la- +bel distribution for data points. +Figure 6: Dashboard widget to monitor the progress +and statistics of the project and aid decision-making. +task definition. To aid such analysis, the dashboard +widget packs common statistics and analytical vi- +sualizations based on a survey of our pilot users. +The “overview” panel shows statistics about overall +progress and per-label class distribution. If multi- +ple annotators are involved, the distribution reflects +the majority vote over annotators 2. The remaining +“annotator” and “projection” panels are hidden due +to space limitations. To help identify problematic +annotators, the annotator pannel shows statistics +like overall individual contribution and disagree- +ment scores with others. The projection panel pro- +vides customizable visualization to project data +points to a two-dimensional visual space. By de- +fault, we show the t-SNE (Van der Maaten and +Hinton, 2008) projection of sentence bert (Reimers +and Gurevych, 2019) embeddings. +2.3 +MEGAnno APIs +Project management +The management module +provides various interfaces to configure and mon- +itor the annotation project through the Project +class. +import_data loads the data records +from CSV or JSON files into the database. +set_config updates the project configuration as +it evolves. set_meta assigns metadata (e.g.,POS +tags, document embeddings) for each data record +through user-defined functions. get_status re- +turns the status of the project such as the number of +2Users can provide their aggregation function to resovle +conflicts between annotators +annotated data records and detailed statistic about +each label. +A critical feature of MEGAnno is to select in- +teresting subsets of data to show in the widget. +Subsets can be generated in a user-initiative way +via our search engine or a data-driven way via au- +tomated suggestions. +Search for subsets +MEGAnno supports so- +phisticated searches over data records, anno- +tation, and user-defined metadata through the +Project.search API. Users can search data +records by keywords (e.g., +documents men- +tioning “customer service”) or regular expres- +sions to express more complex patterns. +The +users can also search the database based on al- +ready assigned labels (e.g., records with a pos- +itive sentiment label). +As will be explained +later, MEGAnno acknowledges the value of +auxiliary information for ML projects and pro- +vides advanced search functionalities over meta- +data. +For example, users can query with pat- +terns combining regex expressions and POS tags +like project.search("(best|amazing) + ", by="POS"). +Automated subset suggestion +Searches initi- +ated by users can help users explore the dataset +in a controlled way, but the quality of searches +is only as good as users’ knowledge or heuristic +about the data and domain. MEGAnno provides +an automated subset suggestion engine to assist the +exploration. Users can customize the engine by +plugging in external suggestion models as needed. +Currently, the engine provides two types of tech- +niques: +• Embedding-based +suggestions +makes +suggestions +based +on +data +embed- +ding +vectors +provided +by +the +user. +Subset.suggest_similar +sug- +gests neighbors of data in the querying +subset. +Project.suggest_coverage +examines all the data records within the +embedding space in an unsupervised way and +suggests data points from the less annotated +regions to improve annotation coverage of the +corpus. +• Active suggestions utilizes active learning +techniques to recommend the most informa- +tive data for the downstream model. With +libraries like ModAL (Danka and Horvath), + +Reconcile +日 +Reconciling +% +1 +1 +11 data +sen... V +1 + Bulk labe +20 +@VirginAmerica now it's just t-minus 32 minutes I +pos: 100.00% +21 +@SouthwestAir Black History Commercial is really +neg: 50.00% +pos: 50.00% +22 +@UsAirways sits on a throne of lies +neg: 100.00%Dashboard +ii Aggregated +Overview +Annotator +Projection +:. Overall Progress +Annotated 100 | data points (with at least 1 label) out of 201 |total data points + Class Label - Distributions +Label class distribution forsentiment +subtask: aggregated over annotators' votes using (majority_vote). +When there is a tie in the voting, they are categorized under "tied_annotations" class label. +Labels +● neg +● neu +● pos +16 +62users can select from various selection strate- +gies based on model uncertainty and entropy, +etc. Since MEGAnno’s seamless notebook +experience covers the whole loop from anno- +tation to model training and debugging, users +can actively select a subset, annotate the sub- +set, update the model, and test again in the +same notebook without switching environ- +ments. +The output of selection engines are instances +of the Subset class with the following methods: +show returns a notebook widget for interactive ex- +ploration and annotation. batch_annotation +sets the same record-level labels for all data records +in the subset. suggest_similar returns a new +subset of the database containing the most sim- +ilar data for each record in the querying subset +according to some metadata with valid distance +functions. +3 +Use Case: Sentiment Analysis +We present a use case to illustrate how MEGAnno +can support data annotation in NLP researchers and +practitioners’ workflow. Meggie is a data scientist +who wants to train a sentiment-related model for +her project. She obtains a Twitter dataset3about US +airlines and decides to label it using MEGAnno. +Import data +In an empty notebook, Meggie +starts by initializing the project named “Tweet Sen- +timent” using a MEGAnno Python API. She gets +a copy of the data from the product manager who +often uses Google spreadsheet and imports the data +from its published link. +1 from meganno import Project +2 L = Project(, "Tweet Sentiment") +3 L.import_data(, format="csv") +Set up initial schema +Without knowing much +about the data, Meggie decides to start the project +by collecting binary labels and setting up the +project’s schema. Knowing that MEGAnno sup- +ports flexible and editable schema, Meggie does +not worry about getting the perfect schema in the +first round and can start exploring and annotating. +1 label_schema = [{ +2 +"label_name": "sentiment_label", +3The Kaggle dataset (for Everyone library) has ground- +truth sentiment labels available. But for demonstration pur- +poses, we ignore them and assume Meggie only gets the raw +Tweets. The dataset contains 14K tweets about major US +airlines scraped in February 2015. +Figure 7: UI Search by regular expression. Matched +keywords are highlighted. +3 +"level": "record", +4 +"options": ["Positive", "Negative"] +5 }] +6 L.set_config(config1) +Explore and annotate +She starts by exploring +the first 300 data points in the widget’s table view. +Using the search box, she filters the subset with the +keyword “amazing”. As expected, most of the data +records reflect a positive sentiment, so she assigns a +positive label to multiple data items using the “Bulk +label” button. Next, she wants to examine tweets +with hashtags related to failing, so she tries regu- +lar expression search using #fai[^ ]* (Fig. 7). +With better understanding of the dataset, Meggie +chooses to perform more advanced exploration us- +ing part-of-speech metadata. She imports a POS +tagger from spaCy (Honnibal and Montani, 2017) +and retrieves tweets that match interesting patterns +such as best of . +1 import spacy +2 tagger = spacy.load("en_core_web_sm") +3 +4 pos1 = L.search("(best|amazing) < +NOUN>", by="POS", tagger=tagger) +5 pos2 = L.search("best of ", +by="POS", tagger=tagger) +6 meganno.union(pos1, pos2).show() +With such an exploratory approach and the batch +labeling feature, she can annotate much faster and +in a more controlled way. +Candidate suggestion +After a few more heuris- +tics, she runs out of ideas, so she takes advantage +of the suggestion feature. She selects the sentence +bert (Reimers and Gurevych, 2019) encoder as the +meta-data generation function. She first issues a +query with strategy similarity to collect data +points similar to the ones retrieved from the previ- +ous POS search. Finally, she wants to see samples +from less covered areas in the embedding space to + +In［124]: +# iterate by id +L.search_by_id(limit=3oo0, skip = 0).show(widget_config) +showing demo's annotations +Twitter Sentiment +Q /#*fai[^ ]*/ + Single +regex +田 Table +Matching on failures, faith, fail, failing!!, failure., #fail +个 Submit +#fail..., fair, fails, fail., faithful, failed, failscustomers, +1 data +fairs, failure, failures., fairs., #fail., fair." +ent_l... +ulk label +Suggestions +^ey +@USAirways tried twice today on hold for 30 min e +^jh +^jp +^kn +^kp +2 +@SouthwestAir thanks, got put on the am flight tor +Search Mode Tips +3 +@united It is super frustrating that the folks at the United Ticket Counter in Pittsburg aren't honoring their ov +@JetBlue #fail My wife on the phone asking to switch flight times. In mid +4 +Isly? This is why airl +switch gets disconnected. Now Cancelled Flightled! No new time! No call +5 +back +6 +@JetBlue#fail My wife on the phone asking to switch flight times. In mid switch gets disconnected. Now Car +@JetBlue Guys, really bad @JetBlue #fail . Someone better call my wife back to get this handled. 203-382-3 +demo +36 of 2242 showingimprove the diversity of the training data and issues +a coverage query. +1 m = SentenceTransformer("all-MiniLM-L6") +2 # set metadata generation function +3 L.set_meta("bert", lambda x: list(m. +encode(x))) +4 # get more data like the query result in +the previous query(subset_pos) +5 subset_sim = pos2.suggest_similar( +meta_name="bert") +6 # get data from less covered areas in +the embedding space. +7 subset_cov = L.suggest_coverage( +meta_name="bert") +Update the schema +Meggie is now happy with +the collected labels, but she aims to go one step +further to understand which words or phrases +lead to the sentiment judgment. So she updates +her schema by adding a new span-level label +called sentiment_span with label options be- +ing “happy” or “sad”. Each data record can have +arbitrary numbers of such span-level labels. +1 label_schema = [{ +2 +"label_name": "sentiment_span", +3 +"level": "span", +4 +"options": ["Happy", "Sad"] +5 }, ... {# previous label options +6 }] +After updating the schema, she fetches all +records with neutral labels in a widget. To high- +light and annotate spans, she goes into the single +view (as shown in Fig. 4). At any step of the iter- +ation, she could refer to the dashboard widget to +monitor the project progress. After several rounds +of similar iterations, she feels good and concludes +her exploration. Finally, Meggie can export the an- +notated data in JSON or CSV formats for training +or plug in the model directly. +In conclusion, with MEGAnno, Meggie can ex- +plore her dataset using various heuristic-based or +automated search functions and better understand +the data corpus as she labels. She has the flexibil- +ity to iteratively update her schema as the project +evolves. Using the widget, Meggie can finish the +entire ML life cyle in the same Jupyter notebook. +4 +Related Work +There exist numerous annotation tools that can sup- +port NLP tasks, which are extensively surveyed +by Neves and Seva (2019, 2020). In this section, +we focus on works that are closer to our flexible, +exploratory, efficient, and seamless framework. +Flexible schema +Unfortunately, most of existing +tools are not designed for iterative schema devel- +opment, and thus they are not flexible enough for +evolving projects. Felix et al. (2018) and Kulesza +et al. (2014) allow users to progressively define +document classes by inspecting documents that +are assigned to each class. But these works are +more similar to interactive topic modeling or inter- +active classification, where users assign documents +to classes, than document labeling, where users +assign labels to documents. Our tool goes beyond +document label refinement and supports a broader +task of progressively defining annotation schema +(e.g., additionally collecting span-level labels). +Exploratory +labeling +The +concept +of +ex- +ploratory labeling is introduced by Felix et al. +(2018) as using computational techniques to help +users group documents into evolving labels. In +our paper, we use the term “exploratory labeling” +to refer to where exploratory data analysis and +data annotation are iteratively conducted in the +data understanding/exploration loop. Exploratory +labeling can be beneficial because while labeling +data, users gain insight into their dataset (Sun et al., +2017). +Efficient batch/bulk annotation +A few tools of- +fer a functionality to simultaneously assign la- +bels to multiple spans within a record. For ex- +ample, YEDDA (Yang et al., 2018) can anno- +tate multiple span-level labels via command line. +TALEN (Stephen Mayhew, 2018), a named en- +tity tagging tool, has an entity propagation feature +which annotates all mentions of an entity in a doc- +ument at once. In contrast, users can annotate mul- +tiple records simultaneously using our Python API +and a GUI widget. +Notebook widget +Computational notebooks are +frequently used by data analysts to iteratively write +and edit code to understand data, test hypotheses, +and build models (Head et al., 2019; Randles et al., +2017). Following the practice of mage (Kery et al., +2020) which extends Jupyter notebook with GUI +widgets for specific tasks, our widget is designed to +achieve flexible communication with the rest of ML +development codes. Annotation tools which are im- +plemented as Jupyter widgets include Pigeon (pig) +and ipyannotate (ipy), but they only offer a simple +label assignment feature. + +5 +Conclusion +In this paper, we present MEGAnno, an annota- +tion framework designed for NLP researchers and +practitioners. Through MEGAnno’s programmatic +interfaces and interactive widget, users can itera- +tively explore and search for interesting data sub- +sets, annotate data, train models, evaluate and de- +bug models within a Jupyter notebook without the +overhead of context switch or data migration. +References +humanloop.com. https://humanloop.com/. +ipyannotate. +https://github.com/ +ipyannotate/ipyannotate. +pigeon. +https://github.com/ +agermanidis/pigeon. +Tivadar Danka and Peter Horvath. modAL: A modu- +lar active learning framework for Python. Available +on arXiv at https://arxiv.org/abs/1805. +00979. +Cristian Felix, Aritra Dasgupta, and Enrico Bertini. +2018. +The exploratory labeling assistant: Mixed- +initiative label curation with large document collec- +tions. In Proceedings of the 31st Annual ACM Sym- +posium on User Interface Software and Technology, +UIST ’18, pages 153–164, New York, NY, USA. 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In Proceedings of the 2019 Conference on +Empirical Methods in Natural Language Processing. +Association for Computational Linguistics. +Dan Roth Stephen Mayhew. 2018. Talen: Tool for an- +notation of low-resource entities. +In ACL System +Demonstrations. +Yunjia Sun, Edward Lank, and Michael Terry. 2017. +Label-and-learn: Visualizing the likelihood of ma- +chine learning classifier’s success during data la- +beling. +In Proceedings of the 22nd International +Conference on Intelligent User Interfaces, IUI ’17, +pages 523–534, New York, NY, USA. Association +for Computing Machinery. +Laurens Van der Maaten and Geoffrey Hinton. 2008. +Visualizing data using t-sne. +Journal of machine +learning research, 9(11). +Jie Yang, Yue Zhang, Linwei Li, and Xingxuan Li. +2018. +YEDDA: A lightweight collaborative text +span annotation tool. In Proceedings of ACL 2018, + +System Demonstrations, pages 31–36, Melbourne, +Australia. Association for Computational Linguis- +tics. + diff --git a/_NE1T4oBgHgl3EQfUwPr/content/tmp_files/load_file.txt b/_NE1T4oBgHgl3EQfUwPr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8305aa277a2274457d5176ec473192115d9f8e2c --- /dev/null +++ b/_NE1T4oBgHgl3EQfUwPr/content/tmp_files/load_file.txt @@ -0,0 +1,449 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf,len=448 +page_content='MEGAnno: Exploratory Labeling for NLP in Computational Notebooks Dan Zhang∗, Hannah Kim∗, Rafael Li Chen, Eser Kandogan, Estevam Hruschka Megagon Labs {dan_z,hannah,rafael,eser,estevam}@megagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='ai Abstract We present MEGAnno, a novel exploratory annotation framework designed for NLP re- searchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Unlike existing la- beling tools that focus on data labeling only, our framework aims to support a broader, iter- ative ML workflow including data exploration and model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' With MEGAnno’s API, users can programmatically explore the data through sophisticated search and auto- mated suggestion functions and incrementally update labeling schema as their project evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Combined with our widget, the users can inter- actively sort, filter, and assign labels to mul- tiple items simultaneously in the same note- book where the rest of the NLP project re- sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' We demonstrate MEGAnno’s flexible, exploratory, efficient, and seamless labeling experience through a sentiment analysis use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 Introduction Data labeling is an important step in the machine learning life cycle since the quality and quantity of training data directly affect the model perfor- mance (Geiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Unfortunately, existing annotation tools tend to consider the data labeling step in isolation from the broader ML life cycle, ignoring the iterative workflow of researchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' However, activities such as data se- lection, exploratory data analysis, data labeling, model training, and evaluation do not happen se- quentially (Rahman and Kandogan, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Instead, continuous iterations are required to improve data, annotation, and models (Hohman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To further investigate this gap, we examine the data annotation practices within the ML model de- velopment life cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Based on a formative study with six researchers from our organization, we char- acterize their annotation practices as a “dual-loop” model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' After data preprocess- ∗ Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Raw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Corpus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Unlabeled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Pre-processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Labeling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Debugging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Annotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Schema ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Downstream ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Traditional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Labeling Tools ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='MEGAnno ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Defining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Labeling Task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Understanding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='& Exploration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Loop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Evaluation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Loop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Debugging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='Figure 1: Dual-loop model for data annotation: (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='data understanding/exploration loop (yellow): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='itera- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='tively update annotation schema while exploring and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='annotating data and (2) model evaluation loop (green): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='train and improve a downstream model over iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='by debugging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Most tools are focused on the la- beling step only (red box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' MEGAnno aims to cap- ture both loops seamlessly within the framework (green box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' ing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 1⃝), researchers define their annota- tion schema in terms of what labels to collect, how many data points are needed, and so on (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 2⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' As they explore and annotate the data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 3⃝), they often go back and refine the annotation schema due to their improved understanding of the data and updated mental models for the tasks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 4⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' For example, in a document classification task, a user may start with loosely defined category labels and add more choices as she discovers relevant documents (Felix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Throughout this paper, we refer to this cycle as data understand- ing/exploration loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Next, the labeled data is exported from the annotation tool and is used to train a model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 5⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' However, in practice, ML model training is rarely completed in one pass and usually goes through many iterations of labeling, training, data, and model debugging (Pustejovsky and Stubbs, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 6⃝ refers to the cases where researchers may need to collect more data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', for the less represented classes due to a sub- optimal prediction performance of the downstream model) or further refine the schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' We call this cy- cle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', model training, evaluating and debugging, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='03095v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='HC] 8 Jan 2023 collecting more data, and training again) model evaluation loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' We find that the iterative dual-loop workflow of NLP researchers and practitioners is rarely sup- ported in most existing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' More specifically, we identified three main challenges towards supporting the full annotation life cycle: Gaps between ML toolings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Most of the ex- isting tools are standalone and designed for a specific ML step, which results in frequent context switching and data migration over- head in the researchers’ daily workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Lack of customizable and granular con- trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Not all data points are equally impor- tant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' There are often cases where users might want to prioritize a particular batch (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', to achieve better class or domain coverage or focus on the data points that the downstream model cannot predict well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Although some recent active learning based tools (Montani and Honnibal, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' hua) can provide sugges- tions for the next batch, most tools do not offer customizable and fine-grained control with or with a downstream model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', covering both loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Lack of support for project evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Cur- rent annotation tools usually work with the assumption that the data collection task is well-defined and immutable and ignore that annotation projects can evolve as explorations happen and insights are gathered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Thus they lack the support to help users make evolution decisions, and their immutable nature makes it hard to apply these changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To address the challenges, we present MEGAnno, a flexible, exploratory, efficient, and seamless labeling framework for NLP researchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' It provides a seamless expe- rience where data pre-processing, annotation, analysis, model development and evaluation can happen in the same notebook, a popular daily working environment for data science practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' MEGAnno provides customizable interfaces to help users drive their project to the desired directions through rich heuristic-based search, automatic suggestion, and active learning based suggestions of the next data batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' With project evolution in mind, MEGAnno is designed to work with flexible task schema and provides a built-in analysis dashboard to aid decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To our knowledge, MEGAnno is the first flexible, exploratory labeling framework that can support ML workflow seamlessly in computational notebooks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1: green box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 2 MEGAnno 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='1 Framework Overview Database server Web Server Jupyter Server Jupyter Notebook Python API Direct REST requests MEGAnno service MEGAnno toolkit In [1]: MEGAnno toolkit Widget Figure 2: The MEGAnno framework provides ex- ploratory annotation services through a toolkit (instal- lable as Python libraries) providing programmatic inter- faces, a web server providing language-agnostic REST APIs, and an internal database to store data, annotation, and related artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Solid lines show programmatic interactions through Python APIs calls and REST calls delegated by our notebook widget or directly issued by authenticated applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Dotted lines show internal interfaces where MEGAnno toolkit handles communi- cation with the database and are hidden from the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' MEGAnno provides service through 1) an inter- nal database that stores the data, annotations, and various artifacts produced in the annotation pro- cess, 2) a MEGAnno toolkit that provides python API for programmatic and visual data exploration and labeling, and 3) a web server that provides language-agnostic REST APIs (Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' After in- stalling the toolkit on a Jupyter server, users will have access to our Python APIs and React-based widget to manage their project, explore and anno- tate their data from any connected notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Data model A Data record refers to an item in the pre-processed data corpus for labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' It can be a sentence, a paragraph, a document, or a flattened text from multiple texts such as a question- answer pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' A Label is the smallest unit of user labeling output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' MEGAnno currently supports record-level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', topics for document or sentence) and span-level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', named entities) labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' An Annotation is a set of labels given to a data record by an annotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Metadata refers to addi- tional information related to the content of a data record (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', externally generated part-of-speech tags, embeddings) or of an annotation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', time taken to label, disagreement among annotators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' File Edit View Insert Cell Kernel Widgets Help + % 心 个 Run c Codesubset = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content="search(3o0, by='id order') subset." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='show() MEGAnno Q Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=" Single Table Submit Clear filter 1 1 data sentiment_label Bulk label 0 @JetBlueI'vehad no shampoo sad no winter coat sad no deodorant sad no flatiron sad and the rest if n neg @JetBlue right now completely pissed off sad neg 2 didn't go through." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Was tol neg @SouthwestAir no self help way to put in tsa pre check number for 3 existing reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Very annoying sad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 80o is not reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' neg 300 showingjupyterFigure 3: The table view to show multiple data records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Hovering over a data record shows its full text in a pop- up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' This view allows exploration by searching, sorting, and filtering over labels and single/bulk annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Such information can be helpful in various steps of the ML iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' A Subset is a slice of the data records in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Subsets can be of random data records or can be generated through search queries that match certain characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Task schema We support a wide variety of tasks through our customizable schema in JSON format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To collect a label, users need to specify the level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', record or span) and provide a list of options to choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' For a task, MEGAnno supports arbitrary numbers of both types of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' We’ll see concrete schema examples for a sentiment pre- diction and extraction task in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' At any stage, users can always update the schema to re- flect the evolution of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' There are certain constraints to schema updates to keep the consis- tency of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Adding new labels or new label options will always be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Removal of la- bels and options will trigger a database query and MEGAnno will warn the user if there exist such labeled instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='2 MEGAnno Jupyter Notebook Widget MEGAnno’s interactive notebook widget features 1) our novel table view to facilitate exploratory and batch labeling and 2) the single view, which is similar to traditional labeling UIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Table view The table view (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 3) shows data items in a Subset and their annotations if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Each record-level label is shown as a column, and span-level labels are shown together with the high- lighted textual span in the data column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Users can hover over an item to see full text in a pop-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' The search box supports three types of search (exact, fuzzy, and regex-based) to filter the data subset fur- ther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Users can sort and filter rows based on any 1MEGAnno provides an option to clean up legacy labels and retry automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' record-level labels using the dropdown menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To assign a record-level label, the users can click on the cell’s arrow button and select from the drop- down options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Alternatively, the users can assign the same label to multiple records simultaneously by selecting those records and clicking the bulk label button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Single view The table view is good for explo- ration, but the limited space makes span-level an- notation cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' So we also provide a single view where users can have a more zoomed-in ex- perience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' By clicking the “Single” button on the top-right corner or double-clicking on a data record, the widget switches to the single view (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In this view, users can assign record-level labels on the right side and span-level labels on the left side by selecting/dragging target spans and choosing the label from the options drop-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Users can loop through the subset using the prev/next button based on the order specified in the table view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' At any time, users can switch between the two views by clicking the top right buttons, and the widget preserves all uncommitted annotations during view changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Figure 4: The single view to annotate data one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In this view, users can drag and label spans for extrac- tion tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Working with multiple annotators Annotation is rarely done by a single person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' As an initial step towards collaborative annotation, MEGAnno provides virtually separated namespaces for each annotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Users identify themselves by a unique authentication token while connecting to the ser- vice and only update their own labels through the widgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' MEGAnno provides a reconciliation view (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 5) to look at labels from different individuals and resolve potential conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Dashboard MEGAnno also provides a built- in visual monitoring dashboard (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' As projects evolve, users would need to understand the project’s status to make decisions about the next steps, like collecting more data points with certain characteristics or adding a new class to the In[109]: subset = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content="search(3o0, by='id_order') subset." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='show() MEGAnno Q Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=" Single Table + Submit Clear filter Search box Sorting & filtering ::8: :8: data V 1 sentiment_label Bulk label assignment sentiment_label 0 @JetBlue I've had no shampoo sad , no winter coat sad no deodorant sad no flatiron sadand the res neg 12 Sort asc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' ↑ Sort desc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=" 1 @JetBlue right now completely pissed off sad neg Filter by 2 didn't go through." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Wa neg @SouthwestAir no self help way to put in tsa pre check number for Positive pos 3 existing reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Very annoying sad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 8oo is not reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' neg Negative neg 4 @SouthwestAir no self help way to put in tsa pre check number for existing reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Very annoying sad neg Neutral neu 5 @united is there a referral program for the Milageplus explorer card?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 300 showingMEGAnno Single Table ← Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' ← Xan sentiment_label @SouthwestAir no self help way to put in tsa pre check number for existing reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' O Positive (pos) Very annoying sad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 800 is not reachable Negative (neg) Label as Neutral (neu) Happy hap Sad sad 300 showing Remove labelFigure 5: Reconcilation view showing the existing la- bel distribution for data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Figure 6: Dashboard widget to monitor the progress and statistics of the project and aid decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' task definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To aid such analysis, the dashboard widget packs common statistics and analytical vi- sualizations based on a survey of our pilot users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' The “overview” panel shows statistics about overall progress and per-label class distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' If multi- ple annotators are involved, the distribution reflects the majority vote over annotators 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' The remaining “annotator” and “projection” panels are hidden due to space limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To help identify problematic annotators, the annotator pannel shows statistics like overall individual contribution and disagree- ment scores with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' The projection panel pro- vides customizable visualization to project data points to a two-dimensional visual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' By de- fault, we show the t-SNE (Van der Maaten and Hinton, 2008) projection of sentence bert (Reimers and Gurevych, 2019) embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='3 MEGAnno APIs Project management The management module provides various interfaces to configure and mon- itor the annotation project through the Project class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' import_data loads the data records from CSV or JSON files into the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' set_config updates the project configuration as it evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' set_meta assigns metadata (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=',POS tags, document embeddings) for each data record through user-defined functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' get_status re- turns the status of the project such as the number of 2Users can provide their aggregation function to resovle conflicts between annotators annotated data records and detailed statistic about each label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' A critical feature of MEGAnno is to select in- teresting subsets of data to show in the widget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Subsets can be generated in a user-initiative way via our search engine or a data-driven way via au- tomated suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Search for subsets MEGAnno supports so- phisticated searches over data records, anno- tation, and user-defined metadata through the Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='search API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Users can search data records by keywords (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', documents men- tioning “customer service”) or regular expres- sions to express more complex patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' The users can also search the database based on al- ready assigned labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', records with a pos- itive sentiment label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' As will be explained later, MEGAnno acknowledges the value of auxiliary information for ML projects and pro- vides advanced search functionalities over meta- data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' For example, users can query with pat- terns combining regex expressions and POS tags like project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='search("(best|amazing) ", by="POS").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Automated subset suggestion Searches initi- ated by users can help users explore the dataset in a controlled way, but the quality of searches is only as good as users’ knowledge or heuristic about the data and domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' MEGAnno provides an automated subset suggestion engine to assist the exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Users can customize the engine by plugging in external suggestion models as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Currently, the engine provides two types of tech- niques: Embedding-based suggestions makes suggestions based on data embed- ding vectors provided by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='suggest_similar sug- gests neighbors of data in the querying subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='suggest_coverage examines all the data records within the embedding space in an unsupervised way and suggests data points from the less annotated regions to improve annotation coverage of the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Active suggestions utilizes active learning techniques to recommend the most informa- tive data for the downstream model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' With libraries like ModAL (Danka and Horvath), Reconcile 日 Reconciling % 1 1 11 data sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=" V 1 Bulk labe 20 @VirginAmerica now it's just t-minus 32 minutes I pos: 100." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='00% 21 @SouthwestAir Black History Commercial is really neg: 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='00% pos: 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='00% 22 @UsAirways sits on a throne of lies neg: 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='00%Dashboard ii Aggregated Overview Annotator Projection :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=" Overall Progress Annotated 100 | data points (with at least 1 label) out of 201 |total data points Class Label - Distributions Label class distribution forsentiment subtask: aggregated over annotators' votes using (majority_vote)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' When there is a tie in the voting, they are categorized under "tied_annotations" class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Labels neg neu pos 16 62users can select from various selection strate- gies based on model uncertainty and entropy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Since MEGAnno’s seamless notebook experience covers the whole loop from anno- tation to model training and debugging, users can actively select a subset, annotate the sub- set, update the model, and test again in the same notebook without switching environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' The output of selection engines are instances of the Subset class with the following methods: show returns a notebook widget for interactive ex- ploration and annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' batch_annotation sets the same record-level labels for all data records in the subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' suggest_similar returns a new subset of the database containing the most sim- ilar data for each record in the querying subset according to some metadata with valid distance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 3 Use Case: Sentiment Analysis We present a use case to illustrate how MEGAnno can support data annotation in NLP researchers and practitioners’ workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Meggie is a data scientist who wants to train a sentiment-related model for her project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' She obtains a Twitter dataset3about US airlines and decides to label it using MEGAnno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Import data In an empty notebook, Meggie starts by initializing the project named “Tweet Sen- timent” using a MEGAnno Python API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' She gets a copy of the data from the product manager who often uses Google spreadsheet and imports the data from its published link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 from meganno import Project 2 L = Project(, "Tweet Sentiment") 3 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='import_data(, format="csv") Set up initial schema Without knowing much about the data, Meggie decides to start the project by collecting binary labels and setting up the project’s schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Knowing that MEGAnno sup- ports flexible and editable schema, Meggie does not worry about getting the perfect schema in the first round and can start exploring and annotating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 label_schema = [{ 2 "label_name": "sentiment_label", 3The Kaggle dataset (for Everyone library) has ground- truth sentiment labels available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' But for demonstration pur- poses, we ignore them and assume Meggie only gets the raw Tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' The dataset contains 14K tweets about major US airlines scraped in February 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Figure 7: UI Search by regular expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Matched keywords are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 3 "level": "record", 4 "options": ["Positive", "Negative"] 5 }] 6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='set_config(config1) Explore and annotate She starts by exploring the first 300 data points in the widget’s table view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Using the search box, she filters the subset with the keyword “amazing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' As expected, most of the data records reflect a positive sentiment, so she assigns a positive label to multiple data items using the “Bulk label” button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Next, she wants to examine tweets with hashtags related to failing, so she tries regu- lar expression search using #fai[^ ]* (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' With better understanding of the dataset, Meggie chooses to perform more advanced exploration us- ing part-of-speech metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' She imports a POS tagger from spaCy (Honnibal and Montani, 2017) and retrieves tweets that match interesting patterns such as best of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 import spacy 2 tagger = spacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='load("en_core_web_sm") 3 4 pos1 = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='search("(best|amazing) < NOUN>", by="POS", tagger=tagger) 5 pos2 = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='search("best of ", by="POS", tagger=tagger) 6 meganno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='union(pos1, pos2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='show() With such an exploratory approach and the batch labeling feature, she can annotate much faster and in a more controlled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Candidate suggestion After a few more heuris- tics, she runs out of ideas, so she takes advantage of the suggestion feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' She selects the sentence bert (Reimers and Gurevych, 2019) encoder as the meta-data generation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' She first issues a query with strategy similarity to collect data points similar to the ones retrieved from the previ- ous POS search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Finally, she wants to see samples from less covered areas in the embedding space to In［124]: # iterate by id L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='search_by_id(limit=3oo0, skip = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content="show(widget_config) showing demo's annotations Twitter Sentiment Q /#*fai[^ ]*/ Single regex 田 Table Matching on failures, faith, fail, failing!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', #fail 个 Submit #fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', fair, fails, fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', faithful, failed, failscustomers, 1 data fairs, failure, failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', fairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', #fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='" ent_l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=" ulk label Suggestions ^ey @USAirways tried twice today on hold for 30 min e ^jh ^jp ^kn ^kp 2 @SouthwestAir thanks, got put on the am flight tor Search Mode Tips 3 @united It is super frustrating that the folks at the United Ticket Counter in Pittsburg aren't honoring their ov @JetBlue #fail My wife on the phone asking to switch flight times." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In mid 4 Isly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' This is why airl switch gets disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Now Cancelled Flightled!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' No new time!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' No call 5 back 6 @JetBlue#fail My wife on the phone asking to switch flight times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In mid switch gets disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Now Car @JetBlue Guys, really bad @JetBlue #fail .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Someone better call my wife back to get this handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 203-382-3 demo 36 of 2242 showingimprove the diversity of the training data and issues a coverage query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 m = SentenceTransformer("all-MiniLM-L6") 2 # set metadata generation function 3 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='set_meta("bert", lambda x: list(m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' encode(x))) 4 # get more data like the query result in the previous query(subset_pos) 5 subset_sim = pos2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='suggest_similar( meta_name="bert") 6 # get data from less covered areas in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 7 subset_cov = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='suggest_coverage( meta_name="bert") Update the schema Meggie is now happy with the collected labels, but she aims to go one step further to understand which words or phrases lead to the sentiment judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' So she updates her schema by adding a new span-level label called sentiment_span with label options be- ing “happy” or “sad”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Each data record can have arbitrary numbers of such span-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 1 label_schema = [{ 2 "label_name": "sentiment_span", 3 "level": "span", 4 "options": ["Happy", "Sad"] 5 }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' {# previous label options 6 }] After updating the schema, she fetches all records with neutral labels in a widget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To high- light and annotate spans, she goes into the single view (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' At any step of the iter- ation, she could refer to the dashboard widget to monitor the project progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' After several rounds of similar iterations, she feels good and concludes her exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Finally, Meggie can export the an- notated data in JSON or CSV formats for training or plug in the model directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In conclusion, with MEGAnno, Meggie can ex- plore her dataset using various heuristic-based or automated search functions and better understand the data corpus as she labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' She has the flexibil- ity to iteratively update her schema as the project evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Using the widget, Meggie can finish the entire ML life cyle in the same Jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 4 Related Work There exist numerous annotation tools that can sup- port NLP tasks, which are extensively surveyed by Neves and Seva (2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In this section, we focus on works that are closer to our flexible, exploratory, efficient, and seamless framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Flexible schema Unfortunately, most of existing tools are not designed for iterative schema devel- opment, and thus they are not flexible enough for evolving projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Felix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' (2018) and Kulesza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' (2014) allow users to progressively define document classes by inspecting documents that are assigned to each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' But these works are more similar to interactive topic modeling or inter- active classification, where users assign documents to classes, than document labeling, where users assign labels to documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Our tool goes beyond document label refinement and supports a broader task of progressively defining annotation schema (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', additionally collecting span-level labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Exploratory labeling The concept of ex- ploratory labeling is introduced by Felix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' (2018) as using computational techniques to help users group documents into evolving labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In our paper, we use the term “exploratory labeling” to refer to where exploratory data analysis and data annotation are iteratively conducted in the data understanding/exploration loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Exploratory labeling can be beneficial because while labeling data, users gain insight into their dataset (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Efficient batch/bulk annotation A few tools of- fer a functionality to simultaneously assign la- bels to multiple spans within a record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' For ex- ample, YEDDA (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', 2018) can anno- tate multiple span-level labels via command line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' TALEN (Stephen Mayhew, 2018), a named en- tity tagging tool, has an entity propagation feature which annotates all mentions of an entity in a doc- ument at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In contrast, users can annotate mul- tiple records simultaneously using our Python API and a GUI widget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Notebook widget Computational notebooks are frequently used by data analysts to iteratively write and edit code to understand data, test hypotheses, and build models (Head et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Randles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Following the practice of mage (Kery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=', 2020) which extends Jupyter notebook with GUI widgets for specific tasks, our widget is designed to achieve flexible communication with the rest of ML development codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Annotation tools which are im- plemented as Jupyter widgets include Pigeon (pig) and ipyannotate (ipy), but they only offer a simple label assignment feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 5 Conclusion In this paper, we present MEGAnno, an annota- tion framework designed for NLP researchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Through MEGAnno’s programmatic interfaces and interactive widget, users can itera- tively explore and search for interesting data sub- sets, annotate data, train models, evaluate and de- bug models within a Jupyter notebook without the overhead of context switch or data migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' References humanloop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' https://humanloop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' ipyannotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='com/ ipyannotate/ipyannotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' pigeon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content='com/ agermanidis/pigeon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Tivadar Danka and Peter Horvath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' modAL: A modu- lar active learning framework for Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Available on arXiv at 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+page_content=' In Proceedings of the 31st Annual ACM Sym- posium on User Interface Software and Technology, UIST ’18, pages 153–164, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' As- sociation for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Crowdflower Data for Everyone library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Twitter US Airline Sentiment(Version 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Stuart Geiger, Dominique Cope, Jamie Ip, Marsha Lotosh, Aayush Shah, Jenny Weng, and Rebekah Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' "garbage in, garbage out" revisited: What do machine learning application papers report about human-labeled training data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Quantitative Science Studies, pages 1–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Andrew Head, Fred Hohman, Titus Barik, Steven M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Drucker, and Robert DeLine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Managing Messes in Computational Notebooks, pages 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Fred Hohman, Kanit Wongsuphasawat, Mary Beth Kery, and Kayur Patel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Understanding and visualizing data iteration in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' In Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems, pages 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Matthew Honnibal and Ines Montani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' spaCy 2: Natural language understanding with Bloom embed- dings, convolutional neural networks and incremen- tal parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Mary Beth Kery, Donghao Ren, Fred Hohman, Do- minik Moritz, Kanit Wongsuphasawat, and Kayur Patel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' mage: Fluid moves between code and graphical work in computational notebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Todd Kulesza, Saleema Amershi, Rich Caruana, Danyel Fisher, and Denis Charles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' 2014.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} +page_content=' Association for Computational Linguis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfUwPr/content/2301.03095v1.pdf'} diff --git a/aNE3T4oBgHgl3EQfdAo0/content/tmp_files/2301.04530v1.pdf.txt b/aNE3T4oBgHgl3EQfdAo0/content/tmp_files/2301.04530v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e273d4a8cc17df30a893f2023d39b82e10390c96 --- /dev/null +++ b/aNE3T4oBgHgl3EQfdAo0/content/tmp_files/2301.04530v1.pdf.txt @@ -0,0 +1,2546 @@ +arXiv:2301.04530v1 [math.DS] 11 Jan 2023 +ACTION ON THE CIRCLE AT INFINITY OF FOLIATIONS OF R2 +CHRISTIAN BONATTI +Abstract. This paper provides a canonical compactification of the plane R2 by adding +a circle at infinity associated to a countable family of singular foliations or laminations +(under some hypotheses), generalizing an idea by Mather [Ma]. Moreover any homeo- +morphism of R2 preserving the foliations extends on the circle at infinity. +Then this paper provides conditions ensuring the minimality of the action on the +circle at infinity induced by an action on R2 preserving one foliation or two transverse +foliations. +In particular the action on the circle at infinity associated to an Anosov flow X on a +closed 3-manifold is minimal if and only if X is non-R-covered. +Keywords: Foliation of the plane, Anosov flow, compactification. +Codes AMS: 37D20-37E10-37E35-37C86 +January 12, 2023 +1. Introduction +1.1. General presentation. There are many ways to compactify the plane R2, the simplest one being +the Alexandrov compactification by point at infinity, and R2 ∪ {∞} is the topological sphere S2. This +compactication is canonical and does not depend on any extra structure on R2. That is its strength, but +also its weakness as it does not bring any informations on any structure we endow R2. +Another very natural and usual compactification of R2 is by adding a circle at infinity, so that R2 ∪ S1 +is the disc D2. This compactification is not canonical: it consists in a homeomorphism h: R2 → ˚ +D2, where +˚ +D2 is the open disc. Two homeomorphisms h1, h2 define the same compactification if h2 ◦ h−1 +1 : ˚ +D2 → ˚ +D2 +extends on S1 = ∂D2 as a homeomorphism of D2. There are uncountably many such a compactification. +Here, we start be recalling Mather [Ma] canonical compactification of the plane R2, endowed with a +foliation F, by a circle at infinity S1 +F. Then we explore the flexibility of this contruction for extending +it to more general objects. Thus, we provide an elementary (nothing sophisticated), simple (nothing too +complicated), and unified construction which associates a compactification D2 +F of the plane R2 by the +disc D2 to a countable family F = {Fi} of foliations, non-singular or with singular points of saddle type, +which are pairwise transverse or at least have some kind of weak transversality condition at infinity, see +the precise statements below. The boundary ∂D2 +F is called the circle at infinity of F and is denoted by +S1 +F. This compactification is unique, in the sense that the identity on R2 extends as a homeomorphism +on the circles at infinity of two such compactifications. +For giving a concrete example, Corollary 5.1 builds this canonical compactification D2 +F associated to +any countable family F = {Fi} of singular foliations, where each Fi is directed by a polynomial vector +field on R2 whose singular points are hyperbolic saddles. +The uniqueness of the compactification implies that any homeomorphism of R2 preserving F (that is, +permuting the Fi) extends as an homeomorphism of the compactification D2 +F , inducing a homeomorphism +of the circle at infinity S1 +F. +1 + +2 +CHRISTIAN BONATTI +1.2. Mather idea for building the circle at infinity. The common setting for this unified construc- +tion are families of rays, where a ray is a proper topological embedding of [0, +∞) on R2. We require +that the germs of the rays in the family are pairwize disjoint, meaning that the intersection between any +two distinct rays is compact. The key idea is that a set of rays in R2 whose germs are pairwize disjoint +is totally cyclically ordered, and we will use this cyclic order for building the circle at infinity. +The key technical result (essentially due to [Ma]) is: +Theorem 1. Let R be a family of rays in R2 whose germs are pairwise disjoint. Let E ⊂ R be a countable +subset which is separating for the cyclic order, that is, any non-degenerate interval contains a point in E +(see Definition 2.1). +Then there is a compactification of R2 by the disc D2 so that: +• any ray of R tends to a point of the circle at infinity ∂D2 = S1. +• any two distinct rays of R tend to distinct points of S1 +• the points of S1 which are the limit point of a ray in R are dense in S1. +Furthermore, this compactification is unique up to a homeomorphism of D2 and does not depend on the +separating countable set E. +Then Theorem 6 provides such a canonical compactification for a countable union R = � Ri, i ∈ I ⊂ N +of families of rays, assuming that the germs of rays in R are pairwise disjoint and each Ri admits a +countable separating subset Ei. The difficulty here is that R by itself may not admit any separating +family. The idea for solving this problem consists in considering a natural equivalence relation on R, +identifying the rays which cannot be separated. +1.3. Countable families of transverse foliations. A natural setting where we will apply this general +construction are (at most countable) families of transverse foliations on the plane R2. Notice that any +half leaf of a (non-singular) foliation of R2 is a ray. An end of leaf is the germ at infinity of an half leaf. +In this setting we get: +Theorem 2. Let F = {Fi}i∈I⊂N be an at most countable family of pairwise transverse foliations on the +plane R2. +There is a compactification D2 +F ≃ D2 of R2 by adding a circle S1 +F = ∂D2 +F with the following properties: +• Any end of leaf tends to a point of the circle at infinity S1 +F, +• The set of ends of leaves tending to a same points of S1 +F is at most countable, +• For any non-empty open subset O ⊂ S1 +F the set of ends of leaves having their limit in O is +uncountable. +This compactification with these three properties is unique, up to a homeomorphism of D2 +F. +The circle S1 +F is called the circle at infinity of the family F = {Fi}i∈I⊂N. +Remark 1. The countablity of the set of ends tending to the same point implies that +• the two ends of a given leaf always have distinct limits on S1 +F. +• if two leaves L1, L2 of the same foliation Fi have the same pair of limits of ends, they are equal +(see Lemma 4.2). +Recall that foliations of R2 may have leaves which are not separated one from the other. The leaves +which are separated from any other leaves are called regular leaves. At most countably many leaves are +not regular (see here Lemma 3.2). We will see that, +Proposition 1.1. Let F = {Fi}i∈I⊂N be an at most countable family of pairwise transverse foliations +on the plane R2. Any two distinct ends of regular leaves of the same foliation Fi tend to two distinct +points of S1 +F. +Now, in the setting of Theorem 2 we can apply this theorem to each foliation Fi, i ∈ I so that we get +a family of compactifications D2 +Fi. In fact, we get a compactification D2 +J for any subfamily J ⊂ I leading +to an uncountable set of (maybe distinct) compactifications of R2 by the disc D2 (Example 5 provides + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +3 +a simple example where these compactifications D2 +J, for J ⊂ I, are pairwize distincts and uncountably +many). +These compactifications are easily related : for any subfamily J ⊂ I the identity map on R2 extends +in a unique way by continuity as a projection ΠI,J : D2 +F = D2 +I → D2 +J, which simply consists in colapsing +the intervals in S1 +I which do not contain any limit of an end of a leaf of a foliation Fj, j ∈ J. +We will also see in a simple example that the assumption of at most countability of the family I of +foliations cannot be erased: for instance, the conclusion Theorem 2 is false for the family of all afine +foliations (by parallel straight lines) of R2, parametrized by RP1 (see Example 4). +Example 8 and Lemma 4.8 present a simple example where generic points (i.e. points in a residual +set) of the circle at infinity S1 +F of a foliation F are not the limit of any end of leaf of F. In this example, +at the contrary, points in a dense subset of S1 +F are limit of 2 distinct ends of leaves. +Lemma 4.4 and 4.5 caracterize the points p at the circle at infinity S1 +F, where F is a foliation of R2, +which are limit of several ends of leaves: the rays arriving at p are ordered as an interval of Z and two +successive ends bound a hyperbolic sector. +Corollary 5.3 generalizes Lemma 4.4 and 4.5 to the case of a countable family F = {Fi} of transverse +foliations and gives a complete description of the points in S1 +F which are limit of several ends of leaves of +the same Fi. +1.4. Countable families of non-transverse or singular foliations. This construction can be gen- +eralized easily to the setting of families of non transverse or singular foliations. Let us present the most +general setting we consider here. +The foliations we consider admit singular points which are saddle point with k-separatrices (also called +k-prongs singularity), k > 1, the case k = 2 corresponding to non-singular points. +In this setting an end of leaf is a ray of R2 disjoint from the singular points and contained in a leaf. +Theorem 3. Let F = {Fi}, i ∈ I ⊂ N be a family of singular foliations of R2 whose singular points are +each a saddle with k-separatrices with k > 2. We assume that, given any two ends L1, L2 of leaves we +have the following alternative: +• either the germs of L1 and L2 are disjoints +• or the germs of L1 and L2 coincide. +Then there is a compactification D2 +F ≃ D2 of R2 by adding a circle S1 +F = ∂D2 +F with the following properties: +• Any end of leaf tends to a point of the circle at infinity S1 +F, +• The set of ends of leaves tending to a same points of S1 +F is at most countable, +• For any non-empty open subset O ⊂ S1 +F the set of ends of leaves having their limit in O is +uncountable. +This compactification with these three properties is unique, up to a homeomorphism of D2 +F. +The hypothesis that the germs of ends of leaves are either equal or disjoint means that if the intersection +of two leaves is not bounded, then these two leaves coincide on an half leaf. One easily checks that +transverse foliations satisfy this hypothesis so that Theorem 2 is a straightforward corollary of Theorem 3. +As a simple and natural example, we will see that any countable family F = {Fi} of singular foliations, +directed by polynomial vector fields on R2 whose singular points are hyperbolic saddles, satisfies the +hypotheses of Theorem 3: this will prove Corollary 5.1 already mentioned above. +1.5. Laminations. The construction of the circle at infinity for foliations cannot be extended without +hypotheses to the case of laminations, as leaves of laminations may fail to be lines, and can even be +recurrent, see for instance example 13. +Theorems 8 and 10 provide a generalisation of this construction to closed orientable laminations with +no compact leaves and with uncountably many leaves. This generalisation is not as satifactory as in +the case of foliations, and we discuss some of the issues in Section 6. In particular Theorem 9 provides +another canonical compactification, which holds also for countable oriented laminations with no compact +leaves. + +4 +CHRISTIAN BONATTI +1.6. Minimality of the action on the circle at infinity. Then we consider group actions H ⊂ +Homeo(R2) on R2 preserving 1 or 2 transverse foliations Fi. The action of H extends canonically on the +circle at infinity and we will consider the following question: +Question 1.1. Under what conditions on H and on the foliations Fi can we ensure that the action +induced on S1 +{Fi} is minimal? +Our main result, for the case of 1 foliation is the following: +Theorem 4. Let F be a foliation of R2 and H ⊂ Homeo(R2) be a group of homeomorphisms preserving +F. We assume that for any leaf L, the union of its images H(L) is dense in R2. +Then the two following properties are equivalent +(1) the action induced by H on the circle at infinity is minimal +(2) there are pairs of distinct leaves (L1, L2) and (L3, L4) so that L1 and L2 are not separated from +above and L3 and L4 are not separated from below. +We will also generalize Theorem 4 for families of transverse foliations. +1.7. Action on the circle at infinity of an Anosov flow. Finally, we will consider the setting of an +Anosov flow X on a closed 3-manifold M. +Remark 2. In this setting it is known that π1(M) acts on S1 by orientation preserving homeomorphisms, +see Calegari Dunfield [CaDu] inspirated by an unpublished work of Thurston [Th]. This works follows +completely distinct ideas that those presented here. +Another construction of this circle at infinity (called ideal circle boundary) is given in [Fe4] for pseudo- +Anosov flows. +Barbot and Fenley [Ba1, Fe1] show that the lift ˜X of X is conjugated to the constant vector field +∂ +∂x +on R3, so that the ˜X-orbit space is a plane PX ≃ R2. This plane PX is endowed with two transverse +foliations F s, F u which are the projection of the stable and unstable foliations of X lifted on R3. Thus +(PX, F s, F u) is the bifoliated plane associated to X. Furthermore, the fundamental group π1(M) acts on +PX and its action preserves both foliations F s and F u. This action induces a natural action of π1(M) +on the circles at infinity S1 +F s, S1 +F u, and S1 +F s,F u. +A folklore conjecture asserts that two Anosov flows are orbitaly equivalent if and only if they induces +the same action on the circle at infinity of {F s, F u}, see [Ba1] for a result in this direction. This conjecture +as been recently announced to be proved in [BFM]. +[Ba1, Fe1] show that every leaf of F s is regular if and only if every leaf of F u is regular, and then the +Anosov flow X is called R-covered. Our main result in that setting is +Theorem 5. Let X be an Anosov flow on a closed 3-manifold and (PX, F s, F u) its bifoliated plane. Let +D2 +F s,F u, D2 +F s, and D2 +F u be the compactifications associated to, respectively, the pair of foliations F s, F u, +the foliation F s and the foliation F u. Then +(1) D2 +F s,F u = D2 +F s = D2 +F u unless X is orbitally equivalent to the suspension of an Anosov diffeomor- +phism of the torus T2. +(2) the action of π1(M) on the circles at infinity S1 +F s,F u,(or equvalently S1 +F s or S1 +F u) is minimal if +and only if X is not R-covered. +When X is assumed to be transitive, this result is a simple consequence of Theorem 4 above and a +result by Fenley [Fe3] ensuring that, assuming X is non-R-covered, then F s and F u admit non-separated +leaves from above and non-separated leaves form below. The proof of Theorem 5, when X is not assumed +to be transitive, is certainly the most technically difficult argument of the paper, and is based on a +description of hyperbolic basic sets for flows on 3-manifolds. +Theorem 5 implies that the minimality of the action on the circle at infinity is not related with the +transitivity of the flow. However, according to [BFM] the action on the circle at infinity charaterizes the +dynamics of the flow. This leads to the following question: + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +5 +Question 1.2. What property of the action of π1(M) on the circle at infinity S1 +F s,F u implies the transi- +tivity of X? +Can we find the transverse tori by looking at the action of π1(M) on the circle at infinity? +1.7.1. Aknowledgments. I would thank Sebastien Alvarez who invited me to present the results in this +paper as a mini-course in Montevideo. This mini-course has been a motivation for ending this paper. +I would also thanks Kathrin Mann for indicating me that the argument of Theorem 1 is essentially +contained in [Ma], and Michele Triestino for the statement and reference of Cantor-Bendixson theorem. +2. Circles at infinity for families of rays on the plane +2.1. Cyclic order. Let X be a set. A total cyclic order on X is a map θ: X3 → {−1, 0, +1} with the +following properties +• θ(x, y, z) = 0 if and only if x = y or y = z or x = z. +• θ(x, y, z) = −θ(y, x, z) = −θ(x, z, y) for every (x, y, z) +• for every x ∈ X the relation on X \ {x} defined by +y < z ⇔ θ(x, y, z) = +1 +is a total order. +The emblematic example is: +Example 1. The oriented circle S1 = R/Z is totally cyclically ordered by the relation θ defined as follows: +θ(x, y, z) = +1 if and only if the y belongs to the interior of the positively oriented simple arc staring at +x and ending at z. +If θ is a total cyclic order then for x ̸= z we define the interval (x, y) by +(x, z) = {y, θ(x, y, z) = 1}. +We define the semi closed and closed intervals [x, z),(x, z], and [x, z] by adding the corresponding +extremities x or z to the interval (x, z). +We say that y is between x and z is y ∈ (x, z). +The following notion of separating set will be fundamental all along this work: +Definition 2.1. Let X be a set endowed with a total cyclic order. A subset E ⊂ X is said separating if +given any distinct x, z ∈ X there is y ∈ E (distinct from x and z), between x and z. +We will use the following easy exercize of topology of R and S1: +Proposition 2.1. Let X be a set endowed with a total cyclic order. Assume that there is a countable +subset E ⊂ X which is separating. +Then there is a bijection ϕ of X on a dense subset Y ⊂ S1 which is strictly increasing for the cyclic +orders of X and of S1. Furthermore this bijection is unique up to a composition by a homeomorphism of +S1. +The argument is classical but short and beautiful and I have no references for this precise statement. +So let me present it: +Proof. One builds a bijection φ of E to a contable dense subset D ⊂ S1 by induction, as follows: one +choose an indexation of E = {ei, i ∈ N} and of D = {di, i ∈ N}. One defines +• φ(e0) = d0, φ(e1) = d1 i(0) = j(0) = 0 i(1) = j(1) = 1 +• consider e2, it belongs either in (e0, e1) or in (e1, e0) and we chose φ(e2) being dj(2) where j(2) +is the infimum of the di in the corresponding interval (d0, d1) or (d1, d0). One denotes i(2)=2. +• consider now j(3) = inf N \ {0, 1, i(2)} and define φ−1(dj(3)) = ei(3) where i(3) is the infimum of +the i /∈ {0, 1, 2} so that the position of ei(3) with respect to e0, e1, e2 is the same as the position +of dj(3) with repsect to d0, d1, dj(2). +• . . . + +6 +CHRISTIAN BONATTI +• choose i(2n) = inf N\ {i(k), k < 2n} and φ(ei(2n) is dj(2n) where j(2n) is the infimum of the j so +that dj as the same position with respect to the dj(k), k < 2n as ei(2n) with respect to the ei(k). +• choose j(2n + 1) = inf N \ {j(k), k < 2n + 1} and φ−1(dj(2n+1) is ei(2n+1) where i(2n + 1) is the +infimum of the i so that ei as the same position with respect to the ei(k), k < 2n + 1 as dj(2n+1) +with respect to the dj(k). +At each step of this construction one uses the separation property of E and D for ensuring the existence +of the point announced in the same position. +Once we built φ on E, it extends in a unique increasing way on X. Then the separation property of E +implies that this extension is injective. +□ +Remark 3. Assume that Z, θ is a set endowed with a total cyclic order, and E ⊂ X ⊂ Z are subsets so +that E is separating for X, θ. +Let ϕ: X → Y be the map given by Proposition 2.1. Then φ extends in a unique way as an (not +strictly) increasing map Φ: Z → S1: Φ(y) is between Φ(x) and Φ(z) only if y is between x and z. +The non-injectivity of the map Φ is determined as follows. Consider distinct points x ̸= y of Z, then +Φ(x) = Φ(y) if and only if either (x, y) or (y, x) contains no more than 1 element of X +2.2. Cyclic order on families of rays. A line is a proper embedding of R in R2. A line L cuts R2 in +two half plane. If L is oriented, then there is an orientation preserving homeomorphism h of R2 mapping +L on the oriented x-axis of R2 (endowed with the coordinates (x, y)). This allows us to defined the upper +and lower half-planes ∆+ +L and ∆− +L as the pre-images by h of {y ≥ 0} and {y ≤ 0} respectively. +A ray is a proper embedding of [0, +∞) in R2. Two rays define the same germ of ray if their images +coincide out of a compact ball. Two germs of rays are said disjoint if they admit disjoint realisations. +Example 2. +(1) If F is a foliation of R2, every leaf defines to germs of rays called the ends of the +leaf. By fixing an orientation of F we will speak of the right and left ends of a leaf. +(2) If {Fi}i∈I is a family of pairwise transverse foliations of R2 then the set of all ends of leafs of +the foliations Fi is a family of pairwise disjoint germs of rays. +(3) Consider the set S of all germs of rays γ which are contained in an orbit of an affine (polynomial +of degree = 1) vector field of saddle type. Then S is a family of pairwise disjoint germs of rays. +Next lemmas are simple exercizes of plane topology: +Lemma 2.1. Let γ0, γ1, γ2 be three disjoint rays. +Assume that C1 and C2 are simple closed curves on the plane R2 so that γi ∩Cj is a unique point pi,j, +i ∈ {0, 1, 2}, j ∈ {1, 2}. We endow Ci with the boundary-orientation corresponding to the compact disc +bounded by Ci. Then the cyclic order of the 3 points p0,1, p1,1, p2,1 for the orientation of C1 is the same +as the cyclic order of the 3 points p0,2, p1,2, p2,2 for the orientation of C2. +We call it the cyclic order on the rays γ0, γ1, γ2. +Lemma 2.2. The cyclic order on three disjoint germs of rays R0, R1, R2 does not depend on the choice +of disjoint rays γ0, γ1, γ2 realizing the germs R0, R1, R2. +Corollary 2.1. Let γ0, γ1, γ2 be three disjoint rays and C be any simple close curve, oriented as the +boundary of the compact disc bounded by C, and having a non-empty intersection with every γi. +Let pi be the last point of γi in C. Then the cyclic order of the γi coincides with the cyclic order of +the pi for the orientation of C. +Corollary 2.2. Let R0, R1, R2 be three disjoint germs of rays. Let L be an oriented line whose right end +is R0 and whose left end is R2. Then R1 is between R0 and R2 for the cyclic order defined above (we +denote R1 ∈ (R0, R2)) if and only if it admits a realization contained in the upper half-plane ∆+ +L bounded +by L. +Next proposition summerizes what we have got with this sequence of easy lemmas. +Proposition 2.2. Consider R a family of pairwise disjoint germs of rays. Then R is totally cyclically +ordered by the following relation: + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +7 +given three distinct germs of rays R0, R1, R2 ∈ R, the germ R2 is between R1 and R3 if it admits a +realisation contained in the upper-half plane ∆+ +L where L an oriented line whose right end is R0 and and +whose left end is R3. +2.3. Compactification of a family of rays by a circle at infinity. In this paper a compactification +of the plane R2 by the disc D2 is by definition a homeomorphism between R2 and the open disc ˚D2. +The aim of this section is the proof of Theorem 1 which build a canonical compactification of R2 +associated to a family R of rays, assuming it admits a countable separating (for the cyclic order) subset +E ⊂ R. One of the main ingredients for the proof of Theorem 1 is the following lemma which is an easy +exercize of plane topology. +Lemma 2.3. Let γ0, γ1, . . . , γn be n disjoint rays, n > 0, and K ⊂ R2 be a compact set. Then there is +a simple closed curve C disjoint from K bounding a compact disc D containing K in its interior and so +that C ∩ γi consists in a unique point pi, i ∈ {0, . . . , n}. +Proof. Just notice that there is a homeomorphism of R2 mapping γi, i ∈ {1, . . . , n} on radial (half-straight +lines) rays. Then the proof is trivial. +□ +sketch of proof of Theorem 1. We consider the set of rays endowed with the cyclic order and we embedd +it in the circle S1 by Proposition 2.1. We denote by E ⊂ S1 the dense countable subset corresponding +to E. We define a topology on R2 � S1 by choosing a basis of neighborhood of the points in S1 as the +halph planes bounded by lines L whose both ends are rays R−, R+ in E (each half plane correspond to a +segment in S1 \ {R−, R+}. +This topology does not depend of the choice of the countable separating subset E: if ˜E is another +countable separating subset, each neighborhood of a point of S1 obtain by using one family contains a +neighborhood obtained by using the other family. +Now one builds a map from R2 on the interior of D2 as follow: +(1) one considers the circles Cn, n ≥ 1, of radius ρn = 1 − +1 +n+1 (that is , Cn = ρn · S1) endowed with +the finite set of point ρn · x1, . . . , ρn · xn, where E = {xn, n ≥ 1} is a choice of indexation of the +countable set E. +(2) one choses by induction a realisation Rn of the rays in E and a family of simple closed loops γn +with the following properties: +• γn is the boundary of a compact disc Dn containing Dn−1 in its interior and containing +the disk of radius n of R2. In particular, � +n Dn = R2. +• γn cuts the rays Rm, m < n in a unique point. +• one choses a representative of Rn , disjoint from Rm, m < n, with origin on γn and with +no other intersection point with γn. +Then, by definition of the cyclic order on the rays, the points γn ∩ Ri, i ≤ n, are cyclically +ordered on γn as the points ρn · x1, . . . , ρn · xn on Cn +(3) this allows us to choose a homeomorphism of R2 to the interior of D2 sending the loops γn on +the circles Cn and the rays Rn on the segments [ρn, 1) · xn. +This homeomorphisms extends on the circle at infinity S1 to ∂D2. +□ +2.4. Union of countably many families of rays: the circle. +Proposition 2.3. Let {Xi, i ∈ I}, I ⊂ N be a finite or countable family of sets so that � +i Xi is endowed +with a total cyclic order. Assume that, for every i, there exist Ei ⊂ Xi countable separating subset. +On the union X = � +i Xi we consider the relation +x ∼ y ⇔ ([x, y] ∩ Ei is finite for every i, or [y, x] ∩ Ei is finite for every i). +In other words, x ∼ y if one of the two segments (for the cyclic order) bounded by x and y meets each +family Ei in at most finitely many points. +Then ∼ is an equivalence relation and every class contains at most 1 point in each Xi. + +8 +CHRISTIAN BONATTI +Let denote +π: X → X = +� +i +Xi/ ∼ . +We denote by E the projection π(E) of E = � Ei on X. +Then ∼ provides a complete cyclic order on X and E is a countable separating subset. +Proof. The fact that ∼ is an equivalence relation is quite easy, as the union of two intervals meeting Xi +on finite sets meets Xi on a finite set. +Note that, assuming x ∼ y, the interval [x, y] or [y, x] (meeting every Ei in finitely many points) is +contained in the class of x and y. Thus the class [x]∼ is a (proper) interval for the cyclic order. +Consider x, y ∈ X and assume that [x, y] ∩ Ej is finite for every j. Assume that there is i and distinct +z, t ∈ [x, y]∩Xi. Then the separating property of Ei for Xi ensures that [x, y]∩Ei is infinite contradicting +the choice of the interval [x, y]. We deduces that every class meets every Xi in at most 1 point. +Notice that this implies that the projection of Ei on X is injective. +Consider x, y, z ∈ X whose classes are distinct, and assume z ∈ (x, y). Consider now a ∼ x, b ∼ y and +c ∼ z. Let Ia, Ib, Ic be the intervals [x, a] or [a, x], [y, b] or [b, y], [z, c] or [c, z] with finite intersections +with the Ei, respectively. Then these intervals are disjoints as there a contained in disjoint equivalence +classes. Thus the cyclic order for point in Ia, Ib, Ic does not depend on the point in Ia, Ib, Ic and thus +c ∈ (a, b). +This shows that the quotient X is endowed with a total cyclic order. +Consider now two distinct classes [x]∼, [y]∼ ∈ X of points x, y ∈ X. Thus there is i so that [x, y] ∩ Xi +is infinite Now the separating property of Ei implies that [x, y] ∩ Ei is infinite. +As π is injective on Ei one gets that ([x]∼, [y]∼) ∩ π(Ei) is infinite and thus ([x]∼, [y]∼) ∩ E is infinite. +One proved that E is separating for X, ending the proof. +□ +2.5. Union of countably many families of rays: the compactification. +Theorem 6. Let R = � +i∈I Ri, I ⊂ N, be a family of rays in R2 whose germs are pairwise disjoint. +Assume that for every i ∈ I there is a countable subset Ei ⊂ Ri which is separating for Ri. +Then, there is a compactification of R2 by the disc D2 so that: +• any ray of R tends to a point of the circle at infinity ∂D2 = S1. +• for every i, any two distinct rays of Ri tend to distinct points of S1 +• for any non-empty open interval J ⊂ S1 there is i ∈ I so that at least 2 rays in Ri have there +limit point in J. +Furthermore, this compactification is unique up to a homeomorphism of D2 and does not depend on the +separating countable sets Ei. +Let us discuss item 3, whose formulation may be surprising. +Remark 4. +(1) The third item implies that the points of S1 which are the limit point of a ray in +� Ei are dense in S1. +(2) if I is finite, item 3 is equivalent to the density of points in S1 which are limit of rays. +(3) item 3 is necessary when there is an uncountable set of equivalence classe [c], for the equivalence +relation ∼ (defined at Section 2.4), which are infinite (necessarily countable) and contain a set +C([c]) which is separating for the cyclic order. In that case the uniqueness property announced +in Theorem 6 would be wrong if we replace item 3 by the density of points in S1 which are limit +of rays. Example 3 provides a simple illustration of this trouble. +Proof of Theorem 6. Let us denote X = R, Xi = Ri, and E = � Ei. Let ∼ be the equivalence relation +defined in Proposition 2.3 on X = R and let π benote the projection π: X → X = X/ ∼. We choose a +subset Y ⊂ X with the following properties +• each equivalence class [γ] for ∼ contains exactly 1 point yγ in Y +• if a class for ∼ contains a point in E, then yγ ∈ E. +The existence of such subset Y is certainly implied by the choice axiom but this existence does not require +this axiom. For instance we can fix + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +9 +• if [γ]∼ ∈ π(� Ei) then yγ is the unique point in Ei in the [γ]∼ where i is the smallest index for +which [γ]∼ ∈ π(Ei). +• if [γ]∼ /∈ π(E) then yγ is the unique point in Xi ∩ [γ]∼ where i is the smallest index for which +[γ]∼ ∩ Xi) ̸= ∅. +We denote F = Y ∩ E. Notice that π(F) = π(E) by construction. +Now the projection π: Y → X is a bijection which is strictly increasing for the cyclic order. +As +E = π(E) is separating for X (see Proposition 2.3) one gets that F is separating for Y . +We can now apply Theorem 1 to the set of rays Y and the countable separating subset F. One gets a +compactification of R2 as a disc D2 so that every ray in Y tends to a point at the circle at infinity, two +distinct rays in Y tends to two distinct points, and the set of points at infinity limit of rays in F is dense +in the circle at infinity. +Let us check now that every ray γ ∈ R tends to a point at infinity. By construction of Y there is +σ ∈ R so that σ ∼ γ. We will prove that γ tends to the limit point s ∈ S1 of σ. For that we recall that +a basis of neighborhood of s is given by the half planes ∆n bounded by lines Ln whose both ends are +σ− +n , σ+ +n ∈ Y so that σ ∈ (σ− +n , σ+ +n ). Note that both intervals (σ, σ+ +n ) and (σ− +n , σ) are infinite when one of +the intervals (σ, γ) or (γ, σ) is finite. One deduces that γ ∈ (σ− +n , σ+ +n ) and thus the end of γ is contained +in ∆n. Thus γ tends to s. +Now consider two distinct rays γ, γ′ ∈ Ri. As every class for ∼ contains at most 1 point of Ri the +classes of γ and γ′ are distincts. Thus there are σ ̸= σ′ ∈ Y which are equivalent to γ and γ′, respectively. +The limit points of γ and γ′ are those of σ and σ′ respectively, which are distinct. We just checked that +distincts rays in Ri tends to distinct point at infinity. +Claim 1. Let ϕ: R2 → D2 be a compactification satisfying the announced properties. Then 2 rays in R +tends to the same point of the circle at infinity of the compactification if and only if they are equivalent +for ∼ +Proof. If two rays a, b are not equivalent then each of (a, b) and (b, a) contains infinitely many rays in the +same of the sets Ri, by definition of ∼. As the limits of distinct rays in the same Ri are different, one +deduces that the limits of a and b are distinct. +Conversely, if a and b have different limit points r and s in S1 for the compactification then item 3 +implies that there is i ∈ I (resp. j ∈ I so that , (a, b) (resp. (b, a)) contains the ends of at least 2 rays in +Ri (resp. Rj). As Ri and Rj admits separating subsets, this implies that both (a, b)∩Ri and (b, a)∩Rj +are infinite, so that a ≁ b. +□ +Consider now a non-empty open interval J of the circle at infinity. We announced that there is i for +which J contains at least 2 limits of rays in Ri. Recall that, according to Theorem 1 the points in J +which are limit of rays in Y are dense in J. Thus there are at least 2 points in J which are limit of rays +R1, R2 in Y . This implies that, up to exchange R1 and R2 any ray in R between R1 and R2 tend to a +point in J. Now, the rays R1, R2 are not equivalent for ∼ according to Claim 1. By definition of ∼, there +is i so that there are infinitely rays between R1 and R2. This proves Item 3 of Theorem 6. +Assume now that one has another compactification ψ: R2 → D2 satisfying also the announced proper- +ties. One deduces from Claim 1 the fact that the images by ψ of two distinct rays in the set Y (that we +used for building the first compactification ϕ) have two distinct limit points and that the limit points of +the image by ψ of rays in Y are dense in S1. Thus this new compactification satisfies the same property +on the set of rays Y as the one we built. Now Theorem 1 asserts that these compactifications differs from +ϕ by a homeomorphism of D2, concluding the proof. +□ +Lemma 2.4. Assume that R satisfies the hypotheses of Theorem 2.3, and let ˜R be a set of rays to that +the germs of rays in R ∪ ˜R are pairwise disjoint. +Let R2 ֒→ D2 be a compactification given by Theorem 6 applied to R. Then any ray ˜γ in ˜R tends to a +point at infinity. +Proof. The candidate for the limit the intersection of all the closed intervals in S1, bounded by limit of +rays a, b ∈ R, so that ˜γ ∈ (a, b). The basis of neighborhood of this point that we exhibit implies that +indeed ˜γ tends to that point at infinity. + +10 +CHRISTIAN BONATTI +□ +2.6. An example with uncountably many compactifications. The example below shows that, in +the case of a infinite countable family R = {Ri}, i ∈ N, the compactification announced by Theorem 6 +would not be unique if we replace the item 3 of the conclusion by the density in S1 +R of the limits of the +rays in R. In the example below, +Example 3. Let B ⊂ R2 be the open strip {(x, y) ∈ R2, |x − y| < 1}. Let I be the set of linear lines with +a rational inclination ̸= 1. For any i ∈ I, let Fi be the restriction to B of the trivial foliation by parallel +straight lines directed by i ∈ RP1. For any i, let Ri be the set of ends of leaf of Fi. Each Ri admits a +countable separating subset. Thus R = � +i∈I Ri satisfies the hypotheses of Theorem 6. +Then there are uncountably many distinct compactifications of R2 for which +• any ray of R tends to a point of the circle at infinity ∂D2 = S1. +• for every i, any two distinct rays of Ri tend to distinct points of S1 +• the points of S1 which are the limit point of a ray in � Ei are dense in S1. +Proof. Let D2 +R be the compactification of B ≃ R2 by adding the circle at infinity S1 +R. +Every class C for ∼ contains exactly 1 ray in Ri for any i ∈ I. The rays in C are ordered,for the cyclic +order, as the points of I in RP1. So, C is a separating set for itself. +By construction of S1 +R, the class C corresponds to a point c ∈ S1 +R. We can build another circle S1 +R,C +by opening the point c in a segment IC. Then, we can build a compactification D2 +R,C so that the rays in +C tend to distinct points dense in IC. In particular IC contains exactly 1 limit point of a ray in Ri, for +any i. +We can repeat this argument opening not just a point in S1 +R but a countable subset C of classes for +∼: we build a compactification D2 +R,C where the circle at infinity contains disjoint intervals IC, C ∈ C, so +that each IC contains exactly 1 limit point of a ray in Ri for any i, ans these points are dense in IC. +As there are uncountably many such countable subsets C, this provides an uncountable family of +pairwise distinct compactifications of B satisfying the 2 first items and the density of the limit points of +rays. +□ +This shows that the uniqueness part of Theorem 6 becomes wrong if we replace item 3 by the density +in S1 of the set of limits of rays. +2.7. Uncountable families of families of rays. Theorem 6 is wrong for the union of an uncountable +family of sets of rays, as shows the Example 4 below. +Example 4. We consider R2 endowed with all constant foliations Fθ, θ ∈ RP1, where Fθ is the foliation +whose leaves are the straight lines parallels to θ. +Then given any compactification of R2 by D2 for which every end of leaf tends to a point at infinity, +then for all but a countable set of θ the right ends of the leaves of Fθ tends to the same point at infinity. +Proof. The ends of leaves F + +θ at the right, and those at the left F − +θ of the foliation Fθ are disjoint interval +depending on the uncountable parameter theta. On the circle at most countably many disjoint intervals +can be non trivial, ending the proof. +□ +2.8. Projection on the compactifications associated to each families. Let us start with a very +easy example, showing the at the circles at infinity associated to the subsets of a countable family +of transverse foliations may lead to uncountabily many distinct compactifications, all quotient of the +compactification associated to the whole family. +Example 5. Consider now a infinite countable subset I ⊂ RP1 and consider the family RI of the leaves +of the constant foliations Fθ, θ ∈ I on R2 as already considered in example 4. +Now the set of ends of leaves of each foliation Fθ corresponds to 2 (because each leaf has 2 ends) +non-empty open intervals in S1 +I, and these intervals do not contain any end of leaf of any other foliation. + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +11 +Thus if J, K ⊂ I are distinct subsets, the circles at infinity S1 +J and S1 +K are obtained by collapsing +distinct intervals of S1 +I and they are different. +As the set P(I) of all subset of I in uncountable, this leads to an uncountable family of compatifications +{D2 +J}J∈P(I) of R2 by a circle at infinity. +This situation is quite general. +Let R = R1 +� · · · � Rk be a family of rays in R2 whose germs are pairwise disjoint. Assume that for +every i ∈ {1, . . . , k} there is a countable subset Ei ⊂ Ri which is separating. +Thus for every subset I ⊂ {1, . . ., k}, Theorem 6 provides a compactification D2 +I of R2, by the circle +at infinity corresponding to the rays in Ri, i ∈ I. +Proposition 2.4. If J ⊂ I then the identity map on R2 extend by continuity as a projection ΠI,J : D2 +I → +D2 +J. This projection consists in collapsing intervals of S1 +I = ∂D2 +I which do not contain any limit points of +ray is Rj, j ∈ J. +Furthermore if K ⊂ J then +ΠI,K = PJ,K ◦ PI,J. +Proof. We first define a projection πI,J : S1 +I → S1 +J by using Remark 3: the subset RJ ⊂ S1 +I of limit of +rays of RJ is is a strcitly increasing bijection with RJ. Thus the increasing bijection of RJ on a dense +subset of S1 +J induces an increasing bijection of RJ in this dense subset of S1 +J. Now Remark 3 asserts that +this bijection extends on the whole S1 +I in a not-stricly increasing map πI,J : S1 +I → S1 +J. An increasing map +with dense image is always continuous, so that πI,J is continuous. +Finaly Remark 3 asserts that the non-injectivity of πI,J consist in collapsing intervals of S1 +I with at +most 1 point in RJ, which is the same topological operation as collapsing intervals with no points in RJ. +For ending the proof, we will check that πI,J is the extension by continuity of the identity map of R2 +to the circles at infinity. +Recall that we defined a basis of neighborhood of each point of the circle at infinity S1 +I (resp. S1 +J ) as +the half-planes ∆+ +L bounded by lines whose both ends are in RI (resp. RJ). In particular, as J ⊂ I, the +neighborhoods of points at infinity in DD2 +J are still neighborhoods of points at infinity for D2 +I, proving +that the map which is the identity from R2 = ˚D2 +I to R2 = ˚D2 +J and is πI,J from S1 +I to S1 +J is continuous. +This ends the proof. +□ +3. backgroung on foliations: regular leaves, non-separated leaves +3.1. Non-singular foliations. Let F be a foliation of R2. Then +(1) as R2 is simply connected, F is orientable and admits a transverse orientation. Let us fix an +orientation of F and a transverse orientation. +(2) every leaf is a line (i.e. a proper embedding of R in R2). +(3) a basis of neighborhoods of a leaf L is obtained by considering the union of leaves through a +transverse segment σ through a point of L. +Definition 3.1. +• two leaves L1, L2 are not separated one from the other if they do not admit +disjoint neighborhood. +• A leaf L is called not separated or not regular if there is a leaf L′ which is not separated from +L. +• A leaf is called regular if it is separated from any other leaf. +We will need some times to be somewhat more specific. +Let L1 and L2 be distinct leaves of F. Consider two segments σi : [−1, 1] transverse to F, positively +oriented for the transverse orientation of F, and so that σi(0) ∈ Li, i = 1, 2. Then L1 is not separated +from L2 means that there are sequences ti +n, i = 1, 2 tending to 0 as n → +∞ so that σ1(t1 +n) and σ2(t2 +n) +belongs to the same leaf Ln. Then + +12 +CHRISTIAN BONATTI +• as F is transversely oriented and the σi are positively oriented, one gets that t1 +n has the same +sign as t2 +n, for every n. Futhermore all the ti +n have the same sign. +One says that L1 and L2 are not separated from above (resp. from below) if the ti +n are positive +(resp. negative). +• By shirinking the segments σi if necessary one may assume that they are disjoint. Now, up to +exchange L1 with L2 we may assume that σ1(tn) is at the left of σ2(tn) in the oriented leaf Ln. +We say that L1 (resp. L2) is not separated from L2 at its right (resp. at its left). +Consider a leaf L and σ: [−1, 1] → R2 a transverse segment (positively oriented) with σ(0) ∈ L. Let Lt +be the leaf through σ(t). Let Ut, t ∈ (0, 1), be the closure of the connected component of R2 \ (Lt ∪ L−t) +containing L. Then +Lemma 3.1. The leaf L is regular if and only if +� +t +Ut = L. +The intersection � +t Ut does not depend on the segment σ and is denoted U(L). +If L is not regular, U(L) as non-empty interior, and the leaves which are not separated from L are +precisely the leaves in the boundary of U(L). +Proof. A leaf ˜L not separated from L is contained in every Ut and is accumulated by leaves Ltn in the +boundary of Utn. Thus ˜L is contained in the boundary of U(L) = � +t Ut. Furthermore one of the two half +planes bounded by ˜L is contained in Ut and therefore in U(L). +Conversely,� +t Ut consist in entire leaves of F and so does its boundary. Now any transverse segment +through a leaf in the boundary of � +t Ut crosses the boundary Lt ∪ L−t of Ut for t small: that is the +definition of being not sepatated from L. +□ +Lemma 3.2. Let F be a foliation of R2. The set of not separated leaves is at most countable. +Proof. We consider a countable family of transverse lines Σ whoses union cuts every leaf of F. It is enough +to proof that such a tranverse line Σ cuts at most a countable set of non-regular leaves L admiting a non +separated leaf ˜L from below. +For that just notice that the U(L) for L∩Σ ̸= ∅ are pairwise disjoint. Thus,there are at most countably +many of them with non-empty interior, ending the proof. +□ +Note that L cuts the strip Ut, t ∈ (0, 1] in two strips U + +t and U − +t +bounded respectively by Lt ∪ L and +by L−t ∪ L, and we denote +U+(L) = +� +t +U + +t +and U−(L) = +� +t +U − +t +Then +Lemma 3.3. L is non-separated from above (resp. from below) if and only if U+(L) ̸= L (resp. U−(L) ̸= +L) and if and only if U+(L) (resp. U−(L)) has non-emptyinterior. +In the same spirit, σ cuts the strip Ut in two half strips U left +t +and U right +t +according to the orientation +of F. Then one says that the right end L+ (resp. left endL−) of L is regular if +Uright = (L) +� +t +U right +t += L+ (resp. Uleft(L) = +� +t +U left +t += L−). +We can be even more precise by considering the 4 quadrants U +,right +t +, U +,left +t +, U −,right +t +, U −,left +t +obtained +by considering the intersections of U + +t +and U − +t +with U right +t +and U left +t +. This allows us to speak on right +or left ends of leaves non separated from above or from below, in the obvious way. + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +13 +3.2. Singular foliations: saddles with k-separatrices. A singular foliation F on R2 is a foliation +on R2 \ Sing(F) where Sing(F) is a closed subset of R2. +A leaf of F is a leaf of the restriction of +F to R2 \ Sing(F). Let us now recall the notion of saddles with k-separatrices, also called k-prongs +singularities. +We denote by A0 the quotient of [−1, 1]2 by the involution (x, y) �→ (−x, −y). The projection of (0, 0) +on A0 is still called 0, 0. Note that the horizontal foliation (whose leaves are the segments [−1, 1] × {t} is +invariant by (x, y) �→ (−x, −y), and therefore passes to the quotient on A0 \ (0, 0) and we denote by H1 +the induced foliation on A0 \ {(0, 0)}. +A 1-prong singular point p of F is a point of Sing(F) which admits a neighborhood U and a homeo- +morphism h from U to A0 so that h(p) = (0, 0) and h maps F on H1. +We denote by Ak, Hk the cyclic ramified cover of A0 at the point (0, 0) with k leaves, endowed with +the lift of H1. +A k-prongs singular point p, equivalently a sadlle point with k separatrices of F is a singular point +admiting a homeomorphism of a neighborhood onto Ak mapping p on (0, 0) and F on Hk. A separatrix +of the saddle point p is the leaf of F containing a connected component of the lift of ]0, 1] × {0}. +Remark 5. +• If p is a 2-prongs singular point of F, then the foliation F can be extended on p so +that p is not singular. +• The Poincar´e-Hopf index of a k-prongs singular point is 1 − k +2. +A foliation with singularities of saddle type on R2 is a singular foliation for which each singular point +is a saddle with k separatrices, k > 2. +3.3. Leaves of singular foliations. +Lemma 3.4. Let F be a foliation on R2 with singular points of saddle type. Let σ: [0, 1] → R2 \Sing(F) +be a segment transverse to F. Then for every leaf γ one has +#σ ∩ γ ≤ 1, +where # denotes the cardinal. +Proof. Assume (arguing by contradiction) that σ ∩ γ ≥ 2. Let x, y be two successive (for the parametri- +sation of γ) intersection points with σ. The concatenation of the segments [x, y]γ and [y, x]σ is a simple +closed curve c in R2 \Sing(X). By Jordan theorem c bounds a disc D in R2 and the Poincar´e Hopf index +of F on D is either equal to 1, if γ cuts σ with the same orientation at x and y, or 1 +2 otherwise: anyway +this index is strictly positive. However, this index is the sum of the Poincar´e Hopf index of the singular +points of F contained in D. As each of them is negative, that is a contradiction, ending the proof. +□ +The same argument shows that +Lemma 3.5. Let F be a foliation on R2 with singular points of saddle type. Then F has no compact +leaves +Proof. The index of F on the disc bounded by a compact leaf whould be 1 which is impossible with +singular points with negative index. +□ +Corollary 3.1. Let F be a singular foliation of R2 whose singular points are all saddle points with at +least 3 separatrices. Then every half leaf of F is either a ray or tends to a singular point p of F and is +contained in a separatrix of p. +Proof. Consider the Alexandrov compactification of R2 by a point at infinity. Consider a leaf γ and +choose a parametrisation γ(t). Consider +lim sup +t→+∞ γ(t) = +� +t>0 +γ([t, +∞), +where the closure is considered in R2 ∪ {∞}. It is a decreasing intersection of connected compact sets, +and hence it is a non-empty connected compact set. + +14 +CHRISTIAN BONATTI +If lim supt→+∞ γ(t) is not just a point, if contains a regular point x of F, hence it cuts infinitely many +times any transverse segment through x, which is forbidden by Lemma 3.4. +Now lim supt→+∞ γ(t) is either the point ∞ or is a singular point of F, which is the announced +alternative. +□ +3.4. Regular leaves of singular foliations. Let F be a foliation with singular points of saddle type, +L0 a leaf of F and σ be a transverse segment through the point σ(0) ∈ L0. +The set of t so that σ(t) is contained in a separatrix of a singular point is at most countable. For any t +so that σ(t) and σ(−t) are not in a separatrix of a singular point, the leaves Lt and L−t through σ(t) and +σ(−t) are disjoint lines and therefore cut R2 in 3 connected components. We denote by Ut the closure of +the connected component of R2 \ (Lt ∪ L−t) containing L0. Notice that Ut is a strip (homeomorphic to +R × [−1, 1]) bounded by Lt ∪ L−t and saturated for F. +Lemma 3.6. With the notation above � +t Ut is a non-empty closed subset of R2 saturated for F and we +have the following alternative: +• either � +t Ut = L0 and L0 is a non-singular leaf of F, +• or � +t Ut has non-empty interior. +Furthermore, � +t Ut does not depend on the choice of the transverse segment σ through L0 and is +denoted U(L0). +Proof. U(L0) is saturated for F. If it contains a non-singular leaf, it contains one of the half planes +bounded by this leaf. If it contains a singular leaf, it contains the corresponding singular point, and then +it contains at least one of the sectors bounded by the separatrices. +□ +Definition 3.2. With the notation above, the leaf L0 is called regular if U(L0) = L0, and will be called +non-regular otherwise. +Remark 6. If L0 is a separatrix of a singular point, then it is non-regular. +As in the case of non-singular foliations we have: +Proposition 3.1. Let F be a foliation with singular points of saddle type. Then the set of non-regular +leaves is at most countable. +Proof. For any transverse segment σ let denote by Lt the leaf through σt. Then by construction the +closed sets U(Lt) are pairwise disjoint. Thus at most countably many of them may have non-empty +interior, that is, at most countably many of leaves Lt are non-regular. We conclude the proof by noticing +that F admits a countable family of transverse segment σn, n ∈ N every leaf of F cuts at least 1 segment +σn. +□ +The leaves of a foliations have two ends, and the notion of regular leaves can be made more precise, +looking at each of its ends. +More precisely, let L0,+ be an half leaf of F, and let σ be a transverse segment so that σ(0) is the +initial point of L0,+. For any t so that σ(t) and σ(−t) do not belong to a separatrix of a singular point, +we consider Lt,+ and Lt,− the half leaves starting at σ(t) and σ(−t) in the same side of σ as L0,+. We +denote by Ut(L0,+) ⊂ R2 the closed half plane containing L0,+ and bounded by the line of R2 obtained +by concatenation of Lt,+ , σ([−t, t]) and Lt,−. We denote U(L0,+) = � +t Ut(L0,+). Then : +• either U(L0,+) = L0,+ and one says that the half leaf L0,+ (or equivalently, the end of L0 +corresponding to L0,+) is regular +• or U(L0,t) ̸= L0,t is a closed subset with non-empty interior. +A leaf is regular if and only if its two ends are regular, and the set of non-regular ends of leaves is at +most countable. + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +15 +3.5. Orientations. A foliation with singular points of saddle type is locally orientable (and transversely +orientable) in a neighborhood of a singular point x if and only if the number of separatrices of x is even. +Thus a foliation of R2 whose singular points are sadlles with even numbers of separatrices is locally +orientable and transversely orientable, and therefore is globally orientable and transversely orientable, as +R2 is simply connected. +Let F be a foliation with singular points of saddle type with even numbers of separatrices, and fix an +orientation and transverse orientation of F. +Thus every leaf L have a right and left end. We defined Uright(L) and Uleft(L) so that L can be +regular at the right or at the left. +If L0 is a leaf which is not a separatrix and σ be a transverse segment with σ(0) ∈ L0. One defines in +the same way the notions of being regular from above and from below, for L0 or for each of its two ends. +For instance Lright +0 +is regular from above if U+(Lright +0 +) = � +t Ut,+(Lright +0 +) = Lright +0 +where Ut,+(Lright) +is bounded by Lright +0 +, σ([0, t]) and Lright +t +. +4. The circle at infinity of a family of foliations +4.1. The circle at infinity of a foliation of R2: statement. The aim of this section is to recall the +following result essentially due to [Ma] and to present a short proof of it. +Theorem 7. Let F be a foliation of the plane R2, possibly with singularities of saddle type. Then there +is a compactification D2 +F ≃ of R2 by adding a circle at infinity S1 +F = ∂D2 +F with the following property: +• any half leaf tends either to a saddle point or to a point at infinity. +• given a point θ ∈ S1 +F the set of ends of leaves tending to θ is at most countable. +• the subset of S1 +F corresponding to limits of regular ends of leaves is dense in S1 +F. +Furthermore this compactification of R2 by D2 with these three properties is unique, up to a homeo- +morphisms of the disk D2. +Remark 7. If L+ +1 ̸= L+ +2 are two ends of leaves tending to the same point θ ∈ S1 +F, then L2 ⊂ U+. In +particular, the ends L+ +1 and L+ +2 are not regular. +Corollary 4.1. If a homeomorphisms f of the plane R2 preserves the foliation F then it extends in a +unique way as a homeomorphism F of the compactification D2 +F. +Furhtermore the restriction of F to S1 +F is the identity map if and only if f preserves every leaf of F +and preserves the orientation on each leaf. +Proof. The first part is, as already noted, a straightforward consequence of the uniqueness of the com- +pactification. +If f preserves every leaf and preserves the orientation of the leaves, then it preserves every end of leaf. +Thus the extension F fixes every point of S1 +F which is limit of an end of leaf. As the limit points of end +of leaves are dense in S1 +F one deduces that the restriction of F to S1 +F is the identity map. +Conversely, assume that F is the identity on S1 +F. If θ ∈ S1 +F is the limit of an unique end of leaf L+ +then L+ is preserved by f. +Thus f preserves every regular end of leaf. As the regular leaves are dense in R2, one deduces that f +preserves every oriented leaf, concluding. +□ +4.2. Proof of Theorem 7. We denote by Reg(F) the set of regular leaves of F and by R(F) the set +of ends of regular leaves (any non singular leaf and in particular any regular leaf has two ends). Recall +that R(F) is a family of disjoint rays of R2 and therefore is cyclically ordered. +Lemma 4.1. If D is a family of regular leaves whose union in dense in R2, then the set D of ends of +the leaves in D is a separating family for the set of ends of regular leaves R(F). +Proof. Let L0 be a regular leaf of F, σ: [−1, 1] → R2 a segement transverse to F with σ(0) ∈ L0 and Ut +the family of neighborhoods of L0 associated to the transverse segment σ. Our assumption implies that +for a dense subset of t ∈ [−1, 1], the leaf Lt belongs to D. Consider a sequence tn ∈ [−1, 1], n ∈ Z so that + +16 +CHRISTIAN BONATTI +• Ln = Ltn ∈ D +• tn → 0 as |n| → ∞ +• tn as the same signe as n ∈ Z +Let L+ +n and L− +n be the half leaves of Ln (for the orientation given by the transverse orientation induced +by σ). As L0 is regular one gets U(L+ +0 ) = L+ +0 . This implies that L+ +0 (resp. L− +0 ) is the intersection of the +intervals (for the cyclic order) [L+ +−n, L+ +n ] (resp.[L− +n , L− +−n]) for n > 0. In other words, the rays L+ +−n, L+ +n +(resp. L− +n , L− +−n) are separating the ray L+ +0 (resp. L− +0 ) from any other ray in R(F) (and indeed from any +other ray of leaf, regular or not ), concluding the proof. +□ +We are now ready to prove Theorem 7 +Proof of Theorem 7. We chose a countable set E of regular leaves whose union is dense in R2. According +to Lemma 4.1 the set E of ends of leaves in E is a countable separating subset of R(F). Thus we may +apply Theorem 1. +One gets a compactification of R2 by the disc D2 +F ≃ D2, so that every two distinct ends of regular +leaves tend to two distinct points at the circle at infinity S1 +F and these points are dense on the circle and +this compactification does not depend of the choice of the family. This prove the items 2 and 3 of the +theorem, and also proves that these two items are enough for the uniqueness of this compactification. +It remains to prove the first item, that is to show that the rays contained in non-regular leaves also +tend to points on S1 +F. That is done by Lemma 2.4. +□ +Remark 8. Let F be a foliation (possibly with saddles). Then every line L transverse to F has 2 distinct +limit points at infinity corresponding to its 2 ends. +Proof. The two ends of L are rays disjoint from the ends in R(F) (that is of the ends of leaves of F), as +any transverse segment intersects any leaf in at most 1 point. Now Lemma 2.4 implies that the ends of L +tends to points on S1 +F. These points are distinct because the regular half leaves through L are between +these two ends. +□ +Lemma 4.2. Let F be a foliation (possibly with saddles). Given any two (non-singular) leaves L1, L2, +if the ends of L1 and L2 tend to the same 2 points in S1 +F then L1 = L2. +Proof. Assume L1 ̸= L2 share the same end points. Then the leaves in the strip bounded by L1 ∪ L2 +would have their ends on the same points in S1 +F contradicting the fact that at most contably many ends +of leaves share the same end point on S1 +F. +□ +As a by-product of the proof of Lemma 4.1 we get the following: +Lemma 4.3. Let F be a foliation (maybe with saddle-like singular points) and le σ: [−1, 1] → R2 \ +Sing(F) be a transverse segment. Let {L+ +t } and {L− +t } be the half leaves starting at σ(t). Consider the +map associating to t ∈ (−1, 1) the limit point of L+ +t on S1 +F. Then t is a continuous point of this map if +and only if L+ +t is a regular end. +4.3. Points at S1 +F limit of several ends of leaves: hyperbolic sectors. +Lemma 4.4. Let A and B be distinct ends of leaves. Then the following properties are equivalent +• There are no end of regular leaf between A and B. +• The set of ends of leaves between A and B is at most countable. +• The set of ends of leaves between A and B is finite. +Proof. First assume that there is an end L+ of a regular leaf L between A and B. We will prove that +the interval (A, B) is uncountable. +Consider the neighborhood Ut of L associated to a transverse segment σ with σ(0) ∈ L. As L is +regular, one gets that U(L) = � +t Ut = L. As a consequence there is t so that A and B are out of Ut. +First assume that A and B are in the same connected component of R2 \ Ut. Then there is a line L +whose left end is B and whose right end is A and which is disjoint from Ut. One deduces that one of + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +17 +the interval (A, B) and (B, A) contains no end of leaf in Ut (this cannot be (A, B) which contains L+ by +assumption) and the other contains all ends of leaves in Ut, so (A, B) contains ucountably many ends of +leaves as announced. +Now assume that A and B are in distinct connected components of R2 \ Ut. Then there is a line Γ +whose left end is B, whose right end is A and whose intersection with Ut is σ([−t, t]). As L+ is in the +interval (A, B) so that L+ ⊂ ∆+ +Γ , one deduces that all the positive half leaves L+ +r , r ∈ [−t, t] are contained +in the upper half plane ∆+ +Γ and therefore are between A and B. So the interval (A, B) (and also (B, A)) +is uncoutable which is what we announced. +Conversely, if there are uncountably many ends in (A, B) one of them is the end of a regular leaf as +non-regular leaves are countably many. +This proves the equivalence of the two first items. The third items implies trivialy the second, so we +now prove that the second implies the third. +Let A and B be two ends of leaves so that (A, B) is at most countable. We consider a line δ with the +folowing properties: +• A and B are the right and left ends of δ, respectively, +• δ \ (B ∪ A) is a segment σ, consisting in finitely many transverse segments a0, . . . ak and finitely +many leaf segments b1, . . . , bk, with a0(0) ∈ B and ak(1) ∈ A. +Let ∆ = ∆+(δ) be the upper half plane bounded by δ and corresponding to the interval (A, B). +Notice that no entire leaf may be contained in ∆ otherwhise there would be uncountably many ends +between A and B. +We consider the half leaves L+ +0,t entering in ∆ through a0(t). As there are only countably many end +between A and B, there is a sequence of tn → 0 so that L+ +0,tn goes out of ∆ through a point σ(sn). Note +that the half leaves L+ +0,t, t ∈ [tn+1, tn] need to go out of ∆. +Thus every L+ +0,t, t ≤ t0 goes out of ∆ at a point σ(s(t)), where t �→ s(t) is a decreasing function. Let +s0 be the limit +s0 = lim +t→0 s(t). +Notice that a half leaf entering in ∆ though a0 cannot go out ∆ through a0 because a transverse +segment cuts a leaf in at most a point. Thus we deduce that s0 belongs to some ai, i > 0. +We consider the compact segments It ⊂ L+ +0,t joining a0(t) to σ(s(t)). We consider +lim sup +t→0 +It. +It is a closed subset of R2 consisting on B and of whole leaves contained in ∆ and of a half leaf ˜B1 ending +at σ(s0). We already noticed that no entire leaves may be contained in ∆. Thus this limit consists in +B ∪ ˜B1. As a consequence, the ends B and ˜B1 are successive ends, ˜B1 ∈ (A, B) and thus ( ˜B1, A) is at +most countable too. +We consider B1 ⊂ ˜B1 the half leaf starting at the last intersection point of ˜B1 with σ. Note that B1 +starts at a point of some segment ai, with i > 0. +Thus, if B1 ̸= A one may iterate the argument, getting successive half leaves Bi starting at points of +some transverse segment aj(i), where i �→ j(i) is stricly increasing. As there are finitely many segments +ai one gets that this inductive argument needs to stop. In other words, there is i with Bi = A, ending +the proof: [A, B] = A = Bi, Ai−1, . . . , B1, B. +This proves that the second item is equivalent to the third. +□ +The proof of Lemma 4.4 proved, as a by product, the following: +Lemma 4.5. Assume that A and B are successive ends of leaves, that is: the interval (A, B) is empty. +Then, there is an embedding of ψ: [−1, 1] × [0, 1] → D2 +F so that: +• the segments ψ([−1, 1] × {t}), 0 ≤ t < 1, are leaf segment +• A = ψ([−1, 0) × {1}) and B = ψ((0, 1] × {1}) +• the point ψ(0, 1) is the point is S1 +F end of both end A and B. +Definition 4.1. The embedding ψ: [−1, 1] × [0, 1] → D2 +F is called a hyperbolic sector. + +18 +CHRISTIAN BONATTI +We say that two half leaves A, B are asymptotic if [A, B] or [B, A] does not contain any end of regular +leaf. We already proved next lemma: +Lemma 4.6. To be asymptotic is an equivalence relation in the set of ends of leaves of F. +Each equivalence class is either finite or countable and is, as an ordered set, isomorphic to an interval +of (Z, <). +There are at most countably many non-trivial classes. +We also already proved: +Lemma 4.7. Let F be a foliation (possibly with singular points of saddle type). Then two half leaves +tend to the same point θ ∈ S1 +F if and only if they are asymptotic, and every half leaf arriving to θ belongs +to their asymptotic class. +In particular, if a point of S1 +F is the limit of a regular end of leaf, it is the limit of a unique end of leaf. +Notice that points at infinity which are limit of a unique end of leaf may be the limit of a non-separated +end of leaf as shows next example: +Example 6. Let K ⊂ R be a Cantor set and consider +PK = R2 \ (K × [0, +∞)). +Thus PK is homeomorphic to R2. +Let FK be the restriction to PK of the horizontal foliation on R2 (whose leaves are the R× {y}). Thus +all the leaves of the form I × {0} where I is a connected component of R \ K are pairwise non separated +from below. +However, any two distinct ends of leaves of FK tend to distinct points in S1 +FK. +Remark 9. Assume that F is oriented. +If A0, . . . Ak are successive ends of leaves, and assuming A0 is a right half leaf, then A1 is a left half +leaf and A0 and A1 are not separated from above. +Then A2 is a right half leaf and A1 and A2 are not separated from below, and so on. +Thus, each non trivial classes of the asymptotic relation consists in alternately right and left ends of +non-separated leaves, alternately from above and from below. +4.4. Points at infinity which are not limit of leaves: center-like points. In this section, foliations +are assumed to be non-singular. +Remark 10. Let F be a foliation of R2 and o ∈ S1 +F be a point so that o = � +n(an, bn), n ∈ N where +an, bn are the limit points of the two ends of a same leaf Ln. +Then an and bn tends to o and o is not a limit point of an end of leaf of F. +Proof. Consider ∆n being the compact disc of D2 +F whose boudary (as a disc) is Ln∪[an, bn]. Then the ∆n +are totally ordered by the inclusion and o ∈ � +n ∆n. If a leaf L had an end on o, it should be contained +in every ∆n and hence contained in � +n ∆n. Thus the two ends of L are distinct points in � +n[an, bn] +contradicting the hypothesis. +□ +We say that a point o ∈ S1 +F satisfying the hypothese of Remark 10 is a center-like point. +Here is a very simple example with this situation: +Example 7. The trivial horizontal foliation H admits two center-like points at infinity which are the +limit points of the (vertical) y axis (transverse to H). +It is indeed easy to check that: +Remark 11. Given any foliation F of R2, S1 +F carries at least 2 center-like points. To see that, just +consider the (decreasing) intersection of the closure in D2 +F of the half planes ∆± +L for a maximal chain +(given by Zorn lemma) for the inclusion. +But the situation may be much more complicated, as shows next example. + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +19 +Example 8. Consider a simple closed curve γ = γ+ ∪ γ− of R2 where γ+ and γ− are the graphs of +continuous functions ϕ: [−1, 1] → [0, 1] and −ϕ, respectively, where +• ϕ(−1) = ϕ(1) = 0, +• ϕ(t) > 0 for t ∈ (−1, 1), +• the local maxima and minima of ϕ are dense in [−1, 1] (some kind of Weierstrass function). +Let ∆ be the open disc bounded by γ and endowed with the constant horizontal foliation F. +Then S1 +F = γ and any local maximum point of γ+ and any local minimum of γ− are center-like points +of S1 +F +The aim of this section is to show that the situation of Example 8 is in fact very common. +Lemma 4.8. Le F be a foliation on R2. Assume that the union of leaves which are non separated at +their right side is dense in R2, and in the same way, that the union of leaves which are non separated at +their left side is dense in R2. +Then the set of center-like points on S1 +F is a residual subset of S1 +F. +Proof. Fix a metric on S1 +F. Let On ⊂ S1 +F be the set of points belonging to an interval (a, b) of length less +than 1 +n where a, b are both ends of a same leaf of F. +We will proof that On is a dense open subset of S1 +F. Then � +n On will be the announced residual +subset. +The fact that On is open is by definition. We just need to prove the density of On. +Recall that the ends of regular leaves are dense in S1 +F. Thus we just need to prove that the ends of +regular leaves are contained in the closure of On. +Let L be a regular leaf and σ: [−, 1, 1] → R2 be a positively oriented transverse segment with σ0 ∈ L. +We denote Lt the leaf through σt and ve recall that, as L is regular, the rignt and left ends L+ +t , L− +t of Lt +tend to the right and left ends L+ annd L−, respectively, as t → 0. +Given any r < s ∈ [−1, 1], we denote by Ur,s, U right +r,s +, and U left +r,s +the strip bounded by Lr and Ls, +and the two closed half strips obtained by cutting Ur,s along the segment σ([r, s]). Let Iright +r,s +⊂ S1 +F and +Ileft +r,s +⊂ S1 +F be the corresponding intervals on S1 +F. Notice that, as L is regular, these interval have a length +smaller than 1 +n if r, s close to 0. +Our hypotheses imply that there are tright, tleft ∈ (r, s) so that Ltright is non-separated at the right, +and Ltleft is non-separated at the right. +This implies that both U right +r,s +, and U left +r,s +contain entire leaves. Thus Iright +r,s +and Ileft +r,s +contain intervals +whose both extremal points are ends of the same leaf. +Taking r, s small enough, these intervals are +contained in On showing that the points of S1 +F corresponding to L+ and L− are in the closure of On. +This ends the proof. +□ +However, not every point o which is not limit of an end of leaf is center-like. +Example 9. Let FK be the foliation defined in Example 6 by restriction of the horizontal foliation on +R2 \ (K × [0, +∞)) where K is a Cantor set K ⊂ R. Consider a point x ∈ K which are not the end point +of a component of R \ K. Then the point (x, 0) corresponds to a point in S1 +F which is not limit of an end +of leaf, and is not center-like. +Consider a point o ∈ S1 +F and assume it is not the limit point of any end of leaf. For any leaf L we +denote by ∆L ⊂ D2 +F the compact disk containing o and whose frontier in D2 +F is the segment ¯L closure of +L. Then ∆L ∩ S1 +L is a segment IL whose end points are the limit points of the ends of L. Note that the +closed segment IL are totally ordered for the inclusion, and so does the disks ∆L. Let denote +Io = +� +L +IL and ∆o = +� +L +∆L +Then +• if o = Io then o is a center-like point. +• Otherwise, ∂∆o ∩ ˚ +D2 +F consists in infinitely (countably) many leaves pairwise not separated and +there is a subsequence of them whose limit is o. + +20 +CHRISTIAN BONATTI +5. The circle at infinity of a countable family of foliations +The aim of this section is the proof of Theorem 3, that is to build the compactification associated to +a countable family of foliations with saddles and prove its uniqueness. +Remark 12. Example 4 already shown us that Theorem 3 is wrong for uncountable families. +The new difficulty in comparison to Theorem 7 is that there are no more separating families for the +set of ends of all the foliations. +Example 10. Consider the restriction of the constant horizontal and vertical foliations to the strip +{(x, y), |x − y| < 1}, so important for the study of Anosov flows. Then every end of horizontal leaf has a +unique successor or predecessor which is the end of a vertical leaf. Thus no family can be separating. +For by-passing this difficulty, we will apply Theorem 6 instead of Theorem 1. +Proof of Theorem 3. The ends of regular leaves R = � +i∈I Ri of all the foliations Fi, i ∈ I is a family of +disjoints ends of rays. +We have seen that for every foliation Fi the set of ends of regular leaves Ri admits a countable +separating family, for instance by considering regular leaves through a dense subset in R2. +Thus Theorem 6 provides a compactification of R2 by D2 satisfying the announced properties for the +regular leaves, that is, items 2 and 3. +For item 1 one need to see that even the ends of non regular leaves tends to points at infinity. That +is given by Lemma 2.4. +The uniqueness comes from the uniqueness in Theorem 6, ending the proof. +□ +5.1. Example: Countable families of polynomial vector fields. +Corollary 5.1. Let F = {Fi}i∈I, I ⊂ N be a countable family of foliations directed by polynomial vector +fields on R2 whose singular points are all of saddle type. Then the ends of leaves either are disjoint or +coincide. +Thus, according to Theorem 3, there is a unique compactification D2 +F = R2 ∪ S1 +F for which the ends of +regular leaves of the same foliation tend to pairwise distinct points at the circle at infinity, and this ends +of leaves are dense in S1 +F. +Proof. We just need to prove it for 2 such distinct foliations F and G. Consider the tangency locus of F +and G. That is an algebraic set in R2 which is either R2 (so that F = G contradicting the assumption) +or is at most 1-dimensional. Thus it consist in the union of a compact part and a family of disjoint rays +δ1, . . . , δk. +If it is compact, then every end of leaf of F is transverse to G and therefore cuts every leaf of G in at +most 1 point: the ends are disjoints. +Otherwize, each ray δi either is tangent to both foliations and is therefore a comon leaf (which is one +of the announced possibilities) or is transverse to F and to G out of a finite set (because the tangencies +on δi are a algebraic subset). +Thus up to shrink the non-tangent δi, we assume that they are transverse to both foliations therefore +cut every leaf of F in at most 1 point. This implies that every end of leaf of F which is not an end of G +is transverse to G and thus is disjoint from any end of leaf of G, concluding. +□ +Remark 13. The compactification in Example 5.1 is in general distinct from the algebraic extension of +the Fi on RP2: for instance, consider the trivial example of R2 endowed with the horizontal and vertical +foliations. In this case the compactification by the algebraic extension, all the leaves of the horizontal +(resp. vertical) foliations tend to the same point at RP1 (which corresponds to 2 points for the circle at +infinity). + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +21 +5.2. Projections of D2 +F on D2 +Fi and center-like points on the circle at infinity. +Example 11. Consider R2 endowed with the trivial horizontal and vertical foliation, H and V respec- +tively. +Then the compactification D2 +H,V is conjugated to the square [−1, 1]2 endowed with the trivial +horizontal and vertical foliation. Every point p ∈ S1 +H,V, but the four vertices, are limit of exactly 1 end +of leaf, either horizontal (for p in the vertical sides) or vertical (for p in the horizontal sides). +The projection ΠH : D2 +H,V → D2H consists in colapsing the two horizontal sides, which are tranformed +in the center-like points of S1 +H. +The projection ΠH : D2 +H,V → D2V consists in colapsing the two vertical sides, which are tranformed in +the center-like points of S1 +H. +Example 12. Consider the strip {(x, y) ∈ R2, |x − y| < 1} endowed with the horizontal and vertical +foliations H and V respectively.. Then D2 +H,V = D2 +H = D2 +V and consists in adding to two points ±∞ to the +closed strip {(x, y) ∈ R2, |x − y| ≤ 1}. Every point in the sides |x − y| = 1 are the limit of exactly 1 end +of leaf of H and 1 end of leaf of V, and the points ±∞ are center like for both foliations. +These two examples show that pairs of very simple foliations may lead to different behavior of the +projection of the compactification associated to the pair on the compactification of each foliation. +Proposition 5.1 below shows that, for complicated foliations, the compactification of the pair of folia- +tions in general coincides with the compactification of each foliations. +Proposition 5.1. Let F, G be two transverse foliations on R2. Assume that +• the union of leaves of G which are not separated at their right from an other leaf is dense in R2 +• the union of leaves of G which are not separated at their left from an other leaf is dense in R2. +Then the identity map on R2 extend as a homeomorphism from D2 +F,G → D2 +F: in other words D2 +F,G = D2 +F. +Proof. Assume that there is an open interval I of S1 +F,G corresponding to no end of leaf of F. Then the +ends of leaves of G are dense in I, and therefore the projection of I on S1 +G is injective. +Now Lemma 4.8 implies that there are leaves L of G having both ends on I. Thus up to change positive +in negative, every positive half leaf of F through L has its end on I contradicting the definition of I. +So the points of S1 +F,G corresponding to ends of leaves of F are dense. Thus S1 +F,G = S1 +F, concluding the +proof. +□ +As a direct corollary of Proposition 5.1 and Lemma 4.8 one gets +Corollary 5.2. Let F, G be two transverse foliations on R2 so that both F and G have density of leaves +non separated at the right and of leaves non separated at the left. +Then generic points in S1 +F,G = S1 +F = S1 +G are center-like for both foliations. +5.3. Hyperbolic sectors. In the case of 1 foliation we have seen that, if several ends of leaves have the +same limit points on the circle at infinity, then they are ordered as a segment of Z and two succesive +ends bound a hyperbolic sector. These hyperbolic sectors have a very precise model, which allows us to +understand the position of a transverse foliation. +Lemma 5.1. Let F and G be two transverse foliations on R2. and consider πF : D2 +F,G → D2 +F. Assume +that p ∈ D2 +F is the corner of a hyperbolic sector bounded by the ends A and B of leaves of F. +Then there is a non-empty interval IG of ends of leaves of G ending at p in D2 +F. Furthermore +• either IG consist in a unique end of leaf C of G and A, B, C tend to tend same point at infinity +in D2 +F,G +• or π−1 +F (IG) is a closed interval on the circle S1 +F,G whose interior consist in regular ends of leaves +of G. +Proof. Just use the model [−1, 1]×[0, 1] where p is the point (0, 1), A = [−1, 0)×{1} and B = (0, 1]×{1}, +and the horizontal segment [−1, 1] × {t}, 0 ≤ t < 1 are F-leaf segments. We can choose this model so +that the vertical sides {−1}×[0, 1] and {1}×[0, 1] are leaves segments of G. Consider the G-leaves throug +[−1, 1] × {0}. The leaves reaching A and the leaves reaching B are two non-empty intervals, open in + +22 +CHRISTIAN BONATTI +[−1, 1] and disjoint. By connectedness, there are leaves, corresponding to an closed interval of [−1, 1] +which reach neither A nor B and these leaves end at p ∈ S1 +F. +Assume that this interval is not reduced to a single end of leaf of G and consider an end C in the +interior of this interval and assume that C is, for instance, a right end. Consider the neighborhoods +U right +t +of C defined in Section 3. Then � +t U right +t +consist in C and in a (maybe empty) set of entire leaves +of G contained in the hyperbolic sector. Assume that this set is not empty and let D be such a leaf of G. +Every leaf of F cutting D has an half leaf contained in the hyperbolic sector, contradicting the definition +of hyperbolic sector. Thus C = � +t U right +t +, meaning that C is a regular end of leaf of G, ending the proof. +□ +As a straightforward consequence, one gets: +Corollary 5.3. Let F = {Fi}i∈I, I ⊂ N be an at most countable family of pairwise transverse foliations +on R2. Consider a point p ∈ D2 +F. Then +• either at most 1 end of leaf of each Fi has p as its limit. +• or the set of ends tending to p is ordered as an interval of Z and, between any two succesive ends +of leaves of the same Fi, there is exactly 1 end of a leaf of each Fj, j ̸= i. +6. +The circle at infinity for orientable laminations. +6.1. The circle at infinity of a lamination. The way we proposed to compactify R2 can be generalized +for any object providing a family of rays admitting a separating set. +For instance, what about laminations? The theory cannot be extended without hypotheses. An evident +obstruction is that the leaves can be too few for going to a dense subset of a circle at infinity. But there +are more subtle issues as shows Example 13 below. +Example 13. There are closed laminations whose leaves are recurrent. For instance consider a Plykin +attractor on R2: it is a compact minimal lamination (by the unstable manifolds). +If you consider now a Plykyn attractor on S2 = R2 ∪ {∞} where ∞ belongs to the attractor, we get a +closed lamination on R2 where every leaf is unbounded but recurrent. +Notice that the recurrent lamination in Example 13 are not orientable. Let me show that Poincar´e +Bendixson argument applies on orientable laminations: +Lemma 6.1. Let L be a closed orientable lamination of R2. Given any leaf L, either the closure ¯L +contains a compact leaf or L is a line. +Proof. If ¯L = L then L is either a compact leaf or is a line (i.e. is properly embedded in R2). Assume +now that ¯L\L contains a point x ∈ R2. We fix an orientation of L. Chose a segment σ: [−1, 1] transverse +to L so that σ(0) = x and so that σ cuts positively every leaf. The hypothesis implies that L cuts σ in +an infinite set. Consider 2 successive (for the order in the leaf L) intersection points z0, z1. Then one +gets a simple closed curve δ in R2 by concatenation of the segments [z0, z1]L and [z1, z0]σ joining z0 to +z1, in L and in σ respectively. +Consider the disc ∆ bounded by δ. Every leaf cuts δ with the same orientation, that is, either every +leaf enter in ∆ or every leaf goes out of ∆. Up to reverse the orientation one may assume that every leaf +enters in ∆ and in particular L enters in ∆. In other words, there is a positive half leaf L+ included in +∆. This half leaf cannot be reccurent (otherwize it would cut again [z0, z1]σ and for that it needs to go +out of ∆). Futhermore: +Claim 2. No other leaf L′ ̸= L can accumulate on L: L ∩ ¯L′ = ∅ if L′ ̸= L. +Proof. If L′ accumulates on L, it cuts [z0, z1]σ on an infinite set. +□ +Thus the ω-limit set is ω(L) = ¯L+ \ L+ and is not empty. Consider y ∈ ¯L+ \ L+. The leaf Ly is +contained in ∆. Either Ly is compact and ω(L) = Ly and we are done, or ¯Ly \ Ly ̸ ∅. In that case, +Claim 2 implies that Ly is not accumulated by any leaf, in particular by L, contradicting the definition +of Ly. +□ + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +23 +We are now ready to extend Theorem 7 to the case of orientable laminations: +Theorem 8. Let L be a closed orientable lamination of R2 with no compact leaf and assume that the set +of leaves of L is uncountable. Then there is a compactification D2 +L ≃ of R2 by adding a circle at infinity +S1 +L = ∂D2 +F with the following properties: +• any half leaf tends to a point at infinity. +• given a point θ ∈ S1 +L the set of ends of leaves tending to θ is at most countable. +• for any non-empty open subset I of S1 +L the set of points in I corresponding to limits of ends of +leaves is uncountable. +Furthermore this compactification of R2 by D2 with these three properties is unique, up to a homeo- +morphism of the disk D2. +Let me just given a sketch of proof. +Proof. The lamination L is assumed to be oriented and without compact leaves, so that every leaf is a +line, according to Lemma 6.1. +According to Cantor-Bendixson theorem, see for instance [Ke], the lamination L can be written in a +unique way as union L = L0 ∪ L1 of two disjoint laminations, where L0 is a closed lamination with no +isolated leaves and L1 consists in a countable set of leaves. +A leaf L ∈ L0 is called regular if it is accumulated on both sides by leaves in L0 and is separated from +any other leaf of L0. The same proof as for foliations shows that the set of leaves in L0 which are not +regular are at most countable. +Finally, as for foliations, one consider the set R of germs of rays contained in regular leaves of L0. We +consider a countable set D of regular leaves, whose union is dense in L0, and as for the case of foliations, +one proves that the rays in D are separating for R. +Then we apply Theorem 1 and we get the announced canonical compactification. +□ +When L is transversely a perfect compact set (that is, there is a transverse segment σ through every +point x ∈ L so that σ ∩ L is a compact set without isolated points), then the compactification given by +Theorem 8 seems very natural: any homeomorphism h of R2 preserving L extends on S1 +L as a homeo- +morphism H of D2 +L, and the restriction H|S1 +L is the identity map if and only if h preserves every leaf of +L. That is no more the case if L has isolated leaves. +For lamination with isolated leaves, Theorem 8 just ignores the countable part L1 of L (in the Cantor +Bendixson decomposition of L). +We will now propose a cannonical compactification which takes in +account this countable part. +We start by looking at two very different examples of countable oriented laminations. +Example 14. Consider the lamination L = R × Z. Then L does not admit any compactification by a +circle at infinity so that any homeomorphism h preserving L extends on the circle at infinity. +Example 15. Consider a hyperbolic surface S of finite volume and consider a set ℓ of essential disjoint +simple closed geodesic on S. Then the lift L of ℓ on the universal cover ˜S = ˚D2 is a countable, discrete +lamination by geodesic of the Poincar´e disc so that the ends of leaves tend each to points on the circle +S1 = ∂D2, and the set of such limit points is dense on S1 as the action of π1S on S1 is minimal. +In this example, the lamination is transversely discrete, but the set of ends of leaves is a separating set +for himself for the cyclic order. +In the example 15 above, what implies the existence of a separating set is the minimality of the action +on the circle at infinity of a natural compactification. +In order to propose a cannonical compactification for a closed, oriented lamination without compact +leaves we need to determine what part of a cyclically totally ordered set admits separating subset. That +is what is done in next easy proposition whose proof is let to the reader. +Proposition 6.1. Let X be a set endowed with a total cyclic order. Consider the relation on X defined +as follows: x ≈ y if one of the intervals [x, y] or [y, x] does not contained any self-separating subset E +(with #E ≥ 2). Then + +24 +CHRISTIAN BONATTI +• the relation ≈ is an equivalence relation +• each equivalence class is an interval +• the cyclic order on X induces a total cyclic order on the quotient X/ ≈ +Futhermore, X/ ≈ is either a single point or is an infinite self-separating set. +Note that any two distinct points in a self separated set belong to distinct classes, so that #(X/ ≈) = 1 +if and only if X does not contain any (non-trivial) self-separating subsets. Otherwise #(X/ ≈) = ∞. +The canonical compactification is now given by Theorem 9 below: +Theorem 9. Let L be a closed oriented lamination of the plane R2, with no compact leaf, and let R +be the set of ends of leaves of L. As the ends of leaves are disjoints rays the set R is totally cyclically +ordered. Assume that #(R/ ≈) > 1. +Then there is a unique compactification D2 +L of R2 by adding a circle at infinity S1 +L so that +• any end of leaf of L tends to a point in S1 +L +• the set of points in S1 +L end of an end of leaf is dense in S1 +• two ends of leaves tend to the same point in S1 +L if and only if they belong to the same class in +R/ ≈. +Proof. Just apply Theorem 1 to a subset E ⊂ R containing exactly 1 representative in each class of ≈. +One checks that the compactification obtained satisfies the announced properties and does not depend +on the choice of E. +□ +Remark 14. Every class of ≈ in R is at most countable because the set of ends of regular leaves in the +perfect part L0 is self-separating. +This compactification takes in account more leaves that the compactification given by Theorem 8, but +it is still may have unexpected behaviours: +Example 16. Consider a non-compact hyperbolic surface S of finite volume and consider a closed lam- +ination ℓ defined by two disjoint freely homotopic essential closed curves and a closed (but non-compact) +leaf whose both ends tend to the same puncture of S. +Then the lift L of ℓ on the universal cover ˜S = ˚D2 is a countable, discrete lamination of the Poincar´e +disc so that the ends of leaves tend each to points on the circle S1 = ∂D2, the set of such limit points is +dense on S1 (again as the action of π1S on S1 is minimal). +In this example, however, there are pairs of leaves which share the same limits of their both ends, and +there are leaves whose both ends tend to the same point. +Given a closed oriented lamination L with no compact leaves and its Cantor-Bendixson decomposition +L = L0 ∪ L1 (L0 is a closed lamination without isolated leaves and L1 is countable), Theorem 9 takes +in account the part of the ends of leaves in L1 with separating subsets, in contrast with Theorem 8. For +my personal taste, the main issue in Theorem 9 is that I did not found any natural criterion to calculate +the equivalence classes of ≈ in L1. In fact Lemma 6.2 below seems to present as paradoxal the fact that +L1 may have separating subsets: +Lemma 6.2. Let D ⊂ R be a countable compact subset, ordered by R. Then D does not contain any self +separating subsets E ⊂ D(that is, #E > 2 and for every x < z, x, z ∈ E there is y ∈ E with x < y < z) . +Proof. If E is a non-trivcial sel-spearting subset, then there is an increasing bijection from E\{min E, max E} +to Q ∩ (0, 1). This increasing bijection extends in a unique way in a (non-strictly) increasing map from +R → [0, 1]. This map is continuous and the image of D is [0, 1]. +Thus D is not countable. +□ +Lemma 6.2 tell us that the separating property of a closed countable lamination cannot be obtained +locally (in foliated charts of the lamination). One deduces: +Proposition 6.2. Let L be a closed countable orientd lamination of ˚ +D2 so that every end of leaf tends +to a point on S1 and the set of such limit points are dense in S1. + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +25 +Then given any non-empty open interval I ⊂ S1 there is L ∈ L whose both ends have their limits in I. +More precisely, any neighborhood of I in D2 contains an entire leaf of L. +Proof. One consider a neighborhood of I bounded by two half leaves whose limits are points x ̸= y ∈ I +and a segment σ transverse to L and joining this two leaves. If no leaves is contained in this neighborhood, +then every leaf having an end in I cuts σ. +On the other hand any dense subset of an interval J of R is self separating. One deduces that σ ∩ L +contains a self separating subset, but is is a countable compact set, and this contradicts Lemma 6.2. +□ +This proposition says that the separating property for a countable oriented lamination is obtained by +leaves in small neighborhoods of the points at infinity. +6.2. Families of transverse laminations. Transversality does not imply in general the compactness +of the intersection of two leaves of transverse laminations. But this compactness is our main hypothesis +for the compactification associated to families of foliations. +However, if two lines L1, L2 ⊂ R intersect always with the same orientation, then #L1 ∩ L2 ≤ 1. One +deduces that Theorem 2 extends without difficulties to countable families of oriented closed laminations +intersecting pairwise transversely and with always the same orientation +Theorem 10. Let L = {Li}, i ∈ I ⊂ N be a family of closed orientable laminations of R2 with no +compact leaves and so that the set of leaves of Li is uncountable. We assume that the laminations are +pairwize transverse with constant orientation of the intersections. Then there is a compactification D2 +L ≃ +of R2 by adding a circle at infinity S1 +L = ∂D2 +F with the following properties: +• any half leaf tends to a point at infinity. +• given a point θ ∈ S1 +L the set of ends of leaves tending to θ is at most countable. +• for any non-empty open subset I of S1 +L the set of points in I corresponding to limits of ends of +leaves is uncountable. +Furthermore this compactification of R2 by D2 with these three properties is unique, up to a homeo- +morphism of the disk D2. +7. Actions on a bifoliated plane +We have seen than any homeomorphism h ∈ Homeo(R2) preserving a at most countable family of +transverse foliations F admits a unique extension as an homeomorphism on the compactification D2 +F. +Thus if H ֒→ Homeo(R2) is a group acting on R2 and preserving the (at most countable) family of +transverse foliations F then this action extends in an action on D2 +F. By restriction to the circle at infinity, +one gets an action of H on S1 +F. +If H ֒→ Homeo(R2) is a group acting on R2 and preserving a family of foliations F, we say that the +action is minimal on the leaves of F if H(L) is dense of R2 for every leaf L of a foliation of the family F. +7.1. Faithfullness. +Proposition 7.1. Let F be a foliation, and h ∈ Homeo(R2) be a homeomorphism preserving F. Then +the action of h on S1 +F is the identity map if and only if h(L) = L for any leaf L, and h preserves the +orientation of the leaves. +Proof. If h preserves every leaf and its orientation, then it preserves any limit of its ends. As these limit +of ends are dense in S1 +F one gets that the homeomorphism induced by h on S1 +F is the identity map. +Conversely, if h induces the identity on SS1 +F then for every leaf L the leaf h(L) have the same limit of +ends as L. According to Lemma 4.2 this implies h(L) = L as announced. +□ +Corollary 7.1. Let F = {Fi}, i ∈ I ⊂ N be a family of at least 2 transverse foliations. +Let h ∈ +Homeo(R2) be a homeomorphism preserving each foliation Fi. Then the action of h on S1 +F is the identity +map if and only if h itself is the identity map. + +26 +CHRISTIAN BONATTI +Proof. If the induced homeomorphism induced by h on S1 +F is the identity map then the same happens +to homeomorphism induced by h on every S1 +Fi (because they are quotient of S1 +F). Thus Proposition 7.1 +implies that h preserves each leaf of each Fi. As every point of R2 is the unique intersection point of the +leaves through it, one deduces that every point of R2 is fixed by h and h is the identity map. +□ +7.2. Orientations and injectivity of the projections. Let F be a foliation of the plane R2, endowed +with an orientation and a transverse orientation. Let G ⊂ Homeo(R2) be a group of homeomorphisms +preserving (globally) F. Let G+ (resp. G+) be the index at most 2 subgroup consisting of the elements +of G preserving the orientation (reps. the transverse orientation)of F, and G+ ++ = G+ ∩ G+) the subgroup +of elements preserving both orientations. Then: +Lemma 7.1. If one of the groups G, G+, G+, G+ ++ acts minimally on the leaves of F, then so does each of +these 4 groups. +We will indeed prove Lemma 7.2 for which Lemma 7.1 is a particular case. +Lemma 7.2. Let G be a group acting minimally on the leaves of a foliation F of R2, and H ⊂ G be a +subgroup of finite index. Then H acts minimally on the leaves of F. +Proof. G acts minimally on the leaves of F, and consider such a leaf L. As H is a finite index subgroup, +there are g1, .., gn ∈ G so that for any g ∈ G there is i ∈ {1, . . ., n} with g.H = giH. Let denote Hi = gi·H. +In particular G = � +i Hi, and then R2 = � +i Hi(L). +Consider any open subset O of R2. +O = O ∩ +� +i +Hi(L) = +� +i +(O ∩ Hi(L)) +The open set O is a baire space so that the union of finitely many closed sets with empty interior has +empty interior: one deduce that at least one of the O ∩Hi(L) have non empty interior. One deduces that +the union � +i +˚ +Hi(L) of the interiors of the Hi(L) is dense in R2. +Notice that for every i and every g there is j so that g(Hi(L)) = Hj(L). +Consider R2 \ � +i +˚ +Hi(L). It is a G-invariant closed set, saturated for the foliation F, and with empty +interior. As every G-orbit is dense, one deduces that this set is empty. +Thus +R2 = +� +i +˚ +Hi(L). +The open sets +˚ +Hi(L) are images on from the other by homeomorphisms in G, and in particular they +are all non-empty. +As R2 is connected, one deduces that the open sets +˚ +Hi(L) are not pairwise disjoint. Let k ∈ {1, . . ., n} +be the maximum number so that there are distinct i1, . . . , ik with +k� +1 +˚ +Hij(L) ̸= ∅. +As the +˚ +Hi(L) are not pairwise disjoint, we know that k ≥ 2. We will prove, arguing by contradiction: +Claim 3. k = n. +Proof. For that we assume that k < n. +Then we consider the union of all the intersections of k of these open sets. This union is a F-saturated +G-invariant non-empty set and hence is dense. Its complement is an F-saturated invariant closed set with +empty interior, and therefore is empty. +Thus R2 is the union of these open sets. Now again the connexity of R2 implies that these open sets +are not pairwise disjoint. This provides a non-empty intersection of 2 distinct of these sets, that is, a non + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +27 +empty intersection of more than k of the +˚ +Hi(L), contradicting the choice of k. This shows k = n proving +the claim. +□ +Thus +n� +1 +˚ +Hi(L). +is an non-empty, G-invariant open set saturated for the foliation F, and thus it is dense in R2. +We just proved that H(L) is dense in R2, concluding the proof. +□ +We will use the next straightforward corollary of Lemma 7.1 +Corollary 7.2. Let H ⊂ Homeo(R2) be a group preserving a foliation F and acting minimally on the +leaves. Assume that L is a leaf which is not separated at the right and from below. Then the union of the +leaves h(L), h ∈ H, which are non-separated at the right and from below is dense in R2 (the same holds +changing right by left and/or below by above). +As a direct consequence of Proposition 5.1 and Corollary 7.2 we get: +Proposition 7.2. Let F,G be two transverse foliations of R2 and H ⊂ Homeo(R2) preserving F and G. +Assume that the orbit of every leaf of G in dense in R2. +If G has a non-separated leaf, then the projection of ΠF : D2 +F,G → D2 +F is injective. +Proof. If G has a non separated leaf L1 at the right, it is non separated from a leaf L2 which is non +separated at the left. Now Corollary 7.2 asserts that the leaves of G non separated at the left as well +as the leaves non separated at the right are dense in R2. Now Proposition 5.1 asserts that ΠF is a +homeomorphism, concluding. +□ +7.3. Minimality of the action on the circle at infinity. +Theorem 11. Let F be a foliation on the plane R2 and H ⊂ Homeo(R2) preserving the foliation F. +(1) If the action of H on S1 +F is minimal then the foliation F admits non separated leaves from above +and non separated leaves from below. +(2) Conversely if the foliation F admits non separated leaves from above and non separated leaves +from below and if the orbit of every leaf is dense in R2 then the action of H on S1 +F is minimal. +We will see with Theorem 14 that the minimality of the action on the leaves is not a necessary condition +for the minimality of the action on the circle at infinity. +Item 1 of Theorem 11 is a consequence of Proposition 7.3 below. +Proposition 7.3. Let F be a foliation of R2 and assume that F has no non-separated leaves from below. +Given any leaf L we denote by ∆+ +L the closure on D2 +F of the upper half plane of R2 bounded by L. +Then � +L∈F ∆+ +L is non empty and consists in an unique point OF on S1 +F. As a consequence, any +h ∈ Homeo(D2 +F) preserving F admits OF as a fixed point: +h(OF) = OF. +Proof. We introduce a relation on the set L of leaves of F as follows: L1 ⪯ L2 if there is a positively +oriented (for a transverse orientation of F) transverse segment σ starting at L1 and ending at L2. One +easily checks that ⪯ is a partial order relation on L. +Due to the connexity of R2, one gets: +Claim 4. given any leaves L, ˜L ∈ L there is k ≥ 0 and L0, . . . , Lk ∈ L so that, +• for any i ∈ {0, . . ., k − 1} the leaves Li and Li+1 are comparable for ⪯ (that is Li ⪯ Li+1 or +Li+1 ⪯ Li) +• L = L0 and L′ = Lk + +28 +CHRISTIAN BONATTI +Proof. There is a countable family of segments in R2 transverse to F and so that every leaf L cuts at +least one of these segments. The set of leaves cutting a given segment induces a connected open set of +R2. Given any two points in R2 one considers a compact path joining these two points. By compacity, it +is covered by a finite family of these open sets. One concludes easily. +□ +We denote ≺ L, L′ ≻∈ N the minimum value of such a number k. One easily checks that ≺ ·, · ≻ is a +distance on the set of leaves L. +Up to now, this could be done for any foliation F. In this setting, our hypothesis that F does not +admit leaves which are non-spearate from below is translated as follows: +Claim 5. Assume that L0, L1, L2 ∈ L are three leaves so that L0 ⪯ L1 and L0 ⪯ L2. Then L1 and L2 +are comparable for ⪯. +Proof. We assume that the leaves Li are distinct, otherwise there is nothing to do. Let σi : [0, 1] → R2, +i = 1, 2 transverse to F and positively oriented so that σi(0) ∈ L0 and σi(1) ∈ Li. +Let I = {t ∈ [0, 1], L(σ1(t)) ∩ σ2 ̸= ∅} and J = {t ∈ [0, 1], L(σ1(t)) ∩ σ2 ̸= ∅}. As R2 is simply +connected, one shows that I and J are connected and each of them contains 0. +Let t1 = sup I and t2 = sup J For any t ∈ [0, t1) let ˜t ∈ J so that L(σ1(t)) = L(σ2(˜t). In particular, ˜t +tends to t2 as t tends to t1. +Thus the leaves L(σ1(t1) and L(σ2(t2) are accumulated from below by the leaves L(σ1(t)) = L(σ2(˜t), +thus are non separated from below. By assumption on F this implies that they are equal: +L(σ1(t1) = L(σ2(t2) +If t1 < 1 and t2 < 1 then the leaf L(σ1(t) for t > t1 close to t cuts σ1 at a point σ2(˜t) with ˜t > t2, close +to t2. This contradicts our choice of t1 and t2. +Thus t1 = 1 or (non exclusive) t2 = 1. In the first case L1 = L(σ1(t1)) cuts σ2 and then L1 ⪯ L2 and +in the second case L2 cuts σ1 and L2 ⪯ L1. This ends the proof. +□ +As a consequence of Claims 4 and 5 one deduces +Claim 7.1. Given any two leaves L, ˜L there is a leaf ˆL so that L ⪯ ˆL and ˜L ⪯ ˆL. In particular, the +distance ≺ ·, · ≻ is bounded by 2. +Proof. Consider a finite sequence of leaves L = L0, . . . , Lk = ˜L, k =≺ L, ˜L ≻, and Li comparable with +Li+1. +The minimality of k implies that Li−1 and Li+1 are not comparable (otherwise one could delete Li +geting a strictly smaller sequence). +Assume that there is i ∈ {1, . . .k − 1} so that Li−1 ⪰ Li. If Li ⪰ Li+1 then Li−1 ⪰ Li+1 which is +forbidden by the observation above. Thus Li ⪯ Li+1 and Clain 5 implies again that Li−1 and Li+1 are +comparable, which again is impossible. This proves that +∀i ∈ {1, . . . k − 1}, Li−1 ⪯ Li +□ +As a consequence one deduces +Claim 6. There is a increasing sequence Li ≺ Li+1 i ∈ N, Li ∈ L so that, given any leaf L ∈ L there is +n with L ≺ Ln. +Proof. One chose a countable set of compact positively oriented segments σi : [0, 1] → R2 transverse to +F to that any leaf cuts one of the σi (and thus is less than L(σi(1) for ⪯). Then one builds inductively +the sequence Li: Li+1 is obtained by applying Claim 7.1 to the leaves Li and L(σi(1). +□ +Claim 7. The compact discs ∆+ +L are decreasing with L for ≺: more precisely, if L ≺ ˜L then +∆+ +˜L ⊂ ˚ +∆+ +L, + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +29 +where ˚ +∆+ +L denotes the interior for the topology of D2 +F ( it does not means the open disc). +Proof. The hypothesis implies that Li+1 is contained in the interior of ∆+ +Li, so that ∆+ +Li+1∩R2 is contained +in the interior of ∆+ +Li. We need to prove that ∆+ +Li+1 ∩ S1 +F is contained in the interior of ∆+ +Li ∩ S1 +F. In +other words, we need to prove that the ends of Li+1 do not share a limit with the ends of Li. +Recall that Li ≺ Li+1 that is, there is a segment σ: [0, 1] → R2 transverse to F and positively +oriented, with σ(0) ∈ Li and σ(1) ∈ Li+1. If Li share with Li+1 a limit point in S1 +F so does any leaf +L(σ(t)) contradicting the fact that points in S1 +F are limits of at most a countable set of ends of leaves. +This ends the proof. +□ +Thus Claims 6 and 7 implies +� +L∈F +∆+ +L = +� +i∈N +∆+ +Ln. +Now � +L∈F ∆+ +L = � +i∈N ∆+ +Ln is a decreasing sequence of connected compact metric sets, saturated for F +and therefore is a non-empty connected compact sets saturated for F. As it does not contain any leaf of +F one deduces that � +L∈F ∆+ +L ∩ R2 = ∅ that is � +L∈F ∆+ +L is a compact interval U in S1 +F. +It remains to show that this interval U = � +L∈F ∆+ +L is reduced to a point. Otherwise, there is and half +leaf L+ whose limit belongs to the interior of U. According to Claim 7, this implies that L+ ∩ ∆+ +Li ̸= ∅ +for every i. This contradics the fact that, for n large enough, the leaf Ln is larger (for ≺) than the leaf +L carrying the half leaf L+ and thus L ∩ ∆+ +Ln = ∅. +This contradiction ends the proof of Proposition 7.3. +□ +Proof of item 1 of Theorem 11. Assume that the action of H on S1 +F is minimal. The foliation F cannot +be conjugated to the trivial foliation otherwise the set {N, S} (unique points in S1 +F which are not limit +of leaves) would be H-invariant. +Thus F admits non-separated leaves, and we can assume it is from above (up to change the transverse +orientation of F). If it do not admit non-separated leaves from below then the point OF in S1 +F given by +Proposition 7.3 is a global fix point of H. +□ +Proof of the Item 2 of Theorem 11. We assume that H is a group acting minimally on the leaves of +a foliation F having non-separated leaves, some of them from above and some of them from below. +According to lemma 7.1 up to consider a finite index subgroup of H, acting minimally on the leaves of +F, one may assume that H preserves the orientation and transverse orientation of F. +Recall that the ends of regular leaves are dense in S1 +F. Thus it is enough to check that any neighborhood +of any end of a regular leaf contains points in the orbit for H of any point of S1 +F. Consider a regular +leaf L and σ: [−1, 1] → R2 a segment transverse to F with σ(0) ∈ L. We will show that the end of L+ +belongs to the closure of any H-orbit (the same argument holds for the end of L−). +We denote by Lt the leaf through t. Consider the basis of neighborhood U + +t +of the end L+ given by +the compact discs in D2 +F closure of the half plane bounded by L+ +−t, σ([−t, t]), and L+ +t . +Our hypothesis implies +Claim 8. There is a dense subset of values of t so that Lt is not separated at the right. As a consequence +for every t the topological disc U + +t +contains entire leaves. +Proof. The first sentence is directly implied by the existence of leaves which are non-separated at the +right, the fact that H preserves the orientations of the leaves and acts minimaly on the leaves of F. +The second sentence have been seen in Section 4. +□ +Any leaf L cuts D2 +F in two discs, ∆+ +L and ∆− +L (following the tranverse orientation of F) whose union +∆+ +L ∪ ∆−L is D2 +F. +Claim 9. Under the hypotheses, given any L there are g1, g2 ∈ H so that g1(∆+ +L) ⊂ ˚∆− +L and g2(∆− +L) ⊂ +˚∆+ +L + +30 +CHRISTIAN BONATTI +As a consequence both ∆+ +L and ∆− +L contains points in any H-orbit of point in D2 +F. +Proof. We prove the first inclusion, the other is obtained by reversing the transverse orientation of F. +Considers L a leaf and σ: [−1, 1] → R2 a segment transverse to F (positively oriented for the transverse +orientation of F) so that σ(0) ∈ L. There is −t ∈ [−1, 0) so that the leaf L−t is non-separated from +below from a leaf L2, because the leaves non-separated from below are dense in R2, due to the minimality +of the action of H on the leaves, and the fact that H preserves the transverse orientation of F. Thus +L−t ⊂ ∆− +L, L2 ⊂ ∆− +L. Furthermore ∆− +L2 contains L−t and thus contains L. One deduces: +∆+ +L2 ⊂ ∆− +L. +Now, there is h ∈ H so that h(L2) = L−s.with −s ∈ (−1, 0). In particular one gets that ∆+ +L ⊂ ˚∆+(h(L2) +and thus +h−1∆+ +L = ∆+ +h−1(L) ⊂ ˚∆+(L2) ⊂ ˚∆− +L. +This concludes the proof. +□ +We are ready to conclude the proof of Theorem 11: Any neighborhood in D2 +F of any point of S1 +F +contains an entire leaf L (claim 8 above), and thus contains either ∆+ +L or ∆− +L. According to Claim 9 this +neighborhood contains points in any H-orbit of points in D2 +F. This shows the minimality of the action +of H on S1 +F, concluding. +□ +Theorem 12. Let F, G be two transverse foliations on the plane R2. Let H ⊂ Homeo(R2) be a group +preserving both foliations F and G. +(1) If the action of H on S1 +F,G is minimal then both foliations F G have non-separated leaves from +above and non separated leaves from below. +(2) Conversely, if both foliations F G have non-separated leaves from above and non separated leaves +from below and if the orbit of every leaf of F and G is dense R2, then the action of H on S1 +F,G +is minimal. +Proof. For item 1, if the action of H on S1 +F,G is minimal then both actions of H on S1 +F and S1 +G are +minimal. Thus item 1 follows from Item 1 of Theorem 11. +Conversely, as the action on the leaves of F and G is assumed to be minimal, and they have non- +separated leaves, then Proposition 7.2 implies that both projections ΠF and ΠG are injective. That is +S1 +F,G = S1 +F = S1 +G. Now the minimality of the action of H on this circle at infinity is given by item 2 of +Theorem 11. +□ +8. Action of the fundamental group on the bifoliated plane of an Anosov flow +8.1. The bifoliated plane associated to a Anosov flow. Let X be an Anosov flow on a closed 3- +manifold M. Then Fenley and Barbot show that the lift of X on the universal cover of M is conjugated +to R3, ∂ +∂x; in particular the space of orbits of this lifted flow is a plane PX ≃ R2. Then, the center-stable +and center-unstable foliations of X induce (by lifitng on the universal cover and projecting on PX) a pair +of transverse foliations Fs, Fu on the plane PX. The triple (PX, Fs, Fu) is called the bi-foliated plane +associated to X. Finally, the natural action of the fundamental group π1(M) on the universal cover of +M projects on PX in an action preserving both foliations Fs and Fu. +Fenley and Barbot proved that, if one of the foliation Fs, Fu is trivial (that is, has no non-spearated +leaf and therefore is conjugate to an affine foliation by parallel straight lines) then the other is also trivial. +In that case, one says that X is R-covered. In that case the bifoliated plane is conjugated to one of the +two possible models: +• the plane R2 endowed with the trivial horizontal and vertical foliations; Solodov proved that +this is equivalent to the fact that X is orbitally equivalent to the suspension flow of a linear +automorphism of the torus T2; +• the restriction of the trivial horizontal and vertical foliation to the strip |x − y| < 1. + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +31 +8.2. Injectivity of the projection of D2 +F s,F u on D2 +F s and D2 +F u. The aim of this section is to prove +Theorem 5 which is restated as Proposition 8.1 below and Theorem 13 (in next section). +Proposition 8.1. Let X be an Anosov flow on a 3-manifold. Then: +• Either X is topologically equivalent to the suspension flow of a hyperbolic element of SL(2, Z) +• Or both projections of the compactification D2 +F s,F u on D2 +F s and D2 +F u are homeomorphisms. +Proof. Assume that the projection on D2 +F u is not injective. Thus there is a non trivial open interval I +of S1 +F s,F u whose point are not limit of end of leaves of Fu. Thus a dense subset of point in I are limit +of ends of leaves of Fs. Furthermore, every end of leaf of Fs in I is a regular end. Consider a regular +end (for instance, a right end) of leaf Ls +right of Fs whose limit is in the interior of I. Then there is a +small unstable segment σ through a point of Ls +right so that every right half leaf Ls +right,t of Fs is regular +and has its limits in I. Then the union of all these half leaves is what Fenley called a product region , in +[Fe2]. Now [Fe2, Theorem 5.1] asserts that any Anosov flow admiting a product region is a suspension +flow, concluding. +□ +8.3. Minimality of the action on the circle at infinity. In order to prove Theorem 5 it remais to +prove Theorem 13 below: +Theorem 13. Let X a flot d’Anosov on a closed 3-manifold M. Then X is non-R-covered if and only +if the action of π1(M) on the circle S1 +F s,F u at infinity is minimal. +Remark 15. If the manifold M is not orientable and if X is R-covered, then [Fe1] noticed that X is a +suspension flow. Thus, on non-orientable manifolds M, Theorem 13 asserts the minimality of the action +on the circle at infinity, excepted if M is a suspension manifold. +Remark 16. The bifoliated plane (PX, Fs, Fu) remains unchanged if we consider a lift of X on a finite +cover. Thus it is enough to prove Theorem 13 in the case where M is oriented and the action of π1(M) +preserves both orientation and transverse orientation of both foliations Fs, Fu. +Thus, up to now we will assume that M is oriented and the action of π1(M) preserves both orientations +and transverse orietations of both foliations Fs, Fu. +Remark 17. If X is R-covered, then S1 +F s has exactly 2 center-like points, which are therefore preserved +by the action of π1(M) on S1 +F s: this action is not minimal, and thus the action on S1 +F s,F u is not minimal. +Thus we are left to prove Theorem 13 in the case where X is not R-covered.We will start with the +easier case, when X is assumed to be transitive. The non-transitive case will be done in the whole next +section. +Proof of Theorem 13 when X is transitive. When X is non-R-covered and transitive, then [Fe3] proved +that Fs and Fu admits non-separated leaves from above and non-separated leaves from below. As (up +to consider a finite cover of M), the action of π1(M) preserves the orientation and tranverse orientation +of Fs, and the action is minimal on the set of leaves of Fs thus Theorem 11 asserts that the action of +π1(M) on S1 +F s is minimimal. As S1 +F s = S1 +F s,F u, this concludes the proof. +□ +9. Minimality of the action on the circle at infinity for non-transitive Anosov flows: +ending the proof of Theorem 5 +For ending the proof of Theorem 5, we are left to prove : +Theorem 14. Let X be a non-transitive Anosov flow on a closed connected 3-manifold M. Then the +action of the fundamental group of M on the circle at infinity is minimal. +This result is somewhat less intuitive, as the action of the fundamental group π1(M) on the leaves of +M is not minimal, and even, if X has several attractors, may fail to admit a leaf whose orbit is dense. +The proof of the minimality of the action on the circle at infinity will require some background on +Anosov flow, in particular on non-transitive Anosov flows. In the whole section, X is a non-transitive + +32 +CHRISTIAN BONATTI +Anosov flow on an orientable closed connected manifold M and the natural action of π1(M) on the +bifoliated plane (PX, Fs, Fu) preserves the orientations and transverse orientations of both foliations. +Recall that we have seen that the compactification of both foliations coincide with the one of each +foliation. We will denote by D2 +X, S1 +X this compactification and the corresponding circle at infinity. We +refer by ∗ for this package of hypotheses and notations. +9.1. Background on non-transitive Anosov flows. Let X be a non-transitive Anosov flow. Thus, +according to [Fe1, Ba1] X is not R-covered. +The flow X is a structurally stable flow, so that Smale spectral decomposition theorem splits the +non-wandering set of X in basic pieces ordered by Smale order: a basic piece is upper another if its +unstable manifolds cuts the stable manifold of the other. For this order, the maximal basic pieces are the +repellers and the minimal are the attractors. In [Br], Brunella noticed that the basic pieces are separated +by incompressible tori transverse to the flow. +Consider an attractor A of X. It is a compact set consisting in leaves of the unstable foliation of X, +hence it is a compact lamination by unstable leaves. Furthermore the intersection of A with a transverse +segment σ is a Cantor set. An unstable leaf W u in A is called of boundary type if W u ∩ σ belongs to the +boundary of a connected component of σ \ A. +A classical results from hyperbolic theory (see for instance [BeBo]) asserts that the unstable leaves in +A of boundary type are the unstable manifolds of a finite number of periodic orbits called periodic orbits +fo boundary type. +The same happens for repellers R: they are compact laminations by stable leaves, tranversally a +Cantor sets, and they admits finitely many boundary leaves, stable manifolds of finitely many periodic +orbits called of boundary type. +In this section, we will focus on attractors and repellers. Consider an attractor A of X, its lift ˜A on +the universal cover, and consider the projection of A on the bi-foliatioed plane PX. This projection is a +closed lamination by leaves of Fs and its cuts every tranverse curve along a Cantor set. By a pratical +abuse of notation we will still denote by A this lamination of PX: thus A denotes at the same time a +2-dimensional lamination on M and a 1-dimensional lamination on PX. +The same happens for repeller. +Let A ⊂ PX and R ⊂ PX be the unstable and stable laminations (respectively) corresponding to an +attractor and a repeller of X. Then +• A ∩ R = ∅. This seems obvious, but it will be a crucial property for us: given an unstable leaf +Lu and a stable leaf Ls, this will be our unique criterion for knowing that they don’t intersect. +• the periodic point contained in A (reps. R) are dense in A (resp. R). +• Each periodic orbit of X has a discrete π1(M)-orbit in PX +• the periodic orbits of boundary types are the π1(M)-orbits of finitely many X-orbits, and there- +fore are a discrete set in PX. +• Fenley [Fe2] shows that the non-separated stable leaves of Fs (resp. Fu) correspond to finitely +many orbits of X, and hence to a discrete set of periodic points in PX +• thus the periodic points p in A (resp. R) which are not of boundary type and whose unstable +(resp. stable) leaf is regular are dense in A. +• If A1, . . . , Ak are the attractors of X then the union of the stable leaves of Fs through the +laminations A1, . . . , AK of PX are dijoint open seubsets of PX whose union is dense in PX. The +same holds for the repellers. +As a straightforward consequence one gets: +Lemma 9.1. There is a dense subset of PX of points x whose stable leaf Ls(x) contains a periodic point +p in an attractor A, not of boundary type and so that Lu(p) is regular. A symmetric statement holds for +repellers. +9.2. Proof of Theorem 13. The two main steps of the proof of Theorem 13 are Propositions 9.1 and 9.2 +below. + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +33 +Proposition 9.1. Let Lu be a leaf of Fu corresponding to an unstable leaf of X contained in a attractor +of X. Let ∆+ and ∆− be the closures in D2 +X of the half planes in R2 bounded by Ls. Then there are +g+, g− ∈ π1(M) so that g−(∆−) ⊂ ∆+ and g+(∆+) ⊂ ∆−. +The same statement holds for stable leaves in the repellers. +Corollary 9.1. Let Ls and Lu be leaves of Fs and Fu in a repeller and in an attractor, respectively. +Let I ⊂ S1 +X be a segment with non empty interior and whose end points are the limit of both ends of the +same leaf, Ls or Lu. +Then every orbit of the action of π1(M) contains points in I. +Proof. According to Proposition 9.1 there is g ∈ π1(M) so that g(S1 \ I) ⊂ I, ending the proof. +□ +Proposition 9.2. Given any non-empty open interval J ⊂ S1 +X, there is a L which is either a leaf of Fs +in a repeller or a leaf of Fu in an attractor whose both ends have limits in J. +Proof of Theorem 14 assuming Propositions 9.1 and 9.2. According to Proposition 9.2, every interval J +with non-empty interior contains an interval I whose end points are both limit point of the end of a +stable or unstable leaf in an a repeller or an attractor, respectively. Now, according to Corollary 9.1, the +interval I contains a point in every π1(M) orbit in S1 +X. Thus any π1(M) orbit in S1 +X has points in any +interval with non-empty interior: in other words, every π1(M) orbit si dense in S1 +X, or else, the action of +π1(M) on S1 +X is minimal, ending the proof. +□ +9.3. Proof of Proposition 9.1. Let Lu +0 be an unstable leaf in an attractor A0, and ∆+ +0 be the closure +of the upper half plane bounded by Lu +0. For proving Proposition 9.1 we want to prove that there is +f ∈ π1(M) so that f(∆− +0 ) ⊂ ∆+ +0 (the other announced inclusion is identical). +Consider a point p0 ∈ Lu +0 and Ls +0 the stable leaf through p0. +Claim 10. There is an unstable leaf Lu +1 with the following property: +• Lu +1 ⊂ ∆+ +0 +• Lu +1 is contained in the basin of a repeller R1 +• Lu +1 contains a non-boundary periodic point p1 ∈ Lu +1 of the repeller R1. +• Lu +1 cuts the stable leaf Ls +0 in a point Lu +1 ∩ Ls +0 = q0. +Proof. The union of unstable leaves in the basin of a repeller and carrying a non-boundary periodic point +of this repeller is dense in R2. We can therefore choose such a leaf in ∆+ +0 and cutting Ls +0. +□ +Let Ls +1 be the stable leaf through p1. It is a non-boundary stable leaf contained in the repeller R1. +Note that Ls +1 is disjoint from the attractor A0. Thus +• Ls +1 is disjoint from Lu +0 ∈ A0. +• the stable leaf Ls +1 is distinct, and therefore disjoint from the stable leaf Ls +0. +In other words, the union Ls +0 ∪ Lu +0 divides PX in 4 quadrants and Ls +1 contained in one of this quadrants. +Let us denote by C±,± these 4 quadrants so that ∆+ +0 = C−,+ ∪ C+,+ and Ls +1 ⊂ C+,+. +Let denote by ∆+ +1 = ∆+(Ls +1) the closure of the half plane bounded by Ls +1 and contained in ∆+ +0 . Thus +∆+ +1 is contained in the same quadrant C+,+ as Ls +1. We denote by ∆− +1 the closure of the other half plane +bounded by Ls +1. Note that ∆− +1 contains the 3 other quadrants, in particular it contains ∆− +0 and C−,+. +As the leaf Ls +1 is (by assumption) not a boundary leaf of R1 it is accumulated on both sides by its +π1(M)-orbit. Thus there is a leaf Ls +2 = g(Ls +1) in its orbit, cutting Lu +1 at a point x ∈ ∆− +1 arbitrarilly +close to p1 and hence x ∈ C+,+. Notice that Ls +2 is contained in the repeller R1 and thus is disjoint from +Lu +0 ∪ Ls +0. Thus it is contained in one quadrant. As it contained x ∈ C+,+ one has +Ls +2 ⊂ C+,+. +Let h ∈ π1(M) be the generator of the stabilizer p1 so that Lu +1 is expanded by h. We consider the +sequence of leaves hn(Ls +2) which cut Lu +1,− at the point hn(x). +Claim 11. For n large enough hn(Ls +2) is contained in the quadrant C−,+. + +34 +CHRISTIAN BONATTI +Proof. Each leaf hn(Ls +2) intersects Lu +1 ⊂ ∆+ +0 and is disjoint from Lu +0 (because hn(Ls +2) is contained in the +repeller). Hence hn(Ls +2) is contained in ∆+ +0 , and is distinct and therefore disjoint from Ls +0. Thus hn(Ls +2) +is contained in one of the quadrants C+,+ of C−,+. +The point xn tends to infinity in Lu +1 and so goes further q0 = Ls +0 ∩ Lu +1. Thus for n large enough +xn ∈ C−,+. We proved the for n large enough hn(Ls +2) ⊂ C−,+, proving the claim. +□ +We conclude the proof of proposition 9.1 by proving : +Claim 12. Consider n large enough so that hn(Lu +2) ⊂ C−,+. +Then either g(∆− +1 ) ⊂ C+,+ ⊂ ∆+ +0 or hng(∆− +1 ) ⊂ C−,+ ⊂ ∆+ +0 +As ∆− +1 contains ∆− +0 the claim implies that either g(∆− +0 ) ⊂ ∆+ +0 or hng(∆− +0 ) ⊂ ∆+ +0 which concludes +the proof of Proposition 9.1. +Proof of the claim. Assume g(∆− +1 ) is not contained in C+,+. As g(∆− +1 ) is one of the half plane bounded +by g(Lu +1) = Lu +2 ⊂ C+,+ one gets that g(∆+ +1 ) is the half plane bounded by Lu +2 and contained in C+,+. +In particular,g(∆+ +1 ) does not contain q0. As p1 and q0 are on distinct sides of Lu +2 one deduces that +p1 ∈ g(∆+ +1 ). +As p1 is the fixed point of h one deduces +p1 ∈ hng(∆+ +1 ) +Thus hng(∆+ +1 ) is the half plane bounded by hn(Lu +2) which is not contained in the quadrant C−,+. Thus +hng(∆− +1 ) is the other half plane bounded by hn(Lu +2) and is contained in C−,+, ending the proof. +□ +9.4. Proof of Proposition 9.2. We want to prove that any open interval I in the circle S1 +X contains +the two ends of an unstable leaf in an attractor or the two ends of a stable leaf of a repeller. +Lemma 9.2. Assuming ∗, there are dense subsets Es +0, Eu +0 of S1 +X so that +• any p ∈ Es +0 is the limit of a regular leaf of Fs containing a periodic point x which belongs to an +attractor A(p), and is not of boundary type. +• any q ∈ Eu +0 is the limit of a regular leaf of Fu containing a periodic point y which belongs to an +repeler R(p), and is not of boundary type. +Proof. According to Lemma 9.1 the union of regular stable leaves containing periodic point of non- +boundary type of an attractors are dense in PX. +This family is therefore separating, according to +Lemma 3.1. Thus the limits of their ends is a dense subset of S1 +X, as announced. +□ +Lemma 9.3. Assuming ∗, there is a dense subset E ⊂ S1 +X so that every x ∈ E is the limit of the end a +regular leaf of Fs (resp. Fu) contained in a repeller R (resp. an attractor A), and carrying a periodic +point of non-boundary type. +Proof. Consider a non empty open interval I ⊂ S1 +X. According to Lemma 9.2 there is a point x ∈ I which +is the limit of an end Ls ++(p0) of a regular leave of Fs carrying a periodic point p0 in a non-boundary type +unstable leaf Lu(p0) of a attractor A. +The point p0 is accumulated on both sides by periodic points in A. We chose p1 so that the limit y +of Ls ++(p1) belongs to I (that is possible because Ls ++(p0) is regular) and Ls ++(p1) intersects Lu(p0) at a +point q1. Thus let J ⊂ I be the segment contained in I and whose end points are x and y.Notice that +y ̸= x,that is J has non-empty interior, as Ls(p0) is a regular leaf. +Now Lu(p0) is accumulated on both sides by regular unstable leaves contained in the attractor A and +containing periodic point of non-boundary type. Let Lu +0 be such a leaf, with non empty intersection with +Ls ++(p0). +If Lu +0 does not cut Ls ++(p1), then one ends is contained in the half strip bounded by Ls ++(p0), the segment +of [p0, q1]u and Lu ++(q1). As a consequence, the limit of this end belongs to I and we are done. +Thus we may assume now that Lu +0 cuts Ls ++(p1). + +CIRCLE AT INFINITY OF FOLIATIONS OF R2 +35 +Let h0 and h1 be the generators of the stabilizers of p0 and p1, respectively, so that h0 expands Ls ++(p0) +and h1 expands Ls ++(p1). +We consider the images {hn +0(Lu +0), hn +1, n ∈ N(Lu +0)} of the leaf Lu +0 by the positive iterates of h0 and +h1. Each of these images is an regular unstable leaf in A, and has a non-empty intersection with either +Ls ++(p0) or Ls ++(p1). If one of these leaves does not cross both Ls ++(p0) and Ls ++(p1), then it has an end in +the segment J ⊂ I, and we are done. +Assume now that every leaf in {hn +0(Lu +0), hn +1 , n ∈ N(Lu +0)} crosses both Ls ++(p0) and Ls ++(p1). These images +are leaves of Fu, and therefore they are either disjoint or equal. For L ∈ {hn +0(Lu +0), hn +1, n ∈ N(Lu +0)}, let +D(L) ⊂ D2 +F be the disk obtained as follows: one cuts along L the strip bounded by Ls ++(p0) and Ls ++(p1), +one gets two components; one considers the closure in D2 +F of these components; now D(L) is the one +containing the segment J ⊂ S1 +F. +The disks D(L) are naturally totally ordered by the inclusion and we fix the indexation {hn +0(Lu +0), hn +1(Lu +0), n ∈ +N} = {Lu +n} according to this order: D(Lu +n+1 ⊂ D(Lu +n). +Consider D = � +n(D(Lu +n). It is a compact subset of D2 +F whose intersection with S1 +F is the segment J. +Claim 13. D ∩ (Ls ++(p0) ∪ Ls ++(p1)) = ∅ +Proof. The leaves hn +0(Lu +0) have their intersection with Ls(p0) tending to x as n → ∞: one deduces +that D ∩ Ls(p0) = ∅. The leaves hn +1(Lu +1) have their intersection with Ls(p1) tending to y, and thus +D ∩ Ls(p1) = ∅. +□ +Claim 14. D \ S1 +F ̸= ∅. +Proof. there is a point z in the interior of J which is the limit of an end of leaf of Fu. Thus there is an +half unstable leaf Lu ++ contained in the strip bounded by Ls ++(p0) and Ls ++(p1), and whose limit is z. Now +Lu ++ is disjoint from all the Lu +n, and therefore +Lu ++ ⊂ D(Lu +n), ∀n +This concludes the proof of the claim. +□ +Consider now a point t ∈ D \ S1 +F. The leaf Lu(t) is disjoint from the leaves Lu +n for any n. Thus it has +an empty intersection with (Ls ++(p0) ∪ Ls ++(p1)). As the consequence one gets +Lu(t) ⊂ D +In particular, Lu(t) has both ends on J. +Suppose now that the point t ∈ D \ S1 +X as been chosen on the boundary of D. Thus t is a limit of +points in Lu +n ⊂ A. As A is a closed subset of R2 = ˚ +D2 +F one deduces that t ∈ A, and so Lu(t) ⊂ A. +One just found a leaf Lu(t) contained in A and having both ends in J ⊂ I. Let Dt ⊂ D be the disc +bounded by Lu(t).We are not yet done, because Lu +t may fail to be a regular leaf. +Now Proposition 9.1 implies that every unstable leaf has an image by an element k ∈ π1(M) which +is contained in Dt, for instance Lu +0. Now k(Lu +0) is a regular unstable leaf in an attractor which has the +limits of its both ends contained in J ⊂ I, ending the proof. +□ +We are now ready for ending the proof of Proposition 9.2, and therefore of Theorem 14 which ends +the proof of Theorem 13 and Theorem 5. +Proof of Proposition 9.1. Let I ⊂ S1 +X be a non-empty open interval. According to Lemma 9.3 there is +a regular unstable leaf Lu +0, contained in an attractor A and containing a periodic point of non-boudary +type p0, and having an end, say Lu +0,+, whose limit is a point x ∈ I. +As Lu +0 is not a boundary leaf of A there are unstable leaves in A arbitrarily close to Lu +0, on both sides +of Lu +0. As furthermore Lu +0 is a regular leaf, one can chose a leaf Lu +1 ⊂ A so that +• the limit of the end Lu +1,+ is a point y ∈ I with [x, y] ⊂ I. +• there is a segment σ of a stable leaf having both ends a and b on Lu +0 and Lu +1 repsectively. + +36 +CHRISTIAN BONATTI +We denote by Dσ the disc in D2 +X bounded by σ, [x, y] , Lu ++(a) ⊂ Lu +0 and Lu ++(b) ⊂ Lu +1. +Now according to Lemma 9.2 there is a point z ∈ [x, y] which is limit of the end Lu ++ of a unstable +leaf Lu which carries a periodic point q in a repeller R, and q is not of boundary type. We denote by +h ∈ π1(M) the generator of the stabilizer of q which is expanding along Lu. +The stable leaf Ls(q) is contained in the repeller R and is accumulated on both sides by stable leaves +in R. We denote by Ls +0 a stable leaf in R crossing Lu ++ at a point x0. +We consider Ls +n = hn(Ls +0). It is a stable leaf in R which cuts Lu ++ at the point xn = hn(x0). +Not that xn → z as n → +∞. In particular, xn belongs to the disc Dσ for n large. +As R ∩ A = ∅ the leaves Ls +n are disjoint from Lu +0 and Lu +1. As two distinct stable leaves are disjoint +they are (all but at most one of them) disjoint from σ. +So for large n, the leaf Ls +n is contained in Dσ and therefore as its both ends on [x, y] ⊂ I. +We just exhibit a stable leaf in a repeller, whose both ends are in I, that is we ended the proof of +Proposition 9.2. +□ +References +[Ba1] Barbot, Thierry. Caract´erisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles, +Ergodic Theory Dynam. Systems 15 (1995) 247-270. +[BFM] Barthelm´e, Thomas; Frankel, Steven; Mann, Kathryn Orbit equivalences of pseudo-Anosov flows +preprint arXiv:2211.10505 +[BeBo] B´eguin, Fran¸cois, Bonatti Christian. Flots de Smale en dimension 3: pr´esentations finies de +voisinages invariants d’ensembles selles. (French) [Smale flows in dimension 3: finite presentations +of invariant neighborhoods of saddle sets] Topology 41 (2002), no. 1, 119-162. +[Br] +Brunella, Marco. Separating the basic sets of a nontransitive Anosov flow. Bull. London Math. +Soc. 25 (1993), no. 5, 487-490. +[CaDu] Calegari, Danny; Dunfield, Nathan M. Laminations and groups of homeomorphisms of the circle. +Invent. Math. 152 (2003), no. 1, 149–204. +[Fe1] +Fenley, Sergio. Anosov flows in 3-manifolds, Ann. of Math. (2) 139 (1) (1994) 79-115. +[Fe2] +Fenley, Sergio. The structure of branching in Anosov flows of 3-manifolds, Comment. Math. Helv. +73 (2) (1998) 259-297. +[Fe3] +Fenley, Sergio. One sided branching in Anosov foliations, Comment. Math. Helv. 70 (2) (1995) +248-266. +[Fe4] +Fenley, Sergio. Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with con- +nections and applications to large scale geometry, Geom. Topol. 16 (2012), no. 1, 1–110. +[Ke] +Alexander S. Kechris, Classical Descriptive Set Theory, Berlin, New York, Springer-Verlag, 1995, +p. 150. +[Ma] +Mather, J. Foliations of surfaces I : an ideal boundary Annales inst. Fourier 32 (1982), page 235-261 +[Th] +Thurston, William P. Three-manifolds, Foliations and Circles, II. Unfinished manuscript, 1998 +Christian Bonatti bonatti@u-bourgogne.fr +Institut de Math´ematiques de Bourgogne1, UMR 5584 du CNRS, Universit´e de Bourgogne, +21000, Dijon, France. +1The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002) + diff --git a/aNE3T4oBgHgl3EQfdAo0/content/tmp_files/load_file.txt b/aNE3T4oBgHgl3EQfdAo0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3a5b70fa56e2b36b2f072e25b8d2703a40617c2 --- /dev/null +++ b/aNE3T4oBgHgl3EQfdAo0/content/tmp_files/load_file.txt @@ -0,0 +1,1676 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf,len=1675 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='04530v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='DS] 11 Jan 2023 ACTION ON THE CIRCLE AT INFINITY OF FOLIATIONS OF R2 CHRISTIAN BONATTI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This paper provides a canonical compactification of the plane R2 by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather [Ma].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Moreover any homeo- morphism of R2 preserving the foliations extends on the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on R2 preserving one foliation or two transverse foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular the action on the circle at infinity associated to an Anosov flow X on a closed 3-manifold is minimal if and only if X is non-R-covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Keywords: Foliation of the plane, Anosov flow, compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Codes AMS: 37D20-37E10-37E35-37C86 January 12, 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' General presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There are many ways to compactify the plane R2, the simplest one being the Alexandrov compactification by point at infinity, and R2 ∪ {∞} is the topological sphere S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This compactication is canonical and does not depend on any extra structure on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' That is its strength, but also its weakness as it does not bring any informations on any structure we endow R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Another very natural and usual compactification of R2 is by adding a circle at infinity, so that R2 ∪ S1 is the disc D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This compactification is not canonical: it consists in a homeomorphism h: R2 → ˚ D2, where ˚ D2 is the open disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Two homeomorphisms h1, h2 define the same compactification if h2 ◦ h−1 1 : ˚ D2 → ˚ D2 extends on S1 = ∂D2 as a homeomorphism of D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There are uncountably many such a compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Here, we start be recalling Mather [Ma] canonical compactification of the plane R2, endowed with a foliation F, by a circle at infinity S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then we explore the flexibility of this contruction for extending it to more general objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus, we provide an elementary (nothing sophisticated), simple (nothing too complicated), and unified construction which associates a compactification D2 F of the plane R2 by the disc D2 to a countable family F = {Fi} of foliations, non-singular or with singular points of saddle type, which are pairwise transverse or at least have some kind of weak transversality condition at infinity, see the precise statements below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The boundary ∂D2 F is called the circle at infinity of F and is denoted by S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This compactification is unique, in the sense that the identity on R2 extends as a homeomorphism on the circles at infinity of two such compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For giving a concrete example, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 builds this canonical compactification D2 F associated to any countable family F = {Fi} of singular foliations, where each Fi is directed by a polynomial vector field on R2 whose singular points are hyperbolic saddles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The uniqueness of the compactification implies that any homeomorphism of R2 preserving F (that is, permuting the Fi) extends as an homeomorphism of the compactification D2 F , inducing a homeomorphism of the circle at infinity S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 1 2 CHRISTIAN BONATTI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Mather idea for building the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The common setting for this unified construc- tion are families of rays, where a ray is a proper topological embedding of [0, +∞) on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We require that the germs of the rays in the family are pairwize disjoint, meaning that the intersection between any two distinct rays is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The key idea is that a set of rays in R2 whose germs are pairwize disjoint is totally cyclically ordered, and we will use this cyclic order for building the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The key technical result (essentially due to [Ma]) is: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let R be a family of rays in R2 whose germs are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let E ⊂ R be a countable subset which is separating for the cyclic order, that is, any non-degenerate interval contains a point in E (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a compactification of R2 by the disc D2 so that: any ray of R tends to a point of the circle at infinity ∂D2 = S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' any two distinct rays of R tend to distinct points of S1 the points of S1 which are the limit point of a ray in R are dense in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore, this compactification is unique up to a homeomorphism of D2 and does not depend on the separating countable set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then Theorem 6 provides such a canonical compactification for a countable union R = � Ri, i ∈ I ⊂ N of families of rays, assuming that the germs of rays in R are pairwise disjoint and each Ri admits a countable separating subset Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The difficulty here is that R by itself may not admit any separating family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The idea for solving this problem consists in considering a natural equivalence relation on R, identifying the rays which cannot be separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Countable families of transverse foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A natural setting where we will apply this general construction are (at most countable) families of transverse foliations on the plane R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that any half leaf of a (non-singular) foliation of R2 is a ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' An end of leaf is the germ at infinity of an half leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this setting we get: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F = {Fi}i∈I⊂N be an at most countable family of pairwise transverse foliations on the plane R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There is a compactification D2 F ≃ D2 of R2 by adding a circle S1 F = ∂D2 F with the following properties: Any end of leaf tends to a point of the circle at infinity S1 F, The set of ends of leaves tending to a same points of S1 F is at most countable, For any non-empty open subset O ⊂ S1 F the set of ends of leaves having their limit in O is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This compactification with these three properties is unique, up to a homeomorphism of D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The circle S1 F is called the circle at infinity of the family F = {Fi}i∈I⊂N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The countablity of the set of ends tending to the same point implies that the two ends of a given leaf always have distinct limits on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' if two leaves L1, L2 of the same foliation Fi have the same pair of limits of ends, they are equal (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Recall that foliations of R2 may have leaves which are not separated one from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The leaves which are separated from any other leaves are called regular leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' At most countably many leaves are not regular (see here Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will see that, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F = {Fi}i∈I⊂N be an at most countable family of pairwise transverse foliations on the plane R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Any two distinct ends of regular leaves of the same foliation Fi tend to two distinct points of S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now, in the setting of Theorem 2 we can apply this theorem to each foliation Fi, i ∈ I so that we get a family of compactifications D2 Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In fact, we get a compactification D2 J for any subfamily J ⊂ I leading to an uncountable set of (maybe distinct) compactifications of R2 by the disc D2 (Example 5 provides CIRCLE AT INFINITY OF FOLIATIONS OF R2 3 a simple example where these compactifications D2 J, for J ⊂ I, are pairwize distincts and uncountably many).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' These compactifications are easily related : for any subfamily J ⊂ I the identity map on R2 extends in a unique way by continuity as a projection ΠI,J : D2 F = D2 I → D2 J, which simply consists in colapsing the intervals in S1 I which do not contain any limit of an end of a leaf of a foliation Fj, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will also see in a simple example that the assumption of at most countability of the family I of foliations cannot be erased: for instance, the conclusion Theorem 2 is false for the family of all afine foliations (by parallel straight lines) of R2, parametrized by RP1 (see Example 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 8 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='8 present a simple example where generic points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' points in a residual set) of the circle at infinity S1 F of a foliation F are not the limit of any end of leaf of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this example, at the contrary, points in a dense subset of S1 F are limit of 2 distinct ends of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='5 caracterize the points p at the circle at infinity S1 F, where F is a foliation of R2, which are limit of several ends of leaves: the rays arriving at p are ordered as an interval of Z and two successive ends bound a hyperbolic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3 generalizes Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='5 to the case of a countable family F = {Fi} of transverse foliations and gives a complete description of the points in S1 F which are limit of several ends of leaves of the same Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Countable families of non-transverse or singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This construction can be gen- eralized easily to the setting of families of non transverse or singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let us present the most general setting we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The foliations we consider admit singular points which are saddle point with k-separatrices (also called k-prongs singularity), k > 1, the case k = 2 corresponding to non-singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this setting an end of leaf is a ray of R2 disjoint from the singular points and contained in a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F = {Fi}, i ∈ I ⊂ N be a family of singular foliations of R2 whose singular points are each a saddle with k-separatrices with k > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We assume that, given any two ends L1, L2 of leaves we have the following alternative: either the germs of L1 and L2 are disjoints or the germs of L1 and L2 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a compactification D2 F ≃ D2 of R2 by adding a circle S1 F = ∂D2 F with the following properties: Any end of leaf tends to a point of the circle at infinity S1 F, The set of ends of leaves tending to a same points of S1 F is at most countable, For any non-empty open subset O ⊂ S1 F the set of ends of leaves having their limit in O is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This compactification with these three properties is unique, up to a homeomorphism of D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The hypothesis that the germs of ends of leaves are either equal or disjoint means that if the intersection of two leaves is not bounded, then these two leaves coincide on an half leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One easily checks that transverse foliations satisfy this hypothesis so that Theorem 2 is a straightforward corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As a simple and natural example, we will see that any countable family F = {Fi} of singular foliations, directed by polynomial vector fields on R2 whose singular points are hyperbolic saddles, satisfies the hypotheses of Theorem 3: this will prove Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 already mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Laminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The construction of the circle at infinity for foliations cannot be extended without hypotheses to the case of laminations, as leaves of laminations may fail to be lines, and can even be recurrent, see for instance example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Theorems 8 and 10 provide a generalisation of this construction to closed orientable laminations with no compact leaves and with uncountably many leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This generalisation is not as satifactory as in the case of foliations, and we discuss some of the issues in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular Theorem 9 provides another canonical compactification, which holds also for countable oriented laminations with no compact leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 4 CHRISTIAN BONATTI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Minimality of the action on the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then we consider group actions H ⊂ Homeo(R2) on R2 preserving 1 or 2 transverse foliations Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The action of H extends canonically on the circle at infinity and we will consider the following question: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Under what conditions on H and on the foliations Fi can we ensure that the action induced on S1 {Fi} is minimal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Our main result, for the case of 1 foliation is the following: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation of R2 and H ⊂ Homeo(R2) be a group of homeomorphisms preserving F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We assume that for any leaf L, the union of its images H(L) is dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the two following properties are equivalent (1) the action induced by H on the circle at infinity is minimal (2) there are pairs of distinct leaves (L1, L2) and (L3, L4) so that L1 and L2 are not separated from above and L3 and L4 are not separated from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will also generalize Theorem 4 for families of transverse foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Action on the circle at infinity of an Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Finally, we will consider the setting of an Anosov flow X on a closed 3-manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this setting it is known that π1(M) acts on S1 by orientation preserving homeomorphisms, see Calegari Dunfield [CaDu] inspirated by an unpublished work of Thurston [Th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This works follows completely distinct ideas that those presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Another construction of this circle at infinity (called ideal circle boundary) is given in [Fe4] for pseudo- Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Barbot and Fenley [Ba1, Fe1] show that the lift ˜X of X is conjugated to the constant vector field ∂ ∂x on R3, so that the ˜X-orbit space is a plane PX ≃ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This plane PX is endowed with two transverse foliations F s, F u which are the projection of the stable and unstable foliations of X lifted on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus (PX, F s, F u) is the bifoliated plane associated to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore, the fundamental group π1(M) acts on PX and its action preserves both foliations F s and F u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This action induces a natural action of π1(M) on the circles at infinity S1 F s, S1 F u, and S1 F s,F u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A folklore conjecture asserts that two Anosov flows are orbitaly equivalent if and only if they induces the same action on the circle at infinity of {F s, F u}, see [Ba1] for a result in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This conjecture as been recently announced to be proved in [BFM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' [Ba1, Fe1] show that every leaf of F s is regular if and only if every leaf of F u is regular, and then the Anosov flow X is called R-covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Our main result in that setting is Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be an Anosov flow on a closed 3-manifold and (PX, F s, F u) its bifoliated plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let D2 F s,F u, D2 F s, and D2 F u be the compactifications associated to, respectively, the pair of foliations F s, F u, the foliation F s and the foliation F u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then (1) D2 F s,F u = D2 F s = D2 F u unless X is orbitally equivalent to the suspension of an Anosov diffeomor- phism of the torus T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (2) the action of π1(M) on the circles at infinity S1 F s,F u,(or equvalently S1 F s or S1 F u) is minimal if and only if X is not R-covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' When X is assumed to be transitive, this result is a simple consequence of Theorem 4 above and a result by Fenley [Fe3] ensuring that, assuming X is non-R-covered, then F s and F u admit non-separated leaves from above and non-separated leaves form below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The proof of Theorem 5, when X is not assumed to be transitive, is certainly the most technically difficult argument of the paper, and is based on a description of hyperbolic basic sets for flows on 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Theorem 5 implies that the minimality of the action on the circle at infinity is not related with the transitivity of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' However, according to [BFM] the action on the circle at infinity charaterizes the dynamics of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This leads to the following question: CIRCLE AT INFINITY OF FOLIATIONS OF R2 5 Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' What property of the action of π1(M) on the circle at infinity S1 F s,F u implies the transi- tivity of X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Can we find the transverse tori by looking at the action of π1(M) on the circle at infinity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Aknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' I would thank Sebastien Alvarez who invited me to present the results in this paper as a mini-course in Montevideo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This mini-course has been a motivation for ending this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' I would also thanks Kathrin Mann for indicating me that the argument of Theorem 1 is essentially contained in [Ma], and Michele Triestino for the statement and reference of Cantor-Bendixson theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Circles at infinity for families of rays on the plane 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A total cyclic order on X is a map θ: X3 → {−1, 0, +1} with the following properties θ(x, y, z) = 0 if and only if x = y or y = z or x = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' θ(x, y, z) = −θ(y, x, z) = −θ(x, z, y) for every (x, y, z) for every x ∈ X the relation on X \\ {x} defined by y < z ⇔ θ(x, y, z) = +1 is a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The emblematic example is: Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The oriented circle S1 = R/Z is totally cyclically ordered by the relation θ defined as follows: θ(x, y, z) = +1 if and only if the y belongs to the interior of the positively oriented simple arc staring at x and ending at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If θ is a total cyclic order then for x ̸= z we define the interval (x, y) by (x, z) = {y, θ(x, y, z) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We define the semi closed and closed intervals [x, z),(x, z], and [x, z] by adding the corresponding extremities x or z to the interval (x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We say that y is between x and z is y ∈ (x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The following notion of separating set will be fundamental all along this work: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be a set endowed with a total cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A subset E ⊂ X is said separating if given any distinct x, z ∈ X there is y ∈ E (distinct from x and z), between x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will use the following easy exercize of topology of R and S1: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be a set endowed with a total cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that there is a countable subset E ⊂ X which is separating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a bijection ϕ of X on a dense subset Y ⊂ S1 which is strictly increasing for the cyclic orders of X and of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore this bijection is unique up to a composition by a homeomorphism of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The argument is classical but short and beautiful and I have no references for this precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' So let me present it: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One builds a bijection φ of E to a contable dense subset D ⊂ S1 by induction, as follows: one choose an indexation of E = {ei, i ∈ N} and of D = {di, i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One defines φ(e0) = d0, φ(e1) = d1 i(0) = j(0) = 0 i(1) = j(1) = 1 consider e2, it belongs either in (e0, e1) or in (e1, e0) and we chose φ(e2) being dj(2) where j(2) is the infimum of the di in the corresponding interval (d0, d1) or (d1, d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One denotes i(2)=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' consider now j(3) = inf N \\ {0, 1, i(2)} and define φ−1(dj(3)) = ei(3) where i(3) is the infimum of the i /∈ {0, 1, 2} so that the position of ei(3) with respect to e0, e1, e2 is the same as the position of dj(3) with repsect to d0, d1, dj(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 6 CHRISTIAN BONATTI choose i(2n) = inf N\\ {i(k), k < 2n} and φ(ei(2n) is dj(2n) where j(2n) is the infimum of the j so that dj as the same position with respect to the dj(k), k < 2n as ei(2n) with respect to the ei(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' choose j(2n + 1) = inf N \\ {j(k), k < 2n + 1} and φ−1(dj(2n+1) is ei(2n+1) where i(2n + 1) is the infimum of the i so that ei as the same position with respect to the ei(k), k < 2n + 1 as dj(2n+1) with respect to the dj(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' At each step of this construction one uses the separation property of E and D for ensuring the existence of the point announced in the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Once we built φ on E, it extends in a unique increasing way on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the separation property of E implies that this extension is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that Z, θ is a set endowed with a total cyclic order, and E ⊂ X ⊂ Z are subsets so that E is separating for X, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let ϕ: X → Y be the map given by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then φ extends in a unique way as an (not strictly) increasing map Φ: Z → S1: Φ(y) is between Φ(x) and Φ(z) only if y is between x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The non-injectivity of the map Φ is determined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider distinct points x ̸= y of Z, then Φ(x) = Φ(y) if and only if either (x, y) or (y, x) contains no more than 1 element of X 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Cyclic order on families of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A line is a proper embedding of R in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A line L cuts R2 in two half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If L is oriented, then there is an orientation preserving homeomorphism h of R2 mapping L on the oriented x-axis of R2 (endowed with the coordinates (x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This allows us to defined the upper and lower half-planes ∆+ L and ∆− L as the pre-images by h of {y ≥ 0} and {y ≤ 0} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A ray is a proper embedding of [0, +∞) in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Two rays define the same germ of ray if their images coincide out of a compact ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Two germs of rays are said disjoint if they admit disjoint realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (1) If F is a foliation of R2, every leaf defines to germs of rays called the ends of the leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By fixing an orientation of F we will speak of the right and left ends of a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (2) If {Fi}i∈I is a family of pairwise transverse foliations of R2 then the set of all ends of leafs of the foliations Fi is a family of pairwise disjoint germs of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (3) Consider the set S of all germs of rays γ which are contained in an orbit of an affine (polynomial of degree = 1) vector field of saddle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then S is a family of pairwise disjoint germs of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Next lemmas are simple exercizes of plane topology: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let γ0, γ1, γ2 be three disjoint rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that C1 and C2 are simple closed curves on the plane R2 so that γi ∩Cj is a unique point pi,j, i ∈ {0, 1, 2}, j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We endow Ci with the boundary-orientation corresponding to the compact disc bounded by Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the cyclic order of the 3 points p0,1, p1,1, p2,1 for the orientation of C1 is the same as the cyclic order of the 3 points p0,2, p1,2, p2,2 for the orientation of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We call it the cyclic order on the rays γ0, γ1, γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The cyclic order on three disjoint germs of rays R0, R1, R2 does not depend on the choice of disjoint rays γ0, γ1, γ2 realizing the germs R0, R1, R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let γ0, γ1, γ2 be three disjoint rays and C be any simple close curve, oriented as the boundary of the compact disc bounded by C, and having a non-empty intersection with every γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let pi be the last point of γi in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the cyclic order of the γi coincides with the cyclic order of the pi for the orientation of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let R0, R1, R2 be three disjoint germs of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L be an oriented line whose right end is R0 and whose left end is R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then R1 is between R0 and R2 for the cyclic order defined above (we denote R1 ∈ (R0, R2)) if and only if it admits a realization contained in the upper half-plane ∆+ L bounded by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Next proposition summerizes what we have got with this sequence of easy lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider R a family of pairwise disjoint germs of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then R is totally cyclically ordered by the following relation: CIRCLE AT INFINITY OF FOLIATIONS OF R2 7 given three distinct germs of rays R0, R1, R2 ∈ R, the germ R2 is between R1 and R3 if it admits a realisation contained in the upper-half plane ∆+ L where L an oriented line whose right end is R0 and and whose left end is R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Compactification of a family of rays by a circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this paper a compactification of the plane R2 by the disc D2 is by definition a homeomorphism between R2 and the open disc ˚D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The aim of this section is the proof of Theorem 1 which build a canonical compactification of R2 associated to a family R of rays, assuming it admits a countable separating (for the cyclic order) subset E ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One of the main ingredients for the proof of Theorem 1 is the following lemma which is an easy exercize of plane topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let γ0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , γn be n disjoint rays, n > 0, and K ⊂ R2 be a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a simple closed curve C disjoint from K bounding a compact disc D containing K in its interior and so that C ∩ γi consists in a unique point pi, i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Just notice that there is a homeomorphism of R2 mapping γi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , n} on radial (half-straight lines) rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the proof is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ sketch of proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider the set of rays endowed with the cyclic order and we embedd it in the circle S1 by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by E ⊂ S1 the dense countable subset corresponding to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We define a topology on R2 � S1 by choosing a basis of neighborhood of the points in S1 as the halph planes bounded by lines L whose both ends are rays R−, R+ in E (each half plane correspond to a segment in S1 \\ {R−, R+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This topology does not depend of the choice of the countable separating subset E: if ˜E is another countable separating subset, each neighborhood of a point of S1 obtain by using one family contains a neighborhood obtained by using the other family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now one builds a map from R2 on the interior of D2 as follow: (1) one considers the circles Cn, n ≥ 1, of radius ρn = 1 − 1 n+1 (that is , Cn = ρn · S1) endowed with the finite set of point ρn · x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , ρn · xn, where E = {xn, n ≥ 1} is a choice of indexation of the countable set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (2) one choses by induction a realisation Rn of the rays in E and a family of simple closed loops γn with the following properties: γn is the boundary of a compact disc Dn containing Dn−1 in its interior and containing the disk of radius n of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular, � n Dn = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' γn cuts the rays Rm, m < n in a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' one choses a representative of Rn , disjoint from Rm, m < n, with origin on γn and with no other intersection point with γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then, by definition of the cyclic order on the rays, the points γn ∩ Ri, i ≤ n, are cyclically ordered on γn as the points ρn · x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , ρn · xn on Cn (3) this allows us to choose a homeomorphism of R2 to the interior of D2 sending the loops γn on the circles Cn and the rays Rn on the segments [ρn, 1) · xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This homeomorphisms extends on the circle at infinity S1 to ∂D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Union of countably many families of rays: the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let {Xi, i ∈ I}, I ⊂ N be a finite or countable family of sets so that � i Xi is endowed with a total cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that, for every i, there exist Ei ⊂ Xi countable separating subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' On the union X = � i Xi we consider the relation x ∼ y ⇔ ([x, y] ∩ Ei is finite for every i, or [y, x] ∩ Ei is finite for every i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In other words, x ∼ y if one of the two segments (for the cyclic order) bounded by x and y meets each family Ei in at most finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then ∼ is an equivalence relation and every class contains at most 1 point in each Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 8 CHRISTIAN BONATTI Let denote π: X → X = � i Xi/ ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by E the projection π(E) of E = � Ei on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then ∼ provides a complete cyclic order on X and E is a countable separating subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The fact that ∼ is an equivalence relation is quite easy, as the union of two intervals meeting Xi on finite sets meets Xi on a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that, assuming x ∼ y, the interval [x, y] or [y, x] (meeting every Ei in finitely many points) is contained in the class of x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus the class [x]∼ is a (proper) interval for the cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider x, y ∈ X and assume that [x, y] ∩ Ej is finite for every j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that there is i and distinct z, t ∈ [x, y]∩Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the separating property of Ei for Xi ensures that [x, y]∩Ei is infinite contradicting the choice of the interval [x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We deduces that every class meets every Xi in at most 1 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that this implies that the projection of Ei on X is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider x, y, z ∈ X whose classes are distinct, and assume z ∈ (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider now a ∼ x, b ∼ y and c ∼ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Ia, Ib, Ic be the intervals [x, a] or [a, x], [y, b] or [b, y], [z, c] or [c, z] with finite intersections with the Ei, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then these intervals are disjoints as there a contained in disjoint equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus the cyclic order for point in Ia, Ib, Ic does not depend on the point in Ia, Ib, Ic and thus c ∈ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This shows that the quotient X is endowed with a total cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider now two distinct classes [x]∼, [y]∼ ∈ X of points x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus there is i so that [x, y] ∩ Xi is infinite Now the separating property of Ei implies that [x, y] ∩ Ei is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As π is injective on Ei one gets that ([x]∼, [y]∼) ∩ π(Ei) is infinite and thus ([x]∼, [y]∼) ∩ E is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One proved that E is separating for X, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Union of countably many families of rays: the compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let R = � i∈I Ri, I ⊂ N, be a family of rays in R2 whose germs are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that for every i ∈ I there is a countable subset Ei ⊂ Ri which is separating for Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then, there is a compactification of R2 by the disc D2 so that: any ray of R tends to a point of the circle at infinity ∂D2 = S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' for every i, any two distinct rays of Ri tend to distinct points of S1 for any non-empty open interval J ⊂ S1 there is i ∈ I so that at least 2 rays in Ri have there limit point in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore, this compactification is unique up to a homeomorphism of D2 and does not depend on the separating countable sets Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let us discuss item 3, whose formulation may be surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (1) The third item implies that the points of S1 which are the limit point of a ray in � Ei are dense in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (2) if I is finite, item 3 is equivalent to the density of points in S1 which are limit of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (3) item 3 is necessary when there is an uncountable set of equivalence classe [c], for the equivalence relation ∼ (defined at Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4), which are infinite (necessarily countable) and contain a set C([c]) which is separating for the cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In that case the uniqueness property announced in Theorem 6 would be wrong if we replace item 3 by the density of points in S1 which are limit of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 3 provides a simple illustration of this trouble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let us denote X = R, Xi = Ri, and E = � Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let ∼ be the equivalence relation defined in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3 on X = R and let π benote the projection π: X → X = X/ ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We choose a subset Y ⊂ X with the following properties each equivalence class [γ] for ∼ contains exactly 1 point yγ in Y if a class for ∼ contains a point in E, then yγ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The existence of such subset Y is certainly implied by the choice axiom but this existence does not require this axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For instance we can fix CIRCLE AT INFINITY OF FOLIATIONS OF R2 9 if [γ]∼ ∈ π(� Ei) then yγ is the unique point in Ei in the [γ]∼ where i is the smallest index for which [γ]∼ ∈ π(Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' if [γ]∼ /∈ π(E) then yγ is the unique point in Xi ∩ [γ]∼ where i is the smallest index for which [γ]∼ ∩ Xi) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote F = Y ∩ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that π(F) = π(E) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now the projection π: Y → X is a bijection which is strictly increasing for the cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As E = π(E) is separating for X (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3) one gets that F is separating for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We can now apply Theorem 1 to the set of rays Y and the countable separating subset F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One gets a compactification of R2 as a disc D2 so that every ray in Y tends to a point at the circle at infinity, two distinct rays in Y tends to two distinct points, and the set of points at infinity limit of rays in F is dense in the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let us check now that every ray γ ∈ R tends to a point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By construction of Y there is σ ∈ R so that σ ∼ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will prove that γ tends to the limit point s ∈ S1 of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For that we recall that a basis of neighborhood of s is given by the half planes ∆n bounded by lines Ln whose both ends are σ− n , σ+ n ∈ Y so that σ ∈ (σ− n , σ+ n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that both intervals (σ, σ+ n ) and (σ− n , σ) are infinite when one of the intervals (σ, γ) or (γ, σ) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One deduces that γ ∈ (σ− n , σ+ n ) and thus the end of γ is contained in ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus γ tends to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now consider two distinct rays γ, γ′ ∈ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As every class for ∼ contains at most 1 point of Ri the classes of γ and γ′ are distincts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus there are σ ̸= σ′ ∈ Y which are equivalent to γ and γ′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The limit points of γ and γ′ are those of σ and σ′ respectively, which are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We just checked that distincts rays in Ri tends to distinct point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let ϕ: R2 → D2 be a compactification satisfying the announced properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then 2 rays in R tends to the same point of the circle at infinity of the compactification if and only if they are equivalent for ∼ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If two rays a, b are not equivalent then each of (a, b) and (b, a) contains infinitely many rays in the same of the sets Ri, by definition of ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As the limits of distinct rays in the same Ri are different, one deduces that the limits of a and b are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Conversely, if a and b have different limit points r and s in S1 for the compactification then item 3 implies that there is i ∈ I (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' j ∈ I so that , (a, b) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (b, a)) contains the ends of at least 2 rays in Ri (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As Ri and Rj admits separating subsets, this implies that both (a, b)∩Ri and (b, a)∩Rj are infinite, so that a ≁ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Consider now a non-empty open interval J of the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We announced that there is i for which J contains at least 2 limits of rays in Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Recall that, according to Theorem 1 the points in J which are limit of rays in Y are dense in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus there are at least 2 points in J which are limit of rays R1, R2 in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This implies that, up to exchange R1 and R2 any ray in R between R1 and R2 tend to a point in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now, the rays R1, R2 are not equivalent for ∼ according to Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By definition of ∼, there is i so that there are infinitely rays between R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This proves Item 3 of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume now that one has another compactification ψ: R2 → D2 satisfying also the announced proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One deduces from Claim 1 the fact that the images by ψ of two distinct rays in the set Y (that we used for building the first compactification ϕ) have two distinct limit points and that the limit points of the image by ψ of rays in Y are dense in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus this new compactification satisfies the same property on the set of rays Y as the one we built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Theorem 1 asserts that these compactifications differs from ϕ by a homeomorphism of D2, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that R satisfies the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3, and let ˜R be a set of rays to that the germs of rays in R ∪ ˜R are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let R2 ֒→ D2 be a compactification given by Theorem 6 applied to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then any ray ˜γ in ˜R tends to a point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The candidate for the limit the intersection of all the closed intervals in S1, bounded by limit of rays a, b ∈ R, so that ˜γ ∈ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The basis of neighborhood of this point that we exhibit implies that indeed ˜γ tends to that point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 10 CHRISTIAN BONATTI □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' An example with uncountably many compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The example below shows that, in the case of a infinite countable family R = {Ri}, i ∈ N, the compactification announced by Theorem 6 would not be unique if we replace the item 3 of the conclusion by the density in S1 R of the limits of the rays in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In the example below, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let B ⊂ R2 be the open strip {(x, y) ∈ R2, |x − y| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let I be the set of linear lines with a rational inclination ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For any i ∈ I, let Fi be the restriction to B of the trivial foliation by parallel straight lines directed by i ∈ RP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For any i, let Ri be the set of ends of leaf of Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Each Ri admits a countable separating subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus R = � i∈I Ri satisfies the hypotheses of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there are uncountably many distinct compactifications of R2 for which any ray of R tends to a point of the circle at infinity ∂D2 = S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' for every i, any two distinct rays of Ri tend to distinct points of S1 the points of S1 which are the limit point of a ray in � Ei are dense in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let D2 R be the compactification of B ≃ R2 by adding the circle at infinity S1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Every class C for ∼ contains exactly 1 ray in Ri for any i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The rays in C are ordered,for the cyclic order, as the points of I in RP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' So, C is a separating set for itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By construction of S1 R, the class C corresponds to a point c ∈ S1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We can build another circle S1 R,C by opening the point c in a segment IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then, we can build a compactification D2 R,C so that the rays in C tend to distinct points dense in IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular IC contains exactly 1 limit point of a ray in Ri, for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We can repeat this argument opening not just a point in S1 R but a countable subset C of classes for ∼: we build a compactification D2 R,C where the circle at infinity contains disjoint intervals IC, C ∈ C, so that each IC contains exactly 1 limit point of a ray in Ri for any i, ans these points are dense in IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As there are uncountably many such countable subsets C, this provides an uncountable family of pairwise distinct compactifications of B satisfying the 2 first items and the density of the limit points of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ This shows that the uniqueness part of Theorem 6 becomes wrong if we replace item 3 by the density in S1 of the set of limits of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Uncountable families of families of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Theorem 6 is wrong for the union of an uncountable family of sets of rays, as shows the Example 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider R2 endowed with all constant foliations Fθ, θ ∈ RP1, where Fθ is the foliation whose leaves are the straight lines parallels to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then given any compactification of R2 by D2 for which every end of leaf tends to a point at infinity, then for all but a countable set of θ the right ends of the leaves of Fθ tends to the same point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The ends of leaves F + θ at the right, and those at the left F − θ of the foliation Fθ are disjoint interval depending on the uncountable parameter theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' On the circle at most countably many disjoint intervals can be non trivial, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Projection on the compactifications associated to each families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let us start with a very easy example, showing the at the circles at infinity associated to the subsets of a countable family of transverse foliations may lead to uncountabily many distinct compactifications, all quotient of the compactification associated to the whole family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider now a infinite countable subset I ⊂ RP1 and consider the family RI of the leaves of the constant foliations Fθ, θ ∈ I on R2 as already considered in example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now the set of ends of leaves of each foliation Fθ corresponds to 2 (because each leaf has 2 ends) non-empty open intervals in S1 I, and these intervals do not contain any end of leaf of any other foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 11 Thus if J, K ⊂ I are distinct subsets, the circles at infinity S1 J and S1 K are obtained by collapsing distinct intervals of S1 I and they are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As the set P(I) of all subset of I in uncountable, this leads to an uncountable family of compatifications {D2 J}J∈P(I) of R2 by a circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This situation is quite general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let R = R1 � · · · � Rk be a family of rays in R2 whose germs are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , k} there is a countable subset Ei ⊂ Ri which is separating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus for every subset I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=', k}, Theorem 6 provides a compactification D2 I of R2, by the circle at infinity corresponding to the rays in Ri, i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If J ⊂ I then the identity map on R2 extend by continuity as a projection ΠI,J : D2 I → D2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This projection consists in collapsing intervals of S1 I = ∂D2 I which do not contain any limit points of ray is Rj, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore if K ⊂ J then ΠI,K = PJ,K ◦ PI,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We first define a projection πI,J : S1 I → S1 J by using Remark 3: the subset RJ ⊂ S1 I of limit of rays of RJ is is a strcitly increasing bijection with RJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus the increasing bijection of RJ on a dense subset of S1 J induces an increasing bijection of RJ in this dense subset of S1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Remark 3 asserts that this bijection extends on the whole S1 I in a not-stricly increasing map πI,J : S1 I → S1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' An increasing map with dense image is always continuous, so that πI,J is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Finaly Remark 3 asserts that the non-injectivity of πI,J consist in collapsing intervals of S1 I with at most 1 point in RJ, which is the same topological operation as collapsing intervals with no points in RJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For ending the proof, we will check that πI,J is the extension by continuity of the identity map of R2 to the circles at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Recall that we defined a basis of neighborhood of each point of the circle at infinity S1 I (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' S1 J ) as the half-planes ∆+ L bounded by lines whose both ends are in RI (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' RJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular, as J ⊂ I, the neighborhoods of points at infinity in DD2 J are still neighborhoods of points at infinity for D2 I, proving that the map which is the identity from R2 = ˚D2 I to R2 = ˚D2 J and is πI,J from S1 I to S1 J is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' backgroung on foliations: regular leaves, non-separated leaves 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Non-singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then (1) as R2 is simply connected, F is orientable and admits a transverse orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let us fix an orientation of F and a transverse orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (2) every leaf is a line (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' a proper embedding of R in R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (3) a basis of neighborhoods of a leaf L is obtained by considering the union of leaves through a transverse segment σ through a point of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' two leaves L1, L2 are not separated one from the other if they do not admit disjoint neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A leaf L is called not separated or not regular if there is a leaf L′ which is not separated from L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A leaf is called regular if it is separated from any other leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will need some times to be somewhat more specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L1 and L2 be distinct leaves of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider two segments σi : [−1, 1] transverse to F, positively oriented for the transverse orientation of F, and so that σi(0) ∈ Li, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then L1 is not separated from L2 means that there are sequences ti n, i = 1, 2 tending to 0 as n → +∞ so that σ1(t1 n) and σ2(t2 n) belongs to the same leaf Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then 12 CHRISTIAN BONATTI as F is transversely oriented and the σi are positively oriented, one gets that t1 n has the same sign as t2 n, for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Futhermore all the ti n have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One says that L1 and L2 are not separated from above (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' from below) if the ti n are positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By shirinking the segments σi if necessary one may assume that they are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now, up to exchange L1 with L2 we may assume that σ1(tn) is at the left of σ2(tn) in the oriented leaf Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We say that L1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' L2) is not separated from L2 at its right (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' at its left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a leaf L and σ: [−1, 1] → R2 a transverse segment (positively oriented) with σ(0) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Lt be the leaf through σ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Ut, t ∈ (0, 1), be the closure of the connected component of R2 \\ (Lt ∪ L−t) containing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The leaf L is regular if and only if � t Ut = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The intersection � t Ut does not depend on the segment σ and is denoted U(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If L is not regular, U(L) as non-empty interior, and the leaves which are not separated from L are precisely the leaves in the boundary of U(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A leaf ˜L not separated from L is contained in every Ut and is accumulated by leaves Ltn in the boundary of Utn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus ˜L is contained in the boundary of U(L) = � t Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore one of the two half planes bounded by ˜L is contained in Ut and therefore in U(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Conversely,� t Ut consist in entire leaves of F and so does its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now any transverse segment through a leaf in the boundary of � t Ut crosses the boundary Lt ∪ L−t of Ut for t small: that is the definition of being not sepatated from L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The set of not separated leaves is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider a countable family of transverse lines Σ whoses union cuts every leaf of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is enough to proof that such a tranverse line Σ cuts at most a countable set of non-regular leaves L admiting a non separated leaf ˜L from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For that just notice that the U(L) for L∩Σ ̸= ∅ are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus,there are at most countably many of them with non-empty interior, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Note that L cuts the strip Ut, t ∈ (0, 1] in two strips U + t and U − t bounded respectively by Lt ∪ L and by L−t ∪ L, and we denote U+(L) = � t U + t and U−(L) = � t U − t Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' L is non-separated from above (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' from below) if and only if U+(L) ̸= L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' U−(L) ̸= L) and if and only if U+(L) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' U−(L)) has non-emptyinterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In the same spirit, σ cuts the strip Ut in two half strips U left t and U right t according to the orientation of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then one says that the right end L+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' left endL−) of L is regular if Uright = (L) � t U right t = L+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Uleft(L) = � t U left t = L−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We can be even more precise by considering the 4 quadrants U +,right t , U +,left t , U −,right t , U −,left t obtained by considering the intersections of U + t and U − t with U right t and U left t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This allows us to speak on right or left ends of leaves non separated from above or from below, in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Singular foliations: saddles with k-separatrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A singular foliation F on R2 is a foliation on R2 \\ Sing(F) where Sing(F) is a closed subset of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A leaf of F is a leaf of the restriction of F to R2 \\ Sing(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let us now recall the notion of saddles with k-separatrices, also called k-prongs singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by A0 the quotient of [−1, 1]2 by the involution (x, y) �→ (−x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The projection of (0, 0) on A0 is still called 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that the horizontal foliation (whose leaves are the segments [−1, 1] × {t} is invariant by (x, y) �→ (−x, −y), and therefore passes to the quotient on A0 \\ (0, 0) and we denote by H1 the induced foliation on A0 \\ {(0, 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A 1-prong singular point p of F is a point of Sing(F) which admits a neighborhood U and a homeo- morphism h from U to A0 so that h(p) = (0, 0) and h maps F on H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by Ak, Hk the cyclic ramified cover of A0 at the point (0, 0) with k leaves, endowed with the lift of H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A k-prongs singular point p, equivalently a sadlle point with k separatrices of F is a singular point admiting a homeomorphism of a neighborhood onto Ak mapping p on (0, 0) and F on Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A separatrix of the saddle point p is the leaf of F containing a connected component of the lift of ]0, 1] × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If p is a 2-prongs singular point of F, then the foliation F can be extended on p so that p is not singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The Poincar´e-Hopf index of a k-prongs singular point is 1 − k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A foliation with singularities of saddle type on R2 is a singular foliation for which each singular point is a saddle with k separatrices, k > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Leaves of singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation on R2 with singular points of saddle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let σ: [0, 1] → R2 \\Sing(F) be a segment transverse to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then for every leaf γ one has #σ ∩ γ ≤ 1, where # denotes the cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume (arguing by contradiction) that σ ∩ γ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let x, y be two successive (for the parametri- sation of γ) intersection points with σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The concatenation of the segments [x, y]γ and [y, x]σ is a simple closed curve c in R2 \\Sing(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By Jordan theorem c bounds a disc D in R2 and the Poincar´e Hopf index of F on D is either equal to 1, if γ cuts σ with the same orientation at x and y, or 1 2 otherwise: anyway this index is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' However, this index is the sum of the Poincar´e Hopf index of the singular points of F contained in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As each of them is negative, that is a contradiction, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ The same argument shows that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation on R2 with singular points of saddle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then F has no compact leaves Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The index of F on the disc bounded by a compact leaf whould be 1 which is impossible with singular points with negative index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a singular foliation of R2 whose singular points are all saddle points with at least 3 separatrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then every half leaf of F is either a ray or tends to a singular point p of F and is contained in a separatrix of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the Alexandrov compactification of R2 by a point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a leaf γ and choose a parametrisation γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider lim sup t→+∞ γ(t) = � t>0 γ([t, +∞), where the closure is considered in R2 ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is a decreasing intersection of connected compact sets, and hence it is a non-empty connected compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 14 CHRISTIAN BONATTI If lim supt→+∞ γ(t) is not just a point, if contains a regular point x of F, hence it cuts infinitely many times any transverse segment through x, which is forbidden by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now lim supt→+∞ γ(t) is either the point ∞ or is a singular point of F, which is the announced alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Regular leaves of singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation with singular points of saddle type, L0 a leaf of F and σ be a transverse segment through the point σ(0) ∈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The set of t so that σ(t) is contained in a separatrix of a singular point is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For any t so that σ(t) and σ(−t) are not in a separatrix of a singular point, the leaves Lt and L−t through σ(t) and σ(−t) are disjoint lines and therefore cut R2 in 3 connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by Ut the closure of the connected component of R2 \\ (Lt ∪ L−t) containing L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that Ut is a strip (homeomorphic to R × [−1, 1]) bounded by Lt ∪ L−t and saturated for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' With the notation above � t Ut is a non-empty closed subset of R2 saturated for F and we have the following alternative: either � t Ut = L0 and L0 is a non-singular leaf of F, or � t Ut has non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore, � t Ut does not depend on the choice of the transverse segment σ through L0 and is denoted U(L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' U(L0) is saturated for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If it contains a non-singular leaf, it contains one of the half planes bounded by this leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If it contains a singular leaf, it contains the corresponding singular point, and then it contains at least one of the sectors bounded by the separatrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' With the notation above, the leaf L0 is called regular if U(L0) = L0, and will be called non-regular otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If L0 is a separatrix of a singular point, then it is non-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As in the case of non-singular foliations we have: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation with singular points of saddle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the set of non-regular leaves is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For any transverse segment σ let denote by Lt the leaf through σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then by construction the closed sets U(Lt) are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus at most countably many of them may have non-empty interior, that is, at most countably many of leaves Lt are non-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We conclude the proof by noticing that F admits a countable family of transverse segment σn, n ∈ N every leaf of F cuts at least 1 segment σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ The leaves of a foliations have two ends, and the notion of regular leaves can be made more precise, looking at each of its ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' More precisely, let L0,+ be an half leaf of F, and let σ be a transverse segment so that σ(0) is the initial point of L0,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For any t so that σ(t) and σ(−t) do not belong to a separatrix of a singular point, we consider Lt,+ and Lt,− the half leaves starting at σ(t) and σ(−t) in the same side of σ as L0,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by Ut(L0,+) ⊂ R2 the closed half plane containing L0,+ and bounded by the line of R2 obtained by concatenation of Lt,+ , σ([−t, t]) and Lt,−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote U(L0,+) = � t Ut(L0,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then : either U(L0,+) = L0,+ and one says that the half leaf L0,+ (or equivalently, the end of L0 corresponding to L0,+) is regular or U(L0,t) ̸= L0,t is a closed subset with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A leaf is regular if and only if its two ends are regular, and the set of non-regular ends of leaves is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A foliation with singular points of saddle type is locally orientable (and transversely orientable) in a neighborhood of a singular point x if and only if the number of separatrices of x is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus a foliation of R2 whose singular points are sadlles with even numbers of separatrices is locally orientable and transversely orientable, and therefore is globally orientable and transversely orientable, as R2 is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation with singular points of saddle type with even numbers of separatrices, and fix an orientation and transverse orientation of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus every leaf L have a right and left end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We defined Uright(L) and Uleft(L) so that L can be regular at the right or at the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If L0 is a leaf which is not a separatrix and σ be a transverse segment with σ(0) ∈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One defines in the same way the notions of being regular from above and from below, for L0 or for each of its two ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For instance Lright 0 is regular from above if U+(Lright 0 ) = � t Ut,+(Lright 0 ) = Lright 0 where Ut,+(Lright) is bounded by Lright 0 , σ([0, t]) and Lright t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The circle at infinity of a family of foliations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The circle at infinity of a foliation of R2: statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The aim of this section is to recall the following result essentially due to [Ma] and to present a short proof of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation of the plane R2, possibly with singularities of saddle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a compactification D2 F ≃ of R2 by adding a circle at infinity S1 F = ∂D2 F with the following property: any half leaf tends either to a saddle point or to a point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' given a point θ ∈ S1 F the set of ends of leaves tending to θ is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' the subset of S1 F corresponding to limits of regular ends of leaves is dense in S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore this compactification of R2 by D2 with these three properties is unique, up to a homeo- morphisms of the disk D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If L+ 1 ̸= L+ 2 are two ends of leaves tending to the same point θ ∈ S1 F, then L2 ⊂ U+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular, the ends L+ 1 and L+ 2 are not regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If a homeomorphisms f of the plane R2 preserves the foliation F then it extends in a unique way as a homeomorphism F of the compactification D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furhtermore the restriction of F to S1 F is the identity map if and only if f preserves every leaf of F and preserves the orientation on each leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The first part is, as already noted, a straightforward consequence of the uniqueness of the com- pactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If f preserves every leaf and preserves the orientation of the leaves, then it preserves every end of leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus the extension F fixes every point of S1 F which is limit of an end of leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As the limit points of end of leaves are dense in S1 F one deduces that the restriction of F to S1 F is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Conversely, assume that F is the identity on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If θ ∈ S1 F is the limit of an unique end of leaf L+ then L+ is preserved by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus f preserves every regular end of leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As the regular leaves are dense in R2, one deduces that f preserves every oriented leaf, concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by Reg(F) the set of regular leaves of F and by R(F) the set of ends of regular leaves (any non singular leaf and in particular any regular leaf has two ends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Recall that R(F) is a family of disjoint rays of R2 and therefore is cyclically ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If D is a family of regular leaves whose union in dense in R2, then the set D of ends of the leaves in D is a separating family for the set of ends of regular leaves R(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L0 be a regular leaf of F, σ: [−1, 1] → R2 a segement transverse to F with σ(0) ∈ L0 and Ut the family of neighborhoods of L0 associated to the transverse segment σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Our assumption implies that for a dense subset of t ∈ [−1, 1], the leaf Lt belongs to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a sequence tn ∈ [−1, 1], n ∈ Z so that 16 CHRISTIAN BONATTI Ln = Ltn ∈ D tn → 0 as |n| → ∞ tn as the same signe as n ∈ Z Let L+ n and L− n be the half leaves of Ln (for the orientation given by the transverse orientation induced by σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As L0 is regular one gets U(L+ 0 ) = L+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This implies that L+ 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' L− 0 ) is the intersection of the intervals (for the cyclic order) [L+ −n, L+ n ] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' [L− n , L− −n]) for n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In other words, the rays L+ −n, L+ n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' L− n , L− −n) are separating the ray L+ 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' L− 0 ) from any other ray in R(F) (and indeed from any other ray of leaf, regular or not ), concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ We are now ready to prove Theorem 7 Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We chose a countable set E of regular leaves whose union is dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 the set E of ends of leaves in E is a countable separating subset of R(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus we may apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One gets a compactification of R2 by the disc D2 F ≃ D2, so that every two distinct ends of regular leaves tend to two distinct points at the circle at infinity S1 F and these points are dense on the circle and this compactification does not depend of the choice of the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This prove the items 2 and 3 of the theorem, and also proves that these two items are enough for the uniqueness of this compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It remains to prove the first item, that is to show that the rays contained in non-regular leaves also tend to points on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' That is done by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation (possibly with saddles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then every line L transverse to F has 2 distinct limit points at infinity corresponding to its 2 ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The two ends of L are rays disjoint from the ends in R(F) (that is of the ends of leaves of F), as any transverse segment intersects any leaf in at most 1 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4 implies that the ends of L tends to points on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' These points are distinct because the regular half leaves through L are between these two ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation (possibly with saddles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given any two (non-singular) leaves L1, L2, if the ends of L1 and L2 tend to the same 2 points in S1 F then L1 = L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume L1 ̸= L2 share the same end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the leaves in the strip bounded by L1 ∪ L2 would have their ends on the same points in S1 F contradicting the fact that at most contably many ends of leaves share the same end point on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ As a by-product of the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 we get the following: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation (maybe with saddle-like singular points) and le σ: [−1, 1] → R2 \\ Sing(F) be a transverse segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let {L+ t } and {L− t } be the half leaves starting at σ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the map associating to t ∈ (−1, 1) the limit point of L+ t on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then t is a continuous point of this map if and only if L+ t is a regular end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Points at S1 F limit of several ends of leaves: hyperbolic sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let A and B be distinct ends of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the following properties are equivalent There are no end of regular leaf between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The set of ends of leaves between A and B is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The set of ends of leaves between A and B is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' First assume that there is an end L+ of a regular leaf L between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will prove that the interval (A, B) is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the neighborhood Ut of L associated to a transverse segment σ with σ(0) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As L is regular, one gets that U(L) = � t Ut = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As a consequence there is t so that A and B are out of Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' First assume that A and B are in the same connected component of R2 \\ Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a line L whose left end is B and whose right end is A and which is disjoint from Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One deduces that one of CIRCLE AT INFINITY OF FOLIATIONS OF R2 17 the interval (A, B) and (B, A) contains no end of leaf in Ut (this cannot be (A, B) which contains L+ by assumption) and the other contains all ends of leaves in Ut, so (A, B) contains ucountably many ends of leaves as announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now assume that A and B are in distinct connected components of R2 \\ Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a line Γ whose left end is B, whose right end is A and whose intersection with Ut is σ([−t, t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As L+ is in the interval (A, B) so that L+ ⊂ ∆+ Γ , one deduces that all the positive half leaves L+ r , r ∈ [−t, t] are contained in the upper half plane ∆+ Γ and therefore are between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' So the interval (A, B) (and also (B, A)) is uncoutable which is what we announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Conversely, if there are uncountably many ends in (A, B) one of them is the end of a regular leaf as non-regular leaves are countably many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This proves the equivalence of the two first items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The third items implies trivialy the second, so we now prove that the second implies the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let A and B be two ends of leaves so that (A, B) is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider a line δ with the folowing properties: A and B are the right and left ends of δ, respectively, δ \\ (B ∪ A) is a segment σ, consisting in finitely many transverse segments a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' ak and finitely many leaf segments b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , bk, with a0(0) ∈ B and ak(1) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let ∆ = ∆+(δ) be the upper half plane bounded by δ and corresponding to the interval (A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that no entire leaf may be contained in ∆ otherwhise there would be uncountably many ends between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider the half leaves L+ 0,t entering in ∆ through a0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As there are only countably many end between A and B, there is a sequence of tn → 0 so that L+ 0,tn goes out of ∆ through a point σ(sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that the half leaves L+ 0,t, t ∈ [tn+1, tn] need to go out of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus every L+ 0,t, t ≤ t0 goes out of ∆ at a point σ(s(t)), where t �→ s(t) is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let s0 be the limit s0 = lim t→0 s(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that a half leaf entering in ∆ though a0 cannot go out ∆ through a0 because a transverse segment cuts a leaf in at most a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus we deduce that s0 belongs to some ai, i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider the compact segments It ⊂ L+ 0,t joining a0(t) to σ(s(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider lim sup t→0 It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is a closed subset of R2 consisting on B and of whole leaves contained in ∆ and of a half leaf ˜B1 ending at σ(s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We already noticed that no entire leaves may be contained in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus this limit consists in B ∪ ˜B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As a consequence, the ends B and ˜B1 are successive ends, ˜B1 ∈ (A, B) and thus ( ˜B1, A) is at most countable too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider B1 ⊂ ˜B1 the half leaf starting at the last intersection point of ˜B1 with σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that B1 starts at a point of some segment ai, with i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus, if B1 ̸= A one may iterate the argument, getting successive half leaves Bi starting at points of some transverse segment aj(i), where i �→ j(i) is stricly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As there are finitely many segments ai one gets that this inductive argument needs to stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In other words, there is i with Bi = A, ending the proof: [A, B] = A = Bi, Ai−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , B1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This proves that the second item is equivalent to the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ The proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4 proved, as a by product, the following: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that A and B are successive ends of leaves, that is: the interval (A, B) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then, there is an embedding of ψ: [−1, 1] × [0, 1] → D2 F so that: the segments ψ([−1, 1] × {t}), 0 ≤ t < 1, are leaf segment A = ψ([−1, 0) × {1}) and B = ψ((0, 1] × {1}) the point ψ(0, 1) is the point is S1 F end of both end A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The embedding ψ: [−1, 1] × [0, 1] → D2 F is called a hyperbolic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 18 CHRISTIAN BONATTI We say that two half leaves A, B are asymptotic if [A, B] or [B, A] does not contain any end of regular leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We already proved next lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' To be asymptotic is an equivalence relation in the set of ends of leaves of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Each equivalence class is either finite or countable and is, as an ordered set, isomorphic to an interval of (Z, <).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There are at most countably many non-trivial classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We also already proved: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation (possibly with singular points of saddle type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then two half leaves tend to the same point θ ∈ S1 F if and only if they are asymptotic, and every half leaf arriving to θ belongs to their asymptotic class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular, if a point of S1 F is the limit of a regular end of leaf, it is the limit of a unique end of leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that points at infinity which are limit of a unique end of leaf may be the limit of a non-separated end of leaf as shows next example: Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let K ⊂ R be a Cantor set and consider PK = R2 \\ (K × [0, +∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus PK is homeomorphic to R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let FK be the restriction to PK of the horizontal foliation on R2 (whose leaves are the R× {y}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus all the leaves of the form I × {0} where I is a connected component of R \\ K are pairwise non separated from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' However, any two distinct ends of leaves of FK tend to distinct points in S1 FK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that F is oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If A0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Ak are successive ends of leaves, and assuming A0 is a right half leaf, then A1 is a left half leaf and A0 and A1 are not separated from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then A2 is a right half leaf and A1 and A2 are not separated from below, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus, each non trivial classes of the asymptotic relation consists in alternately right and left ends of non-separated leaves, alternately from above and from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Points at infinity which are not limit of leaves: center-like points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this section, foliations are assumed to be non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation of R2 and o ∈ S1 F be a point so that o = � n(an, bn), n ∈ N where an, bn are the limit points of the two ends of a same leaf Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then an and bn tends to o and o is not a limit point of an end of leaf of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider ∆n being the compact disc of D2 F whose boudary (as a disc) is Ln∪[an, bn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the ∆n are totally ordered by the inclusion and o ∈ � n ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If a leaf L had an end on o, it should be contained in every ∆n and hence contained in � n ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus the two ends of L are distinct points in � n[an, bn] contradicting the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ We say that a point o ∈ S1 F satisfying the hypothese of Remark 10 is a center-like point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Here is a very simple example with this situation: Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The trivial horizontal foliation H admits two center-like points at infinity which are the limit points of the (vertical) y axis (transverse to H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is indeed easy to check that: Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given any foliation F of R2, S1 F carries at least 2 center-like points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' To see that, just consider the (decreasing) intersection of the closure in D2 F of the half planes ∆± L for a maximal chain (given by Zorn lemma) for the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' But the situation may be much more complicated, as shows next example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 19 Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a simple closed curve γ = γ+ ∪ γ− of R2 where γ+ and γ− are the graphs of continuous functions ϕ: [−1, 1] → [0, 1] and −ϕ, respectively, where ϕ(−1) = ϕ(1) = 0, ϕ(t) > 0 for t ∈ (−1, 1), the local maxima and minima of ϕ are dense in [−1, 1] (some kind of Weierstrass function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let ∆ be the open disc bounded by γ and endowed with the constant horizontal foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then S1 F = γ and any local maximum point of γ+ and any local minimum of γ− are center-like points of S1 F The aim of this section is to show that the situation of Example 8 is in fact very common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Le F be a foliation on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that the union of leaves which are non separated at their right side is dense in R2, and in the same way, that the union of leaves which are non separated at their left side is dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the set of center-like points on S1 F is a residual subset of S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Fix a metric on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let On ⊂ S1 F be the set of points belonging to an interval (a, b) of length less than 1 n where a, b are both ends of a same leaf of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will proof that On is a dense open subset of S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then � n On will be the announced residual subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The fact that On is open is by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We just need to prove the density of On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Recall that the ends of regular leaves are dense in S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus we just need to prove that the ends of regular leaves are contained in the closure of On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L be a regular leaf and σ: [−, 1, 1] → R2 be a positively oriented transverse segment with σ0 ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote Lt the leaf through σt and ve recall that, as L is regular, the rignt and left ends L+ t , L− t of Lt tend to the right and left ends L+ annd L−, respectively, as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given any r < s ∈ [−1, 1], we denote by Ur,s, U right r,s , and U left r,s the strip bounded by Lr and Ls, and the two closed half strips obtained by cutting Ur,s along the segment σ([r, s]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Iright r,s ⊂ S1 F and Ileft r,s ⊂ S1 F be the corresponding intervals on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that, as L is regular, these interval have a length smaller than 1 n if r, s close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Our hypotheses imply that there are tright, tleft ∈ (r, s) so that Ltright is non-separated at the right, and Ltleft is non-separated at the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This implies that both U right r,s , and U left r,s contain entire leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus Iright r,s and Ileft r,s contain intervals whose both extremal points are ends of the same leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Taking r, s small enough, these intervals are contained in On showing that the points of S1 F corresponding to L+ and L− are in the closure of On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ However, not every point o which is not limit of an end of leaf is center-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let FK be the foliation defined in Example 6 by restriction of the horizontal foliation on R2 \\ (K × [0, +∞)) where K is a Cantor set K ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a point x ∈ K which are not the end point of a component of R \\ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the point (x, 0) corresponds to a point in S1 F which is not limit of an end of leaf, and is not center-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a point o ∈ S1 F and assume it is not the limit point of any end of leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For any leaf L we denote by ∆L ⊂ D2 F the compact disk containing o and whose frontier in D2 F is the segment ¯L closure of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then ∆L ∩ S1 L is a segment IL whose end points are the limit points of the ends of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that the closed segment IL are totally ordered for the inclusion, and so does the disks ∆L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let denote Io = � L IL and ∆o = � L ∆L Then if o = Io then o is a center-like point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Otherwise, ∂∆o ∩ ˚ D2 F consists in infinitely (countably) many leaves pairwise not separated and there is a subsequence of them whose limit is o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 20 CHRISTIAN BONATTI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The circle at infinity of a countable family of foliations The aim of this section is the proof of Theorem 3, that is to build the compactification associated to a countable family of foliations with saddles and prove its uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 4 already shown us that Theorem 3 is wrong for uncountable families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The new difficulty in comparison to Theorem 7 is that there are no more separating families for the set of ends of all the foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the restriction of the constant horizontal and vertical foliations to the strip {(x, y), |x − y| < 1}, so important for the study of Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then every end of horizontal leaf has a unique successor or predecessor which is the end of a vertical leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus no family can be separating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For by-passing this difficulty, we will apply Theorem 6 instead of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The ends of regular leaves R = � i∈I Ri of all the foliations Fi, i ∈ I is a family of disjoints ends of rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We have seen that for every foliation Fi the set of ends of regular leaves Ri admits a countable separating family, for instance by considering regular leaves through a dense subset in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus Theorem 6 provides a compactification of R2 by D2 satisfying the announced properties for the regular leaves, that is, items 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For item 1 one need to see that even the ends of non regular leaves tends to points at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' That is given by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The uniqueness comes from the uniqueness in Theorem 6, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example: Countable families of polynomial vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F = {Fi}i∈I, I ⊂ N be a countable family of foliations directed by polynomial vector fields on R2 whose singular points are all of saddle type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the ends of leaves either are disjoint or coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus, according to Theorem 3, there is a unique compactification D2 F = R2 ∪ S1 F for which the ends of regular leaves of the same foliation tend to pairwise distinct points at the circle at infinity, and this ends of leaves are dense in S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We just need to prove it for 2 such distinct foliations F and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the tangency locus of F and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' That is an algebraic set in R2 which is either R2 (so that F = G contradicting the assumption) or is at most 1-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus it consist in the union of a compact part and a family of disjoint rays δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , δk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If it is compact, then every end of leaf of F is transverse to G and therefore cuts every leaf of G in at most 1 point: the ends are disjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Otherwize, each ray δi either is tangent to both foliations and is therefore a comon leaf (which is one of the announced possibilities) or is transverse to F and to G out of a finite set (because the tangencies on δi are a algebraic subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus up to shrink the non-tangent δi, we assume that they are transverse to both foliations therefore cut every leaf of F in at most 1 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This implies that every end of leaf of F which is not an end of G is transverse to G and thus is disjoint from any end of leaf of G, concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The compactification in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 is in general distinct from the algebraic extension of the Fi on RP2: for instance, consider the trivial example of R2 endowed with the horizontal and vertical foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this case the compactification by the algebraic extension, all the leaves of the horizontal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' vertical) foliations tend to the same point at RP1 (which corresponds to 2 points for the circle at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Projections of D2 F on D2 Fi and center-like points on the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider R2 endowed with the trivial horizontal and vertical foliation, H and V respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the compactification D2 H,V is conjugated to the square [−1, 1]2 endowed with the trivial horizontal and vertical foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Every point p ∈ S1 H,V, but the four vertices, are limit of exactly 1 end of leaf, either horizontal (for p in the vertical sides) or vertical (for p in the horizontal sides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The projection ΠH : D2 H,V → D2H consists in colapsing the two horizontal sides, which are tranformed in the center-like points of S1 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The projection ΠH : D2 H,V → D2V consists in colapsing the two vertical sides, which are tranformed in the center-like points of S1 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the strip {(x, y) ∈ R2, |x − y| < 1} endowed with the horizontal and vertical foliations H and V respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='. Then D2 H,V = D2 H = D2 V and consists in adding to two points ±∞ to the closed strip {(x, y) ∈ R2, |x − y| ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Every point in the sides |x − y| = 1 are the limit of exactly 1 end of leaf of H and 1 end of leaf of V, and the points ±∞ are center like for both foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' These two examples show that pairs of very simple foliations may lead to different behavior of the projection of the compactification associated to the pair on the compactification of each foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 below shows that, for complicated foliations, the compactification of the pair of folia- tions in general coincides with the compactification of each foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F, G be two transverse foliations on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that the union of leaves of G which are not separated at their right from an other leaf is dense in R2 the union of leaves of G which are not separated at their left from an other leaf is dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the identity map on R2 extend as a homeomorphism from D2 F,G → D2 F: in other words D2 F,G = D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that there is an open interval I of S1 F,G corresponding to no end of leaf of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the ends of leaves of G are dense in I, and therefore the projection of I on S1 G is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='8 implies that there are leaves L of G having both ends on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus up to change positive in negative, every positive half leaf of F through L has its end on I contradicting the definition of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' So the points of S1 F,G corresponding to ends of leaves of F are dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus S1 F,G = S1 F, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ As a direct corollary of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='8 one gets Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F, G be two transverse foliations on R2 so that both F and G have density of leaves non separated at the right and of leaves non separated at the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then generic points in S1 F,G = S1 F = S1 G are center-like for both foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Hyperbolic sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In the case of 1 foliation we have seen that, if several ends of leaves have the same limit points on the circle at infinity, then they are ordered as a segment of Z and two succesive ends bound a hyperbolic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' These hyperbolic sectors have a very precise model, which allows us to understand the position of a transverse foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F and G be two transverse foliations on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' and consider πF : D2 F,G → D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that p ∈ D2 F is the corner of a hyperbolic sector bounded by the ends A and B of leaves of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a non-empty interval IG of ends of leaves of G ending at p in D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore either IG consist in a unique end of leaf C of G and A, B, C tend to tend same point at infinity in D2 F,G or π−1 F (IG) is a closed interval on the circle S1 F,G whose interior consist in regular ends of leaves of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Just use the model [−1, 1]×[0, 1] where p is the point (0, 1), A = [−1, 0)×{1} and B = (0, 1]×{1}, and the horizontal segment [−1, 1] × {t}, 0 ≤ t < 1 are F-leaf segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We can choose this model so that the vertical sides {−1}×[0, 1] and {1}×[0, 1] are leaves segments of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the G-leaves throug [−1, 1] × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The leaves reaching A and the leaves reaching B are two non-empty intervals, open in 22 CHRISTIAN BONATTI [−1, 1] and disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By connectedness, there are leaves, corresponding to an closed interval of [−1, 1] which reach neither A nor B and these leaves end at p ∈ S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that this interval is not reduced to a single end of leaf of G and consider an end C in the interior of this interval and assume that C is, for instance, a right end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the neighborhoods U right t of C defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then � t U right t consist in C and in a (maybe empty) set of entire leaves of G contained in the hyperbolic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that this set is not empty and let D be such a leaf of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Every leaf of F cutting D has an half leaf contained in the hyperbolic sector, contradicting the definition of hyperbolic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus C = � t U right t , meaning that C is a regular end of leaf of G, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ As a straightforward consequence, one gets: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F = {Fi}i∈I, I ⊂ N be an at most countable family of pairwise transverse foliations on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a point p ∈ D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then either at most 1 end of leaf of each Fi has p as its limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' or the set of ends tending to p is ordered as an interval of Z and, between any two succesive ends of leaves of the same Fi, there is exactly 1 end of a leaf of each Fj, j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The circle at infinity for orientable laminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The circle at infinity of a lamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The way we proposed to compactify R2 can be generalized for any object providing a family of rays admitting a separating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For instance, what about laminations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The theory cannot be extended without hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' An evident obstruction is that the leaves can be too few for going to a dense subset of a circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' But there are more subtle issues as shows Example 13 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There are closed laminations whose leaves are recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For instance consider a Plykin attractor on R2: it is a compact minimal lamination (by the unstable manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If you consider now a Plykyn attractor on S2 = R2 ∪ {∞} where ∞ belongs to the attractor, we get a closed lamination on R2 where every leaf is unbounded but recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that the recurrent lamination in Example 13 are not orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let me show that Poincar´e Bendixson argument applies on orientable laminations: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L be a closed orientable lamination of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given any leaf L, either the closure ¯L contains a compact leaf or L is a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If ¯L = L then L is either a compact leaf or is a line (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' is properly embedded in R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume now that ¯L\\L contains a point x ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We fix an orientation of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Chose a segment σ: [−1, 1] transverse to L so that σ(0) = x and so that σ cuts positively every leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The hypothesis implies that L cuts σ in an infinite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider 2 successive (for the order in the leaf L) intersection points z0, z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then one gets a simple closed curve δ in R2 by concatenation of the segments [z0, z1]L and [z1, z0]σ joining z0 to z1, in L and in σ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the disc ∆ bounded by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Every leaf cuts δ with the same orientation, that is, either every leaf enter in ∆ or every leaf goes out of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Up to reverse the orientation one may assume that every leaf enters in ∆ and in particular L enters in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In other words, there is a positive half leaf L+ included in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This half leaf cannot be reccurent (otherwize it would cut again [z0, z1]σ and for that it needs to go out of ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Futhermore: Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' No other leaf L′ ̸= L can accumulate on L: L ∩ ¯L′ = ∅ if L′ ̸= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If L′ accumulates on L, it cuts [z0, z1]σ on an infinite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Thus the ω-limit set is ω(L) = ¯L+ \\ L+ and is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider y ∈ ¯L+ \\ L+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The leaf Ly is contained in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Either Ly is compact and ω(L) = Ly and we are done, or ¯Ly \\ Ly ̸ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In that case, Claim 2 implies that Ly is not accumulated by any leaf, in particular by L, contradicting the definition of Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ CIRCLE AT INFINITY OF FOLIATIONS OF R2 23 We are now ready to extend Theorem 7 to the case of orientable laminations: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L be a closed orientable lamination of R2 with no compact leaf and assume that the set of leaves of L is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a compactification D2 L ≃ of R2 by adding a circle at infinity S1 L = ∂D2 F with the following properties: any half leaf tends to a point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' given a point θ ∈ S1 L the set of ends of leaves tending to θ is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' for any non-empty open subset I of S1 L the set of points in I corresponding to limits of ends of leaves is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore this compactification of R2 by D2 with these three properties is unique, up to a homeo- morphism of the disk D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let me just given a sketch of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The lamination L is assumed to be oriented and without compact leaves, so that every leaf is a line, according to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Cantor-Bendixson theorem, see for instance [Ke], the lamination L can be written in a unique way as union L = L0 ∪ L1 of two disjoint laminations, where L0 is a closed lamination with no isolated leaves and L1 consists in a countable set of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A leaf L ∈ L0 is called regular if it is accumulated on both sides by leaves in L0 and is separated from any other leaf of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The same proof as for foliations shows that the set of leaves in L0 which are not regular are at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Finally, as for foliations, one consider the set R of germs of rays contained in regular leaves of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider a countable set D of regular leaves, whose union is dense in L0, and as for the case of foliations, one proves that the rays in D are separating for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then we apply Theorem 1 and we get the announced canonical compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ When L is transversely a perfect compact set (that is, there is a transverse segment σ through every point x ∈ L so that σ ∩ L is a compact set without isolated points), then the compactification given by Theorem 8 seems very natural: any homeomorphism h of R2 preserving L extends on S1 L as a homeo- morphism H of D2 L, and the restriction H|S1 L is the identity map if and only if h preserves every leaf of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' That is no more the case if L has isolated leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For lamination with isolated leaves, Theorem 8 just ignores the countable part L1 of L (in the Cantor Bendixson decomposition of L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will now propose a cannonical compactification which takes in account this countable part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We start by looking at two very different examples of countable oriented laminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the lamination L = R × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then L does not admit any compactification by a circle at infinity so that any homeomorphism h preserving L extends on the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Example 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a hyperbolic surface S of finite volume and consider a set ℓ of essential disjoint simple closed geodesic on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the lift L of ℓ on the universal cover ˜S = ˚D2 is a countable, discrete lamination by geodesic of the Poincar´e disc so that the ends of leaves tend each to points on the circle S1 = ∂D2, and the set of such limit points is dense on S1 as the action of π1S on S1 is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this example, the lamination is transversely discrete, but the set of ends of leaves is a separating set for himself for the cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In the example 15 above, what implies the existence of a separating set is the minimality of the action on the circle at infinity of a natural compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In order to propose a cannonical compactification for a closed, oriented lamination without compact leaves we need to determine what part of a cyclically totally ordered set admits separating subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' That is what is done in next easy proposition whose proof is let to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be a set endowed with a total cyclic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the relation on X defined as follows: x ≈ y if one of the intervals [x, y] or [y, x] does not contained any self-separating subset E (with #E ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then 24 CHRISTIAN BONATTI the relation ≈ is an equivalence relation each equivalence class is an interval the cyclic order on X induces a total cyclic order on the quotient X/ ≈ Futhermore, X/ ≈ is either a single point or is an infinite self-separating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that any two distinct points in a self separated set belong to distinct classes, so that #(X/ ≈) = 1 if and only if X does not contain any (non-trivial) self-separating subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Otherwise #(X/ ≈) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The canonical compactification is now given by Theorem 9 below: Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L be a closed oriented lamination of the plane R2, with no compact leaf, and let R be the set of ends of leaves of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As the ends of leaves are disjoints rays the set R is totally cyclically ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that #(R/ ≈) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a unique compactification D2 L of R2 by adding a circle at infinity S1 L so that any end of leaf of L tends to a point in S1 L the set of points in S1 L end of an end of leaf is dense in S1 two ends of leaves tend to the same point in S1 L if and only if they belong to the same class in R/ ≈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Just apply Theorem 1 to a subset E ⊂ R containing exactly 1 representative in each class of ≈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One checks that the compactification obtained satisfies the announced properties and does not depend on the choice of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Remark 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Every class of ≈ in R is at most countable because the set of ends of regular leaves in the perfect part L0 is self-separating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This compactification takes in account more leaves that the compactification given by Theorem 8, but it is still may have unexpected behaviours: Example 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a non-compact hyperbolic surface S of finite volume and consider a closed lam- ination ℓ defined by two disjoint freely homotopic essential closed curves and a closed (but non-compact) leaf whose both ends tend to the same puncture of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the lift L of ℓ on the universal cover ˜S = ˚D2 is a countable, discrete lamination of the Poincar´e disc so that the ends of leaves tend each to points on the circle S1 = ∂D2, the set of such limit points is dense on S1 (again as the action of π1S on S1 is minimal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this example, however, there are pairs of leaves which share the same limits of their both ends, and there are leaves whose both ends tend to the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given a closed oriented lamination L with no compact leaves and its Cantor-Bendixson decomposition L = L0 ∪ L1 (L0 is a closed lamination without isolated leaves and L1 is countable), Theorem 9 takes in account the part of the ends of leaves in L1 with separating subsets, in contrast with Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For my personal taste, the main issue in Theorem 9 is that I did not found any natural criterion to calculate the equivalence classes of ≈ in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In fact Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 below seems to present as paradoxal the fact that L1 may have separating subsets: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let D ⊂ R be a countable compact subset, ordered by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then D does not contain any self separating subsets E ⊂ D(that is, #E > 2 and for every x < z, x, z ∈ E there is y ∈ E with x < y < z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If E is a non-trivcial sel-spearting subset, then there is an increasing bijection from E\\{min E, max E} to Q ∩ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This increasing bijection extends in a unique way in a (non-strictly) increasing map from R → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This map is continuous and the image of D is [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus D is not countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 tell us that the separating property of a closed countable lamination cannot be obtained locally (in foliated charts of the lamination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One deduces: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L be a closed countable orientd lamination of ˚ D2 so that every end of leaf tends to a point on S1 and the set of such limit points are dense in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 25 Then given any non-empty open interval I ⊂ S1 there is L ∈ L whose both ends have their limits in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' More precisely, any neighborhood of I in D2 contains an entire leaf of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One consider a neighborhood of I bounded by two half leaves whose limits are points x ̸= y ∈ I and a segment σ transverse to L and joining this two leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If no leaves is contained in this neighborhood, then every leaf having an end in I cuts σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' On the other hand any dense subset of an interval J of R is self separating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One deduces that σ ∩ L contains a self separating subset, but is is a countable compact set, and this contradicts Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ This proposition says that the separating property for a countable oriented lamination is obtained by leaves in small neighborhoods of the points at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Families of transverse laminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Transversality does not imply in general the compactness of the intersection of two leaves of transverse laminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' But this compactness is our main hypothesis for the compactification associated to families of foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' However, if two lines L1, L2 ⊂ R intersect always with the same orientation, then #L1 ∩ L2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One deduces that Theorem 2 extends without difficulties to countable families of oriented closed laminations intersecting pairwise transversely and with always the same orientation Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let L = {Li}, i ∈ I ⊂ N be a family of closed orientable laminations of R2 with no compact leaves and so that the set of leaves of Li is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We assume that the laminations are pairwize transverse with constant orientation of the intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a compactification D2 L ≃ of R2 by adding a circle at infinity S1 L = ∂D2 F with the following properties: any half leaf tends to a point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' given a point θ ∈ S1 L the set of ends of leaves tending to θ is at most countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' for any non-empty open subset I of S1 L the set of points in I corresponding to limits of ends of leaves is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore this compactification of R2 by D2 with these three properties is unique, up to a homeo- morphism of the disk D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Actions on a bifoliated plane We have seen than any homeomorphism h ∈ Homeo(R2) preserving a at most countable family of transverse foliations F admits a unique extension as an homeomorphism on the compactification D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus if H ֒→ Homeo(R2) is a group acting on R2 and preserving the (at most countable) family of transverse foliations F then this action extends in an action on D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By restriction to the circle at infinity, one gets an action of H on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If H ֒→ Homeo(R2) is a group acting on R2 and preserving a family of foliations F, we say that the action is minimal on the leaves of F if H(L) is dense of R2 for every leaf L of a foliation of the family F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Faithfullness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation, and h ∈ Homeo(R2) be a homeomorphism preserving F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the action of h on S1 F is the identity map if and only if h(L) = L for any leaf L, and h preserves the orientation of the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If h preserves every leaf and its orientation, then it preserves any limit of its ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As these limit of ends are dense in S1 F one gets that the homeomorphism induced by h on S1 F is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Conversely, if h induces the identity on SS1 F then for every leaf L the leaf h(L) have the same limit of ends as L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 this implies h(L) = L as announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F = {Fi}, i ∈ I ⊂ N be a family of at least 2 transverse foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let h ∈ Homeo(R2) be a homeomorphism preserving each foliation Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the action of h on S1 F is the identity map if and only if h itself is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 26 CHRISTIAN BONATTI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If the induced homeomorphism induced by h on S1 F is the identity map then the same happens to homeomorphism induced by h on every S1 Fi (because they are quotient of S1 F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 implies that h preserves each leaf of each Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As every point of R2 is the unique intersection point of the leaves through it, one deduces that every point of R2 is fixed by h and h is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Orientations and injectivity of the projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation of the plane R2, endowed with an orientation and a transverse orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let G ⊂ Homeo(R2) be a group of homeomorphisms preserving (globally) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let G+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' G+) be the index at most 2 subgroup consisting of the elements of G preserving the orientation (reps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' the transverse orientation)of F, and G+ + = G+ ∩ G+) the subgroup of elements preserving both orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then: Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If one of the groups G, G+, G+, G+ + acts minimally on the leaves of F, then so does each of these 4 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will indeed prove Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 for which Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 is a particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let G be a group acting minimally on the leaves of a foliation F of R2, and H ⊂ G be a subgroup of finite index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then H acts minimally on the leaves of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' G acts minimally on the leaves of F, and consider such a leaf L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As H is a finite index subgroup, there are g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='., gn ∈ G so that for any g ∈ G there is i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=', n} with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='H = giH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let denote Hi = gi·H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular G = � i Hi, and then R2 = � i Hi(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider any open subset O of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' O = O ∩ � i Hi(L) = � i (O ∩ Hi(L)) The open set O is a baire space so that the union of finitely many closed sets with empty interior has empty interior: one deduce that at least one of the O ∩Hi(L) have non empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One deduces that the union � i ˚ Hi(L) of the interiors of the Hi(L) is dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that for every i and every g there is j so that g(Hi(L)) = Hj(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider R2 \\ � i ˚ Hi(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is a G-invariant closed set, saturated for the foliation F, and with empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As every G-orbit is dense, one deduces that this set is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus R2 = � i ˚ Hi(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The open sets ˚ Hi(L) are images on from the other by homeomorphisms in G, and in particular they are all non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As R2 is connected, one deduces that the open sets ˚ Hi(L) are not pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=', n} be the maximum number so that there are distinct i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , ik with k� 1 ˚ Hij(L) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As the ˚ Hi(L) are not pairwise disjoint, we know that k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will prove, arguing by contradiction: Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' k = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For that we assume that k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then we consider the union of all the intersections of k of these open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This union is a F-saturated G-invariant non-empty set and hence is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Its complement is an F-saturated invariant closed set with empty interior, and therefore is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus R2 is the union of these open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now again the connexity of R2 implies that these open sets are not pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This provides a non-empty intersection of 2 distinct of these sets, that is, a non CIRCLE AT INFINITY OF FOLIATIONS OF R2 27 empty intersection of more than k of the ˚ Hi(L), contradicting the choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This shows k = n proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Thus n� 1 ˚ Hi(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' is an non-empty, G-invariant open set saturated for the foliation F, and thus it is dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We just proved that H(L) is dense in R2, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ We will use the next straightforward corollary of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let H ⊂ Homeo(R2) be a group preserving a foliation F and acting minimally on the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that L is a leaf which is not separated at the right and from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the union of the leaves h(L), h ∈ H, which are non-separated at the right and from below is dense in R2 (the same holds changing right by left and/or below by above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As a direct consequence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 and Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 we get: Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F,G be two transverse foliations of R2 and H ⊂ Homeo(R2) preserving F and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that the orbit of every leaf of G in dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If G has a non-separated leaf, then the projection of ΠF : D2 F,G → D2 F is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If G has a non separated leaf L1 at the right, it is non separated from a leaf L2 which is non separated at the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 asserts that the leaves of G non separated at the left as well as the leaves non separated at the right are dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 asserts that ΠF is a homeomorphism, concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Minimality of the action on the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation on the plane R2 and H ⊂ Homeo(R2) preserving the foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (1) If the action of H on S1 F is minimal then the foliation F admits non separated leaves from above and non separated leaves from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (2) Conversely if the foliation F admits non separated leaves from above and non separated leaves from below and if the orbit of every leaf is dense in R2 then the action of H on S1 F is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will see with Theorem 14 that the minimality of the action on the leaves is not a necessary condition for the minimality of the action on the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Item 1 of Theorem 11 is a consequence of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F be a foliation of R2 and assume that F has no non-separated leaves from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given any leaf L we denote by ∆+ L the closure on D2 F of the upper half plane of R2 bounded by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then � L∈F ∆+ L is non empty and consists in an unique point OF on S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As a consequence, any h ∈ Homeo(D2 F) preserving F admits OF as a fixed point: h(OF) = OF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We introduce a relation on the set L of leaves of F as follows: L1 ⪯ L2 if there is a positively oriented (for a transverse orientation of F) transverse segment σ starting at L1 and ending at L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One easily checks that ⪯ is a partial order relation on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Due to the connexity of R2, one gets: Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' given any leaves L, ˜L ∈ L there is k ≥ 0 and L0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , Lk ∈ L so that, for any i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=', k − 1} the leaves Li and Li+1 are comparable for ⪯ (that is Li ⪯ Li+1 or Li+1 ⪯ Li) L = L0 and L′ = Lk 28 CHRISTIAN BONATTI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There is a countable family of segments in R2 transverse to F and so that every leaf L cuts at least one of these segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The set of leaves cutting a given segment induces a connected open set of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given any two points in R2 one considers a compact path joining these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By compacity, it is covered by a finite family of these open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One concludes easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ We denote ≺ L, L′ ≻∈ N the minimum value of such a number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One easily checks that ≺ ·, · ≻ is a distance on the set of leaves L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Up to now, this could be done for any foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this setting, our hypothesis that F does not admit leaves which are non-spearate from below is translated as follows: Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that L0, L1, L2 ∈ L are three leaves so that L0 ⪯ L1 and L0 ⪯ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then L1 and L2 are comparable for ⪯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We assume that the leaves Li are distinct, otherwise there is nothing to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let σi : [0, 1] → R2, i = 1, 2 transverse to F and positively oriented so that σi(0) ∈ L0 and σi(1) ∈ Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let I = {t ∈ [0, 1], L(σ1(t)) ∩ σ2 ̸= ∅} and J = {t ∈ [0, 1], L(σ1(t)) ∩ σ2 ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As R2 is simply connected, one shows that I and J are connected and each of them contains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let t1 = sup I and t2 = sup J For any t ∈ [0, t1) let ˜t ∈ J so that L(σ1(t)) = L(σ2(˜t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular, ˜t tends to t2 as t tends to t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus the leaves L(σ1(t1) and L(σ2(t2) are accumulated from below by the leaves L(σ1(t)) = L(σ2(˜t), thus are non separated from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By assumption on F this implies that they are equal: L(σ1(t1) = L(σ2(t2) If t1 < 1 and t2 < 1 then the leaf L(σ1(t) for t > t1 close to t cuts σ1 at a point σ2(˜t) with ˜t > t2, close to t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This contradicts our choice of t1 and t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus t1 = 1 or (non exclusive) t2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In the first case L1 = L(σ1(t1)) cuts σ2 and then L1 ⪯ L2 and in the second case L2 cuts σ1 and L2 ⪯ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ As a consequence of Claims 4 and 5 one deduces Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given any two leaves L, ˜L there is a leaf ˆL so that L ⪯ ˆL and ˜L ⪯ ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular, the distance ≺ ·, · ≻ is bounded by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a finite sequence of leaves L = L0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , Lk = ˜L, k =≺ L, ˜L ≻, and Li comparable with Li+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The minimality of k implies that Li−1 and Li+1 are not comparable (otherwise one could delete Li geting a strictly smaller sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that there is i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='k − 1} so that Li−1 ⪰ Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If Li ⪰ Li+1 then Li−1 ⪰ Li+1 which is forbidden by the observation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus Li ⪯ Li+1 and Clain 5 implies again that Li−1 and Li+1 are comparable, which again is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This proves that ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' k − 1}, Li−1 ⪯ Li □ As a consequence one deduces Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There is a increasing sequence Li ≺ Li+1 i ∈ N, Li ∈ L so that, given any leaf L ∈ L there is n with L ≺ Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One chose a countable set of compact positively oriented segments σi : [0, 1] → R2 transverse to F to that any leaf cuts one of the σi (and thus is less than L(σi(1) for ⪯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then one builds inductively the sequence Li: Li+1 is obtained by applying Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 to the leaves Li and L(σi(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The compact discs ∆+ L are decreasing with L for ≺: more precisely, if L ≺ ˜L then ∆+ ˜L ⊂ ˚ ∆+ L, CIRCLE AT INFINITY OF FOLIATIONS OF R2 29 where ˚ ∆+ L denotes the interior for the topology of D2 F ( it does not means the open disc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The hypothesis implies that Li+1 is contained in the interior of ∆+ Li, so that ∆+ Li+1∩R2 is contained in the interior of ∆+ Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We need to prove that ∆+ Li+1 ∩ S1 F is contained in the interior of ∆+ Li ∩ S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In other words, we need to prove that the ends of Li+1 do not share a limit with the ends of Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Recall that Li ≺ Li+1 that is, there is a segment σ: [0, 1] → R2 transverse to F and positively oriented, with σ(0) ∈ Li and σ(1) ∈ Li+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If Li share with Li+1 a limit point in S1 F so does any leaf L(σ(t)) contradicting the fact that points in S1 F are limits of at most a countable set of ends of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Thus Claims 6 and 7 implies � L∈F ∆+ L = � i∈N ∆+ Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now � L∈F ∆+ L = � i∈N ∆+ Ln is a decreasing sequence of connected compact metric sets, saturated for F and therefore is a non-empty connected compact sets saturated for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As it does not contain any leaf of F one deduces that � L∈F ∆+ L ∩ R2 = ∅ that is � L∈F ∆+ L is a compact interval U in S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It remains to show that this interval U = � L∈F ∆+ L is reduced to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Otherwise, there is and half leaf L+ whose limit belongs to the interior of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Claim 7, this implies that L+ ∩ ∆+ Li ̸= ∅ for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This contradics the fact that, for n large enough, the leaf Ln is larger (for ≺) than the leaf L carrying the half leaf L+ and thus L ∩ ∆+ Ln = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This contradiction ends the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Proof of item 1 of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that the action of H on S1 F is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The foliation F cannot be conjugated to the trivial foliation otherwise the set {N, S} (unique points in S1 F which are not limit of leaves) would be H-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus F admits non-separated leaves, and we can assume it is from above (up to change the transverse orientation of F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If it do not admit non-separated leaves from below then the point OF in S1 F given by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3 is a global fix point of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Proof of the Item 2 of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We assume that H is a group acting minimally on the leaves of a foliation F having non-separated leaves, some of them from above and some of them from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 up to consider a finite index subgroup of H, acting minimally on the leaves of F, one may assume that H preserves the orientation and transverse orientation of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Recall that the ends of regular leaves are dense in S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus it is enough to check that any neighborhood of any end of a regular leaf contains points in the orbit for H of any point of S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a regular leaf L and σ: [−1, 1] → R2 a segment transverse to F with σ(0) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will show that the end of L+ belongs to the closure of any H-orbit (the same argument holds for the end of L−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by Lt the leaf through t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider the basis of neighborhood U + t of the end L+ given by the compact discs in D2 F closure of the half plane bounded by L+ −t, σ([−t, t]), and L+ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Our hypothesis implies Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There is a dense subset of values of t so that Lt is not separated at the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As a consequence for every t the topological disc U + t contains entire leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The first sentence is directly implied by the existence of leaves which are non-separated at the right, the fact that H preserves the orientations of the leaves and acts minimaly on the leaves of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The second sentence have been seen in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Any leaf L cuts D2 F in two discs, ∆+ L and ∆− L (following the tranverse orientation of F) whose union ∆+ L ∪ ∆−L is D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Under the hypotheses, given any L there are g1, g2 ∈ H so that g1(∆+ L) ⊂ ˚∆− L and g2(∆− L) ⊂ ˚∆+ L 30 CHRISTIAN BONATTI As a consequence both ∆+ L and ∆− L contains points in any H-orbit of point in D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We prove the first inclusion, the other is obtained by reversing the transverse orientation of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Considers L a leaf and σ: [−1, 1] → R2 a segment transverse to F (positively oriented for the transverse orientation of F) so that σ(0) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There is −t ∈ [−1, 0) so that the leaf L−t is non-separated from below from a leaf L2, because the leaves non-separated from below are dense in R2, due to the minimality of the action of H on the leaves, and the fact that H preserves the transverse orientation of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus L−t ⊂ ∆− L, L2 ⊂ ∆− L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore ∆− L2 contains L−t and thus contains L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One deduces: ∆+ L2 ⊂ ∆− L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now, there is h ∈ H so that h(L2) = L−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='with −s ∈ (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular one gets that ∆+ L ⊂ ˚∆+(h(L2) and thus h−1∆+ L = ∆+ h−1(L) ⊂ ˚∆+(L2) ⊂ ˚∆− L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ We are ready to conclude the proof of Theorem 11: Any neighborhood in D2 F of any point of S1 F contains an entire leaf L (claim 8 above), and thus contains either ∆+ L or ∆− L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Claim 9 this neighborhood contains points in any H-orbit of points in D2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This shows the minimality of the action of H on S1 F, concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let F, G be two transverse foliations on the plane R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let H ⊂ Homeo(R2) be a group preserving both foliations F and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (1) If the action of H on S1 F,G is minimal then both foliations F G have non-separated leaves from above and non separated leaves from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' (2) Conversely, if both foliations F G have non-separated leaves from above and non separated leaves from below and if the orbit of every leaf of F and G is dense R2, then the action of H on S1 F,G is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For item 1, if the action of H on S1 F,G is minimal then both actions of H on S1 F and S1 G are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus item 1 follows from Item 1 of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Conversely, as the action on the leaves of F and G is assumed to be minimal, and they have non- separated leaves, then Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 implies that both projections ΠF and ΠG are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' That is S1 F,G = S1 F = S1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now the minimality of the action of H on this circle at infinity is given by item 2 of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Action of the fundamental group on the bifoliated plane of an Anosov flow 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The bifoliated plane associated to a Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be an Anosov flow on a closed 3- manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then Fenley and Barbot show that the lift of X on the universal cover of M is conjugated to R3, ∂ ∂x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' in particular the space of orbits of this lifted flow is a plane PX ≃ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then, the center-stable and center-unstable foliations of X induce (by lifitng on the universal cover and projecting on PX) a pair of transverse foliations Fs, Fu on the plane PX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The triple (PX, Fs, Fu) is called the bi-foliated plane associated to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Finally, the natural action of the fundamental group π1(M) on the universal cover of M projects on PX in an action preserving both foliations Fs and Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Fenley and Barbot proved that, if one of the foliation Fs, Fu is trivial (that is, has no non-spearated leaf and therefore is conjugate to an affine foliation by parallel straight lines) then the other is also trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In that case, one says that X is R-covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In that case the bifoliated plane is conjugated to one of the two possible models: the plane R2 endowed with the trivial horizontal and vertical foliations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Solodov proved that this is equivalent to the fact that X is orbitally equivalent to the suspension flow of a linear automorphism of the torus T2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' the restriction of the trivial horizontal and vertical foliation to the strip |x − y| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 31 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Injectivity of the projection of D2 F s,F u on D2 F s and D2 F u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The aim of this section is to prove Theorem 5 which is restated as Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 below and Theorem 13 (in next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be an Anosov flow on a 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then: Either X is topologically equivalent to the suspension flow of a hyperbolic element of SL(2, Z) Or both projections of the compactification D2 F s,F u on D2 F s and D2 F u are homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume that the projection on D2 F u is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus there is a non trivial open interval I of S1 F s,F u whose point are not limit of end of leaves of Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus a dense subset of point in I are limit of ends of leaves of Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore, every end of leaf of Fs in I is a regular end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a regular end (for instance, a right end) of leaf Ls right of Fs whose limit is in the interior of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there is a small unstable segment σ through a point of Ls right so that every right half leaf Ls right,t of Fs is regular and has its limits in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the union of all these half leaves is what Fenley called a product region , in [Fe2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now [Fe2, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1] asserts that any Anosov flow admiting a product region is a suspension flow, concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Minimality of the action on the circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In order to prove Theorem 5 it remais to prove Theorem 13 below: Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X a flot d’Anosov on a closed 3-manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then X is non-R-covered if and only if the action of π1(M) on the circle S1 F s,F u at infinity is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If the manifold M is not orientable and if X is R-covered, then [Fe1] noticed that X is a suspension flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus, on non-orientable manifolds M, Theorem 13 asserts the minimality of the action on the circle at infinity, excepted if M is a suspension manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The bifoliated plane (PX, Fs, Fu) remains unchanged if we consider a lift of X on a finite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus it is enough to prove Theorem 13 in the case where M is oriented and the action of π1(M) preserves both orientation and transverse orientation of both foliations Fs, Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus, up to now we will assume that M is oriented and the action of π1(M) preserves both orientations and transverse orietations of both foliations Fs, Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Remark 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If X is R-covered, then S1 F s has exactly 2 center-like points, which are therefore preserved by the action of π1(M) on S1 F s: this action is not minimal, and thus the action on S1 F s,F u is not minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus we are left to prove Theorem 13 in the case where X is not R-covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='We will start with the easier case, when X is assumed to be transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The non-transitive case will be done in the whole next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Theorem 13 when X is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' When X is non-R-covered and transitive, then [Fe3] proved that Fs and Fu admits non-separated leaves from above and non-separated leaves from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As (up to consider a finite cover of M), the action of π1(M) preserves the orientation and tranverse orientation of Fs, and the action is minimal on the set of leaves of Fs thus Theorem 11 asserts that the action of π1(M) on S1 F s is minimimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As S1 F s = S1 F s,F u, this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Minimality of the action on the circle at infinity for non-transitive Anosov flows: ending the proof of Theorem 5 For ending the proof of Theorem 5, we are left to prove : Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be a non-transitive Anosov flow on a closed connected 3-manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then the action of the fundamental group of M on the circle at infinity is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This result is somewhat less intuitive, as the action of the fundamental group π1(M) on the leaves of M is not minimal, and even, if X has several attractors, may fail to admit a leaf whose orbit is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The proof of the minimality of the action on the circle at infinity will require some background on Anosov flow, in particular on non-transitive Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In the whole section, X is a non-transitive 32 CHRISTIAN BONATTI Anosov flow on an orientable closed connected manifold M and the natural action of π1(M) on the bifoliated plane (PX, Fs, Fu) preserves the orientations and transverse orientations of both foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Recall that we have seen that the compactification of both foliations coincide with the one of each foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We will denote by D2 X, S1 X this compactification and the corresponding circle at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We refer by ∗ for this package of hypotheses and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Background on non-transitive Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let X be a non-transitive Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus, according to [Fe1, Ba1] X is not R-covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The flow X is a structurally stable flow, so that Smale spectral decomposition theorem splits the non-wandering set of X in basic pieces ordered by Smale order: a basic piece is upper another if its unstable manifolds cuts the stable manifold of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For this order, the maximal basic pieces are the repellers and the minimal are the attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In [Br], Brunella noticed that the basic pieces are separated by incompressible tori transverse to the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider an attractor A of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is a compact set consisting in leaves of the unstable foliation of X, hence it is a compact lamination by unstable leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Furthermore the intersection of A with a transverse segment σ is a Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' An unstable leaf W u in A is called of boundary type if W u ∩ σ belongs to the boundary of a connected component of σ \\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A classical results from hyperbolic theory (see for instance [BeBo]) asserts that the unstable leaves in A of boundary type are the unstable manifolds of a finite number of periodic orbits called periodic orbits fo boundary type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The same happens for repellers R: they are compact laminations by stable leaves, tranversally a Cantor sets, and they admits finitely many boundary leaves, stable manifolds of finitely many periodic orbits called of boundary type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In this section, we will focus on attractors and repellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider an attractor A of X, its lift ˜A on the universal cover, and consider the projection of A on the bi-foliatioed plane PX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This projection is a closed lamination by leaves of Fs and its cuts every tranverse curve along a Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' By a pratical abuse of notation we will still denote by A this lamination of PX: thus A denotes at the same time a 2-dimensional lamination on M and a 1-dimensional lamination on PX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The same happens for repeller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let A ⊂ PX and R ⊂ PX be the unstable and stable laminations (respectively) corresponding to an attractor and a repeller of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then A ∩ R = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This seems obvious, but it will be a crucial property for us: given an unstable leaf Lu and a stable leaf Ls, this will be our unique criterion for knowing that they don’t intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' the periodic point contained in A (reps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' R) are dense in A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Each periodic orbit of X has a discrete π1(M)-orbit in PX the periodic orbits of boundary types are the π1(M)-orbits of finitely many X-orbits, and there- fore are a discrete set in PX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Fenley [Fe2] shows that the non-separated stable leaves of Fs (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Fu) correspond to finitely many orbits of X, and hence to a discrete set of periodic points in PX thus the periodic points p in A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' R) which are not of boundary type and whose unstable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' stable) leaf is regular are dense in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , Ak are the attractors of X then the union of the stable leaves of Fs through the laminations A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' , AK of PX are dijoint open seubsets of PX whose union is dense in PX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The same holds for the repellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As a straightforward consequence one gets: Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There is a dense subset of PX of points x whose stable leaf Ls(x) contains a periodic point p in an attractor A, not of boundary type and so that Lu(p) is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' A symmetric statement holds for repellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The two main steps of the proof of Theorem 13 are Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 33 Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Lu be a leaf of Fu corresponding to an unstable leaf of X contained in a attractor of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let ∆+ and ∆− be the closures in D2 X of the half planes in R2 bounded by Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then there are g+, g− ∈ π1(M) so that g−(∆−) ⊂ ∆+ and g+(∆+) ⊂ ∆−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The same statement holds for stable leaves in the repellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Ls and Lu be leaves of Fs and Fu in a repeller and in an attractor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let I ⊂ S1 X be a segment with non empty interior and whose end points are the limit of both ends of the same leaf, Ls or Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then every orbit of the action of π1(M) contains points in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 there is g ∈ π1(M) so that g(S1 \\ I) ⊂ I, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Given any non-empty open interval J ⊂ S1 X, there is a L which is either a leaf of Fs in a repeller or a leaf of Fu in an attractor whose both ends have limits in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Theorem 14 assuming Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2, every interval J with non-empty interior contains an interval I whose end points are both limit point of the end of a stable or unstable leaf in an a repeller or an attractor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now, according to Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1, the interval I contains a point in every π1(M) orbit in S1 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus any π1(M) orbit in S1 X has points in any interval with non-empty interior: in other words, every π1(M) orbit si dense in S1 X, or else, the action of π1(M) on S1 X is minimal, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Lu 0 be an unstable leaf in an attractor A0, and ∆+ 0 be the closure of the upper half plane bounded by Lu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For proving Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 we want to prove that there is f ∈ π1(M) so that f(∆− 0 ) ⊂ ∆+ 0 (the other announced inclusion is identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a point p0 ∈ Lu 0 and Ls 0 the stable leaf through p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' There is an unstable leaf Lu 1 with the following property: Lu 1 ⊂ ∆+ 0 Lu 1 is contained in the basin of a repeller R1 Lu 1 contains a non-boundary periodic point p1 ∈ Lu 1 of the repeller R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lu 1 cuts the stable leaf Ls 0 in a point Lu 1 ∩ Ls 0 = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The union of unstable leaves in the basin of a repeller and carrying a non-boundary periodic point of this repeller is dense in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We can therefore choose such a leaf in ∆+ 0 and cutting Ls 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Let Ls 1 be the stable leaf through p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is a non-boundary stable leaf contained in the repeller R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that Ls 1 is disjoint from the attractor A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus Ls 1 is disjoint from Lu 0 ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' the stable leaf Ls 1 is distinct, and therefore disjoint from the stable leaf Ls 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In other words, the union Ls 0 ∪ Lu 0 divides PX in 4 quadrants and Ls 1 contained in one of this quadrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let us denote by C±,± these 4 quadrants so that ∆+ 0 = C−,+ ∪ C+,+ and Ls 1 ⊂ C+,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let denote by ∆+ 1 = ∆+(Ls 1) the closure of the half plane bounded by Ls 1 and contained in ∆+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus ∆+ 1 is contained in the same quadrant C+,+ as Ls 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by ∆− 1 the closure of the other half plane bounded by Ls 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Note that ∆− 1 contains the 3 other quadrants, in particular it contains ∆− 0 and C−,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As the leaf Ls 1 is (by assumption) not a boundary leaf of R1 it is accumulated on both sides by its π1(M)-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus there is a leaf Ls 2 = g(Ls 1) in its orbit, cutting Lu 1 at a point x ∈ ∆− 1 arbitrarilly close to p1 and hence x ∈ C+,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Notice that Ls 2 is contained in the repeller R1 and thus is disjoint from Lu 0 ∪ Ls 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus it is contained in one quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As it contained x ∈ C+,+ one has Ls 2 ⊂ C+,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let h ∈ π1(M) be the generator of the stabilizer p1 so that Lu 1 is expanded by h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider the sequence of leaves hn(Ls 2) which cut Lu 1,− at the point hn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Claim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For n large enough hn(Ls 2) is contained in the quadrant C−,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 34 CHRISTIAN BONATTI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Each leaf hn(Ls 2) intersects Lu 1 ⊂ ∆+ 0 and is disjoint from Lu 0 (because hn(Ls 2) is contained in the repeller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Hence hn(Ls 2) is contained in ∆+ 0 , and is distinct and therefore disjoint from Ls 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus hn(Ls 2) is contained in one of the quadrants C+,+ of C−,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The point xn tends to infinity in Lu 1 and so goes further q0 = Ls 0 ∩ Lu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus for n large enough xn ∈ C−,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We proved the for n large enough hn(Ls 2) ⊂ C−,+, proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ We conclude the proof of proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 by proving : Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider n large enough so that hn(Lu 2) ⊂ C−,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Then either g(∆− 1 ) ⊂ C+,+ ⊂ ∆+ 0 or hng(∆− 1 ) ⊂ C−,+ ⊂ ∆+ 0 As ∆− 1 contains ∆− 0 the claim implies that either g(∆− 0 ) ⊂ ∆+ 0 or hng(∆− 0 ) ⊂ ∆+ 0 which concludes the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume g(∆− 1 ) is not contained in C+,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As g(∆− 1 ) is one of the half plane bounded by g(Lu 1) = Lu 2 ⊂ C+,+ one gets that g(∆+ 1 ) is the half plane bounded by Lu 2 and contained in C+,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular,g(∆+ 1 ) does not contain q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As p1 and q0 are on distinct sides of Lu 2 one deduces that p1 ∈ g(∆+ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As p1 is the fixed point of h one deduces p1 ∈ hng(∆+ 1 ) Thus hng(∆+ 1 ) is the half plane bounded by hn(Lu 2) which is not contained in the quadrant C−,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus hng(∆− 1 ) is the other half plane bounded by hn(Lu 2) and is contained in C−,+, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We want to prove that any open interval I in the circle S1 X contains the two ends of an unstable leaf in an attractor or the two ends of a stable leaf of a repeller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assuming ∗, there are dense subsets Es 0, Eu 0 of S1 X so that any p ∈ Es 0 is the limit of a regular leaf of Fs containing a periodic point x which belongs to an attractor A(p), and is not of boundary type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' any q ∈ Eu 0 is the limit of a regular leaf of Fu containing a periodic point y which belongs to an repeler R(p), and is not of boundary type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 the union of regular stable leaves containing periodic point of non- boundary type of an attractors are dense in PX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' This family is therefore separating, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus the limits of their ends is a dense subset of S1 X, as announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assuming ∗, there is a dense subset E ⊂ S1 X so that every x ∈ E is the limit of the end a regular leaf of Fs (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Fu) contained in a repeller R (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' an attractor A), and carrying a periodic point of non-boundary type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider a non empty open interval I ⊂ S1 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 there is a point x ∈ I which is the limit of an end Ls +(p0) of a regular leave of Fs carrying a periodic point p0 in a non-boundary type unstable leaf Lu(p0) of a attractor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The point p0 is accumulated on both sides by periodic points in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We chose p1 so that the limit y of Ls +(p1) belongs to I (that is possible because Ls +(p0) is regular) and Ls +(p1) intersects Lu(p0) at a point q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus let J ⊂ I be the segment contained in I and whose end points are x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='Notice that y ̸= x,that is J has non-empty interior, as Ls(p0) is a regular leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Lu(p0) is accumulated on both sides by regular unstable leaves contained in the attractor A and containing periodic point of non-boundary type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Lu 0 be such a leaf, with non empty intersection with Ls +(p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If Lu 0 does not cut Ls +(p1), then one ends is contained in the half strip bounded by Ls +(p0), the segment of [p0, q1]u and Lu +(q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As a consequence, the limit of this end belongs to I and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus we may assume now that Lu 0 cuts Ls +(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' CIRCLE AT INFINITY OF FOLIATIONS OF R2 35 Let h0 and h1 be the generators of the stabilizers of p0 and p1, respectively, so that h0 expands Ls +(p0) and h1 expands Ls +(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider the images {hn 0(Lu 0), hn 1, n ∈ N(Lu 0)} of the leaf Lu 0 by the positive iterates of h0 and h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Each of these images is an regular unstable leaf in A, and has a non-empty intersection with either Ls +(p0) or Ls +(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' If one of these leaves does not cross both Ls +(p0) and Ls +(p1), then it has an end in the segment J ⊂ I, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Assume now that every leaf in {hn 0(Lu 0), hn 1 , n ∈ N(Lu 0)} crosses both Ls +(p0) and Ls +(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' These images are leaves of Fu, and therefore they are either disjoint or equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' For L ∈ {hn 0(Lu 0), hn 1, n ∈ N(Lu 0)}, let D(L) ⊂ D2 F be the disk obtained as follows: one cuts along L the strip bounded by Ls +(p0) and Ls +(p1), one gets two components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' one considers the closure in D2 F of these components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' now D(L) is the one containing the segment J ⊂ S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The disks D(L) are naturally totally ordered by the inclusion and we fix the indexation {hn 0(Lu 0), hn 1(Lu 0), n ∈ N} = {Lu n} according to this order: D(Lu n+1 ⊂ D(Lu n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Consider D = � n(D(Lu n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is a compact subset of D2 F whose intersection with S1 F is the segment J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Claim 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' D ∩ (Ls +(p0) ∪ Ls +(p1)) = ∅ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The leaves hn 0(Lu 0) have their intersection with Ls(p0) tending to x as n → ∞: one deduces that D ∩ Ls(p0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The leaves hn 1(Lu 1) have their intersection with Ls(p1) tending to y, and thus D ∩ Ls(p1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Claim 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' D \\ S1 F ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' there is a point z in the interior of J which is the limit of an end of leaf of Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus there is an half unstable leaf Lu + contained in the strip bounded by Ls +(p0) and Ls +(p1), and whose limit is z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Lu + is disjoint from all the Lu n, and therefore Lu + ⊂ D(Lu n), ∀n This concludes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ Consider now a point t ∈ D \\ S1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The leaf Lu(t) is disjoint from the leaves Lu n for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus it has an empty intersection with (Ls +(p0) ∪ Ls +(p1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As the consequence one gets Lu(t) ⊂ D In particular, Lu(t) has both ends on J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Suppose now that the point t ∈ D \\ S1 X as been chosen on the boundary of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Thus t is a limit of points in Lu n ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As A is a closed subset of R2 = ˚ D2 F one deduces that t ∈ A, and so Lu(t) ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' One just found a leaf Lu(t) contained in A and having both ends in J ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let Dt ⊂ D be the disc bounded by Lu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='We are not yet done, because Lu t may fail to be a regular leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1 implies that every unstable leaf has an image by an element k ∈ π1(M) which is contained in Dt, for instance Lu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now k(Lu 0) is a regular unstable leaf in an attractor which has the limits of its both ends contained in J ⊂ I, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ We are now ready for ending the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2, and therefore of Theorem 14 which ends the proof of Theorem 13 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Let I ⊂ S1 X be a non-empty open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' According to Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='3 there is a regular unstable leaf Lu 0, contained in an attractor A and containing a periodic point of non-boudary type p0, and having an end, say Lu 0,+, whose limit is a point x ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As Lu 0 is not a boundary leaf of A there are unstable leaves in A arbitrarily close to Lu 0, on both sides of Lu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As furthermore Lu 0 is a regular leaf, one can chose a leaf Lu 1 ⊂ A so that the limit of the end Lu 1,+ is a point y ∈ I with [x, y] ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' there is a segment σ of a stable leaf having both ends a and b on Lu 0 and Lu 1 repsectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 36 CHRISTIAN BONATTI We denote by Dσ the disc in D2 X bounded by σ, [x, y] , Lu +(a) ⊂ Lu 0 and Lu +(b) ⊂ Lu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Now according to Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2 there is a point z ∈ [x, y] which is limit of the end Lu + of a unstable leaf Lu which carries a periodic point q in a repeller R, and q is not of boundary type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by h ∈ π1(M) the generator of the stabilizer of q which is expanding along Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' The stable leaf Ls(q) is contained in the repeller R and is accumulated on both sides by stable leaves in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We denote by Ls 0 a stable leaf in R crossing Lu + at a point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We consider Ls n = hn(Ls 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' It is a stable leaf in R which cuts Lu + at the point xn = hn(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Not that xn → z as n → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' In particular, xn belongs to the disc Dσ for n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As R ∩ A = ∅ the leaves Ls n are disjoint from Lu 0 and Lu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' As two distinct stable leaves are disjoint they are (all but at most one of them) disjoint from σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' So for large n, the leaf Ls n is contained in Dσ and therefore as its both ends on [x, y] ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' We just exhibit a stable leaf in a repeller, whose both ends are in I, that is we ended the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' □ References [Ba1] Barbot, Thierry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Caract´erisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles, Ergodic Theory Dynam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Systems 15 (1995) 247-270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' [BFM] Barthelm´e, Thomas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Frankel, Steven;' 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and applications to large scale geometry, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 16 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 1, 1–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' [Ke] Alexander S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Kechris, Classical Descriptive Set Theory, Berlin, New York, Springer-Verlag, 1995, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' [Ma] Mather, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Foliations of surfaces I : an ideal boundary Annales inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Fourier 32 (1982), page 235-261 [Th] Thurston, William P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Three-manifolds, Foliations and Circles, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' Unfinished manuscript, 1998 Christian Bonatti bonatti@u-bourgogne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content='fr Institut de Math´ematiques de Bourgogne1, UMR 5584 du CNRS, Universit´e de Bourgogne, 21000, Dijon, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} +page_content=' 1The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE3T4oBgHgl3EQfdAo0/content/2301.04530v1.pdf'} diff --git a/cNE4T4oBgHgl3EQfow2y/content/tmp_files/load_file.txt b/cNE4T4oBgHgl3EQfow2y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4cb318ff4f089b0aece6a0cf1b974c93c6c563c4 --- /dev/null +++ b/cNE4T4oBgHgl3EQfow2y/content/tmp_files/load_file.txt @@ -0,0 +1,968 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf,len=967 +page_content='WIRE: Wavelet Implicit Neural Representations Vishwanath Saragadam, Daniel LeJeune, Jasper Tan, Guha Balakrishnan, Ashok Veeraraghavan, Richard G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Baraniuk Rice University https://vishwa91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='io/wire Abstract Implicit neural representations (INRs) have recently ad- vanced numerous vision-related areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' INR performance depends strongly on the choice of the nonlinear activation function employed in its multilayer perceptron (MLP) net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A wide range of nonlinearities have been explored, but, unfortunately, current INRs designed to have high ac- curacy also suffer from poor robustness (to signal noise, pa- rameter variation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Inspired by harmonic analysis, we develop a new, highly accurate and robust INR that does not exhibit this tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Wavelet Implicit neural REpresen- tation (WIRE) uses a continuous complex Gabor wavelet activation function that is well-known to be optimally con- centrated in space-frequency and to have excellent biases for representing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A wide range of experiments (im- age denoising, image inpainting, super-resolution, com- puted tomography reconstruction, image overfitting, and novel view synthesis with neural radiance fields) demon- strate that WIRE defines the new state of the art in INR accuracy, training time, and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Introduction Implicit neural representations (INRs), which learn a continuous function over a set of data points, have emerged as a promising general-purpose signal processing frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' An INR consists of a multilayer perceptron (MLP) with alternating linear layers and element-wise nonlinear activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Thanks to the MLP, INRs do not share the locality biases that can limit the performance of convo- lutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Consequently, INRs have advanced the state of the art in numerous vision-related ar- eas, including computer graphics [24, 29, 30], image pro- cessing [11], inverse problems [43], and signal representa- tions [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Currently, INRs still face a number of obstacles that limit their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' First, for some applications, especially those with high-dimensional data such as 3D volumes, fitting an INR to high accuracy can still take too long (tens of seconds) for WIRE Gauss SIREN Ground truth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='0 Spatially spread- out error Spatially compact but large error Spatially compact and small error Error map SIREN sin(𝜔0𝑥) Gauss 𝑒− 𝑠0𝑥 2 WIRE 𝑒j𝜔0𝑥𝑒−|𝑠0𝑥|2 𝜎 𝑥 𝜎 𝑥 Re 𝜎 𝑥 𝑥 𝑥 𝑥 Nonlinearity for Implicit Neural Representations Approximation accuracy with various nonlinearities Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Wavelet implicit neural representation (WIRE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We propose a new nonlinearity for implicit neural representations (INRs) based on the continuous complex Gabor wavelet that has high representation capacity for visual signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The top row vi- sualizes two commonly used nonlinearities: SIREN with sinu- soidal nonlinearity and Gaussian nonlinearity, and WIRE that uses a continuous complex Gabor wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE benefits from the fre- quency compactness of sine, and spatial compactness of a Gaus- sian nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The bottom row shows error maps for approx- imating an image with strong edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' SIREN results in global ringing artifacts while Gaussian nonlinearity leads to compact but large error at edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE produces results with the smallest and most spatially compact error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' This enables WIRE to learn repre- sentations rapidly and accurately, while being robust to noise and undersampling of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' real time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Second, INRs are not robust to sig- nal noise or insufficient measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Indeed, most works on INRs in the literature assume virtually no signal noise and large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We find in our own experiments that current INR methods are ineffective for tasks such as denoising or super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Finally, INRs still have room for improvement in representational accuracy, especially for fine details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='05187v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='CV] 5 Jan 2023 In this paper, we develop a new, faster, more accurate, and robust INR that addresses these issues and takes INR performance to the next level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' To achieve this, we take in- spiration from harmonic analysis and reconsider the nonlin- ear activation function used in the MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Recent work has shown that an INR can be interpreted as a structured signal representation dictionary [53], where the activation nonlin- earity dictates the atoms of the dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For example, the sine activation creates a pseudo-Fourier transform rep- resentation of the signal that is maximally concentrated in the frequency domain [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' An important conclusion one can draw from the past four decades of harmonic analysis research is that Fourier meth- ods are suboptimal for representing the kinds of signals that feature in typical vision tasks [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' These kinds of signals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=', natural images from photographs, are much more con- cisely and robustly represented using wavelet atoms that are optimally concentrated in space–frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Sparse compo- sitions of wavelet atoms are known to have excellent bi- ases for representing images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' the seminal work in com- puter vision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=', Laplacian pyramid), computational neu- roscience [32], and the JPEG2000 compression standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In this paper, we introduce Wavelet Implicit neural REpresentation (WIRE), a new INR based on a complex Gabor wavelet activation function (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Through a wide range of experiments, we demonstrate that WIRE defines the new state of the art in INR accuracy, train- ing time, and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We showcase that WIRE’s in- creased robustness is particularly useful for solving difficult vision inverse problems, including image denoising (robust- ness), image inpainting and super-resolution (superior inter- polation), and 2D computed tomography (CT) reconstruc- tion (solving higher-dimensional inverse problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE also outperforms other INRs for signal representation tasks such as overfitting images and learning point cloud occu- pancy volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Finally, we show that WIRE enables faster, more robust novel view synthesis with neural radiance fields (NeRF) [29] from critically few training views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Prior Work Regularization for inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Inverse problems involve estimating a signal from a linear or nonlinear set of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Inevitably, the measurements are de- graded by noise (such as camera readout or photon noise), or the problem is ill-conditioned, necessitating regulariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' There are many forms of regularization, including ridge regression, Lasso [46], total variation (TV) [10], and sparsity-based [7] techniques that seek to penalize the ℓ1 norm the signal or some transform thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In the past decade, data-driven regularization, including overcomplete dictionary-based [4] and generative network-based [31, 36, 37] ones, have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The classical model-based approaches are inadequate for severely ill-conditioned prob- lems, while the data-driven ones critically depend on data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' CNNs, the most popular neural network architectures in computer vi- sion for the past decade, have been shown to exhibit strong implicit biases that favor image-like signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' This has been demonstrated with works like deep image prior (DIP) [48] and its variations [15, 21] that produce remarkable results on image-related linear inverse problems without any prior training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' However, such CNN-based priors are tied to a discrete grid-like signal representation which is not applica- ble to problems such as novel view synthesis, or for solving ordinary and partial differential equations, and not scalable for very high dimensional signals such as 3D tomographic volumes, gigapixel images, or large point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Deep image prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Neural networks, and particularly CNNs, exhibit implicit biases due to their specific archi- tectures (such as a UNet [38]), implying that even untrained neural networks can be used for regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' This was leveraged to build a deep image prior (DIP) [48] that pro- duces outputs that tend to look like images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The key idea is to recast regularization as optimizing for the weights of the network for each instance of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The perfor- mance of DIPs is considerably superior to classical regu- larization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' However, DIPs exhibit good per- formance only when over-parameterized and are tied to a grid-like discretized representation of the signal, implying DIPs do not scale to high dimensional signals such as point clouds with a large number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The issue of compu- tational cost has been addressed to a certain extent by the deep decoder [21] and the DeepTensor [39], but they still need the signal to be defined as a regular data grid such as a 2D matrix or 3D tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Implicit representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' INRs are continuous learned function approximators based on multilayer perceptrons (MLPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The continuous nature of INRs is particularly ap- pealing when dealing with irregularly sampled signals such as a point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Since its first widespread usage in novel view synthesis in graphics [29], INRs have pervaded nearly all fields of vision and signal processing including render- ing [24], computational imaging [6, 12], medical imag- ing [51], and virtual reality [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The popular choice of the ReLU nonlinearity in stan- dard neural networks has been empirically shown to result in poor approximation accuracy in INRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' This has been remedied by several modifications to the MLP including the so-called positional encoding [30, 44], as well as var- ious choices of nonlinearity such as the sinusoidal func- tion [42] and the Gaussian function [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A closely related work is the Gabor wavelet-based multiplicative filter net- works (MFN), where the output after each layer is multi- 2 plied by a Gabor filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The output then results in a combina- tion of exponentially many Gabor wavelets, thereby result- ing in large capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Numerous architectural changes have also been proposed that leverage multiscale properties of vi- sual signals to accelerate the INR training procedure includ- ing adaptive block decomposition [27], kilo-NeRF [35], and predicting the Laplacian pyramid of the signal [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' INRs can now train on signals nearly instantly [30] thanks to these numerous advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' However, the high ca- pacity of such INRs precludes robustness — implying that the signal representation is brittle, resulting in overfitting to both noise and signal equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In this paper, we propose the complex Gabor wavelet as a nonlinearity, which is uniquely well-suited to induce robustness in INRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Wavelet transform and the Gabor wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The Fourier transform decomposes the signal as a sum of sinusoids with infinite space support, implying that there is no no- tion of spatial compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The wavelet transform reme- dies this by decomposing the signal as a linear combina- tion of translated and scaled versions of a short oscillating pulse called a wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Wavelets typically result in faster ap- proximation rates for signals and images than Fourier trans- form [17], and hence they are often used for image com- pression [18, 41] and as a robust prior for inverse problems of images [20] and videos [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In this paper, we show that wavelets are a universally superior choice for the nonlin- earity in INRs due to their compact support in space and frequency and therefore faster approximation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Wavelet Implicit Representations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' INR details Consider an INR function Fθ : RDi �→ RDo mapping Di input dimensions to Do output dimensions, where θ repre- sents the MLP’s tunable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The goal is to con- struct Fθ such that it approximates a function g(x) of in- terest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=', g(x) ≈ Fθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For example, g(x) may simply be a ground truth image, represented as a function mapping coordinates to pixel values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Modeling Fθ(·) as an M-layer MLP, the output at each layer is given by ym = σ(Wmym−1 + bm), (1) where σ is the nonlinearity (or nonlinear activation func- tion);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Wm, bm are weights and biases for the mth layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' y0 = x ∈ RDi is the input coordinate and yM+1 = WM+1yM + bM+1 is the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The nonlinear activation σ plays a key role in the rep- resentation capacity of the INR (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Two lead- ing choices include the periodic σ(x) = sin(ω0x) used in SIREN [42], and the Gaussian nonlinearity σ(x) = e−(s0x)2 used by Ramasinghe et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' [34]—both result in significantly higher representation accuracy than ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' However, their high representation capacity is also a draw- back, since they can represent noise with nearly equal ac- curacy as an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Our goal is to propose a nonlinearity σ that is well-suited for visual signals such as images, videos, and 3D volumes but poorly fits noise-like signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE Armed with the insight that a Gabor wavelet achieves op- timal time-frequency compactness, we propose the wavelet implicit representation (WIRE) that uses the continuous complex Gabor wavelet ψ for its activation nonlinearity: σ(x) = ψ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' ω0, s0) = ejω0xe−|s0x|2, (2) where ω0 controls the frequency of the wavelet and s0 con- trols the spread (or width).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The first layer activations have the form y1 = ψ(W1x + b1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' ω0, s0), (3) which are copies of the mother Gabor wavelet ψ at scales and shifts determined by W1 and b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Hence the building blocks of WIRE are drawn from a dictionary of wavelet atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We let the weights of the INR as well as the outputs be complex-valued to preserve phase relationships through- out, and we represent real signals by simply taking the real part of the output and discarding the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Just as wavelets combine space and frequency compactness, WIRE enjoys the advantages of periodic nonlinearities such as SIREN due to the complex exponential term and the spa- tial compactness from the Gaussian window term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' recall Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Additionally, unlike SIREN, WIRE does not re- quire a carefully chosen set of initial weights (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 3) due to the Gaussian window, which creates a spatially com- pact output at each layer and produces high quality results with the default neural network initialization of uniformly random weights independent of the parameters ω0, s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Implicit bias of WIRE Neural tangent kernel perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' As stated, we seek an INR that fits visual signals well but fits noise poorly in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Inspired by [53], who proposed to com- pare eigenfunctions of the empirical neural tangent ker- nel (NTK) [22] of INRs to understand their approximation properties, we compare the fitting of noisy natural images using NTK gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The NTK gradient flow of INRs accurately captures the behavior of early training of neural networks, and so in tasks such as denoising where we reg- ularize via early stopping, the early training behavior deter- mines the implicit bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In the lazy training regime of wide neural networks [25], the fit image at time t ≥ 0 has value Fθt(x) = [(I − e−tK)g](x), (4) 3 (a) Implicit bias with NTK gradient flow (b) Implicit bias in standard INRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Implicit bias in denoising (a) The empirical NTK of finite-width INRs provides an insight into the implicit bias of INRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Early trajectories of NTK gradient flow show WIRE converging to the image faster than the noise, outperforming all other nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Bars indicate one standard deviation over the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (b) Early iterations of standard training are reflected well by the relative performances of NTK gradient flow from part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Furthermore, WIRE maintains its advantage against other nonlin- earities throughout the remainder of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' where I is the identity operator, K is the NTK operator on the image’s spatial domain, and g is the image being fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2a, we apply NTK gradient flow using the em- pirical finite-width NTK to a denoising task, fitting the original image with N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='052) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' pixel-wise additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Due to the computational intensity of evaluating the NTK, we evaluate on 64 × 64 × 3 images from Tiny Ima- geNet [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Comparing WIRE to other INRs, we see that, as desired, WIRE prefers to learn the signal in the image early in training rather than the noise, converging orders of mag- nitude faster to essentially any given peak signal-to-noise- ratio (PSNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Empirical evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We perform a denoising task anal- ogous to the NTK-based analysis for real INRs on the 24 768 × 512 × 3 images from the Kodak Lossless True Color Image Suite [1], again with N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='052) additive noise, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We apply the same INRs as in the NTK example, but train with ordinary neural network gradient optimiza- tion instead of NTK gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Again, WIRE drastically outperforms other INRs, converging an order of magnitude faster to the same PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Choosing the parameters ω0, s0 WIRE’s performance is primarily decided by the con- stants ω0, s0 that control frequency of the sinusoid and width of the Gaussian, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE outperforms both the SIREN and Gaussian nonlinearities across a broad range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 3 shows the approximation ac- curacy achieved by WIRE for various parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We set the number of hidden layers to three, and number of hidden SIREN (𝑠0 = 0) Image Representation Image denoising Gauss (𝜔0 = 0) PSNR (dB) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE is robust to the choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The plot above shows accuracy for image representation and denoising with various settings of ω0 and s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The boxes show special cases with ω = 0 corresponding to Gaussian nonlinearity, and s0 = 0 cor- responding to SIREN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE achieves higher accuracy than both SIREN and Gauss on image representation as well as image de- noising tasks (marked by white cross).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Further, WIRE achieves super performance for a large choice of parameters ω0, s0 imply- ing that WIRE is not overly sensitive to the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' features to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' When ω0 = 0, we used a Gaussian non- linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' When s0 = 0, we used a sinusoidal nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' When both parameters were zero, we used a ReLU nonlin- earity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For the denoising task, we added photon noise equiv- alent to a maximum of 50 photons per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 3 that WIRE outperforms SIREN, Gauss, and ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Moreover, the performance is superior for a large swath of values of ω0, s0 for both image representation and denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The reduced sensitivity to the exact values of ω0, s0 implies that WIRE can be used without precise in- formation about image or noise statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Alternate forms of WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For problems where com- plex weights are infeasible, WIRE can be instantiated as the imaginary (or real) part of the complex Gabor wavelet, ψ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' ω0, s0) = sin(ω0x)e−(s0x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Note that setting s0 = 0 results in the sine nonlinearity used in SIREN [42] and ω0 = 0 results in Gaussian nonlinearity [34] implying WIRE inherits the favorable properties of previously pro- posed nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Another embodiment of WIRE is a Constant-Q Gabor wavelet where ω0s0 = Q, which re- sults in constant fractional bandwidth (ω/δω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Constant- Q Gabor wavelets are often used in music analysis [8, 47], wavelet transforms [26], and the Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Hav- ing only a single parameter makes hyperparameter tuning simpler with a fixed Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Multidimensional localization As defined above, WIRE applies the Gabor mother wavelet ψ element-wise to output of the linear transfor- mation Wmym−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Hence, the output of each unit is spa- tially localized only in the single direction determined by the corresponding row of Wm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 11 top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' While such highly anisotropic spatial localization is well-suited for certain kinds of data, many natural data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=', photo- graphic images) are best represented using a combination of atoms with isotropic and anisotropic spatial localization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=', wavelets and curvelets [9] or wavelets and wedgelets [50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' To achieve spatial localization along multiple direc- tions, we augment the Gabor mother wavelet with Dm − 1 additional Gaussian windows: ym = ψ(W (1) m ym−1 + b(1) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' w0, s0) · e− �Dm k=2 |s0(W (k) m ym−1+b(k) m )|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (5) In two-dimensional settings, such as with natural image data, the resulting first-layer activations will resemble a mixture of Gabor wavelets and curvelets (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 11 bot- tom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' As we will see below in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4, the more diverse spatial localization of the resulting 2D WIRE representa- tion significantly benefits its performance (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 12 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Experiments WIRE learns representations for all signal classes faster than state-of-the-art techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In addition, WIRE is well- suited to solve a large class of inverse problems where the number of measurements is far fewer than the dimensional- ity of the signal, or when the measurements are corrupted by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For all the experiments below, we implemented the optimization procedure in PyTorch [33] and used the Adam optimizer [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Code was executed on a system unit equipped with 64GB RAM, and an Nvidia RTX 2080 Ti graphical processing unit (GPU) with 8GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Unless specified, we used an ℓ2 loss function between the measure- ments and the outputs of INR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' No other regularization was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We used a learning rate scheduler which decayed the initial learning rate by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1 at the end of training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Signal representation A common feature enabled by INRs is representation of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We evaluate two tasks for this experiment: repre- senting images and representing occupancy volumes [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In both cases, we used an MLP with three hidden layers with a width of 300 features for all nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For WIRE, we reduced the number of parameters by half to ac- count for the doubling due to real and imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We did so by reducing the number of hidden features by √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The parameters for each nonlinearity and the learning rate (a) Image representation (b) Volume representation Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE learns faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The two plots above show rep- resentation accuracy for an image (top row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 5) and an oc- cupancy volume (bottom row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 5) over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Owing to the high approximation capacity of Gabor wavelets for visual signals, WIRE achieves high accuracy at a faster rate, making it an appro- priate choice for representing visual signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' were chosen to obtain fastest approximation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Specif- ically, we chose ω0 = 20, s0 = 10 for WIRE, ω0 = 40 for SIREN, and s0 = 30 for Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We also compare against multiplicative frequency networks (MFN) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For the occupancy volume, we sampled over a 512 × 512 × 512 grid with each voxel within the volume assigned a 1, and voxels outside the volume assigned a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We evaluated the PSNR and structural similarity (SSIM) [52] for images and intersection over union (IOU) for the occupancy volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 4 shows the approximation accuracy as a func- tion of time for an image (Kodak dataset) and an occupancy volume (Thai statue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE not only achieves the highest accuracy, but it does so at a much faster rate than other ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 5 visualizes the final representation of the example image after 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='6 minutes, and the 3D mesh of the Thai Statue constructed with marching cubes after 30 min- utes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE achieves the highest accuracy both for images (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB) and for the occupancy volume (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='99), underlying our hypothesis that INRs equipped with a Gabor nonlinear- ity have higher approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Solving inverse problems of 2D images WIRE’s inductive bias favors images, and hence can be used for solving linear inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' To demonstrate the advantages of WIRE as a strong prior for images, we showcase its performance on image denoising, single image super resolution, and multiimage super resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Image denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' To evaluate the robustness of INRs for representing noisy signals, we learned a representation on a high resolution color image from the DIV2K dataset [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We simulated photon noise with an independently dis- tributed Poisson random variable at each pixel with a max- imum mean photon count of 30, and a readout count of 2, resulting in an input PSNR of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='6 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We then learned a 5 Ground truth WIRE (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='99) ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='98) SIREN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='97) Gauss (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='97) MFN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='94) Ground truth WIRE (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB) ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB) SIREN (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4dB) Gauss (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='0dB) MFN (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE has high representation capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The results above show image representation in the first row and meshes generated with occupancy volumes in the second row with various nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE achieves highest representation accuracy for both data, underlining its advantages as a signal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' representation on this noisy image with various nonlineari- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In all cases, we chose an MLP with two hidden layers and 256 features per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We also compared the denoising result with deep image prior (DIP) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 6 visualizes the final result for each nonlinearity along with metrices for each result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE produces the sharpest image with least amount of residual noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Qualitatively, WIRE’s result is similar to DIP’s, implying that WIRE enjoys inductive bi- ases that make it a good choice for inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Image super resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' INRs function as interpolatants, and hence super resolution is expected to benefit from INRs with good implicit biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We evaluate this hypothesis by implementing 4× super resolution on a DIV2K image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The forward operator can be cast as y = A4x where A4 imple- ments a 4× downsampling operator (without aliasing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We then solved for the sharp image by modeling x as output of an INR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 7 visualizes the result on super resolution of image of a butterfly with various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE pro- duces the sharpest result with crisp details on the butterfly’s antenna and on the wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' As with denoising, WIRE results are similar to DIP, establishing the ubiquity of WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' INRs are particularly advantageous when data interpola- tion needs to be performed on an irregular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' An example of such settings is multi-image super resolution where the images are shifted and rotated with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 21 shows an example of 4× super resolution with four images (and hence 25% compression) from the Kodak dataset [1] simulated with a small sub-pixel motion between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The forward operator is then yk = Ak 4x where Ak 4 encodes the downsampling, and translation and rotation for the kth image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The visualizations in the figure demonstrate that WIRE achieves the highest accuracy and is qualitatively better at reconstructing high frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In con- trast, the Gaussian nonlinearity leads to a blurry reconstruc- tion, while SIREN results in ringing artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Computed tomography (CT) reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Strong signal priors are critical for solving underconstrained prob- lems, and CT reconstruction is one such example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We em- 6 Ground truth Noisy image MFN DIP WIRE Gauss ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' SIREN 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='6dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='34 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='85 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='93 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='7dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='93 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='89 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='6dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='90 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='93 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE is robust to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A powerful feature uniquely enabled by WIRE is the robustness to noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Here, we show an image representation with added shot noise, resulting in an input PSNR of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='6dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Among the various approaches, WIRE results in the highest PSNR and SSIM of any representation, thereby naturally resulting in denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Ground truth Bilinear interpolation MFN DIP WIRE Gauss ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' SIREN 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='92 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='77 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='93 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='91 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='92 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='6dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='90 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='93 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE for single image super resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The figure above shows results for a 4× single image super resolution with various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Thanks to its strong implicit bias, WIRE results in the sharpest reconstruction with quantitatively higher reconstruction metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' ulated 100 CT measurements of a 256 × 256 x-ray col- orectal image [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 9 shows the final reconstruction with various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE results in the sharpest re- construction with clearly pronounced features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' SIREN per- forms the second best but has striation artifacts that are ex- pected from an unregularized reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The Gaussian nonlinearity results in overly smooth results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE can hence be used as a robust prior for inverse problems with noisy and undersampled measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Learning neural radiance fields INRs have been leveraged successfully for novel-view synthesis with neural radiance fields (NeRF) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Given images from a sparse set of view points, the goal is to render an image from a different view point that is not in the train- ing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' NeRF achieves this by training a common INR that takes 3D location and viewing directions as inputs (hence a 5D input), and produces transmission and color at that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Images are then produced by integrating along lines that pass through each view’s lens (pinhole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The sim- plest NeRF architecture consists of positional encoding, and two MLPs equipped with ReLU for transmission and color values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We show that WIRE without any positional en- coding produces higher quality results within fewer epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We trained NeRFs for reconstruction on the synthetic drum dataset [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Each image was downsampled to a resolution 7 Q00Ground truth Bicubic (4x) WIRE ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc SIREN Gauss 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='8dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='58 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='80 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='7dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='71 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='74 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='75 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Multi-image super resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' INRS are particularly appealing for handling data on an irregular grid, such as images captured with multiple sub-pixel shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The figure above shows 4× super resolution with 4 images captured with varying sub-pixel shifts and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We then solved a joint inverse problem where the high resolution image is modeled as the output of an INR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE produces the best reconstruction both quantitatively and qualitatively, implying that WIRE has favorable interpolation properties for visual signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Ground truth TV-regularized MFN DIP WIRE Gauss ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' SIREN 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='8dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='75 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='23 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='82 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='73 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='71 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='76 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='81 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Computed tomography reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Inverse problems with noisy undersampled data require a strong signal prior for robust reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Here, we show CT-based reconstruction with 100 angles for a 256 × 256 image (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5× compression) with various approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE results in sharp reconstruction, exposing features that are blurry, or with ringing artifacts in reconstructions with other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE is hence a strong signal prior for images, and can solve a large class of inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4dB) ReLU+Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='Enc (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB) SIREN (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB) Gauss (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Novel-view synthesis with neural radiance fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' INRs have shown most promise in novel-view synthesis where the transmit- tance and color at each 3D voxel is modeled as output of INRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Here, we show that WIRE is well-suited for novel-view synthesis with no additional positional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE not only achieves higher accuracy (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB) with fewer epochs, but captures details that are missed out by other nonlinearities, such as the rod connecting the ride cymbal to its stand and the anisotropic reflections on the cymbals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 8 24 23 (dB) 22 PSNR ( 21 20 -WIRE e -ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' -A--SIREN 19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='. Gauss 0 500 1000 1500 EpochReal WIRE Imaginary Real Imaginary 2D WIRE Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' First layer outputs with multi-dimensional WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The figure above shows outputs after first hidden layer with WIRE and 2D WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We observe that 2D WIRE has spatially compact outputs due to the second Gaussian window, while WIRE has elon- gated structures orthogonal to the Gaussian window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' of 200 × 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' To show the advantages of WIRE, we trained the radiance field with only 25 images instead of the default 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We used the “torch-NGP” codebase [45] for training the NeRF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For all experiments, we chose a 4-layered MLP with a width of 128 features for each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Parameters and learning rate were chosen to achieve fastest rate of increase of approximation accuracy on the validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 10 shows results with various nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE produces highest accuracy (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB) with fastest rate of increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE learns features absent in outputs of other nonlinearities such as the rod connecting the ride cymbal to its stand and the anisotropic reflections on the cymbal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Multi-dimensional WIRE comparisons As we discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5, WIRE can be instanti- ated as a multi-dimensional non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' One advantage of a two-dimensional WIRE is that its activations tend to be compact along both spatial axes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' This enables a more accurate fit for signals that are composed of spatially compact structures, such as many natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 12 demonstrates the advantage of 2D WIRE’s more general spatial localization for representing an image of point sin- gularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2D WIRE results in a sharper representation than 1D WIRE, which blurs out some of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (a) Ground truth (b) 2D WIRE (c) WIRE (d) Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' time Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Multi-dimensional localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The spatially com- pact nature of 2D WIRE enables representing sparse images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In this example, we show representation of a 256 × 256 image with 256 non-zero values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2D WIRE represents each dot sharply while WIRE tends to blur the features along one of the two axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Both WIRE and 2D WIRE converge rapidly compared to other ap- proaches, as visualized in the plot in the bottom right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The spatial localization of multi-dimensional WIRE is also of great advantage for solving inverse problems more robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 13 compares WIRE and 2D WIRE on de- noising, CT reconstruction, and 4× super-resolution tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In all tasks, ω0 and s0 of the mother wavelet were chosen for best performance, and the total number of parameters were the same for WIRE and 2D WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Across the board, 2D WIRE achieves higher accuracy than WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2D WIRE also learns sharper features for the CT reconstruction of lungs while WIRE’s tend to be more blurry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Similarly, in the super-resolution task, 2D WIRE learns improved high frequency features to represent the center of the flower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Conclusions We have proposed and validated the advantages of WIRE that equips INRs with a complex Gabor wavelet activation nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We have shown with an extensive set of ex- periments that WIRE (a) has higher representation capac- ity, (b) achieves higher accuracy at a faster rate, and (c) has strong inductive biases that make it compelling for solv- ing challenging inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Other activation non- linearities have largely complementary strengths: SIREN 9 国国国WIRE 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='78 CT with 50 projections 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='84 2D WIRE 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='972 4X Super-resolution 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='974 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='95 Denoising 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='96 Ground truth Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Performance with multi-dimensional WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The figure above shows various linear inverse problems solved with INRs equipped with WIRE and 2D WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Across the board, 2D WIRE achieves higher performance in terms of PSNR and SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Visually, we observe that CT reconstruction and super resolution is significantly sharper with 2D WIRE owing to the compact nature of activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' has high representation capacity and trains fast, but is poor at regularizing inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Positional encoding has lower capacity but is a good choice for novel-view synthe- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Gaussian nonlinearity is more favorable for denoising tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' However, WIRE inherits the best properties of all of the above activation nonlinearities, and hence is the current go-to INR solution for signal representation and solving in- verse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Acknowledgements This work was supported by NSF grants CCF-1911094, IIS-1838177, and IIS-1730574;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' ONR grants N00014-18- 12571, N00014-20-1-2534, and MURI N00014-20-1-2787;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' AFOSR grant FA9550-22-1-0060;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' and a Vannevar Bush Faculty Fellowship, ONR grant N00014-18-1-2047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Appendix 1: Experimental Details A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE initialization INRs like SIREN [42] strongly depend on initialization to obtain accurate representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE does not require any initialization except for the default uniform weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' However, since WIRE consists of a complex sinusoidal term, it marginally benefits from SIREN-like initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Image representation Image denoising Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Effect of initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The plots show approximation accuracy for image representation and image denoising (20dB in- put PSNR) across training epochs with SIREN-like weight initial- ization and standard initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE is robust to the initial weights, but marginally benefits from a SIREN-like initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' To understand the dependence, we evaluated approximation accuracy for image representation (no noise), and image de- noising (20dB image noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Here, a SIREN-like weight initialization implies the first layer weights are drawn from U(−1/N, 1/N) and the weights of the rest of the layers are drawn from U(− � 6/(ω0N), � 6/(ω0N)), where N is the number of input features and U(a, b) is a uniform distribution over [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A normal weight initialization in- volves drawing weights from U(−1/ √ N, 1/ √ N) for all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 14 compares the representation accuracy for SIREN-like and standard initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In both cases, we see that the trends are nearly similar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' SIREN-like initializa- tion results in up to 1dB higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Hence WIRE is largely robust to initial parameters which enables easy tun- ing across a large range of hyperparameters ω0, s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE layer visualizations Gabor wavelets uniquely enable space–frequency local- ization, a property we observe is inherited by WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' To evaluate this hypothesis, we visualized the output WIRE composed of an MLP with two hidden layers and 181 hid- den features each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We then learned a representation for a Siemens star test image that consists of all spatial frequen- cies and orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 15 visualizes the input image real and imaginary outputs of 64 hidden features with least vari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The output of each first layer feature consists of one- dimensional Gabor wavelets at various orientations, while the outputs of second layer consist of sparsely populated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 16 visualizes outputs at each layer for various non- linearities and the final approximated image for the Siemens star test image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The sparse outputs of second layer are ev- idently unique to WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Gauss has outputs that look less spares, while SIREN and ReLU with positional encoding result in dense outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' This has a direct consequence on approximation capacity for high frequency parts of the sig- 10 Real part Layer 1 Imaginary part Layer 2 Input image (256x256) Each patch 256x256 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Layer outputs for WIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The input image is a Siemens star test image that contains all spatial frequencies and all angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The patches show outputs (same size as image) of each hidden feature in layer one and two for a two hidden-layer MLP equipped with WIRE nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE results in sparse images which enables high representational capacity for images, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The final result in the bottom row shows that the sparse nature of outputs of WIRE enables high approximation ac- curacy with qualitatively better features at the center of the image which consists of highest spatial frequenices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Gauss follows next as it results in the second most sparse outputs at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' SIREN and ReLU with positional encoding alike produce blurry outputs at the center, primarily due to the non-compact nature of outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE’s ability to de- compose images as a linear combination of sparse images results in high representational capacity for the same num- ber of parameters, as we verify empirically in the next sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Sensitivity to training parameters WIRE is a promising INR model that achieves high rep- resentation accuracy and is robust to a wide range of train- ing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We demonstrate the efficacy of WIRE in this section with several sensitivity analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Effect of learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE performs well for a large range of learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' To understand the performance trends, we learned an image representation with added noise (20dB input PNSR) with various nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We used a 2-layer MLP with 256 hidden features per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 17 shows the maximum representation PSNR with varying learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE has a stable and significantly higher accuracy compared to other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Interestingly, the highest accuracy is achieved at a high learning rate of 2 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' This behavior is also observed with deep im- age prior [48] where a larger learning rate enabled stronger regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Such a similar behavior implies WIRE en- joys strong inductive biases and hence is amenable to solve inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Effect of number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 18a shows a plot of rep- resentation accuracy of an image for varying number of hidden layers with various nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In each case, the number of hidden features were set to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We re- duced the learning rate with increasing layers to avoid diver- gence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE uniformly outperforms other approaches (ex- cept with 0 hidden features), as is to be expected as Gabor wavelets enable high approximation accuracies for images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Interestingly, for a large number of hidden layers (≥ 3), WIRE performance is similar to SIREN and Gaussian non- linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' This is to be expected as the network has a large capacity with so many layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' However, a large number of layers is computationally expensive and often results in an unstable learning regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE therefore is a reliable choice for small to medium number of hidden layers for most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Effect of number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 18b shows approx- imation accuracy for image representation with varying number of hidden features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In all cases, the number of hid- den layers were fixed to be two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The performance of WIRE is similar to other nonlinearities at very low number of hid- den features, where all models similarly lack sufficient rich- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For higher than 128 features, WIRE outperforms other approaches with MFN [19] coming a close second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Inverse problems Computed tomographic reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We showed in Section 4 that computed tomography (CT) benefits from in- ductive biases of INRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Here, we study the effect of num- ber of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 19a shows the ground truth im- age we used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We denoised the original 512 × 512 image [5] with BM3D [14] (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1) to remove streak artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We then simulated CT reconstruction with varying numbers of projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In each case, we used an MLP with three hidden layers and 256 hidden features per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We sampled the INR on a regular grid to first gen- erate the image, and then use Radon transform to obtain the sinogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' From the accuracy plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 19b, we see that WIRE achieves higher PSNR than any other nonlinear- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 20 visualizes the reconstruction with varying num- ber of projections for each nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The reconstruction is visually superior even with small number of projections, which is particularly beneficial for reducing exposure to x- rays during capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 11 WIRE (real part) Gauss SIREN ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Layer 1 Layer 2 Layer 3/output 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='99 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='98 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='96 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='92 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Visualization of hidden layer outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The figure above visualizes outputs of hidden features in the two layers for the Siemens sector test image shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE uniquely results in sparse images, which enables high accurate representation of high frequency parts of the image (center of the sector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Multi-image super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We showed a result on multi-image super-resolution in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Here, we pro- vide more details about the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 21 shows the 512×768 dimensional ground truth image from the Ko- dak dataset [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We simulated a total of four low-resolution images by modifying each 4× downsampled image by a small translation and rotation, thereby resulting in sub-pixel motion between the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We assumed the transforma- tion Ak between the high-resolution frame x and each low- resolution frame yk was known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We represented the high resolution x as output of an INR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In each case, the INR had three hidden layers with 256 hidden features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We then solved a linear inverse problem to estimate the high reso- lution image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 21 shows the reconstructed output for each nonlinearity and their metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The inset shows re- construction of spokes in the motorcycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Visually, WIRE generates the sharpest features without any ringing artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Moreover, WIRE results in 1dB or better reconstruction ac- curacy, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='04 higher SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Neural radiance fields Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' For all experiments, we used the torch-ngp package [45] that implements a wide va- riety of approaches for training neural radiance fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The architecture consists of two networks that predict transmit- tance (sigma) and the color at each voxel respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Each of the two networks consisted of an MLP with four hid- den layers and 182 hidden features each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The color MLP took position (x,y,z) and direction (θ, φ) as inputs, while the transmittance MLP took only the position as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' As with all other experiments, we used 182/ √ 2 = 128 hidden features for WIRE to account for parameter doubling due 12 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Effect of learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The plot above shows approx- imation accuracy for representing a noisy image (input PSNR of 20dB) with various nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE is robust to learning rate, and produces best results with high learning rate of 2 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (a) PSNR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' number of features (b) PSNR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' number of layers Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Effect of number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The plot above shows approximation accuracy for representing an image with varying number of (a) hidden features and (b) hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE outperforms other nonlinearities with 128 or more hidden features, and one or more layers and is nearly the same as SIREN and Gaussian nonlinearities for more than 3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' to complex weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We downsampled the images by 4× to ensure that the model and training data fit in the graphi- cal processing unit’s (GPU) memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 10, we used a total of 25 randomly chosen images to train the NeRF, and then validated it on 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We used a learning rate of 4×10−4 for WIRE and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5×10−4 for all other nonlinearities and reduced it to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1× initial value over a total of 2500 training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Except for ReLU, we did not use any form of positional encoding with other non- linearities as we wished to demonstrate the capacity of each nonlinearity by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (a) Ground truth image (b) PSNR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' number of prjections Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' CT with varying number of projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (a) shows the 512×512 ground truth x-ray image of lungs [5] we used for our CT experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We denoised the original image to remove streak artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (b) shows accuracy as a function of number of measure- ments with various nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Across the board, WIRE out- performs all other approaches by a considerable margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Effect of number of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 22a shows accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' number of epochs for the drums dataset when trained with all 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE results in highest accuracy within 2500 epochs and converges more rapidly than other ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 22b shows accuracy as a function of num- ber of training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB higher than the next competitor SIREN for 25, 50, and 75 images, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4dB higher when trained with 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 23 visualizes one of the reconstructed views for the drums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' We varied the number of images from 25 to 100 and then rendered the image from a novel view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Visually, WIRE generated the most pleasing results including sharp features of the cymbals and their stands, and the smooth membrane on the drum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In contrast, Gaussian nonlinearity results in cloudy artifacts, while SIREN has high frequency artifacts, especially at lower numbers of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' ReLU+positional encoding requires all 100 images and considerably more than 2500 epochs to reconstruct the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In all, WIRE is a a robust solution for training radiance fields, even with a small number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' References [1] Kodak lossless true color image suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' http://r0k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='us/ graphics/kodak/, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Accessed: 2022-11-09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4, 6, 12 [2] Tiny imagenet visual recognition challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' https:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='com/c/tiny-imagenet, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Ac- cessed: 2022-11-03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4 [3] Eirikur Agustsson and Radu Timofte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' NTIRE 2017 chal- lenge on single image super-resolution: Dataset and study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' In IEEE Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Vision and Pattern Recognition (CVPR), July 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 5 [4] Michal Aharon, Michael Elad, and Alfred Bruckstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' K- svd: An algorithm for designing overcomplete dictionaries 13 WIRE Gauss SIREN ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' MFN 50 projections 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='78 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='8dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='76 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='71 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='8dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='76 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='10 100 projections 200 projections 300 projections 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='5dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='83 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='2dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='79 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='6dB 079 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='3dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='74 Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Effect of number of projections on CT accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The images above visualize reconstruction for the lungs image shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 19a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE outperforms all other approaches even with 50 projections (10% measurements) and is visually pleasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' for sparse representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Signal Processing, 54(11):4311–4322, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2 [5] Samuel G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Armato III, Geoffrey McLennan, Luc Bidaut, Michael F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' McNitt-Gray, Charles R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Laderach, Daniel Max, Richard C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Pais, David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Qing, Rachael Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Roberts, Amanda R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Smith, Adam Starkey, Poonam Batra, Philip Caligiuri, Ali Farooqi, Gregory W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The lung image database consortium (lidc) and image database resource initiative (idri): A com- pleted reference database of lung nodules on ct scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Medi- cal Physics, 38(2):915–931, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 11, 13 [6] Benjamin Attal, Eliot Laidlaw, Aaron Gokaslan, Changil Kim, Christian Richardt, James Tompkin, and Matthew O’Toole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' T¨orf: Time-of-flight radiance fields for dynamic scene view synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Neural Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Processing Systems, 34:26289–26301, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2 [7] Richard G Baraniuk, Volkan Cevher, Marco F Duarte, and Chinmay Hegde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Model-based compressive sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Theory, 56(4):1982–2001, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2 [8] Judith C Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Calculation of a constant q spectral trans- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Acoustical Society of America, 89(1):425–434, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 4 [9] Emmanuel J Cand`es and David L Donoho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' New tight frames of curvelets and optimal representations of objects 14 Ground truth One out of the four images Bilinear interpolation MFN WIRE Gauss ReLU + Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' SIREN 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='8dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='71 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='64 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='9dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='75 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='76 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='6dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='73 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='1dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='71 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='8dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='80 Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Multi-image super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' The figure above visualizes multi-frame super-resolution where each sub-frame was simulated with a small known sub-pixel shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE achieves highest reconstruction accuracy with qualitatively better reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (a) Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' epochs for 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (b) Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' number of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' NeRF accuracy on drums dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (a) shows train- ing accuracy at each epoch for various nonlinearities with neural radiance fields when trained with 100 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' WIRE achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content='4dB higher than the next highest (SIREN) when trained with 100 images and does so in a rapid manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' (b) shows accuracy as a function of number of images with WIRE outperforming other approaches for all number of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' with piecewise c2 singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' Pure and Applied Mathematics, 57(2):219–266, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 5 [10] Antonin Chambolle.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} +page_content=' 2, 3 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE4T4oBgHgl3EQfow2y/content/2301.05187v1.pdf'} diff --git a/d9E4T4oBgHgl3EQfpw0m/content/tmp_files/2301.05194v1.pdf.txt b/d9E4T4oBgHgl3EQfpw0m/content/tmp_files/2301.05194v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..319167f0e66e8371ea0316aa3512917f0627893c --- /dev/null +++ b/d9E4T4oBgHgl3EQfpw0m/content/tmp_files/2301.05194v1.pdf.txt @@ -0,0 +1,894 @@ +arXiv:2301.05194v1 [math.CV] 12 Jan 2023 +Albanese morphism of log smooth klt compact K¨ahler +manifold with nef log anticanonical divisor +Xiaojun WU +13 janvier 2023 +R´esum´e +Let (X, ω) be an n-dimensional compact K¨ahler manifold. Let D = �(1 − βj)Yj = �(1 − +βj)[sj = 0] a divisor with simple normal crossings with βj ∈]0, 1[ such that −(KX + D) is nef. +We show that its Albanese map is submersion outside an analytic set of codimension larger +than two with connected fibres. +1 +Regularity of Monge-Amp`ere equation with conic singu- +larities +The following result on the regularities of the Monge-Amp`ere equation is well-known to experts. +Since we have not found a precise reference of some result, we provide some proof here. Let us +fix some notations. Let (X, D) be a log smooth klt pair, i.e. X is a compact K¨ahler manifold, and +D = �(1 − βk)Yk is a R-divisor with simple normal crossing support such that βk ∈]0, 1[ for all +k. The coefficients (βk) are called standard orbifold is βk = 1/nk for some nk ∈ N∗ for all k. We +denote by ωβ the standard cone metric attached with (Cn, D), i.e. +ωβ := i +d +� +k=1 +dzk ∧ d¯zk +|zk|2(1−βk) + i +n +� +k=d+1 +dzk ∧ d¯zk. +A metric ω is said to have conic singularities if it is quasi-isometric to the model metric with +conic singularities : more precisely, near each point p ∈ Supp(D) where Supp(D) is defined by the +equation {z1 · · · zd = 0} for some holomorphic system of coordinates (zi), we want ω to satisfy +C−1ωβ ≤ ω ≤ Cωβ +for some constant C > 0 and some β (near p). We have the following result due to [CGP13]. +Proposition 1. Let X be a compact K¨ahler manifold and D = �(1 − βj)Yj = �(1 − βj)[sj = 0] +a divisor with simple normal crossings with βj ∈]0, 1[. Let µD = dVω/ � +j |sj|2(1−βj) be a volume +form with conic singularities along D, µ ∈ R, and ω a K¨ahler form on X. Then any (bounded) +solution ϕ of +(ω + i∂∂ϕ)n = eµϕµD +is H¨older-continuous and the metric ω + id∂ϕ has conic singularities along D. +In the following section, we start by considering the standard orbifold case to illustrate the +proof. In this case, we have a more precise regularity of ϕ, and we adapt the notations of [CGP13] +in this case. As written at the beginning of this section, the following Proposition 3 is well-known +to experts. +We fix Dn ⊂ Cn the unit polydisk centered at the origin, and a divisor D = �d +k=1(1 − 1/nk)Dk +where Dk = {zk = 0} for all k, and d ≤ n. +Consider the branched cover +π : +Dd × Dn−d +−→ +Dd × Dn−d +(z1, . . . , zd, zd+1, . . . , zn) +�−→ +(zn1 +1 , . . . , znd +d , zd+1, . . . , zn). +1 + +If we denote by w the coordinates upstairs, +π∗ωβ = i +d +� +k=1 +|nk|2dwk ∧ d ¯wk + i +n +� +k=d+1 +dwk ∧ d ¯wk. +In particular, π∗ωβ is a genuine metric upstairs. Recall that in the standard orbifold case, a function +f is said to have orbifold Ck,α regularity (denoted by Ck,α,β) if its pull-back π∗f by the ramified +cover π is Ck,α in the usual sense. More generally, let τ (resp. σ) be a bounded (1, 0)-form (resp. +(1, 1)-form) on Dn \ D. Then we say that τ ∈ Cα,β if for all k, we have π∗τ( +∂ +∂wk ) ∈ Cα in the usual +sense (resp. σ ∈ Cα,β if for all k and l, we have π∗σ( +∂ +∂wk , +∂ +∂wl ) ∈ Cα in the usual sense). Define +C2,α,β := {f ∈ L∞(Dn); f, ∂f, ∂∂f ∈ Cα,β} +with the associated natural norm. It is checked in Lemma 7.2 [CGP13] that this definition coincides +with the definition of Donaldson [Don12]. +Cover X be open sets Ui so that each open set is biholomorphic to the unit polydisk with +ramified cover associated with D. Denote ˜Ui the ramified cover of πi : ˜Ui → Ui. Let Ni be the +ramified order. We have a more precise regularity result than Proposition 1, also stated in [CGP13]. +Proposition 2. Let (X, D) as in Proposition 1 with standard orbifold coefficients, and let ϕ ∈ +L∞(X) be any solution of +(ω + i∂∂ϕ)n = eµϕµD +Then ϕ belongs to the class C2,α,β. +More precisely, in the standard orbifold case, by elliptic regularity, one can show that π∗ϕ is in +fact smooth. +Proposition 3. Let (X, D) as in Proposition 1 with standard orbifold coefficients, and let ϕ ∈ +L∞(X) be any solution of +(ω + i∂∂ϕ)n = eµϕµD +Then π∗ϕ is smooth in each ramified cover. +D´emonstration. Note that π∗dνD is a smooth volume from upstairs. Without loss of generality, +assume that ω = i∂∂ψ for some smooth function locally. Locally the pullback of the Monge-Amp`ere +equation on the ramified cover can be written as +det( +∂2 +∂wi∂wj +π∗(ψ + ϕ)) = geµπ∗ϕ +for some nowhere vanishing smooth function g. By Proposition 2, π∗(ω +i∂∂ϕ) is a genuine metric +with C0,α coefficients. Denote locally +π∗(ω + i∂∂ϕ) = +� +i,j +hij +√ +−1dwi ∧ dwj. +From the Monge-Amp`ere equation, we have an a priori positive lower bound on the determinant +of hij. In particular, if hij ∈ Ck,α for some k ≥ 0, then det(hij)−1 ∈ Ck,α (hence hij ∈ Ck,α). Take +∂ +∂wl log(∀l) for both sides of equation +hij +∂2 +∂wi∂wj +∂ +∂wl +π∗(ω + i∂∂ϕ) = +∂ +∂wl +logg + µ ∂ +∂wl +π∗ϕ. +Inductively for k ≥ 0, by interior Schauder estimates for linear elliptic equations with coefficients +in Ck,α, we obtain the a priori Ck+3,α norm estimate of π∗ϕ in terms of the Ck+1,α norm of π∗ϕ. +Thus π∗ϕ is smooth in any ramified cover by bootstrapping. +We now recall the definition of a singular metric on a vector bundle according to [Paun16]. +2 + +Definition 1. A singular Hermitian metric h on E is given locally by a measurable, possibly +unbounded map with values in the set of semi-positive Hermitian matrices, such that 0 < deth < ∞ +almost everywhere. +By this definition, a solution of the Monge-Amp`ere equation with conic singularities defines a +singular metric on TX. In particular, the solution also induces a singular metric on any quotient +bundle of TX. We observe that by the Monge-Amp`ere equation, the Ricci curvature of the singular +metric is well defined as a current. However, one can notice that the curvature tensor of TX is not +necessarily well-defined as a current with values in semi-positive, possibly unbounded Hermitian +matrices. +Now we return to the existence and regularity of the Monge-Amp`ere equation for the general +coefficient case. +In fact, the work of [Gue13] and [CGP13] gives the following weak estimate for the following +type of Monge-Amp`ere equation. The theorem is not essentially used for the following section, +but the discussion after the theorem also applies to this slight generalisation. So we still state it +explicitly. +Theorem 1. Let X be an n-dimensional compact K¨ahler manifold, and let D = � +i aiDi, E = +� +j bjEj be two effective R-divisors with simple normal crossing support, such that for all 1 ≤ i ≤ r, +0 < ai < 1. Assume that D and E have no common irreducible component. Let ω be a K¨ahler metric +on X, dV a smooth volume form, and let ε > 0. Then the weak solution of the Monge-Amp`ere +equation +⟨(ω + i +2π ∂∂ϕ)n⟩ = eεϕ +� |tj|2bjdV +� |si|2ai +exists, which is smooth on X ∖ (D ∪ E) and has an upper bound by a metric with conic singularity +along D. Here ⟨•⟩ is the positive intersection product defined in [BEGZ10]. Here si(resp. tj) is the +canonical section of O(Di) (resp. O(Ej)) and |si|2 (resp. |tj|2) is the norm of si (resp. tj) with +respect to some smooth metric. +We observe that the existence of a solution is proved in [BEGZ10]. As a consequence of Theorem +1, there exists C > 0 such that the solution has on X ∖ (D ∪ E) an upper bound +ω + i +2π ∂∂ϕ ≤ +Cω +� +i |si|2ai . +By the Monge-Amp`ere equation, we find on X ∖ (D ∪ E) a lower bound +ω + i +2π ∂∂ϕ ≥ eεϕ +� |tj|2bjω +� |si|2ai ( +C +� +i |si|2ai )−(n−1). +Notice that since the solution is smooth on X ∖ (D ∪ E), the above inequalities are satisfied +pointwise. By the result of [BEGZ10], |ϕ| is uniformly bounded on X. In particular, we have +ω + i +2π ∂∂ϕ ≥ C � |tj|2bjω +� |si|2ai +( +C +� +i |si|2ai )−(n−1). +In conclusion outside D ∪ E, the solution ω + +i +2π∂∂ϕ viewed as a Hermitian form over TX with +respect to ω has positive eigenvalues bounded from above by +C +� +i |si|2ai and bounded from below +by C � |tj|2bj +� |si|2ai ( +C +� +i |si|2ai )−(n−1). +Let us observe that for the singular metric on the determinant line bundle of the quotient +bundle Q given by a short exact sequence of vector bundles +0 → S → TX → Q → 0, +the curvature form is well-defined as a current. We detail the argument below. Suppose we are +in the situation of Theorem 1, with the same notation above. Since the metric is smooth outside +D ∪ E, we only need to study the neighbourhood of D ∪ E. By a C∞ splitting of the exact +sequence, we can view Q as a subbundle of TX. ω + +i +2π∂∂ϕ thus induces a Hermitian form over +3 + +Q which we will denote by ω + +i +2π∂∂ϕ|Q. By the minimax principle, for the induced Hermitian +form on Q, the eigenvalues are bounded from above by +C +� +i |si|2ai and bounded from below by +C � |tj|2bj +� |si|2ai ( +C +� +i |si|2ai )−(n−1). To prove that the curvature of det(Q) is well-defined as a current (not +necessarily positive), it is enough to prove that log(det(ω + +i +2π∂∂ϕ|Q)) ∈ L1 +loc. det(ω + +i +2π∂∂ϕ|Q) +is the product of all eigenvalues of the Hermitian form ω + +i +2π∂∂ϕ|Q. Thus we get the estimate for +the potential +|log(det(ω + i +2π ∂∂ϕ|Q))| ≤ +� +i +Cilog|si|2 + +� +j +Cjlog|tj|2 + C +for some Ci > 0, Cj > 0 and C > 0. In the following, we will refer to this type of control as +potentials possessing at most logarithmic poles along D ∪ E. Notice also that for any i, log|zi| is +locally integrable with respect to the euclidean metric. In particular, the curvature of the induced +metric on det(Q) is well defined as a current since locally it is the i∂∂ of some L1 +loc function. +For any global potential ψ (defined on X) possessing at most logarithmic poles along D ∪ E. +As analogy to Monge-Amp`ere operator in the sense of Bedford-Taylor [BT82], we want to define +i∂∂ψ ∧ ωn−1 +ϕ +. Fix a ∈ [0, 1[, the integration +� +log(r)r1−2a = +1 +2 − 2a(log(r) − +1 +2 − 2a)r2−2a +is finite over [0, 1]. In particular, log|z||z|−2a as a function z ∈ C is locally integrable near 0 with +respect to the Lebesgue measure. Thus the coefficients of ψ ∧ ωn−1 +ϕ +are locally integrable with +respect to the Lebesgue measure. Define +i∂∂ψ ∧ ωn−1 +ϕ +:= i∂∂(ψ ∧ ωn−1 +ϕ +). +By Stokes theorem, we have that +� +X +i∂∂ψ ∧ ωn−1 +ϕ += 0. +2 +Albanese morphism +In this section, we start by generalising the results of [Cao13] to log smooth cases with standard +orbifold coefficients. To start with, we need the following result. +Theorem 2. Let (X, ω) be an n-dimensional compact K¨ahler manifold. Let D = �(1−1/nj)Yj = +�(1 − 1/nj)[sj = 0] a divisor with simple normal crossings with nj ∈ N∗ such that −(KX + D) is +nef. Let +0 = E0 ⊂ E1 ⊂ · · · ⊂ Es = TX +be a filtration of torsion-free subsheaves such that Ei+1/Ei is an ω-stable torsion-free subsheaf of +TX/Ei of maximal slope. Then for any i, the slope of Ei+1/Ei with respect to ωn−1, namely +µ(Ei+1/Ei) := +� +X +c1(Ei+1/Ei) ∧ ωn−1, +is positive. +D´emonstration. By the stability condition, to prove the theorem, it is sufficient to prove that for +any i +� +X +c1(TX/Ei) ∧ ωn−1 ≥ 0. +Let r be the generic rank of TX/Ei for some fixed i. The naturally induced morphism ∧rTX → +det(TX/Ei) corresponds to a section τ ∈ H0(X, det(TX/Ei) ⊗ Ωr +X). τ is non-vanishing outside a +closed analytic set of codimension at least two on which TX/Ei is locally free. Fix an arbitrary +smooth metric h on det(TX/Ei). +The critical step is the existence of positive closed (1, 1)-current in a K¨ahler class which is +smooth outside an SNC divisor and whose Ricci curvature can be taken “arbitrary small” outside +the divisor using the theorems in [GP16] and [CGP13]. +4 + +Let Tε be a sequence of smooth forms in c1(−(KX + D)) such that Tε ≥ −εω whose existence +is ensured by the nefness condition. To get the lower bound, we want to solve the following K¨ahler- +Einstein type of equation +Ric(ωϕε) = −εωϕε + εω + Tε + [D] +where ωϕε := ω + i∂∂ϕε is the unknown. Notice that both sides belong to the class c1(−KX). In +order to solve the K¨ahler-Einstein type of equation, we thus solve the following Monge-Amp`ere +equation by Theorem 1. Let γ be a smooth representative of the class {[D]}, which is induced from +the curvature forms of some smooth metrics (O(Yi), hi). By the ∂∂-lemma, there exists fε ∈ C∞(X) +such that Tε+γ = Ric(ω)+ i +2π∂∂fε. The Monge-Amp`ere equation equivalent to the K¨ahler-Einstein +type of equation can be written as +ωn +ϕε = +ωneεϕǫ−fε +� +i |si|2(1−1/ni) +hi +. +Take the same notations as after Proposition 1. Note that π∗ωϕε is a smooth metric when +pulling back onto the ramified cover by Proposition 3. Calculate on the ramified cover +i∂∂π∗log|τ|2 +ωϕǫ,h = i{D′π∗τ, D′π∗τ} +π∗|τ|2 +− i{D′π∗τ, π∗τ} ∧ {π∗τ, D′π∗τ} +π∗|τ|4 +−π∗iΘ(det(TX/Ei), h) − {π∗τ, iΘ(π∗ωϕǫ)π∗τ} +π∗|τ|2 +where {} is a canonical sesquilinear pairing +C∞(∧pT ∗ +X ⊗ Ωr +X ⊗ det(TX/Ei)) × C∞(∧qT ∗ +X ⊗ Ωr +X ⊗ det(TX/Ei)) → C∞(∧p+qT ∗ +X). +Note that the right-handed term on the first line is positive in the sense of currents by Schwarz +inequality. Observe that the right-handed term is locally integrable (and hence well defines its +product with π∗ωn−1 +ϕǫ +in the sense of currents on the ramified cover). Since π∗ωϕǫ is smooth, +the coefficients of +{π∗τ,iΘ(π∗ωϕǫ)π∗τ} +π∗|τ|2 +is locally bounded. On the other hand, |D′π∗τ|2/|π∗τ|2 is +locally integrable with respect to the Lebesgue measure. To see this, let I be the (local) ideal +defined by the coefficients of π∗τ. Without loss of generality, it is enough to consider the case +that I is non-trivial. By Hironaka’s resolution of singularities [Hir64], there exists a modification +p : U ′ → ˜U such that I · OU′ = O(− � +i λiEi) with � +i λiEi an effective SNC divisor. Note that +since τ is non-vanishing outside a codimension 2 set, p is not an identity map. Denote by DI the +(local) ideal defined by the differentials of the coefficients of π∗τ. Then DI · OU′ is contained in +O(− � +i(λi −1)Ei). KU′/ ˜U = � +i νiEi with νi ≥ 1. In particular, the pullback of Lebesgue measure +can be written as � +i |sEi|2νi times a smooth nowhere vanishing form. Thus the coefficient of the +product of p∗(|D′π∗τ|2/|π∗τ|2) with the pullback of Lebesgue measure is locally bounded on U ′. +In particular, |D′π∗τ|2/|π∗τ|2 is locally integrable. +On the ramified cover, the Chern curvature, which is also the Levi-Civita curvature, satisfies +Ric(π∗ωϕε) = −επ∗ωϕε + π∗(Tǫ + ǫω) ≥ −επ∗ωϕε. +Consider +i∂∂π∗log|τ|2 +ωϕǫ,h ∧ π∗ωn−1 +ϕǫ +≥ (−π∗iΘ(det(TX/Ei), h) − {π∗τ, iΘ(π∗ωϕǫ)π∗τ} +π∗|τ|2 +) ∧ π∗ωn−1 +ϕǫ . +On the other hand, by Proposition 2.7 [Cao13], with local curvature calculations, we have +{π∗τ, iΘ(π∗ωϕǫ)π∗τ} +π∗|τ|2 +∧ π∗ωn−1 +ϕǫ +≤ ǫπ∗ωn +ϕǫ. +Note that log|τ|2 +ωϕǫ,h possesses at most logarithmic poles along D. In fact, the dual metric on T ∗ +X +could only vanish along D when the initial metric on TX has conic singularities. In particular, +i∂∂log|τ|2 +ωϕǫ,h ∧ ωn−1 +ϕǫ +is well-defined as discussed after Theorem 1. +5 + +Let θi be a partition of unity associated with the chosen open cover. +0 = +� +i +� +X +θii∂∂log|τ|2 +ωϕǫ,h ∧ ωn−1 +ϕǫ += +� +i +� +X +i∂∂θilog|τ|2 +ωϕǫ,h ∧ ωn−1 +ϕǫ += +� +i +� +˜Ui +1 +Ni +π∗ +i (i∂∂θilog|τ|2 +ωϕǫ,h ∧ ωn−1 +ϕǫ ) = +� +i +� +˜Ui +1 +Ni +π∗ +i (θii∂∂log|τ|2 +ωϕǫ,h ∧ ωn−1 +ϕǫ ). +Thus we have +� +X +c1(TX/Ei) ∧ ωn−1 = +� +i +� +˜Ui +1 +Ni +π∗ +i (θii∂∂log|τ|2 +ωϕǫ,h ∧ ωn−1 +ϕǫ ) + +� +X +iΘ(det(TX/Ei), h) ∧ ωn−1 +ϕǫ +≥ −ǫ +� +i +� +˜Ui +1 +Ni +π∗ +i (θiωn +ϕǫ) = −ǫ +� +X +ωn +ϕǫ = −ǫ +� +X +ωn. +The conclusion follows by taking ǫ → 0+ +Now we consider the general coefficients case following a suggestion of Mihai P˘aun. +Theorem 3. Let (X, ω) be an n-dimensional compact K¨ahler manifold. Let D = �(1 − bj)Dj = +�(1 − bj)[sj = 0] a divisor with simple normal crossings with bj ∈]0, 1[ such that −(KX + D) is +nef. Let +0 = E0 ⊂ E1 ⊂ · · · ⊂ Es = TX +be a filtration of torsion-free subsheaves such that Ei+1/Ei is an ω-stable torsion-free subsheaf of +TX/Ei of maximal slope. Then for any i, the slope of Ei+1/Ei with respect to ωn−1, namely +µ(Ei+1/Ei) := +� +X +c1(Ei+1/Ei) ∧ ωn−1, +is positive. +D´emonstration. Construct as above proof of Theorem 2 a section τ ∈ H0(X, det(TX/Ei) ⊗ Ωr +X). +As in the above theorem 2, construct a sequence of metrics ωϕǫ with conic singularities by solving +Monge-Amp`ere equations. Consider local coordinate (z1, · · · , zn) such that the support of D is +given by {z1 · · · zd = 0}. In the general case, we do a cut-off in a tubular neighbourhood of D of +radius δ and study the asymptotic behavior as δ → 0+ to get the estimate of the slope from the +integration of i∂∂log|τ|2 +ωϕǫ,h ∧ ωn−1 +ϕǫ +over X \ D. Locally +τ = +� +I,|I|=r +τIdzI +with multi-index I = {i1, · · · , ir} of length r and |τ|2 +ωϕǫ by the conic singularities is locally equi- +valent to � +I,|I|=r +� +j,1≤ij≤d |zij|2−2bij |τI|2. Near the origin, +O(1) ≤ −log|τ|2 +ωϕǫ ≤ −log( +� +I,|I|=r +� +j,1≤ij≤d +|zij|2|τI|2) + O(1) +where O(1) means a bounded term. Thus to study the local integrability of −log|τ|2 +ωϕǫ with +respect to some positive locally finite measure, it is enough to study the local integrability of +−log(� +I,|I|=r +� +j,1≤ij≤d |zij|2|τI|2) with respect to that measure which has the same singularities +as some local ideals. Note that outside the support of D (where the metric is smooth), pointwise, +we have +i∂∂log|τ|2 +ωϕǫ,h ∧ ωn−1 +ϕǫ +≥ (−iΘ(det(TX/Ei), h) − ǫωϕǫ) ∧ ωn−1 +ϕǫ . +However, −log(� +I,|I|=r +� +j,1≤ij≤d |zij|2|τI|2) may have strict positive generic Lelong number +along the support of D which may not be integrable with respect to metric with Poincar´e type +singularities along D. (For example, in the Poincar´e type singularity case, the local integrability is +equivalent to the fact that 1/(r|log(r)|α) is locally integrable near 0 if and only if α > 1.) +6 + +Thus we need to rewrite −log(� +I,|I|=r +� +j,1≤ij≤d |zij|2|τI|2) in a golbal form and subtract the +divisorial part along the support of D. +Let ω⌈D⌉ be a smooth metric on X \ D which is locally quasi-isometric near every point in +⌈D⌉ = {z1 · · · zd = 0} to +ω⌈D⌉ := i +d +� +k=1 +dzk ∧ d¯zk +|zk|2 ++ i +n +� +k=d+1 +dzk ∧ d¯zk. +(For example, this kind of metric can be constructed as follows. Cover X by open coordinate +charts such that ⌈D⌉ can be written as zeros of local coordinates. Assume furthermore that the +coordinate charts on which ⌈D⌉ is not irreducible do not intersect each other, up to taking some +further refinement of the open cover. Construct ω⌈D⌉ by glueing local ones via a partition of unity. +Let us check that the glueing metric has desired properties. +Assume that Uz is a coordinate chart such that ⌈D⌉ ∩ Uz = {z1 · · · zd = 0} and that Uw is +a coordinate chart such that ⌈D⌉ ∩ Uw = {w1 = 0} with non-empty intersection. Over Uz ∩ Uw, +z1/w1 ∈ O∗(Uz ∩ Uw) since they define the same irreducible divisor on the intersection. Thus for +some C > 0 large enough, idz1 ∧ dz1/|z1|2 + Ci �n +k=1 dzk ∧ d¯zk is equivalent to idw1 ∧ dw1/|w1|2 + +i �n +k=2 dwk ∧ d ¯wk over any chosen relative compact subset of Uz ∩ Uw. Thus after the partition of +unity, the glueing metric has desired properties. Of course, the glueing metric is not K¨ahler on X.) +Let I be the ideal sheaf defined (locally) by the coefficients of type � +j,1≤ij≤d zijτI to which +log|τ|2 +ω⌈D⌉ is equivalent to. Note that I is locally defined, but its integral closure is globally defined. +In fact, the germ of the integral closure is the holomorphic function germs f such that locally +log|f|2 ≤ log|τ|2 +ω⌈D⌉ + O(1). +We denote this globally defined integral closure by I. Consider the divisorial valuation associated +with Di +µ(I, Di) := max{m ∈ N, I ⊂ Im +Di} +where IDi is the identification of O(−Di) as an ideal sheaf. Consider +σ := +� +i +sµ(I,Di) +Di +which is a global holomorphic section on X where sDi are the canonical sections of O(Di). Consider +log(|τ|2 +ω⌈D⌉,h/|σ|2) such that any Di is not an irreducible component of its pole set. +Now outside the support of D (where the metric is smooth), pointwise, we have +i∂∂log(|τ|2 +ωϕǫ,h/|σ|2) ∧ ωn−1 +ϕǫ +≥ (−iΘ(det(TX/Ei), h) − ǫωϕǫ + +� +j +iµ(I, Dj)Θ(OX(Dj))) ∧ ωn−1 +ϕǫ . +Note that by choice of σ, the poles of log(|τ|2 +ω⌈D⌉,h/|σ|2) contain no irreducible component of +D. In local coordinates, log(|τ|2 +ω⌈D⌉,h/|σ|2) has the same singularities as ideal quotient J := (I : +� +i Iµ(I,Di) +Di +). In the following, we study the local integrability of log(|τ|2 +ω⌈D⌉,h/|σ|2) with respect +to some volume form of mixed type of Poincar´e type and conic type. +Let χ be a cut-off function [0, ∞[→ [0, 1]. Following section 9 of [CGP13], define +ρ(x) := log(log +1 +� +i |sDi|2 ). +Note that −i∂∂ρ is bounded from above by Poincar´e type singularities along the support of D, +which in local coordinates can be written as +i +d +� +k=1 +dzk ∧ d¯zk +|zk|2log2(|zk|) + i +n +� +k=d+1 +dzk ∧ d¯zk. +We claim that +−log(|τ|2 +ω⌈D⌉,h/|σ|2)i∂∂ρ ∧ ωn−1 +ϕǫ +7 + +is locally integrable. It is enough to show that near the support of D in local coordinates, for any +i0 ∈ [1, d], +−log(|τ|2 +ω⌈D⌉,h/|σ|2)id(zi0) ∧ d(zi0) +|zi0|2log2(|zi0|) ∧ +� +i̸=i0,i≤d +id(zi) ∧ d(zi) +|zi|2(1−βi) +∧ +� +i≥d+1 +idzi ∧ dzi +is locally integrable. To simplify the notations, in the following, we assume that i0 = 1. +Consider the branched cover +π : +Dd × Dn−d +−→ +Dd × Dn−d +(z1, z2, . . . , zd, zd+1, . . . , zn) +�−→ +(z1, zn2 +2 , . . . , znd +d , zd+1, . . . , zn). +with ni(∀i ≥ 2) large enough such that βini ≥ 1. The local integrability is equivalent to the fact +that +π∗� +− log(|τ|2 +ω⌈D⌉,h/|σ|2)id(z1) ∧ d(z1) +|z1|2log2(|z1|) ∧ +� +d≥i≥2 +id(zi) ∧ d(zi) +|zi|2(1−βi) +∧ +� +i≥d+1 +idzi ∧ dzi +� +is locally integrable. +Let fj(1 ≤ j ≤ r) be the local generators of π∗J . A sufficient condition is that +−log( +� +j +|fj|2)id(z1) ∧ d(z1) +|z1|2log2(|z1|) ∧ +� +i≥2 +id(zi) ∧ d(zi) +is locally integrable with the choice of branched cover. +By the construction of J , the restrictions of fj on {z1 = 0} are not all identically zero. By +continuity, on Dr0 × Dn−1 for small radius r0, the restrictions of fj on {z1 = s}(∀|s| < r0) are not +all identically zero. +Consider −log(−log(� +j |fj|2)) as a continuous family of psh functions on Dn−1 parametrized +by z1 ∈ Dr0. Note that the Lelong number of any psh function in this family at any point in Dn−1 +is 0. +By uniform Skoda integrability theorem (cf. e.g. Theorem 2.50 [GZ17]), for any s ∈ D such that +|s| < r0, +� +{s}×Dn−1 −log( +� +j +|fj|2) +� +i≥2 +id(zi) ∧ d(zi) ≤ C +for some C > 0 independent of s by vanishing Lelong number. Since the Poincar´e metric in one +variable is locally integrable, we finish the proof of the claim about the local integrability by the +Fubini theorem. +Similarly, by Schwarz inequalities, we can show that +−log(|τ|2 +ω⌈D⌉,h/|σ|2)i∂ρ ∧ ∂ρ ∧ ωn−1 +ϕǫ +is locally integrable. +For each δ > 0, let θδ : [0, ∞[→ [0, 1] be a smooth function which is equal to zero on the +interval [0, 1/δ], and which is equal to 1 on the interval [1 + 1/δ, ∞]. One may for example define +θδ(x) = χ(x − 1/δ). Consider χδ : X → [0, 1] defined by +χδ(x) := 1 − θδ(ρ(x)). +Consider Stokes’s formula +� +X +χδi∂∂(log(|τ|2 +ωϕǫ,h/|σ|2)) ∧ ωn−1 +ϕǫ += +� +X +∂χδ ∧ i∂log(|τ|2 +ωϕǫ,h/|σ|2) ∧ ωn−1 +ϕǫ += − +� +X +∂χδ ∧ i∂log(|τ|2 +ωϕǫ,h/|σ|2) ∧ ωn−1 +ϕǫ += +� +X +i∂∂χδlog(|τ|2 +ωϕǫ,h/|σ|2) ∧ ωn−1 +ϕǫ . +Note that the left-handed side term is pointwise bigger than χδ(−iΘ(det(TX/Ei), h) − ǫωϕǫ + +� +j iµ(I, Dj)Θ(OX(Dj))) ∧ ωn−1 +ϕǫ . Moreover, the integration of the lower bound over X tends to +� +X +(−c1(TX/Ei) − ǫω + +� +j +µ(I, Dj)c1(O(Dj))) ∧ ωn−1 +8 + +as δ → 0+. +On the order hand, i∂∂χδ = −θ′ +δ(ρ)i∂∂ρ − θ′′ +δ (ρ)i∂ρ ∧ ∂ρ. Note that |θ′′ +δ |, |θ′ +δ| is uniformly +bounded independent of δ. Thus log(|τ|2 +ωϕǫ,h/|σ|2)i∂∂χδ ∧ ωn−1 +ϕǫ +is uniformly bounded by some +integrable function by the previous claim. +By Lebesgue’s dominated convergence, the limit of the right-handed term on the second line is +0 as δ → 0+. +Taking δ → 0+ gives +0 ≥ +� +X +(−c1(TX/Ei) − ǫω + +� +j +µ(I, Dj)c1(O(Dj))) ∧ ωn−1 +which finishes the proof by taking ǫ → 0+. +Notice that a similar result is shown in Theorem 0.4 of [Ou17] under the hypothesis that −KX +is nef and that X is projective with mild singularity. +To illustrate the local integrability in the proof of Theorem 3, we give the following elementary +example. +Example 1. Take the same notations as in the proof of Theorem 3. Assume that the complex +dimension of X is two and the support of D has two irreducible components intersecting transver- +sally at x ∈ X. Assume that J is the maximal ideal at x. Denote (z1, z2) local coordinates near x +such that the support of D is defined by {z1z2 = 0} where x corresponds to the origin. +The local integrability near x is equivalent to showing that in local coordinates, for any i0 ∈ +[1, 2], +−log(|τ|2 +ω⌈D⌉,h/|σ|2)id(zi0) ∧ d(zi0) +|zi0|2log2(|zi0|) ∧ +� +i̸=i0,i≤2 +id(zi) ∧ d(zi) +|zi|2(1−βi) +is locally integrable. +Let π be the blow-up of X at x. It is equivalent to showing that +π∗� +− log(|z1|2 + |z2|2)id(zi0) ∧ d(zi0) +|zi0|2log2(|zi0|) ∧ +� +i̸=i0,i≤2 +id(zi) ∧ d(zi) +|zi|2(1−βi) +� +is locally integrable near the exceptional divisor. Without loss of generality, assume that i0 = 1. +In local coordinates near x, π is given by +π(w1, w2) = (w1ws, · · · , ws−1ws, ws, ws+1ws, · · · , w2ws). +In local coordinates, it is enough to show that +−log|ws|2π∗(id(z1) ∧ d(z1) +|z1|2log2(|z1|) ∧ id(z2) ∧ d(z2) +|z2|2(1−β2) +) +is locally integrable near {ws = 0}. +If s = 2, we have an upper bound for +−log|w2|2 id(w1w2) ∧ d(w1w2) +|w1w2|2log2(|w1w2|) ∧ id(w2) ∧ d(w2) +|w2|2(1−β2) +whose potential with respect to the Lebesgue measure is bounded by +−log|w2|2 × 1/(|w1|2log2(w1)) × 1/(|w2|2(1−β2)) +which is locally integrable. +If s = 1, we have an upper bound for +−log|w1|2 id(w1) ∧ d(w1) +|w1|2log2(|w1|) ∧ id(w1w2) ∧ d(w1w2) +|w1w2|2(1−β2) +whose potential with respect to the Lebesgue measure is bounded by +−log|w1|2 × 1/(|w1|2log2(w1)) × |w1|2β2/(|w2|2(1−β2)) +which is locally integrable. +9 + +Now the arguments of Proposition 5.1 of [Cao13] give the following corollary. +Corollary 1. Let (X, ω) be an n-dimensional compact K¨ahler manifold. Let D = �(1 − βj)Yj = +�(1 − βj)[sj = 0] a divisor with simple normal crossings with βj ∈]0, 1[ such that −(KX + D) is +nef. Then the Albanese morphism αX is surjective with connected fibres. In fact, the Albanese map +is submersion outside an analytic set of codimension larger than 2. +D´emonstration. The proof in [Cao13] only uses the fact that the slopes with respect to ωn−1 of the +sheaves obtained as graded pieces of the Harder-Narasimhan filtration are positive. Hence using +Theorem 3, the result is a direct consequence of his arguments. For the convenience of the readers, +we just give here proof of the fact that the fibres of the Albanese map are connected. We follow +the arguments in the Proposition 3.9 of [DPS94]. +Let X → Y → Alb(X) be the Stein decomposition of the Albanese map with Y = Spec αX∗OX. +Since X is smooth, Y is normal. We claim that the map f : Y → Alb(X) is ´etale. The reason +is as follows. By the arguments in [Cao13], there exists Z an analytic subset in Alb(X) with +codimension at least 2 such that X ∖α−1 +X (Z) → Alb(X)∖Z is submersion (thus a fibration). Thus +Y ∖ f −1(Z) → Alb(X) ∖ Z is ´etale. We denote by F the fibre of the fibration f|Y ∖f −1(Z) which is +finite. By the long exact sequence associated with a fibration, we have +π1(F) → π1(Y ∖ f −1(Z)) → π1(Alb(X) ∖ Z) → π0(F) +where π1(F) = 0 and π0(F) is finite. In particular, π1(Y ∖ f −1(Z)) is a free Abelian group of +rank 2q := 2dimCAlb(X). Notice that by the codimension condition, we have π1(Alb(X) ∖ Z) ∼= +π1(Alb(X)). Alb(X) is isomorphic to the quotient of the universal cover Cq of Alb(X) under the +group action π1(Alb(X)). Define T to be the quotient of Cq under the group action π1(Y ∖f −1(Z)) +with the natural cover p : T → Alb(X). By the homotopy lifting property, there exists a map +g : Y ∖ f −1(Z) → T such that p ◦ g = f|Y ∖f −1(Z). Remark that g is holomorphic since it is given +by the composition of f with the holomorphic local inverse of p. Since Y ∖f −1(Z) → Alb(X)∖Z is +finite, f −1(Z) is of codimension at least 2. Since Y is normal, g extends to a morphism g : Y → T . +Now g is a generically injective morphism between Y and T . Since T is smooth, the inverse map of +T ∖ p−1(Z) → Y also extends across p−1(Z) which gives the inverse morphism of g. In conclusion +g is a biholomorphism between T and Y which proves that f is ´etale. Another way to prove that +f is ´etale is using the purity of branch locus. In fact, f : Y → Alb(X) is a ramified finite cover +with Y normal and Alb(X) smooth. Then by the purity of the branch locus, the branch locus is +either of codimension one or empty. Since f is a submersion outside an analytic set of Alb(X) of +codimension at least 2, the branch locus has to be empty. +In particular, Y is a finite ´etale cover of the torus Alb(X), so Y itself is a torus. By the +universality of the Albanese morphism, there exists a morphism h : Alb(X) → Y such that the +morphism X → Y factorises through h. Since the morphisms X → Y and αX are surjective, we +have h ◦ f = idY and f ◦ h = idAlb(X). Thus f is a biholomorphism, and the Albanese morphism +has connected fibres. +The arguments of [Cao13] combined with Theorem 3 also give the following affirmation of a +conjecture of Mumford. The general conjecture of Mumford states that a projective or compact +K¨ahler manifold X is rationally connected if and only if H0(X, (T ∗ +X)⊗m) = 0 for any m ≥ 1. +Corollary 2. Let (X, ω) be an n-dimensional compact K¨ahler manifold. Let D = �(1 − βj)Yj = +�(1 − βj)[sj = 0] a divisor with simple normal crossings with βj ∈]0, 1[ such that −(KX + D) is +nef. Then the following properties are equivalent : +(1) X is projective and rationally connected. +(2)H0(X, (T ∗ +X)⊗m) = 0 for any m ≥ 1. +(3) For every m ≥ 1 and every finite ´etale cover ˜X of X, one has H0( ˜X, Ωm +˜ +X) = 0. +Acknowledgement I thank Jean-Pierre Demailly, my PhD supervisor, for his guidance, pa- +tience and generosity. I would like to thank my post-doc mentor Mihai P˘aun, for many supports +and very useful suggestions on this objective. I would like to thank Junyan Cao, Henri Guenancia +for some very interesting discussions on this objective. This work is supported by DFG Projekt +Singul¨are hermitianische Metriken f¨ur Vektorb¨undel und Erweiterung kanonischer Abschnitte ma- +naged by Mihai P˘aun. +10 + +R´ef´erences +[BEGZ10] S´ebastien Boucksom, Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi, Monge- +Amp`ere equations in big cohomology classes, Acta Math. 205 (2010), no. 2, 199–262. +[BT82] E. Bedford and B.A. Taylor. A new capacity for plurisubharmonic functions, Acta +Math.,149(1982), 1–41. +[Cao13] Junyan Cao, A remark on compact K¨ahler manifolds with nef anticanonical bundles and +applications, arXiv:1305.4397. +[CGP13] Fr´ed´eric Campana, Henri Guenancia, Mihai Pˇaun, Metrics with cone singularities along +normal crossing divisors and holomorphic tensor fields, Annales scientifiques de l’Ecole Normale +Sup´erieure, S´erie 4, Tome 46 (2013) no. 6, p. 879-916. +[Don12] S. K. Donaldson, “K¨ahler metrics with cone singularities along a divisor”, in Essays in +mathematics and its applications, Springer, Heidelberg, 2012, p. 49–79 +[DPS94] J.-P. Demailly, T. Peternell, M. Schneider : Compact complex manifolds with numerically +effective tangent bundles. J. Algebraic Geom. vol 3, (1994) number 2, pp. 295–345. +[GP16] Henri Guenancia, Mihai Pˇaun, Conic singularities metrics with prescribed Ricci curvature : +General cone angles along normal crossing divisors, J. Differential Geom. Volume 103, Number +1 (2016), 15-57. +[Gue13] Henri Guenancia, K¨ahler-Einstein metrics with cone singularities on klt pairs. Int. J. +Math. 24 1350035 (2013). +[GZ17] Vincent Guedj and Ahmed Zeriahi. Degenerate Complex Monge–Amp`ere Equations. EMS +Tracts in Mathematics, Volume : 26 ; 496 pp ; 2017. +[Hir64] Heisuke Hironaka. Resolution of singularities of an algebraic variety over a field of cha- +racteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109–203; ibid. (2), 79 :205–326, 1964. +[Ou17] Wenhao Ou, On generic nefness of tangent sheaves, arXiv:1703.03175 +[Paun16] Mahai Pˇaun, Singular Hermitian metrics and positivity of direct images of pluricanonical +bundles. Algebraic Geometry : Salt Lake City 2015, pp.519-553. arXiv:1606.00174. +[Zha96] Qi Zhang. On projective manifolds with nef anticanonical bundles. J. Reine Angew.Math., +478 :57–60, (1996). +11 + diff --git a/d9E4T4oBgHgl3EQfpw0m/content/tmp_files/load_file.txt b/d9E4T4oBgHgl3EQfpw0m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cf417ccd220006c500b39dfd01ed53757157dbc --- /dev/null +++ b/d9E4T4oBgHgl3EQfpw0m/content/tmp_files/load_file.txt @@ -0,0 +1,381 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf,len=380 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='05194v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='CV] 12 Jan 2023 Albanese morphism of log smooth klt compact K¨ahler manifold with nef log anticanonical divisor Xiaojun WU 13 janvier 2023 R´esum´e Let (X, ω) be an n-dimensional compact K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let D = �(1 − βj)Yj = �(1 − βj)[sj = 0] a divisor with simple normal crossings with βj ∈]0, 1[ such that −(KX + D) is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We show that its Albanese map is submersion outside an analytic set of codimension larger than two with connected fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 1 Regularity of Monge-Amp`ere equation with conic singu- larities The following result on the regularities of the Monge-Amp`ere equation is well-known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since we have not found a precise reference of some result, we provide some proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let us fix some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let (X, D) be a log smooth klt pair, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' X is a compact K¨ahler manifold, and D = �(1 − βk)Yk is a R-divisor with simple normal crossing support such that βk ∈]0, 1[ for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The coefficients (βk) are called standard orbifold is βk = 1/nk for some nk ∈ N∗ for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We denote by ωβ the standard cone metric attached with (Cn, D), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' ωβ := i d � k=1 dzk ∧ d¯zk |zk|2(1−βk) + i n � k=d+1 dzk ∧ d¯zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' A metric ω is said to have conic singularities if it is quasi-isometric to the model metric with conic singularities : more precisely, near each point p ∈ Supp(D) where Supp(D) is defined by the equation {z1 · · · zd = 0} for some holomorphic system of coordinates (zi), we want ω to satisfy C−1ωβ ≤ ω ≤ Cωβ for some constant C > 0 and some β (near p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We have the following result due to [CGP13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let X be a compact K¨ahler manifold and D = �(1 − βj)Yj = �(1 − βj)[sj = 0] a divisor with simple normal crossings with βj ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let µD = dVω/ � j |sj|2(1−βj) be a volume form with conic singularities along D, µ ∈ R, and ω a K¨ahler form on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then any (bounded) solution ϕ of (ω + i∂∂ϕ)n = eµϕµD is H¨older-continuous and the metric ω + id∂ϕ has conic singularities along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In the following section, we start by considering the standard orbifold case to illustrate the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In this case, we have a more precise regularity of ϕ, and we adapt the notations of [CGP13] in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' As written at the beginning of this section, the following Proposition 3 is well-known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We fix Dn ⊂ Cn the unit polydisk centered at the origin, and a divisor D = �d k=1(1 − 1/nk)Dk where Dk = {zk = 0} for all k, and d ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider the branched cover π : Dd × Dn−d −→ Dd × Dn−d (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' , zd, zd+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' , zn) �−→ (zn1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' , znd d , zd+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' , zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 1 If we denote by w the coordinates upstairs, π∗ωβ = i d � k=1 |nk|2dwk ∧ d ¯wk + i n � k=d+1 dwk ∧ d ¯wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, π∗ωβ is a genuine metric upstairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Recall that in the standard orbifold case, a function f is said to have orbifold Ck,α regularity (denoted by Ck,α,β) if its pull-back π∗f by the ramified cover π is Ck,α in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' More generally, let τ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' σ) be a bounded (1, 0)-form (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' (1, 1)-form) on Dn \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then we say that τ ∈ Cα,β if for all k, we have π∗τ( ∂ ∂wk ) ∈ Cα in the usual sense (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' σ ∈ Cα,β if for all k and l, we have π∗σ( ∂ ∂wk , ∂ ∂wl ) ∈ Cα in the usual sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Define C2,α,β := {f ∈ L∞(Dn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' f, ∂f, ∂∂f ∈ Cα,β} with the associated natural norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' It is checked in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='2 [CGP13] that this definition coincides with the definition of Donaldson [Don12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Cover X be open sets Ui so that each open set is biholomorphic to the unit polydisk with ramified cover associated with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Denote ˜Ui the ramified cover of πi : ˜Ui → Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let Ni be the ramified order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We have a more precise regularity result than Proposition 1, also stated in [CGP13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let (X, D) as in Proposition 1 with standard orbifold coefficients, and let ϕ ∈ L∞(X) be any solution of (ω + i∂∂ϕ)n = eµϕµD Then ϕ belongs to the class C2,α,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' More precisely, in the standard orbifold case, by elliptic regularity, one can show that π∗ϕ is in fact smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let (X, D) as in Proposition 1 with standard orbifold coefficients, and let ϕ ∈ L∞(X) be any solution of (ω + i∂∂ϕ)n = eµϕµD Then π∗ϕ is smooth in each ramified cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that π∗dνD is a smooth volume from upstairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Without loss of generality, assume that ω = i∂∂ψ for some smooth function locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Locally the pullback of the Monge-Amp`ere equation on the ramified cover can be written as det( ∂2 ∂wi∂wj π∗(ψ + ϕ)) = geµπ∗ϕ for some nowhere vanishing smooth function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By Proposition 2, π∗(ω +i∂∂ϕ) is a genuine metric with C0,α coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Denote locally π∗(ω + i∂∂ϕ) = � i,j hij √ −1dwi ∧ dwj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' From the Monge-Amp`ere equation, we have an a priori positive lower bound on the determinant of hij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, if hij ∈ Ck,α for some k ≥ 0, then det(hij)−1 ∈ Ck,α (hence hij ∈ Ck,α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Take ∂ ∂wl log(∀l) for both sides of equation hij ∂2 ∂wi∂wj ∂ ∂wl π∗(ω + i∂∂ϕ) = ∂ ∂wl logg + µ ∂ ∂wl π∗ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Inductively for k ≥ 0, by interior Schauder estimates for linear elliptic equations with coefficients in Ck,α, we obtain the a priori Ck+3,α norm estimate of π∗ϕ in terms of the Ck+1,α norm of π∗ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus π∗ϕ is smooth in any ramified cover by bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We now recall the definition of a singular metric on a vector bundle according to [Paun16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 2 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' A singular Hermitian metric h on E is given locally by a measurable, possibly unbounded map with values in the set of semi-positive Hermitian matrices, such that 0 < deth < ∞ almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By this definition, a solution of the Monge-Amp`ere equation with conic singularities defines a singular metric on TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, the solution also induces a singular metric on any quotient bundle of TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We observe that by the Monge-Amp`ere equation, the Ricci curvature of the singular metric is well defined as a current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' However, one can notice that the curvature tensor of TX is not necessarily well-defined as a current with values in semi-positive, possibly unbounded Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Now we return to the existence and regularity of the Monge-Amp`ere equation for the general coefficient case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In fact, the work of [Gue13] and [CGP13] gives the following weak estimate for the following type of Monge-Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The theorem is not essentially used for the following section, but the discussion after the theorem also applies to this slight generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' So we still state it explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let X be an n-dimensional compact K¨ahler manifold, and let D = � i aiDi, E = � j bjEj be two effective R-divisors with simple normal crossing support, such that for all 1 ≤ i ≤ r, 0 < ai < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Assume that D and E have no common irreducible component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let ω be a K¨ahler metric on X, dV a smooth volume form, and let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then the weak solution of the Monge-Amp`ere equation ⟨(ω + i 2π ∂∂ϕ)n⟩ = eεϕ � |tj|2bjdV � |si|2ai exists, which is smooth on X ∖ (D ∪ E) and has an upper bound by a metric with conic singularity along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Here ⟨•⟩ is the positive intersection product defined in [BEGZ10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Here si(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' tj) is the canonical section of O(Di) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' O(Ej)) and |si|2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' |tj|2) is the norm of si (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' tj) with respect to some smooth metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We observe that the existence of a solution is proved in [BEGZ10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' As a consequence of Theorem 1, there exists C > 0 such that the solution has on X ∖ (D ∪ E) an upper bound ω + i 2π ∂∂ϕ ≤ Cω � i |si|2ai .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the Monge-Amp`ere equation, we find on X ∖ (D ∪ E) a lower bound ω + i 2π ∂∂ϕ ≥ eεϕ � |tj|2bjω � |si|2ai ( C � i |si|2ai )−(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Notice that since the solution is smooth on X ∖ (D ∪ E), the above inequalities are satisfied pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the result of [BEGZ10], |ϕ| is uniformly bounded on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, we have ω + i 2π ∂∂ϕ ≥ C � |tj|2bjω � |si|2ai ( C � i |si|2ai )−(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In conclusion outside D ∪ E, the solution ω + i 2π∂∂ϕ viewed as a Hermitian form over TX with respect to ω has positive eigenvalues bounded from above by C � i |si|2ai and bounded from below by C � |tj|2bj � |si|2ai ( C � i |si|2ai )−(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let us observe that for the singular metric on the determinant line bundle of the quotient bundle Q given by a short exact sequence of vector bundles 0 → S → TX → Q → 0, the curvature form is well-defined as a current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We detail the argument below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Suppose we are in the situation of Theorem 1, with the same notation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since the metric is smooth outside D ∪ E, we only need to study the neighbourhood of D ∪ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By a C∞ splitting of the exact sequence, we can view Q as a subbundle of TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' ω + i 2π∂∂ϕ thus induces a Hermitian form over 3 Q which we will denote by ω + i 2π∂∂ϕ|Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the minimax principle, for the induced Hermitian form on Q, the eigenvalues are bounded from above by C � i |si|2ai and bounded from below by C � |tj|2bj � |si|2ai ( C � i |si|2ai )−(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' To prove that the curvature of det(Q) is well-defined as a current (not necessarily positive), it is enough to prove that log(det(ω + i 2π∂∂ϕ|Q)) ∈ L1 loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' det(ω + i 2π∂∂ϕ|Q) is the product of all eigenvalues of the Hermitian form ω + i 2π∂∂ϕ|Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus we get the estimate for the potential |log(det(ω + i 2π ∂∂ϕ|Q))| ≤ � i Cilog|si|2 + � j Cjlog|tj|2 + C for some Ci > 0, Cj > 0 and C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In the following, we will refer to this type of control as potentials possessing at most logarithmic poles along D ∪ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Notice also that for any i, log|zi| is locally integrable with respect to the euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, the curvature of the induced metric on det(Q) is well defined as a current since locally it is the i∂∂ of some L1 loc function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' For any global potential ψ (defined on X) possessing at most logarithmic poles along D ∪ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' As analogy to Monge-Amp`ere operator in the sense of Bedford-Taylor [BT82], we want to define i∂∂ψ ∧ ωn−1 ϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Fix a ∈ [0, 1[, the integration � log(r)r1−2a = 1 2 − 2a(log(r) − 1 2 − 2a)r2−2a is finite over [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, log|z||z|−2a as a function z ∈ C is locally integrable near 0 with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus the coefficients of ψ ∧ ωn−1 ϕ are locally integrable with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Define i∂∂ψ ∧ ωn−1 ϕ := i∂∂(ψ ∧ ωn−1 ϕ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By Stokes theorem, we have that � X i∂∂ψ ∧ ωn−1 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 2 Albanese morphism In this section, we start by generalising the results of [Cao13] to log smooth cases with standard orbifold coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' To start with, we need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let (X, ω) be an n-dimensional compact K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let D = �(1−1/nj)Yj = �(1 − 1/nj)[sj = 0] a divisor with simple normal crossings with nj ∈ N∗ such that −(KX + D) is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let 0 = E0 ⊂ E1 ⊂ · · · ⊂ Es = TX be a filtration of torsion-free subsheaves such that Ei+1/Ei is an ω-stable torsion-free subsheaf of TX/Ei of maximal slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then for any i, the slope of Ei+1/Ei with respect to ωn−1, namely µ(Ei+1/Ei) := � X c1(Ei+1/Ei) ∧ ωn−1, is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the stability condition, to prove the theorem, it is sufficient to prove that for any i � X c1(TX/Ei) ∧ ωn−1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let r be the generic rank of TX/Ei for some fixed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The naturally induced morphism ∧rTX → det(TX/Ei) corresponds to a section τ ∈ H0(X, det(TX/Ei) ⊗ Ωr X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' τ is non-vanishing outside a closed analytic set of codimension at least two on which TX/Ei is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Fix an arbitrary smooth metric h on det(TX/Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The critical step is the existence of positive closed (1, 1)-current in a K¨ahler class which is smooth outside an SNC divisor and whose Ricci curvature can be taken “arbitrary small” outside the divisor using the theorems in [GP16] and [CGP13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 4 Let Tε be a sequence of smooth forms in c1(−(KX + D)) such that Tε ≥ −εω whose existence is ensured by the nefness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' To get the lower bound, we want to solve the following K¨ahler- Einstein type of equation Ric(ωϕε) = −εωϕε + εω + Tε + [D] where ωϕε := ω + i∂∂ϕε is the unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Notice that both sides belong to the class c1(−KX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In order to solve the K¨ahler-Einstein type of equation, we thus solve the following Monge-Amp`ere equation by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let γ be a smooth representative of the class {[D]}, which is induced from the curvature forms of some smooth metrics (O(Yi), hi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the ∂∂-lemma, there exists fε ∈ C∞(X) such that Tε+γ = Ric(ω)+ i 2π∂∂fε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The Monge-Amp`ere equation equivalent to the K¨ahler-Einstein type of equation can be written as ωn ϕε = ωneεϕǫ−fε � i |si|2(1−1/ni) hi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Take the same notations as after Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that π∗ωϕε is a smooth metric when pulling back onto the ramified cover by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Calculate on the ramified cover i∂∂π∗log|τ|2 ωϕǫ,h = i{D′π∗τ, D′π∗τ} π∗|τ|2 − i{D′π∗τ, π∗τ} ∧ {π∗τ, D′π∗τ} π∗|τ|4 −π∗iΘ(det(TX/Ei), h) − {π∗τ, iΘ(π∗ωϕǫ)π∗τ} π∗|τ|2 where {} is a canonical sesquilinear pairing C∞(∧pT ∗ X ⊗ Ωr X ⊗ det(TX/Ei)) × C∞(∧qT ∗ X ⊗ Ωr X ⊗ det(TX/Ei)) → C∞(∧p+qT ∗ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that the right-handed term on the first line is positive in the sense of currents by Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Observe that the right-handed term is locally integrable (and hence well defines its product with π∗ωn−1 ϕǫ in the sense of currents on the ramified cover).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since π∗ωϕǫ is smooth, the coefficients of {π∗τ,iΘ(π∗ωϕǫ)π∗τ} π∗|τ|2 is locally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' On the other hand, |D′π∗τ|2/|π∗τ|2 is locally integrable with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' To see this, let I be the (local) ideal defined by the coefficients of π∗τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Without loss of generality, it is enough to consider the case that I is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By Hironaka’s resolution of singularities [Hir64], there exists a modification p : U ′ → ˜U such that I · OU′ = O(− � i λiEi) with � i λiEi an effective SNC divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that since τ is non-vanishing outside a codimension 2 set, p is not an identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Denote by DI the (local) ideal defined by the differentials of the coefficients of π∗τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then DI · OU′ is contained in O(− � i(λi −1)Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' KU′/ ˜U = � i νiEi with νi ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, the pullback of Lebesgue measure can be written as � i |sEi|2νi times a smooth nowhere vanishing form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus the coefficient of the product of p∗(|D′π∗τ|2/|π∗τ|2) with the pullback of Lebesgue measure is locally bounded on U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, |D′π∗τ|2/|π∗τ|2 is locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' On the ramified cover, the Chern curvature, which is also the Levi-Civita curvature, satisfies Ric(π∗ωϕε) = −επ∗ωϕε + π∗(Tǫ + ǫω) ≥ −επ∗ωϕε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider i∂∂π∗log|τ|2 ωϕǫ,h ∧ π∗ωn−1 ϕǫ ≥ (−π∗iΘ(det(TX/Ei), h) − {π∗τ, iΘ(π∗ωϕǫ)π∗τ} π∗|τ|2 ) ∧ π∗ωn−1 ϕǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' On the other hand, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='7 [Cao13], with local curvature calculations, we have {π∗τ, iΘ(π∗ωϕǫ)π∗τ} π∗|τ|2 ∧ π∗ωn−1 ϕǫ ≤ ǫπ∗ωn ϕǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that log|τ|2 ωϕǫ,h possesses at most logarithmic poles along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In fact, the dual metric on T ∗ X could only vanish along D when the initial metric on TX has conic singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, i∂∂log|τ|2 ωϕǫ,h ∧ ωn−1 ϕǫ is well-defined as discussed after Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 5 Let θi be a partition of unity associated with the chosen open cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 0 = � i � X θii∂∂log|τ|2 ωϕǫ,h ∧ ωn−1 ϕǫ = � i � X i∂∂θilog|τ|2 ωϕǫ,h ∧ ωn−1 ϕǫ = � i � ˜Ui 1 Ni π∗ i (i∂∂θilog|τ|2 ωϕǫ,h ∧ ωn−1 ϕǫ ) = � i � ˜Ui 1 Ni π∗ i (θii∂∂log|τ|2 ωϕǫ,h ∧ ωn−1 ϕǫ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus we have � X c1(TX/Ei) ∧ ωn−1 = � i � ˜Ui 1 Ni π∗ i (θii∂∂log|τ|2 ωϕǫ,h ∧ ωn−1 ϕǫ ) + � X iΘ(det(TX/Ei), h) ∧ ωn−1 ϕǫ ≥ −ǫ � i � ˜Ui 1 Ni π∗ i (θiωn ϕǫ) = −ǫ � X ωn ϕǫ = −ǫ � X ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The conclusion follows by taking ǫ → 0+ Now we consider the general coefficients case following a suggestion of Mihai P˘aun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let (X, ω) be an n-dimensional compact K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let D = �(1 − bj)Dj = �(1 − bj)[sj = 0] a divisor with simple normal crossings with bj ∈]0, 1[ such that −(KX + D) is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let 0 = E0 ⊂ E1 ⊂ · · · ⊂ Es = TX be a filtration of torsion-free subsheaves such that Ei+1/Ei is an ω-stable torsion-free subsheaf of TX/Ei of maximal slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then for any i, the slope of Ei+1/Ei with respect to ωn−1, namely µ(Ei+1/Ei) := � X c1(Ei+1/Ei) ∧ ωn−1, is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Construct as above proof of Theorem 2 a section τ ∈ H0(X, det(TX/Ei) ⊗ Ωr X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' As in the above theorem 2, construct a sequence of metrics ωϕǫ with conic singularities by solving Monge-Amp`ere equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider local coordinate (z1, · · · , zn) such that the support of D is given by {z1 · · · zd = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In the general case, we do a cut-off in a tubular neighbourhood of D of radius δ and study the asymptotic behavior as δ → 0+ to get the estimate of the slope from the integration of i∂∂log|τ|2 ωϕǫ,h ∧ ωn−1 ϕǫ over X \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Locally τ = � I,|I|=r τIdzI with multi-index I = {i1, · · · , ir} of length r and |τ|2 ωϕǫ by the conic singularities is locally equi- valent to � I,|I|=r � j,1≤ij≤d |zij|2−2bij |τI|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Near the origin, O(1) ≤ −log|τ|2 ωϕǫ ≤ −log( � I,|I|=r � j,1≤ij≤d |zij|2|τI|2) + O(1) where O(1) means a bounded term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus to study the local integrability of −log|τ|2 ωϕǫ with respect to some positive locally finite measure, it is enough to study the local integrability of −log(� I,|I|=r � j,1≤ij≤d |zij|2|τI|2) with respect to that measure which has the same singularities as some local ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that outside the support of D (where the metric is smooth), pointwise, we have i∂∂log|τ|2 ωϕǫ,h ∧ ωn−1 ϕǫ ≥ (−iΘ(det(TX/Ei), h) − ǫωϕǫ) ∧ ωn−1 ϕǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' However, −log(� I,|I|=r � j,1≤ij≤d |zij|2|τI|2) may have strict positive generic Lelong number along the support of D which may not be integrable with respect to metric with Poincar´e type singularities along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' (For example, in the Poincar´e type singularity case, the local integrability is equivalent to the fact that 1/(r|log(r)|α) is locally integrable near 0 if and only if α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=') 6 Thus we need to rewrite −log(� I,|I|=r � j,1≤ij≤d |zij|2|τI|2) in a golbal form and subtract the divisorial part along the support of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let ω⌈D⌉ be a smooth metric on X \\ D which is locally quasi-isometric near every point in ⌈D⌉ = {z1 · · · zd = 0} to ω⌈D⌉ := i d � k=1 dzk ∧ d¯zk |zk|2 + i n � k=d+1 dzk ∧ d¯zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' (For example, this kind of metric can be constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Cover X by open coordinate charts such that ⌈D⌉ can be written as zeros of local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Assume furthermore that the coordinate charts on which ⌈D⌉ is not irreducible do not intersect each other, up to taking some further refinement of the open cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Construct ω⌈D⌉ by glueing local ones via a partition of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let us check that the glueing metric has desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Assume that Uz is a coordinate chart such that ⌈D⌉ ∩ Uz = {z1 · · · zd = 0} and that Uw is a coordinate chart such that ⌈D⌉ ∩ Uw = {w1 = 0} with non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Over Uz ∩ Uw, z1/w1 ∈ O∗(Uz ∩ Uw) since they define the same irreducible divisor on the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus for some C > 0 large enough, idz1 ∧ dz1/|z1|2 + Ci �n k=1 dzk ∧ d¯zk is equivalent to idw1 ∧ dw1/|w1|2 + i �n k=2 dwk ∧ d ¯wk over any chosen relative compact subset of Uz ∩ Uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus after the partition of unity, the glueing metric has desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Of course, the glueing metric is not K¨ahler on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=') Let I be the ideal sheaf defined (locally) by the coefficients of type � j,1≤ij≤d zijτI to which log|τ|2 ω⌈D⌉ is equivalent to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that I is locally defined, but its integral closure is globally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In fact, the germ of the integral closure is the holomorphic function germs f such that locally log|f|2 ≤ log|τ|2 ω⌈D⌉ + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We denote this globally defined integral closure by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider the divisorial valuation associated with Di µ(I, Di) := max{m ∈ N, I ⊂ Im Di} where IDi is the identification of O(−Di) as an ideal sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider σ := � i sµ(I,Di) Di which is a global holomorphic section on X where sDi are the canonical sections of O(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider log(|τ|2 ω⌈D⌉,h/|σ|2) such that any Di is not an irreducible component of its pole set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Now outside the support of D (where the metric is smooth), pointwise, we have i∂∂log(|τ|2 ωϕǫ,h/|σ|2) ∧ ωn−1 ϕǫ ≥ (−iΘ(det(TX/Ei), h) − ǫωϕǫ + � j iµ(I, Dj)Θ(OX(Dj))) ∧ ωn−1 ϕǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that by choice of σ, the poles of log(|τ|2 ω⌈D⌉,h/|σ|2) contain no irreducible component of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In local coordinates, log(|τ|2 ω⌈D⌉,h/|σ|2) has the same singularities as ideal quotient J := (I : � i Iµ(I,Di) Di ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In the following, we study the local integrability of log(|τ|2 ω⌈D⌉,h/|σ|2) with respect to some volume form of mixed type of Poincar´e type and conic type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let χ be a cut-off function [0, ∞[→ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Following section 9 of [CGP13], define ρ(x) := log(log 1 � i |sDi|2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that −i∂∂ρ is bounded from above by Poincar´e type singularities along the support of D, which in local coordinates can be written as i d � k=1 dzk ∧ d¯zk |zk|2log2(|zk|) + i n � k=d+1 dzk ∧ d¯zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We claim that −log(|τ|2 ω⌈D⌉,h/|σ|2)i∂∂ρ ∧ ωn−1 ϕǫ 7 is locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' It is enough to show that near the support of D in local coordinates, for any i0 ∈ [1, d], −log(|τ|2 ω⌈D⌉,h/|σ|2)id(zi0) ∧ d(zi0) |zi0|2log2(|zi0|) ∧ � i̸=i0,i≤d id(zi) ∧ d(zi) |zi|2(1−βi) ∧ � i≥d+1 idzi ∧ dzi is locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' To simplify the notations, in the following, we assume that i0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider the branched cover π : Dd × Dn−d −→ Dd × Dn−d (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' , zd, zd+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' , zn) �−→ (z1, zn2 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' , znd d , zd+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' , zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' with ni(∀i ≥ 2) large enough such that βini ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The local integrability is equivalent to the fact that π∗� − log(|τ|2 ω⌈D⌉,h/|σ|2)id(z1) ∧ d(z1) |z1|2log2(|z1|) ∧ � d≥i≥2 id(zi) ∧ d(zi) |zi|2(1−βi) ∧ � i≥d+1 idzi ∧ dzi � is locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let fj(1 ≤ j ≤ r) be the local generators of π∗J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' A sufficient condition is that −log( � j |fj|2)id(z1) ∧ d(z1) |z1|2log2(|z1|) ∧ � i≥2 id(zi) ∧ d(zi) is locally integrable with the choice of branched cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the construction of J , the restrictions of fj on {z1 = 0} are not all identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By continuity, on Dr0 × Dn−1 for small radius r0, the restrictions of fj on {z1 = s}(∀|s| < r0) are not all identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider −log(−log(� j |fj|2)) as a continuous family of psh functions on Dn−1 parametrized by z1 ∈ Dr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that the Lelong number of any psh function in this family at any point in Dn−1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By uniform Skoda integrability theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='50 [GZ17]), for any s ∈ D such that |s| < r0, � {s}×Dn−1 −log( � j |fj|2) � i≥2 id(zi) ∧ d(zi) ≤ C for some C > 0 independent of s by vanishing Lelong number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since the Poincar´e metric in one variable is locally integrable, we finish the proof of the claim about the local integrability by the Fubini theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Similarly, by Schwarz inequalities, we can show that −log(|τ|2 ω⌈D⌉,h/|σ|2)i∂ρ ∧ ∂ρ ∧ ωn−1 ϕǫ is locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' For each δ > 0, let θδ : [0, ∞[→ [0, 1] be a smooth function which is equal to zero on the interval [0, 1/δ], and which is equal to 1 on the interval [1 + 1/δ, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' One may for example define θδ(x) = χ(x − 1/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider χδ : X → [0, 1] defined by χδ(x) := 1 − θδ(ρ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Consider Stokes’s formula � X χδi∂∂(log(|τ|2 ωϕǫ,h/|σ|2)) ∧ ωn−1 ϕǫ = � X ∂χδ ∧ i∂log(|τ|2 ωϕǫ,h/|σ|2) ∧ ωn−1 ϕǫ = − � X ∂χδ ∧ i∂log(|τ|2 ωϕǫ,h/|σ|2) ∧ ωn−1 ϕǫ = � X i∂∂χδlog(|τ|2 ωϕǫ,h/|σ|2) ∧ ωn−1 ϕǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that the left-handed side term is pointwise bigger than χδ(−iΘ(det(TX/Ei), h) − ǫωϕǫ + � j iµ(I, Dj)Θ(OX(Dj))) ∧ ωn−1 ϕǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Moreover, the integration of the lower bound over X tends to � X (−c1(TX/Ei) − ǫω + � j µ(I, Dj)c1(O(Dj))) ∧ ωn−1 8 as δ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' On the order hand, i∂∂χδ = −θ′ δ(ρ)i∂∂ρ − θ′′ δ (ρ)i∂ρ ∧ ∂ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Note that |θ′′ δ |, |θ′ δ| is uniformly bounded independent of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus log(|τ|2 ωϕǫ,h/|σ|2)i∂∂χδ ∧ ωn−1 ϕǫ is uniformly bounded by some integrable function by the previous claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By Lebesgue’s dominated convergence, the limit of the right-handed term on the second line is 0 as δ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Taking δ → 0+ gives 0 ≥ � X (−c1(TX/Ei) − ǫω + � j µ(I, Dj)c1(O(Dj))) ∧ ωn−1 which finishes the proof by taking ǫ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Notice that a similar result is shown in Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='4 of [Ou17] under the hypothesis that −KX is nef and that X is projective with mild singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' To illustrate the local integrability in the proof of Theorem 3, we give the following elementary example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Take the same notations as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Assume that the complex dimension of X is two and the support of D has two irreducible components intersecting transver- sally at x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Assume that J is the maximal ideal at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Denote (z1, z2) local coordinates near x such that the support of D is defined by {z1z2 = 0} where x corresponds to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The local integrability near x is equivalent to showing that in local coordinates, for any i0 ∈ [1, 2], −log(|τ|2 ω⌈D⌉,h/|σ|2)id(zi0) ∧ d(zi0) |zi0|2log2(|zi0|) ∧ � i̸=i0,i≤2 id(zi) ∧ d(zi) |zi|2(1−βi) is locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let π be the blow-up of X at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' It is equivalent to showing that π∗� − log(|z1|2 + |z2|2)id(zi0) ∧ d(zi0) |zi0|2log2(|zi0|) ∧ � i̸=i0,i≤2 id(zi) ∧ d(zi) |zi|2(1−βi) � is locally integrable near the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Without loss of generality, assume that i0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In local coordinates near x, π is given by π(w1, w2) = (w1ws, · · · , ws−1ws, ws, ws+1ws, · · · , w2ws).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In local coordinates, it is enough to show that −log|ws|2π∗(id(z1) ∧ d(z1) |z1|2log2(|z1|) ∧ id(z2) ∧ d(z2) |z2|2(1−β2) ) is locally integrable near {ws = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' If s = 2, we have an upper bound for −log|w2|2 id(w1w2) ∧ d(w1w2) |w1w2|2log2(|w1w2|) ∧ id(w2) ∧ d(w2) |w2|2(1−β2) whose potential with respect to the Lebesgue measure is bounded by −log|w2|2 × 1/(|w1|2log2(w1)) × 1/(|w2|2(1−β2)) which is locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' If s = 1, we have an upper bound for −log|w1|2 id(w1) ∧ d(w1) |w1|2log2(|w1|) ∧ id(w1w2) ∧ d(w1w2) |w1w2|2(1−β2) whose potential with respect to the Lebesgue measure is bounded by −log|w1|2 × 1/(|w1|2log2(w1)) × |w1|2β2/(|w2|2(1−β2)) which is locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 9 Now the arguments of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='1 of [Cao13] give the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let (X, ω) be an n-dimensional compact K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let D = �(1 − βj)Yj = �(1 − βj)[sj = 0] a divisor with simple normal crossings with βj ∈]0, 1[ such that −(KX + D) is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then the Albanese morphism αX is surjective with connected fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In fact, the Albanese map is submersion outside an analytic set of codimension larger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The proof in [Cao13] only uses the fact that the slopes with respect to ωn−1 of the sheaves obtained as graded pieces of the Harder-Narasimhan filtration are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Hence using Theorem 3, the result is a direct consequence of his arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' For the convenience of the readers, we just give here proof of the fact that the fibres of the Albanese map are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We follow the arguments in the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='9 of [DPS94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let X → Y → Alb(X) be the Stein decomposition of the Albanese map with Y = Spec αX∗OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since X is smooth, Y is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We claim that the map f : Y → Alb(X) is ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The reason is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the arguments in [Cao13], there exists Z an analytic subset in Alb(X) with codimension at least 2 such that X ∖α−1 X (Z) → Alb(X)∖Z is submersion (thus a fibration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus Y ∖ f −1(Z) → Alb(X) ∖ Z is ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' We denote by F the fibre of the fibration f|Y ∖f −1(Z) which is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the long exact sequence associated with a fibration, we have π1(F) → π1(Y ∖ f −1(Z)) → π1(Alb(X) ∖ Z) → π0(F) where π1(F) = 0 and π0(F) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, π1(Y ∖ f −1(Z)) is a free Abelian group of rank 2q := 2dimCAlb(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Notice that by the codimension condition, we have π1(Alb(X) ∖ Z) ∼= π1(Alb(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Alb(X) is isomorphic to the quotient of the universal cover Cq of Alb(X) under the group action π1(Alb(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Define T to be the quotient of Cq under the group action π1(Y ∖f −1(Z)) with the natural cover p : T → Alb(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the homotopy lifting property, there exists a map g : Y ∖ f −1(Z) → T such that p ◦ g = f|Y ∖f −1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Remark that g is holomorphic since it is given by the composition of f with the holomorphic local inverse of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since Y ∖f −1(Z) → Alb(X)∖Z is finite, f −1(Z) is of codimension at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since Y is normal, g extends to a morphism g : Y → T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Now g is a generically injective morphism between Y and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since T is smooth, the inverse map of T ∖ p−1(Z) → Y also extends across p−1(Z) which gives the inverse morphism of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In conclusion g is a biholomorphism between T and Y which proves that f is ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Another way to prove that f is ´etale is using the purity of branch locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In fact, f : Y → Alb(X) is a ramified finite cover with Y normal and Alb(X) smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then by the purity of the branch locus, the branch locus is either of codimension one or empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since f is a submersion outside an analytic set of Alb(X) of codimension at least 2, the branch locus has to be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' In particular, Y is a finite ´etale cover of the torus Alb(X), so Y itself is a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' By the universality of the Albanese morphism, there exists a morphism h : Alb(X) → Y such that the morphism X → Y factorises through h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Since the morphisms X → Y and αX are surjective, we have h ◦ f = idY and f ◦ h = idAlb(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Thus f is a biholomorphism, and the Albanese morphism has connected fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The arguments of [Cao13] combined with Theorem 3 also give the following affirmation of a conjecture of Mumford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' The general conjecture of Mumford states that a projective or compact K¨ahler manifold X is rationally connected if and only if H0(X, (T ∗ X)⊗m) = 0 for any m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let (X, ω) be an n-dimensional compact K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Let D = �(1 − βj)Yj = �(1 − βj)[sj = 0] a divisor with simple normal crossings with βj ∈]0, 1[ such that −(KX + D) is nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Then the following properties are equivalent : (1) X is projective and rationally connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' (2)H0(X, (T ∗ X)⊗m) = 0 for any m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' (3) For every m ≥ 1 and every finite ´etale cover ˜X of X, one has H0( ˜X, Ωm ˜ X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Acknowledgement I thank Jean-Pierre Demailly, my PhD supervisor, for his guidance, pa- tience and generosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' I would like to thank my post-doc mentor Mihai P˘aun, for many supports and very useful suggestions on this objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' I would like to thank Junyan Cao, Henri Guenancia for some very interesting discussions on this objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' This work is supported by DFG Projekt Singul¨are hermitianische Metriken f¨ur Vektorb¨undel und Erweiterung kanonischer Abschnitte ma- naged by Mihai P˘aun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 10 R´ef´erences [BEGZ10] S´ebastien Boucksom, Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi, Monge- Amp`ere equations in big cohomology classes, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 205 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 2, 199–262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' [BT82] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Bedford and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' A new capacity for plurisubharmonic functions, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=',149(1982), 1–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' [Cao13] Junyan Cao, A remark on compact K¨ahler manifolds with nef anticanonical bundles and applications, arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='4397.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content='Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=', 478 :57–60, (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E4T4oBgHgl3EQfpw0m/content/2301.05194v1.pdf'} diff --git a/dNE2T4oBgHgl3EQfGAZK/content/tmp_files/load_file.txt b/dNE2T4oBgHgl3EQfGAZK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed3552692d972fa920509148bf22ed782fb79af0 --- /dev/null +++ b/dNE2T4oBgHgl3EQfGAZK/content/tmp_files/load_file.txt @@ -0,0 +1,862 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf,len=861 +page_content='On The Fragility of Learned Reward Functions Lev McKinney∗ University of Toronto lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='mckinney@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='ca Yawen Duan∗ University of Cambridge yd338@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='uk David Krueger University of Cambridge david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='krueger@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='com Adam Gleave University of California, Berkeley gleave@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='edu Abstract Reward functions are notoriously difficult to specify, especially for tasks with complex goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Reward learning approaches attempt to infer reward functions from human feedback and preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Prior works on reward learning have mainly focused on the performance of policies trained alongside the reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This practice, however, may fail to detect learned rewards that are not capable of training new policies from scratch and thus do not capture the intended behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Our work focuses on demonstrating and studying the causes of these relearning failures in the domain of preference-based reward learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We demonstrate with experiments in tabular and continuous control environments that the severity of relearning failures can be sensitive to changes in reward model design and the trajectory dataset composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Based on our findings, we emphasize the need for more retraining-based evaluations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 1 Introduction Reward functions for most real-world tasks are difficult or impossible to specify procedurally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Specifically, hand-designed reward functions frequently misspecify the task [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The field of reward learning attempts to overcome this challenge by designing algorithms to infer reward functions from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' These learned reward functions aim to succinctly represent the desired behaviors [22], drastically reduce the amount of human feedback required to learn a task [8] and allow practitioners to generalize these behaviors to new environments [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' One of the most promising approaches is to learn reward functions from binary human preferences over trajectory segments where these segments are collected online using a sampler agent trained to optimize the learned reward [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This form of preference-based reward learning is already being used to train large language models to summarize [41] and become more helpful and harmless [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Prior work has typically focused on the performance of the sampler agent [19, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Unfortunately, the sampler agent performing the correct behavior does not guarantee that a robust reward function has been uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In particular, when using reinforcement learning to train a randomly initialized relearner agent on the learned reward, the reward may fail to induce the correct behavior despite the sampler agent behaving well [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' If we only require a policy that works passably well in the exact training environment, this may not be an issue because we can use the sampler agent and throw away the learned reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We argue, however, that such a method cannot be accurately described as learning a reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' At most, it is a preference-based policy learning technique, using reward functions to give a helpful inductive bias during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Moreover, it is desirable for many applications to 1Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Work done during internship at Center for Human-Compatible AI, UC Berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='03652v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='LG] 9 Jan 2023 truly uncover a reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' For example, we might wish to train a new policy using a learned reward function with a more powerful R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' algorithm or different agent architecture than was used during the initial reward learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Past work has preformed preliminary investigations into the robustness of learned rewards in toy environments [28], in Atari [15] and for fine-tuning language models [3, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, these investigations are typically only reported short sections of their respective papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Inspired by this work, our paper empirically examines the relearner performance for learned reward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Since we have access to a ground truth reward in our synthetic experiments, we define poor relearner performance as achieving relatively low ground truth returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Our results show that relearing can produce very different policies than the sampler, frequently achieving low ground-truth returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Thus, we argue that current preference-based reward learning methods may produce reward functions that are not reliable as signals for policy relearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Our paper makes three key contributions: We demonstrate that state-of-the-art reward learning algorithms can produce reward models that fail to train new agents from scratch in tabular and continuous control settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We show that the severity can increase as the trajectory dataset concentrates on high reward regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Finally, as an example of how these relearning failures can be sensitive to changes in reward model design, we demonstrate that reward ensembles can effect relearning failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 2 Related Work Preference-based reward learning Our primary focus is on methods that learn from preference comparisons between two trajectories [2, 36, 29, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Preference comparison is one of the most scalable reward learning methods, successfully applied to fine-tune large transformer language models [24, 32, 21, 3] to enhance their performance at certain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Note that trajectory comparison methods contain more information about the reward than demonstrations, so they tend to produce better results when available [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, note that these methods may still fare poorly when the human preference feedback does not match their model of human rationality [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Other reward learning approaches Many other methods have been developed to learn reward functions from human data [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' One of the most popular is Inverse reinforcement learning (IRL) [22] methods that infer a reward function from demonstrations [1, 27, 39, 38, 40, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' T-REX [6] is a hybrid approach, learning from a ranked set of demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' An alternative approach learns from “sketches” of cumulative reward over an episodeCabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Reward hacking Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' provides the first systematic empirical study of reward hacking: RL agents exploiting misspecified reward functions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Notably, they find that increasing agent capabil- ities, such as by increasing the RL policy’s model size, can sometimes lead to worse performance on the ground truth reward, while performance on the misspecified proxy reward increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In contrast to our work, Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' only study reward hacking in hand-designed rewards designed to illustrate the phenomenon, whereas we investigate this phenomenon in learned rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Reward hacking has also been studied from a theoretical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Under the framework of general principle agent problems Zhuang and Hadfield-Menell examines the case where the agent’s utility function can only account for a limited subspace of the set of attributes that make the true utility [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The authors proceed to show that, within their model, an optimal state under this proxy utility can have arbitrarily low ground truth utility, assuming the attributes that make up the reward exhibit a condition analogous to decreasing marginal utility and increasing opportunity cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Skalse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' instead propose a formal definition of reward hacking [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In our paper, however, we focus on more concrete cases of relearning failures and practically attainable measures, such as relearner performance being close to sampler performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The most closely related work is by Ibarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=', which evaluates their learned reward functions by freezing them and training a new policy, analogous to our relearner evaluations [15, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, this study was only a small, half-page section of their paper, and they did not examine factors that may increase or decrease the chances/severity of relearning failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 2 Another important related work is by Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=', who observes that rewards can fail to generalize due to a lack of informative trajectories in their training data [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' They attempt to ameliorate this by querying humans on diverse hypothetical trajectories generated from a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, their method requires a world model and primarily focuses on taking advantage of this model to improve reward quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In addition, they assume the user provides feedback through quantitative reward labels, whereas we focus on the more realistic and widely used preference comparison setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Finally, past work has found that language models trained on a preference-based reward model can learn to exploit their reward model [32, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In a similar vein, Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' found that the ability of their reward model to correctly predict human preferences over a pair of inputs degraded as those inputs where perceived as more rewarding by the model [3, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, none of these works have offered much analysis of what leads to reward hacking or in general relearning failures, beyond training against the learned reward for to long [32, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='3] or a lack of data from off distribution [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Retraining and transfer in IRL domain There has also been multiple works exploring relearning and transfer when learning rewards learned from expert demonstrations i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' inverse reinforcement learning (IRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' [10] propose an IRL method to learn state-only reward functions disentangled from transition dynamics and preform experiments on transferring their learned rewards to new agents and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' [23] derive an analytic gradient estimator for an arbitrary f-divergence between expert and on policy distributions with respect to the reward functions parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In their relearning evaluations, they find that there method produces relearners that match expert performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Finally, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' [34] borrows methods from random network distillation to directly estimate the expert distribution with only expert data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This process, removes the need for a sampler, obviating the issue of relearning failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In contrast to these IRL methods, our work focuses on the more scalable preference-based reward learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 3 Background Deep RL from Human Preferences We follow the framework of learning a preference model ˆrφ from trajectory segment comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Our method is the closest to deep reinforcement learning from human preferences [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' It consists of four phases iterated: trajectory collection, preference elicitation, reward inference and policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' During trajectory collection, the current policy, initially a random policy, samples rollouts from the environment collecting trajectory segments σi = (s0, a0, s1, a1, · · · , sn) without reward labels and stores them in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In phase two, the algorithm, elicits preferences y ∈ {≻, ≺, ≡} for randomly selected pairs of segments (σ1, σ2) ∈ B from a labeler — human or synthetic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The preferences are then stored in a preference dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The algorithm assumes these preferences have been sampled from the Bradly-Terry model [4], P(σ1 ≻ σ2) = exp �� s,a,s′∈σ1 r(s, a, s′) � exp �� s,a,s′∈σ1 r(s, a, s′) � + exp �� s,a,s′∈σ2 r(s, a, s′) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' (1) a widely used approximate model for human data in the preference based reward learning literature [8, 15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In the third phase reward inference, the reward ˆrφ is fit by using Adam [17] to minimize the negative log likelihood of ˆrφ under D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The fourth and final phase of each iteration consists of policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In this stage, we can apply existing deep reinforcement learning algorithms to improve our policies expected return under the learned reward, and the process repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 4 Training and Evaluation Procedure Reward Learning We train reward models with synthetic data that is sampled from the Bradley- Terry model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 1 with r set to the ground truth reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In the tabular setting, we train the sampler policy using soft Q-Learning [13] and the learned reward networks simply take the current state as a one-hot vector for input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In the continuous control setting, we use soft actor-critic (SAC) [14] from Stable-Baselines3 [26] and the learned reward networks receive the observation, action and next observation as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' See Appendix A for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 1We follow Ibarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' in selecting preference pairs to query uniformly at random [15] 3 0 10 20 30 40 50 Number of Iterations 0 2000 4000 6000 8000 10000 12000 Ground Truth Return RL Budget 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='5M 1M 2M 4M 8M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='0 Number of Timesteps 1e6 0 1000 2000 3000 4000 5000 6000 Ground Truth Return RL Budget 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='5M 1M 2M 4M 8M 0 250 500 750 1000 1250 1500 1750 2000 Number of Preference Pairs 0 2 4 6 8 10 12 14 Segment Pair Average Return RL Budget 1M 8M (a) Reward learning curves (b) Relearning curves (c) Preference datasets Figure 1: Anti-correlated sampeler and relearner ground truth returns in HalfCheetah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' (a) x-axis represents the number of iterations of each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' See section 3 RL budget is the total number of RL timesteps available to the sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' (b) x-axis represents the number of timesteps during relearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In plots (a-b), for each RL budget setting, we performed ten runs of reward learning, and for each of these, we ran five relearning evaluations for a total of 50 relearning runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Solid lines and shaded lines represent the mean and 90% confidence respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' (c) Scatterplot of average ground truth reward of each segment pair in the example preference datasets with 1M and 8M RL budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Reward Ensembles Christiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' and Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' use reward ensembles to estimate the uncertainty of the learned reward [8, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We explore how these ensembles may have another benefit, reducing the variance of off-distribution transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' As in prior works, we train each ensemble member on bootstrapped datasets, normalize their outputs separately and use their mean as the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Relearning We freeze the learned reward and train a new, randomly initialized, relearner policy to evaluate our reward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We evaluate this policy under the ground truth reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This is similar to the method employed by Ibarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' [15, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='2] to study reward hacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In the continuous control settings this consist of training a new agent from the learned reward function using the same R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' algorithm as the sampler, then evaluating it under the ground truth reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In the tabular setting, we simply solve for the soft-optimal policy (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='1) [13] under the learned reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 5 Experiments First, we investigate the occurrence of relearning failures in the continuous control domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We use HalfCheetah environment as our test bed since it has been used in past works on preference-based reward learning [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Here we find that increasing the number of training timesteps the sampler takes between sampling trajectories for labeling increases the severity of relearning failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Next, we focus on the effects of reward model design and observe that reward ensembles may reduce reward hacking in tabular environments by reducing the variance of off-distribution transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' To demonstrate this failure mode we use the stay inside environment which consists of two rooms separated by a wall with a small doorway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The agent receives reward for staying in the inside room see Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='1 Preference Trajectory Dataset Imbalance and Relearning Failure The reward model is a function of the dataset D used to train it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' One of the simplest ways to change the preference dataset is to vary the number of timesteps T spent training the sampler between collecting trajectory fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We call the total number of interactions the sampler has with the environment during reward learning the RL budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Note the RL budget does not affect the number of comparisons collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Figure 1 shows the learning curves of the sampler and relearner experiments in HalfCheetah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We find that despite higher RL budget leading to higher sampler returns during reward learning, the relearners’ performance has the reverse trend;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' increasing sampler RL budget actually decreases relearner ground truth return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We can gain some insight into why this is happening by exploring preference datasets shown in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' First let’s consider the preference dateset produced by one of the runs with the highest- budget (8M timesteps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We find that the trajectory segments contained in this datset are concentrated in high ground truth reward regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' On the other hand, when we consider the low-budget dataset (1M timesteps), the distribution of trajectory segments provides a better coverage across all the ground truth reward scales within the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We hypothesize that having an overwhelming proportion of 4 high-reward trajectory segments in the preference dataset — and little preference data on trajectories in the transition from high to low reward — may cause the reward model to effectively over-fit to the high-reward region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This overfitting leads to poor supervision over randomly initialized policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Overall, we believe this could explain the observed relearning failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' It’s important to note that we did not see a significant increase in relearning failure when increasing the RL budget in the tabular setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='2 Reward Ensembles 1 5 Number of Ensemble Members −500 0 500 1000 1500 2000 Return Sampler Relearner (a) Effect of reward ensembles on sampler and relearner returns (b) Ground truth reward (c) Example no ensemble (d) Example with ensemble 1 5 Number of Ensemble Members 0 5 10 Maximum Reward (e) Max reward Figure 2: Ensembles eliminate relearning failures in the stay inside environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' (b) depicts the ground truth reward in the stay inside environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' (c) shows an example individual learned reward and (d) with a five member ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Finally, (e) shows the distribution of max learned-reward across all states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' All sub-figures come from the same run which included 20 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Our tabular experiments provide a concrete, interpretable example of how relearning failures can be effected by reward model implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In particular, we focus on reward ensembles and observe that they have drastic effects on relearners but leave the sampler’s performance unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In the stay inside environment, when using a reward ensemble of size five, all relearners preform at least as well as their respective samplers, as can be seen in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, if we use only a single reward, the relearners behaviour is inconsistent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' some relearners do substantially better then their respective samplers, but almost as many do substantially worse, getting near zero return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Thus, while adding an ensemble has a minimal effect on the sampler, it changes the behaviour of the relearners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' To understand why this happens we must consider the off-distribution behaviour of our reward models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In the stay inside environment, the samplers typically stay in the inside half of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Thus there is often insufficient coverage of the outside half of the environment in our trajectory dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Thus, the reward off-distribution is largely unconstrained by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This means that small changes in the off-distribution behavior of our reward network can become critically important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Reward models based on neural networks produce spurious high rewards off distribution, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' When these reward delusions are more rewarding than any of the in-distribution transitions, reward hacking can occur and cause relearning failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Reward ensembles tends to have lower variance off distribution than an individual reward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Thus, any reward delusions tend to have a lower reward (according to the reward model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This can be directly seen in Figure 2 (c-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This effect reduces the chance that the optimal policy will be attracted to one of these spurious rewards during relearning, which is what we see in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 6 Limitations and Discussion Our experiments have a few important limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' First, they are limited to simple ground truth reward functions and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' For example, in Half-Cheetah-v3 [5, 11], the reward function is essentially a linear in the observation, action and next observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' While these relearning failures also appear in more complex tasks [15, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='2], it is unclear if it is precisely the same phenomenon 5 10 8 6 4 0 S2 0 2 4 60 1 2that causes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The design decisions that seem to improve retraining performance in small-scale experiments, in our case, reward ensembles and less sampler training, may not be the same as those that address the problem at a larger scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We leave such explorations to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Overall, we have demonstrated that evaluations of relearning performance can differ substantially from the results of simply evaluating the sampler agent trained alongside the reward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We hope to see future works include relearning evaluation as they appear to hold fruitful insights into the quality of the learned reward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 6 Author Contributions Lev McKinney designed and implemented the tabular experiments and wrote the relevant parts of the method and experiments sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In addition, he wrote the introduction, discussion and related works sections of the paper/appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Yawen Duan designed and ran the initial experiments that displayed reward model relearning failure on continuous control environments, and wrote relevant sections of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' David Krueger provided ideas, guidance and general feedback on experiment design and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Adam Gleave provided initial ideas of the project, provided high-level and detailed feedback on experiments and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Acknowledgments and Disclosure of Funding This paper was completed as part of an internship at the Center for Human-Compatible Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Funding for this internships was provided by the Berkeley Existential Risk Initiative.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Fine-tuning language models from human preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' CoRR, abs/1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='08593, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='org/abs/1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='08593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Appendices A Training Details and Hyperparameters A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='1 Reinforcement learning algorithms In the tabular setting we train the sampler policy using soft Q-Learning [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We use soft actor-critic (SAC) [14] implementations of Stable-Baselines3 [26] in the locomotion control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Both algorithms are off-policy and use a replay buffer, which ensures their high sample efficiency compared to on-policy RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Note that the learned reward function ˆrφ changes during training, so we relabel the transitions in the replay buffer after each iteration, similar to PEBBLE [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The main difference between our algorithm and PEBBLE is that we omit the unsupervised pre-training stage used in PEBBLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' We used the implementations from Imitation Learning Baseline Implementations [35] to perform the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='2 Continuous Control Experiments In the tabular setting, all reward networks only take the current state as a one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' They consist of a multi-layer perceptron with two hidden layers of size 256 and ReLU activations, similar to those used in PEBBLE [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Training details For reward learning experiments, we used the implementations of Preference Comparisons Algorithm from Imitation Learning Baseline Implementations [35] with a full list of hyperparameters in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' For the RL component, we used soft actor-critic (SAC) [14] implementa- tions from Stable-Baselines3 [26] in the locomotion control tasks with a list of hyperparameters in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' For retraining evaluations, we use the same hyperparameters for SAC to train new agents against the frozen learned reward models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 10 Reward model The reward model consists of a single multi-layer perceptrons with two hidden layers of size 256 and LeakyReLU activations with slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The input of the model consists of the state, action and next state vectors, and the input vector is normalized by running normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The output the the reward model is normalized by by exponen- tial moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' During relearning experiments, we directly use the raw reward output from the reward network while being normalzed by a VecNormalize layer in Stable-Baselines3 (https://stable-baselines3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='io/en/master/guide/vec_envs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='html#vecenv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Reward normalization We compute a normalized version of the learned reward using an Expo- nential Moving Average to normalize the reward to mean zero and unit standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This normalized reward was then used for policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Note that normalizing the reward does not change the optimal policy, which is invariant to positive affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, it does simplify the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' In particular, a normalized reward is a more stable objective for the critic to learn over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Additionally, RL hyperparameters can depend on the reward scale (for example, learning rate should be set inversely proportional to reward scale) – normalizing the learned reward therefore allows us to use a consistent set of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Hyperparameter Value Segment Length 50 Total Comparisons 2000 Number of Iteration 50 Reward Training Epochs 5 Query Schedule constant Table 1: Reward learning hyperparameters for continuous control experiments Hyperparameter Value Learning Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='0003 Batch Size 256 Discount 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='99 Learning Starts from 10000 Table 2: SAC hyperparameters for continuous control experiments A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='3 Tabular Experiments Similarly to the continuous control experiments we use Imitation’s implementation of preference comparison [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, we use a tabular soft-q learning algorithm with a replay buffer [13] with reward relabling [19] to solve the environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The reward network again uses a similar MLP architecture to the continuous control setting with a sightly smaller hidden size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Finally, we normalize the reward functions before ensembling them using a simple running norm over sampled transitions which is frozen during retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Hyperparamaters can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Tabular Relearning When relearning we solve for the soft-optimal policy under the learned reward function with temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='1 and discount factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' B Environments Locomotion Control Task We ran reward learning and relearning on a MuJoCo locomotion task [33] – HalfCheetah environment from the seals benchmark suite [11], a modification of HalfCheetah-v3 in the gym environment suite which adds the x-coordinate of the robot’s center of mass (COM) to the first dimension of the observation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The ground-truth reward function of the HalfCheetah environment is a linear combination of the x-velocity of the robot’s COM and a control cost dependent on the L2 norm of the action vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Consequently, the reward function in seals HalfCheetah is a function of the observations, which is not strictly true in the original gym [5] environment, avoiding a potential confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 11 Hyper Parameter Value Sampler Soft-Q Learning discount 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='99 learning rate 5e-2 replay buffer capacity ∞ temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='1 samples from buffer per env sample 10 initial soft-q value 200 Reward Learning trajectory fragment length 30 total comparison budget 2,500 RL budget 500,000 frac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' of comparisons from inital random trajs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='1 select fragments for comparison randomly epochs of training per iteration 1 number of iterations 100 query schedule constant reward learning rate 1e-3 Reward Network reward network hidden layers [32, 32] activation function ReLu output normilization Running Norm Table 3: Tabular Experiment Hyperparamerers Tabular Environment We constructed the stay inside environment, which consists of a 20x20 closed grid of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The top "outside” and bottom "inside” halves of the environment are separated by a wall with a narrow two cell gap in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The reward for each state is shown in Figure 2 (a), with reward values ranging from +10 to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' C Epic Distance as an Evaluation Metric As an additional evaluation criterion, we consider using EPIC distance [12] to measure the distance between learned reward functions and the ground truth reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' EPIC works by canonicalizing the rewards to be invariant to potential shaping, normalizing them to be invariant to scale, and then computing the L2 norm of the difference of those functions over a coverage distribution of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Here we consider two coverage distributions: uniform and expert distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The uniform distribution is uniform over feasible transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The expert distribution is the distribution of a soft-optimal policy with a temperature of 10 to give slightly more coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' D Additional Tabular Experiments To study the effects of training the sampler for a more time steps, we first consider a simple environment consisting of a 10x10 grid world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The agent begins in the lower left-hand corner of the environment and gains a ground-truth reward of 10 for reaching the lower right-hand cell, as seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The performance of the sampler and relearner initially increases with more training timesteps, with our relearners generalizing well and achieving slightly higher performance than their respective samplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, it quickly plateaus even though we do not see significant reductions in relearner performance with an increased number of time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The EPIC distances of our learned reward functions from the ground truth reward begin to increase after 400,000 timesteps Figure 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Increasing the number of total training timesteps used for DRLHP does seem to degrade the quality of the reward function according to EPIC distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' However, it does not appear to hurt relearning performance in the same way in this simple tabular environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 12 10 3 10 2 10 1 (a) Example without ensemble (b) Example with ensemble Figure 3: Example on policy distribution Examples of the on policy distributions of the samplers in the stay inside environment, marginalized over the entire training run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 200000 400000 600000 800000 1000000 DRLHP Training Timesteps −200 0 200 400 600 800 Return Agent Sampler Relearner Relearner Minus Sampler 200000 400000 600000 800000 1000000 DRLHP Training Timesteps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='7 EPIC Distance To Ground Truth Reward Coverage Distribution Uniform Expert (a) (b) Figure 4: Increasing the number of time steps of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' training does not seem to significantly effect relearning failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This is a strikingly different effect than we see in HalfCheetah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This may be because in a tabular setting the sampler either finds the optimal policy induced by the learned reward function every iteration, so the sampler and relearner have equal performance, or it insufficiently explores the environment, and reward learning completely fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' This dichotomy leaves little room for the subtle degradation in relearner performance we see in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 13 Figure 5: Tiny room environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' The ground-truth reward in the tiny room environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' Note that the reward only depends on the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} +page_content=' 14 10 8 6 4 2 0' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf'} diff --git a/gtE4T4oBgHgl3EQfrA2b/content/tmp_files/2301.05205v1.pdf.txt b/gtE4T4oBgHgl3EQfrA2b/content/tmp_files/2301.05205v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c5ea839223ca6c3f617ce057818d3c02613ee6c --- /dev/null +++ b/gtE4T4oBgHgl3EQfrA2b/content/tmp_files/2301.05205v1.pdf.txt @@ -0,0 +1,8465 @@ +arXiv:2301.05205v1 [gr-qc] 12 Jan 2023 +Running vacuum in FLRW spacetime: +The dynamics of ρvac(H) from the quantized matter fields +Cristian Moreno-Pulido, Joan Sol`a Peracaula +Departament de F´ısica Qu`antica i Astrof´ısica, +and Institute of Cosmos Sciences, +Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain +Samira Cheraghchi +Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Str. 50, +500091 Brasov, Romania +E-mails: cristian.moreno@fqa.ub.edu, sola@fqa.ub.edu, samira.cheraghchi@unitbv.ro +Abstract. Phenomenological work in the last few years has provided significant support +to the idea that the vacuum energy density (VED) is a running quantity with the cos- +mological evolution and that this running helps to alleviate the cosmological tensions +afflicting the ΛCDM. On the theoretical side, recent devoted studies have shown that +the properly renormalized ρvac in FLRW spacetime adopts the ‘running vacuum model’ +(RVM) form. While in three previous studies by two of us (CMP and JSP) such compu- +tations focused solely on scalar fields non-minimally coupled to gravity, in the present +work we compute the spin-1/2 fermionic contributions and combine them both. The +calculation is performed using a new version of the adiabatic renormalization procedure +based on subtracting the UV divergences at an off-shell renormalization point M. The +quantum scaling of ρvac with M turns into cosmic evolution with the Hubble rate, H. As +a result the ‘cosmological constant’ Λ appears in our framework as the nearly sustained +value of 8πG(H)ρvac(H) around (any) given epoch H, where G(H) is the gravitational +coupling, which is also running, although very mildly (logarithmically). We find that +the VED evolution at present reads δρvac(H) ∼ νeffm2 +Pl +� +H2 − H2 +0 +� +(|νeff| ≪ 1). The +coefficient νeff receives contributions from all the quantized fields, bosons and fermions, +which we compute here for an arbitrary number of matter fields. Remarkably, there +also exist higher powers O(H6) which can trigger inflation in the early universe. Fi- +nally, the equation of state (EoS) of the vacuum receives also quantum corrections from +bosons and fermion fields, shifting its value from -1. The remarkable consequence is +that the EoS of the quantum vacuum may nowadays effectively look like quintessence. +1 + +Contents +1 +Introduction +3 +2 +Vacuum energy of a non-minimally coupled scalar field +7 +2.1 +Zero-point energy and adiabatic expansion . . . . . . . . . . . . . . . . . . . . . . . +9 +2.2 +Renormalized vacuum energy and vacuum pressure . . . . . . . . . . . . . . . . . . +11 +3 +Quantization of a spin-1/2 fermion field in curved spacetime +14 +4 +ZPE and VED for a spin-1/2 field in FLRW spacetime +21 +4.1 +Divergence balance between bosons and fermions in vacuum . . . . . . . . . . . . . +22 +4.2 +Renormalized ZPE for fermions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +4.3 +Renormalized VED +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +4.4 +Renormalization group equation for the vacuum energy density . . . . . . . . . . . +28 +4.5 +Renormalization of the fermionic vacuum pressure +. . . . . . . . . . . . . . . . . . +31 +4.6 +Trace Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +5 +Combined fermionic and bosonic contributions +34 +5.1 +Running vacuum from an arbitrary number of quantized matter fields +. . . . . . . +34 +5.2 +The low energy regime: evolution of ρvac and G in the present universe . . . . . . . +37 +5.3 +Inflation from running vacuum +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +5.4 +Equation of state of the quantum vacuum . . . . . . . . . . . . . . . . . . . . . . . +43 +6 +Conclusions +45 +A Appendix: Conventions and Useful Formulas +48 +B Appendix: Adiabatic expansion of the spin-1/2 field modes +49 +C Appendix: Adiabatic expansion of ⟨Tµν⟩ for spin-1/2 fields +58 +2 + +1 +Introduction +Despite having coexisted for many decades, a completely successful theory of gravity that combines +Quantum Field Theory (QFT) and General Relativity (GR) does not exist yet, unfortunately. +However, a variety of different approaches and techniques are available in the literature which +allow one to study the subject of quantum fields in the gravitational context, and more specifically +the physics of the expanding universe and its current speeding up. Our aim is to understand +such an acceleration on fundamental grounds. To be precise, in this work we will concentrate on +the well-known semiclassical approach which goes under the name of QFT in curved spacetime +[1–3]. This means that gravity is still a classical external (background) field, whereas the matter +fields are quantum field operators obeying suitable commutation or anticommutation relations +[4]. A further step in the path of understanding gravity in the QFT context is quantum gravity +(QG), in which spacetime itself (the metric) is quantized and hence functional integration over +metrics is mandatory, see e.g. [5–7], and the review [8] and references therein. At the same time +a lot of exciting QG phenomenology is being investigated in the current multi-messenger era, +characterized by an outburst of experimental data that are being obtained from the detection +of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from +numerous origins [9]. On the theoretical side, effective field theory methods and the possibility of +quantum gravitational effects leading to quantum hair may provide useful hints of QG which have +been explored recently [10–12]. However, while the QG option has, of course, to be kept in mind +since it can be very important when QG can be (hopefully) formulated in a fully consistent way [8], +QFT in curved spacetime may still be of great help to further describe the role of quantum fields +in a gravitational context. In this work, we will continue dwelling upon these lines and shall focus +exclusively on the, more modest, but effective, semiclassical approach. It goes without saying that +the latter has had also its own problems and successes over the years, and still have [13]. However, +new perspectives have recently been explored in this context concerning the vacuum energy and +the cosmological constant (CC) [14–16] which may be of significance, and for this reason we wish +to further pursue this line of approach here. +The agent responsible for the accelerated cosmic expansion is generically called Dark Energy +(DE), an entity which constitutes a key piece in the cosmological puzzle, but whose fundamental +nature is still undisclosed [17]. Typically, it is associated with the CC in Einstein’s equations, +Λ, as done routinely in the standard or concordance model of cosmology, aka ΛCDM [18–20]. +The model has been a rather successful paradigm for the phenomenological description of the +universe for about three decades, but it became consolidated only in the mid nineties [21,22] and +especially after consistent measurements of Λ made in the last twenty years using independent +cosmological sources, in particular including distant type Ia supernovae (SnIa), baryon-acoustic +oscillations (BAO), the data on large-scale structure formation and of course the anisotropies of +the cosmic microwave background (CMB). All in all they have put the very experimental basis for +the concordance ΛCDM model of cosmology [23–26]. The situation is far from being satisfactory, +though. The problems with the ΛCDM are both of theoretical and observational nature. As for +the theoretical problems, recall that the value of Λ is traditionally associated to a parameter called +the vacuum energy density (VED) in the universe, although in the context of the ΛCDM is nothing +but a name for the following quantity with dimensions of energy density: ρvac = Λ/(8πGN) (GN +being Newton’s constant). Its theoretical significance is not explained at all in the context of the +standard cosmological model. If, however, we take quantum theory seriously, the most universal +contribution to this vacuum energy density is the zero-point energy (ZPE) of the massive quantum +fields in the standard model of particle physics, and in fact in any QFT model. However, it is well- +known that a naive calculation of this quantity leads to very large contributions proportional to +the quartic power of the mass of the particles, ρZPE ∼ m4, which is in blatant discordance with the +3 + +order of magnitude obtained for this quantity from cosmological observations: ρobs +vac ∼ 10−47GeV4 +(expressed in natural units, with ℏ = c = 1). Even taking for instance the electron field, one +finds a mismatch of 34 orders of magnitude: ρobs +vac/ρZPE ∼ 10−34. The huge discrepancy between a +typical Standard Model ZPE and the measured value of VED constitutes the so-called Cosmological +Constant Problem (CCP) [27–29]. See also [30, 31] for a recent account. Despite the enormous +discrepancies between usual theory predictions and factual measurements, estimates on the value +of Λ within the right order of magnitude have been attempted under certain assumptions in the +context of QG in different approaches, see e.g. [32–36]. +While the aforementioned measurements of ρvac indicate that the vacuum can gravitate within +an energy density order of magnitude of ∼ 10−47GeV4, what is difficult to understand theoretically +is why the vacuum can only gravitate in that tiny range, given the fact that any typical quantum +effect rockets its contribution to much larger values. This is of course a rephrasing of the same +puzzle associated to the CCP, expressed in the QFT context. However, new avenues for a possible +solution have been suggested recently. +The renormalization approach presented in the present +work and in the preceding studies [14–16] offers some hope to eschew part of these difficulties. +First and foremost, the renormalized quantum effects found here endow the VED with a mild +dynamical nature. The latter thus appears as a slowly varying function of the cosmic expansion, +specifically of the Hubble rate H, see below. Second, the renormalized VED as reported here +proves well behaved and can perfectly accommodate the measured value of Λ from observational +cosmology without fine-tuning. Technically, this is because the “running” of Λ is proportional to +the tiny values of the β-function coefficients for bosons and fermions, which are responsible for the +renormalization group evolution of the VED. As a result, at any given epoch of the late universe +Λ appears essentially as constant, but it is not strictly so. Finally, a third crucial ingredient of our +approach is that, in the very early universe, the VED becomes, in contrast, very large and fast +evolving. There it can take the capital role of bringing about inflation, as we shall see. +As previously mentioned, in addition to the traditional theoretical problems, other issues of +more practical and mundane nature have been perturbing cosmologists in the last few years, which +put the ΛCDM against the wall. The practical problems are the presently irreconcilable obser- +vational differences between the standard picture and a number of different kinds of observations +involving structure formation data (the so-called σ8 tension, at a moderate level of ∼ 3σ, where σ8 +is the root mean square of fluctuations in density perturbations at the 8h−1 Mpc scale), and above +all the notorious disagreement between the local value of the Hubble parameter, H0, obtained from +the traditional distant ladder techniques, and the value extracted from the early universe using +CMB data, see [37–40] and references therein for extensive reviews. The discrepancy in the latter +case is at the level of ∼ 5σ, hence a tension whose notable severity (and persistence over time) +may well be passing the point of being attributable to a fluke [41, 42]. So the prevailing model +of cosmology may well be facing a crisis. Science, however, thrives on crisis since new ideas are +then stimulated which could help to overcome the crisis and maybe refine some aspects of the +underlying paradigm or even originate a new one subsuming the old paradigm. Many proposals +have indeed been made to alleviate these tensions, which include different forms of DE as well, even +though many of them are essentially ad hoc. As indicated above, clues to eventually substantiate +the nature of the DE may come from a variety of cosmic and even astrophysical messengers [9]. +For instance, the possibility of measuring the bending of light in the Solar System scale has been +proposed [43]. But whatever the nature of the DE might be, we must provide an explanation for +the role played by the vacuum energy in QFT. Indeed, in the absence of a correct understanding +of the VED from first principles many DE proposals may look as an escape forward rather than a +real alternative. This work, in contrast, intends to dwell further on the methods of QFT in curved +spacetime so as to shed some useful light on these difficult problems. Thus we take the Λ term +seriously, not just as a mere fitting parameter but as a formal quantity in the gravitational action +4 + +from which one can determine the VED, a fundamental concept in QFT. In fact, we report here +on progress made along the lines of the preceding comprehensive works [14–16], where a detailed +account was made of the virtual contribution to the VED from quantized scalar matter fields. +We found that these effects, when appropriately renormalized, translate into a (mild) dynamical +evolution of the VED with the cosmic expansion, ρvac = ρvac(H). This is not excluded by the +cosmological principle, as it permits a homogeneous and isotropic dependence in time of physical +quantities. More specifically, it was shown by explicit calculation and by appropriate renormal- +ization that the VED behaves in the characteristic manner of the running vacuum model (RVM), +see [31] for a recent comprehensive review (for a shorter summary, see e.g. [44]). Let us also note +that a similar RVM interpretation of the vacuum energy can be achieved in the string context, +what has been called ‘stringy-RVM’ [45–49] and references therein. The fact that a QFT calcu- +lation and an effective string theory approach can lead to the same kind of RVM solution seems +to indicate that the VED as a rigid concept is not natural and that a dynamical evolution of the +vacuum energy should be more plausible. In actual fact, it has been recently shown that this fact +can help significantly to improve the description of the overall cosmological data and in particular +opens a viable solution to the well-known tensions afflicting the ΛCDM, see particularly the last +phenomenological analysis [50], which was preceded by several other works, such as e.g. [51–55]. +It is also interesting to remark that the RVM structure of the vacuum energy has been successfully +tested against competing models (e.g. ghost models and holographic models of the DE) using cos- +mographical methods, which are essentially model-independent – see e.g. [56–58] for details. The +model has indeed passed a battery of different tests [59,60] and the outcome is that the quality fit +provided by the overall cosmological data is comparable, actually better, than that of the ΛCDM, +if we attend to the verdict of the standard information criteria [50]. +In this paper we continue the task of computing the dynamics of ρvac(H) induced by the +quantum effects of the quantized matter fields in Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) +spacetime, which two of us (CMP and JSP) initiated in previous works [14–16], see also [31] +for a comprehensive review. The result of the present, more complete, calculation (involving for +the first time the fermionic contributions) reconfirms that the combined dynamics of the vacuum +adopts the RVM form indicated below. In these works, the adiabatic regularization prescription +(ARP) was used to compute the renormalized VED, ρvac, for a non-minimally coupled real scalar +field. The method is based on a series expansion in the number of derivatives of the scale factor +which introduces a hierarchy in some physical quantities evolving in a dynamical background [1–3]. +Not only it was shown that the running of the VED was free from dangerous large contributions +proportional to m4 (quartic powers of the mass of the particles), but ρvac was shown to be mildly +evolving with the Hubble rate and hence with the cosmic expansion. In fact, if t1 and t2 are two +particular values of cosmic time, both close to the present, the corresponding values of ρvac were +shown to be related as follows: +ρvac(H2) − ρvac(H1) ≈ νeff m2 +Pl +� +H2 +2 − H2 +1 +� +, +(1) +where H1 ≡ H(t1) and H2 ≡ H(t2) are the values of the Hubble function at times t1 and t2, +respectively and |νeff| ≪ 1 is a small parameter. While the above relation is relevant for the (very +mild) evolution of the VED in the current universe, the corresponding analysis of the early universe +leads to a new mechanism of inflation called ‘RVM-inflation’, which relies on the existence of +quantum effects of 6th adiabatic order, i.e. up to terms ∼ O(H6) which have been first accounted +for scalar fields in [15]. While the family of Running Vacuum Models (RVM) has been in the +literature for quite some time (cf. [30,31,61] and references therein), a full-fledged account based +on QFT principles is much more recent [14–16]. +This work is tightly related to the preceding studies, in which the adiabatic regularization was +applied to the ‘simple’ case of one real scalar field. It was natural to perform the next step and +5 + +check if spin-1/2 fermions do preserve the main conclusions derived from scalars, above all to verify +if the corresponding vacuum fluctuations induce also a running of the VED independent of the +quartic powers of their masses and hence remain also free from the traditional fine-tuning illness. +So the main goal of this paper is to extend the computations done for the scalar field, by considering +the quantization of spin-1/2 fermions in the FLRW background. The extension proves rather non- +trivial since we are in curved spacetime and the computations with fermions are no less involved +owing to the Fermi-Dirac statistics and the formal peculiarities associated with spinor calculus. It +is however reassuring to find that the many new technicalities involved in the calculation do not +alter the main conclusions derived from the calculation with scalars. The RVM form (1) at low +energies is once more attained, but the contribution to the coefficient νeff is, of course, different +and involves non-trivial computational details. Similarly, we compute the fermionic contribution +to the ∼ O(H6) terms which are involved in the RVM mechanism of inflation occurring in the very +early universe. The final results concerning the renormalized VED can be obtained by considering +the contributions from an arbitrary number of quantized scalar and fermion fields. We combine +the two types of effects and present a final formula which we refer to as the bosonic and fermionic +contributions to the VED, with the understanding that additional effects from gauge fields and +their interactions with matter would be necessary in our calculation in a more complete approach. +It is nevertheless not necessary in our study since our fields are free, except for the non-minimal +coupling of the quantized scalar fields with the external (non-quantized) gravitational field and the +necessary spinorial affine connection of the fermion fields. All that said, the computation of the +free field contribution from bosons and fermions in curved spacetime is already a formidable task, +so for the sake of a stepwise and clearer presentation we will address the fermionic contributions +here on equal footing to the presentation of the scalar part in the references [14,15]. +This work is structured as follows: In Sec. 2, we consider a quantized scalar field φ non- +minimally coupled to curvature and review the computation of its energy-momentum tensor (EMT) +and corresponding vacuum expectation value (VEV) induced by the vacuum fluctuations of that +field [14,15]. The 00th component of its VEV constitutes the ZPE of φ. We define also the VED +and the vacuum pressure, Pvac. All these quantities are unrenormalized at this point and hence +UV-divergent. In the same section we review the off-shell adiabatic renormalization method used +in the previous references, which involves the distinctive feature of our adiabatic renormalization +of the VED. In Sec. 3, we review the quantization of a Dirac fermion in a curved background, +the corresponding Dirac equation and its spinor solutions obtained from adiabatic expansion of +the field modes. The computation of the EMT and of its VEV for the case of a free quantized +fermionic field in a spatially flat FLRW background is performed in Sec. 4. The off-shell adiabatic +renormalization of the EMT for spin-1/2 fermions is addressed and we derive the corresponding +renormalized ZPE, VED and Pvac in this context. Additionally, some remarks on the trace anomaly +and its role in our approach are presented. +Sec. 5 contains the combined results from all the +quantized matter fields. Specifically, we display the renormalized VED for a system made of an +arbitrary number of quantized scalar fields non-minimally coupled to curvature (with different +masses and non-minimal couplings) and an arbitrary number of quantized spin-1/2 free fermion +fields. In the same section we compute the corresponding running of the gravitational coupling +G = G(H), which is associated with the running of ρvac(H) in order to preserve the Bianchi +identity. We also discuss the mechanism of ‘RVM-inflation’ with the combined contribution from +all these fields, and compute the equation of state (EoS) of the quantum vacuum for that system of +quantized bosons and fermions fluctuating in it. Remarkably, the vacuum EoS is no longer equal +to wvac = −1, the reason being that the vacuum pressure and the VED are not exactly related in +the usual way Pvac = −ρvac since Pvac and ρvac are independent functions of the Hubble rate H +and its time derivatives owing to the quantum effects. In the current universe, there is still some +remnant of these quantum effects which induce a small (but potentially measurable) departure +6 + +making the quantum vacuum mimic quintessence. The conclusions are delivered in Sec. 6 together +with some additional discussion. Finally, three appendices are included. In AppendixA, we define +our conventions and some useful expressions. The last two appendices, B and C, are rather bulky +since they collect a number of cumbersome formulas related to the adiabatic expansion of the EMT +and the Fourier modes of the fermionic field (computed up to 6th order for the first time in the +literature). +2 +Vacuum energy of a non-minimally coupled scalar field +In this section we summarize the results for the VED from quantized scalar fields in FLRW +spacetime obtained in [14,15] and in passing we introduce some notation which will be useful also +for the fermionic calculation that will be subsequently reported. The Einstein-Hilbert (EH) action +for gravity plus matter reads +SEH+m = SEH + Sm = +ˆ +d4x√−g +� +1 +16πG R − ρΛ +� ++ Sm . +(2) +The term ρΛ has dimension of energy density and sometimes is called the vacuum energy density, +but this is inaccurate in the formal QFT context since renormalization is necessary and the physical +vacuum energy density, ρvac, is not just that term. In fact, ρΛ is at this point just a bare parameter +of the action, as the gravitational coupling G itself. Varying the action with respect to the metric +provides Einstein’s equations +1 +8πGGµν = −ρΛgµν + T m +µν , +(3) +with Gµν = Rµν − (1/2)gµνR the usual Einstein tensor and T m +µν the energy-momentum tensor +(EMT) of matter 1.: +T m +µν = − +2 +√−g +δSm +δgµν . +(4) +The matter action Sm may contain a variety of contributions, including those from incoherent +matter, but it will be enough to focus on fundamental effects from quantized scalar and fermion +fields. Here we shall compute the fermionic contribution to the VED. But let us summarize first +the situation with the scalar field part. +The latter was dealt with in great detail in the two +previous studies [14,15], in which the VED calculation was addressed under the assumption that +no effective potential was present. However, we admitted a non-minimal coupling of the scalar +field to gravity. That calculation in curved spacetime is already sufficiently demanding and in +addition it furnishes the universal part of the VED through the zero-point energy (ZPE) effects +in the curved background, see next section. The classical action associated to a non-minimally +coupled real scalar field is the following: +Sφ = − +ˆ +d4x√−g +�1 +2gµν∂µφ∂νφ + 1 +2 +� +m2 +φ + ξR +� +φ2 +� +, +(5) +where ξ is the non-minimal coupling with gravity. It is well known that this action enjoys (local) +conformal symmetry in the massless case with ξ = 1/6. However, the value of ξ is not fixed in our +computation and in general we do not assume the presence of such a symmetry. Varying the above +action with respect to the scalar field leads to the Klein-Gordon (KG) equation with non-minimal +coupling: +� +□ − m2 +φ − ξR +� +φ2 = 0, +(6) +1Our geometric conventions and other formulas of interest for this calculation are collected in the Appendix A. +7 + +where □φ = gµν∇µ∇νφ = (−g)−1/2∂µ (√−g gµν∂νφ). The corresponding EMT associated to φ +follows from the metric variation of the action (5) according to the recipe (4), and yields +Tµν(φ) = − +2 +√−g +δSφ +δgµν = (1 − 2ξ) ∂µφ∂νφ + +� +2ξ − 1 +2 +� +gµν∂σφ∂σφ +− 2ξφ∇µ∇νφ + 2ξgµνφ + ξGµνφ2 − 1 +2m2 +φgµνφ2 . +(7) +As indicated, we perform the calculation in cosmological (FLRW) spacetime with flat three- +dimensional metric. For convenience we use the conformal frame ds2 = a2(τ)ηµνdxµdxν, with +ηµν = diag(−1, +1, +1, +1) the Minkowski metric in our conventions (cf. AppendixA), a(τ) is the +scale factor and τ the conformal time. Differentiation with respect to τ will be denoted with a +prime, so for example H ≡ a′/a is the corresponding Hubble function in conformal time. We will +perform the explicit calculations using the conformal metric but our final results will eventually be +rendered in terms of the usual Hubble function H(t) = ˙a/a in cosmic time t (where a dot denotes +differentiation with respect to t). Recall that dτ = dt/a and hence H = aH. +If we switch on the quantum fluctuations of the φ field it is natural to considering the following +decomposition: +φ (⃗x, τ) = φb(τ) + δφ (⃗x, τ) , +(8) +in which the background φb and the fluctuating part δφ are understood to be independent. In +particular, this is also the case for the corresponding Fourier decomposition in frequency modes. +The decomposition of the fluctuating part reads +δφ(τ, x) = +1 +(2π)3/2a +ˆ +d3k +� +Akeik·xhk(τ) + A† +ke−ik·xh∗ +k(τ) +� +, +(9) +with the usual commutation relations for the creation and annihilation operators, Ak and A† +k: +[Ak, A′† +k] = δ(k − k′), +[Ak, A′ +k] = 0. +(10) +By using these relations, the KG-equation (6) in terms of the frequency modes can be put as +h′′ +k + Ω2 +k(τ)hk = 0, +(11) +where Ω2 +k ≡ k2 + a2m2 +φ + (ξ − 1/6) R and we recall that ()′ ≡ d/dτ (). The above differential +equation does not possess a close analytic solution for the entire cosmological evolution. However, +it can be solved by means of what is called an adiabatic series expansion, which is essentially +WKB-type solution. First of all, it is necessary to introduce the following ansatz for the mode +functions: +hk(τ) = +1 +� +2Wk(τ) +exp +� +−i +ˆ τ +Wk(˜τ)d˜τ +� +. +(12) +Notice that the modes are normalized through the Wronskian condition +hkh∗′ +k − h′ +kh∗ +k = i , +(13) +which is essential to preserve the canonical commutation relations for the quantized field φ. By +introducing the above ansatz into (11), the function Wk (effective frequency) is the solution of the +(WKB-type) non-linear differential equation +W 2 +k = Ω2 +k − 1 +2 +W ′′ +k +Wk ++ 3 +4 +�W ′ +k +Wk +�2 +. +(14) +8 + +2.1 +Zero-point energy and adiabatic expansion +For a slowly varying effective frequency Ωk(τ) one can proceed to solve this equation perturbatively +with the help of an asymptotic series which can be organized through adiabatic orders. +This +approach constitutes the basis for the aforementioned ARP [62–69], see also [70–75] and [76–80] +for more recent applications and extensions, and the textbooks +[1–3] for a more systematic +presentation. +The quantities k2 and a are taken to be of adiabatic order 0. Of adiabatic order 1 are: a′ and +H. The quantities a′′, a′2, H′ and H2 and linear combinations are taken of adiabatic order 2. It +can be summarized by saying that each time derivative increases one unit the adiabatic order. So, +the ansatz solution for Wk can be written as an adiabatic series expansion: +Wk = W (0) +k ++ W (2) +k ++ W (4) +k ++ W (6) +k ++ · · · , +(15) +in which the superscript indicates the adiabatic order. We note that only even orders are allowed, +which is justified from the general covariance of the result since only tensors of even adiabatic +order can be present in the effective action and the field equations. Explicit calculation indeed +corroborates the absence of the odd adiabatic orders. The “seed” to initiate the adiabatic series +(i.e. the zeroth order contribution to Wk) is +W (0) +k +≡ ωk = +� +k2 + a2M2, +(16) +where M is an arbitrary off-shell scale. Nothing enforces us to take the mass of the particle at +this point, we only need to preserve the adiabaticity of the expansion. The floating quantity M +will play the role of renormalization scale, as it will be seen. In fact, as it was shown in [15], this +parameter can also be used as the renormalization scale in the DeWitt-Schwinger expansion [4] +of the vacuum effective action Weff [1], to wit: the effective action obtained from integrating out +the vacuum fluctuations of the quantized matter fields. From the explicit expression of Weff one +can also derive the VEV of the EMT – denoted ⟨Tµν⟩ and often referred to in this paper as the +‘vacuum EMT’ – by computing its metric functional derivative as follows: +⟨Tµν⟩ = − +2 +√−g +δWeff +δgµν . +(17) +This formula is of course similar to Eq. (4), but for the vacuum effective action. This alternative +method provides exactly the same result as the WKB expansion of the field modes, as outlined +below, and it was illustrated in great detail in [15] for the case of the quantized scalar fields. Such a +cross-check in the determination of the vacuum EMT involves a significant amount of calculations +and provides a nontrivial validation of the entire renormalization procedure. The same holds good +for the case of fermions but we shall not present the details of the DeWitt-Schwinger method for +fermions here. +Next we summarize the mode expansion for scalar fields. By introducing W (0) +k +given above in +the RHS of (14), the terms of adiabatic order 2 can be collected to find the next term in the series +W (2) +k , with the result +W (2) +k += a2∆2 +2ωk ++ a2R +2ωk +� +ξ − 1 +6 +� +− ω′′ +k +4ω2 +k ++ 3 (ω′ +k)2 +8ω3 +k +. +(18) +Here ∆2 ≡ m2−M2 is the difference between the quadratic mass of the field and that of the off-shell +scale, and is of adiabatic order 2 because it is necessary for renormalization. Loosely speaking, +since M2 and ∆2 appear together in the expansion they need to be of different adiabatic order so as +9 + +to obtain a consistent adiabatic expansion exploring the off-shell regime. Since M2 is of adiabatic +order 0, the next-to-leading order for ∆2 to be compatible with general covariance is precisely +order 2. This fact is reconfirmed on using the aforementioned DeWitt-Schwinger expansion [15]. +Introducing W (2) +k +on the RHS of (14) we can obtain W (4) +k , . . . and so on. The higher order +adiabatic terms become progressively more and more cumbersome since the number of terms +involved in the expansion becomes larger and larger. +In our case we reach up to order 6, i.e. +we compute the series up to W (6) +k . This was done for the first time in the literature in [15] for +scalar fields, and we will also be done here to order W (6) +k +for the first time for fermions, see Sec. 4. +Notice that attaining the order W (6) +k +is indispensable in order to study RVM-inflation in the early +universe [15]. Extensive use of Mathematica [81] has been made to handle these bulky calculations. +Once obtained the expansion of Wk we can compute the mode functions hk and other physical +quantities such as the EMT to the suitable order, in particular the EMT trace, which can be used +to compute the vacuum pressure (see below). The technical details for the scalars are not going to +be repeated here2, but it is important to remark that these quantities present divergent terms up +to 4th adiabatic order (in 4-dimensional spacetime). The approach we adopt here is the same as +that which was proposed and amply tested in [14,15], namely we define the renormalized vacuum +expectation value (VEV) of the EMT (or renormalized vacuum EMT) by taking the on-shell value +(at an arbitrary adiabatic order) and subtracting from it the divergent orders at an arbitrary scale, +which we denote M: +� +T δφ +µν +� +ren (M) = +� +T δφ +µν +� +(mφ) − +� +T δφ +µν +�(0−4) +(M). +(19) +Here the superscript (0 − 4) refers to the UV-divergent subtracted orders, i.e. from 0th up to 4th +adiabatic order, all of them being UV-divergent (the higher adiabatic orders being all finite in n = 4 +spacetime). Notice that for M = mφ the above definition provides the natural generalization of the +subtraction of divergent constants performed to obtain finite results on trivial backgrounds (such as +Minkowski spacetime). However, in curved backgrounds the mode by mode subtraction implied in +the above prescription is not just a constant term; and moreover for arbitrary M the corresponding +renormalized result allows us to test the evolution of the VED with the scale M. As previous +indicated, this feature obviously offers a floating scale which is characteristic of the renormalization +group (RG) analysis in cosmology [30, 31]. Let us clarify, however, that we distinguish M from +the ’t Hooft’s mass unit µ in dimensional regularization (DR), which will not be used in this +work at any point, although it can be invoked as an intermediate regularization procedure (not +at all as renormalization though) if one likes [14,15]. The parameter µ is unphysical and is used +in the minimal subtraction scheme (MS) with DR to define the renormalization point [82]. We +should emphasize that we do not use MS at all in the present work, although one could make +(optional) use of DR in intermediate steps. In these cases, the quantity µ always cancels out and +the final renormalized expressions depend on M only, as it is the case e.g. of the effective action of +vacuum. But the full effective action (which involves the classical and quantum parts) is of course +independent of M as well, as the running of the couplings exactly compensates for the explicit +M-dependence of the quantum effects. This is, of course, the standard lore of the RG program, +see [15] for detailed considerations along these lines and making use of the effective action. +We refrain from writing out the unrenormalized expression for the EMT in the case of scalar +fields, see [14,15] for full details. Let us however quote the renormalized result emerging from the +ARP prescription (19). Expressing the final result in terms of the cosmic time and the correspond- +ing Hubble function H = ˙a/a, we find for the 00-component of the vacuum EMT the following +2They are provided in [14,15]. Analogous computations will be reported in Sec. 3 and Sec. 4 for the case of the +fermion field. +10 + +result: +� +T δφ +00 +� +ren (M) = +a2 +128π2 +� +−M4 + 4m2 +φM2 − 3m4 +φ + 2m4 +φ ln +m2 +φ +M2 +� +− +� +ξ − 1 +6 +� 3a2H2 +16π2 +� +m2 +φ − M2 − m2 +φ ln +m2 +φ +M2 +� ++ +� +ξ − 1 +6 +�2 9a2 +16π2 +� +6H2 ˙H + 2H ¨H − ˙H2� +ln +m2 +φ +M2 + O +� +H6 +m2 +φ +� +, +(20) +and similarly for the VEV of its trace +� +T δφ� +ren (M) = +1 +32π2 +� +3m4 +φ − 4m2 +φM2 + M4 − 2m2 +φ ln +m2 +φ +M2 +� ++ 3 +� +ξ − 1 +6 +� +8π2 +� +2H2 + ˙H +� � +m2 +φ − M2 − m2 +φ ln +m2 +φ +M2 +� +− +9 +8π2 +� +ξ − 1 +6 +�2 � +12H2 ˙H + 4 ˙H2 + 7H ¨H + +... +H +� +ln +m2 +φ +M2 + O +� +H6 +m2 +φ +� +. +(21) +We used the notation O(H6/m2 +φ) to collectively refer to the terms of adiabatic order 6 (consisting +of 6 time derivatives of the scale factor). It may include terms such as H6/m2 +φ, but also many +other combinations such as ˙H3/m2 +φ, H2... +H/m2 +φ, . . . These terms are rather lengthy and have been +computed and reported explicitly in [15]. We refrain from writing them down again here and invite +the reader to check the aforementioned paper for more details. We will report explicitly on the +6th-order adiabatic terms only in the case of fermions (cf. Sec. 4) since these are computed for the +first time in this work. +2.2 +Renormalized vacuum energy and vacuum pressure +We are now ready to compute the vacuum EMT, ⟨Tµν⟩, which will lead us to the VED, ρvac, and +vacuum’s pressure, Pvac. As in [15], we write the vacuum EMT as the sum of the renormalized +parameter ρΛ and the renormalized ZPE, which embodies the finite form of the adiabatically +renormalized vacuum fluctuations: +� +T vac +µν +� += −ρΛ(M)gµν + +� +T δφ +µν +� +ren (M) . +(22) +Since the vacuum is expected to be a most symmetric state free of any new parameter, this +expression must take on the form of a perfect fluid: ⟨T vac +µν ⟩ = Pvacgµν + (ρvac + Pvac) uµuν, where +uµ is the 4-velocity (uµuµ = −1). +In conformal coordinates in the comoving (FLRW) frame, +uµ = (−a, 0, 0, 0). Taking the 00th and iith-component (any i = 1, 2, 3 is good owing to isotropy, +so we take i = 1) one finds the precise form of the vacuum energy density and pressure [15]: +ρvac(M) = ⟨T vac +00 ⟩ +a2 += ρΛ(M) + +� +T δφ +00 +�ren +(M) +a2 +, +(23) +11 + +Pvac(M) = ⟨T vac +11 ⟩ +a2 += −ρΛ(M) + +� +T δφ +11 +�ren +(M) +a2 += −ρΛ(M) + 1 +3 + + +� +T δφ�ren +(M) + +� +T δφ +00 +�ren +(M) +a2 + + += −ρvac(M) + 1 +3 + + +� +T δφ�ren +(M) + 4 +� +T δφ +00 +�ren +(M) +a2 + + , +(24) +where isotropy allows to express the result in terms of the trace T δφ of the fluctuating part, if +desired. Notice that ρΛ(M) in the above expressions is the renormalized form of the corresponding +bare parameter appearing in the EH action (2) and it has units of energy density. The VED, +however, is not just this renormalized parameter but the renormalized sum (23). +Although is +tantalizing to call ρΛ(M) “the CC density”, and in fact this has been common in the literature +(especially when the discussion is strictly classical without considering quantum effects), this is +not strictly correct since the physical CC is not simply 8πGρΛ but 8πGρvac, that is, the physical +vacuum energy density is connected with the physical Λ through ρvac = Λ/(8πG). The parameter Λ +which is measured in the observations is indeed defined through this expression, which is precisely +computable in QFT from Eq. (23). We shall show once more for fermions (as we did for scalar +fields in the previous works +[14, 15]), that the adiabatically renormalized form of the running +VED is free from the huge ∼ m4 contributions that are usually attributed to the VED in other +(inappropriate) renormalization schemes, and therefore the renormalized expression that we will +obtain can be perfectly consistent with the measured Λ. To be sure, it is not our aim to predict +this value but rather to show that the theoretical formula points naturally to a value as small (in +natural units) as measured by the observations. +A simple way to condense these ideas is to say that the VED is related with the ZPE and +ρΛ as follows: “VED = ρΛ + ZPE”, i.e. Eq. (23). Parameter ρΛ is initially just a bare coupling +in the effective action and it has no direct phenomenological interpretation, not even after renor- +malization. On the other hand, the ZPE embodies the quantum fluctuations of the massive fields +and calls also for renormalization since it is originally UV-divergent. The physical VED in this +context is then the renormalized sum of these two contributions, and it can not be split apart +since the separate terms make no sense in isolated way. Observations are sensitive only to the +sum. Furthermore, as we shall see explicitly for the fermionic case, there is a crucial cancellation +of the quartic mass terms when we consider the evolution of the sum ρΛ + ZPE as a function of +the renormalization point, which does not occur if the two terms are dealt with separately. This +was already pinpointed for the case of scalar fields in [14–16] . +With these provisos, the expression for the VED associated to the scalar field can be obtained. +Notwithstanding, the final renormalized result still requires a physical interpretation since it de- +pends on the renormalization scale M. In fact, recall that the result depends on both the values of +M and H (and corresponding time derivatives), which are independent arguments. The scale M +can not be left arbitrary at this point since we wish to provide an estimate of the VED at a given +expansion epoch. As previously indicated, the vacuum effective action Weff is explicitly dependent +on M despite the full effective action is of course RG-independent. Thus, following , [14,15] an ap- +propriate choice of the renormalization point M is to select it equal to the value of H at the epoch +under consideration. This corresponds to choose the RG scale around the characteristic energy +scale of FLRW spacetime at any given moment, and hence it should have physical significance. +In actual fact this is in analogy with the standard practice in ordinary gauge theories, where the +choice of the renormalization group scale is made near the typical energy of the process. In what +follows we derive the ‘low energy’ form of the VED along these lines. This is actually the form that +applies for the current universe. Subsequently we will focus on the running gravitational coupling +G(M) and its relation with the running ρvac(M). +12 + +We should also point out that the reach of our considerations concern the calculation of the +evolution (or ‘running’) of the VED only, rather than predicting its current value. Given that +value, however, we can predict how it evolves with H around our epoch, or any other epoch. +Now in the absence of an observational input at some expansion epoch H(t) we cannot compute +ρvac(H) at other values of H (i.e. +at other epochs of the cosmic evolution). +To compute the +value of the VED at present is out of the scope of the renormalization program since the latter is +based on the RG flow, which requires a boundary condition. This is exactly the same situation +as in any renormalization calculation, we need the input values of the parameters at one scale to +predict some observable (e.g. a cross-section) at another scale. The truly relevant feature of our +calculational approach, as it should be clear at this point, is that the RG-flow is completely smooth. +It only depends on the evolution of H and is completely free from spurious effects associated to the +large contributions from the quartic masses of the fields. These quartic terms are the traditional +kind of undesirable effects which spoil the physical interpretation of the renormalization program +concerning the CC and the VED. They are typically found in calculations of the VED whose +renormalization is based on, say, the MS scheme. Most existing approaches to the CC problem +in the literature exhibit this unwanted feature, which is already at the basis of the Minkowskian +calculation and is of course unacceptable in curved spacetime [31]. A similar situation is found in +Schwarzschild and de Sitter backgrounds, see e.g. [83,84]. +Bearing in mind the above considerations, the final result for the running VED at low energies +(specifically the part triggered by the quantized scalar fields), can be best written in terms of the +evolution between two expansion history times. It is natural that we choose the current epoch +(characterized by the value H0 of the Hubble parameter) and relate it with the value of the VED +at a nearby epoch H of the cosmic evolution3. The approximate final result can be rendered in a +very compact form as follows: +ρvac(H) = ρ0 +vac + 3νeff +8π +� +H2 − H2 +0 +� +m2 +Pl, +(25) +where +νeff ≡ 1 +2π +� +ξ − 1 +6 +� m2 +φ +m2 +Pl +ln +m2 +φ +H2 +0 +(26) +and ρ0 +vac ≡ ρvac(H0), is the current value of the vacuum energy (accessible by observations) with +H0 the current value of the Hubble function. It is necessary to remark that νeff is an effective +parameter expected to be small due to its proportionality to m2 +φ/m2 +Pl. Remarkably, the above +dynamical form of the VED turns out to adopt the RVM form, see +[31] and references therein. +Phenomenological studies based on fitting the above RVM formula to the overall cosmological +data indeed provide an estimate for νeff at the level of νeff ∼ 10−3 and positive [50]. The order +of magnitude is reasonable if we take into account that the masses involved here pertain of course +to the scale of a typical Grand Unified Theory (GUT) where, in addition, a large factor must +be included to account for the large multiplicity of heavy particles. In Sec. 5 we provide a more +general formula where an arbitrary number of species of bosons and fermion fields are included. It +is worth noticing that the order of magnitude of νeff picked out in the mentioned study is perfectly +compatible with the result recently obtained from the Big Bang nucleosynthesis (BBN) bound +in [85], although in the latter case the bound was not sensitive to the sign of νeff. +On the other hand, the computation of the pressure in an analogous way (we refrain from +providing more details on the scalar field contribution, see once more [15] and [16] for a full- +fledged account) enables us to write an explicit expression for the equation of state (EoS) of the +3In this context, a nearby epoch does not necessarily mean that it is very close to the current epoch, it rather +refers to any cosmic span for which the VED running is still driven by the ∼ H2 terms and not by higher powers. +The higher powers are only relevant for the very early universe, namely during the inflationary time, and hence the +low-energy formula applies virtually to any (post-inflationary) epoch. For more details, see +[15]. +13 + +vacuum [16]. The leading expression for the current universe is the following: +wvac = Pvac(H) +ρvac(H) ≈ −1 − νeff +˙Hm2 +Pl +4πρ0vac +. +(27) +For very low redshift z and in terms of the current cosmological parameters Ω0 +i = ρ0 +i /ρ0 +c = +8πGNρ0 +i /(3H2 +0) the above expression reduces to +wvac(z) ≈ −1 + νeff +Ω0 +vac +Ω0m +(1 + z)3. +(28) +This result is especially remarkable since it predicts a small departure from -1 which could be +measured around the present time. Recall that the traditional value associated to a Cosmological +Constant is just −1. This means that the EoS for the quantum vacuum receives small quantum +effects which trigger a departure from −1. For instance, if we adopt the positive sign for νeff, as +obtained from the latest fitting analysis to a large set of different kinds of observational data [50], +then Eq. (28) predicts that the vacuum energy behaves as quintessence around the current time. +As noted, the EoS formula (28) is valid only for small values of the redshift z, but one can show +that the departure is even bigger in the past, adopting a kind of chameleonic behaviour by which +the EoS of the quantum vacuum tracks the EoS of matter at high redshifts, see [16] and Sec. 5.4 +for more details. All in all, these surprising results have been predicted from first principles, nearly +from explicit QFT calculations in the FLRW background. In particular, the fact that the quantum +vacuum may currently mimic quintessence is truly remarkable since the result does not rely on +ad-hoc fields or on any other phenomenological ansatz. +3 +Quantization of a spin-1/2 fermion field in curved spacetime +As indicated in the introduction, the main goal of this work is to extend the QFT results for the +VED obtained for quantized scalar fields, which we have summarized in the previous section, to the +case of quantized spin-1/2 Dirac fermion fields and then combine the two types of contributions in +closed form. The calculation of the renormalized VED for fermions is again nontrivial and requires +a devoted study, which we present here in detail (see also the appendices provided at the end). +While the QFT treatment is analogous to the case of scalars the technicalities are quite different +and no less intricate, but fortunately the final result proves to be in consonance with the one +derived for the scalars, so it is perfectly possible to furnish a combined contribution to the VED +which involves an arbitrary number of scalar and fermion fields, cf. Sec. 5. +The study of the solutions of the Dirac equation in curved spacetime goes back to the works +from many decades ago by Fock, Tetrode, Schr¨odinger, McVittie, Bargmann, Wheeler and others: +see e.g. [86–90], where the relevant historical references are given and different aspects of spin-1/2 +fermions in curved spacetime are studied, including a detailed account for the solutions in FLRW +spacetime – see also the review [91], with a rather complete list of references. On the other hand, +the subject of adiabatic regularization for fermions has been previously treated in the literature +in different applications, see e.g. [67] as well as the more recent papers [72–75] where emphasis is +made on exact solutions e.g. in de Sitter spacetime. The calculation of the renormalized VED in +FLRW spacetime, however, can be appropriately performed using an off-shell variant of the ARP +framework [14, 15]. which leads to the RVM behavior of the vacuum energy [30, 31]. The RVM +framework has proven successful to mitigate the cosmological tensions [39,40], as shown in different +phenomenological analyses [50–55]. On the theoretical side, attempts at computing the VED with +other procedures has led to the traditional calamity with the quartic powers of the masses. Here +we will show that using the ARP to tackle the VED contribution from fermions generates a result +14 + +which is free from these difficulties and fully along the lines of what has been obtained for the scalar +fields in the previous sections and originally in [14,15]. Therefore, the combined contribution from +fermions and scalar fields to the VED is compatible with a smooth running of the cosmological +vacuum energy and is consistent with the aforementioned phenomenological analysis of the RVM +as a possible solution to the cosmological tensions. +Since we will use the formalism employed in some of the aforementioned papers to treat fermions +within the adiabatic approach, it is convenient to summarize first the necessary aspects of that +formalism before we put forward our main results concerning the VED for fermions. This will be +useful also to fix some notation. Once more we perform the calculations in FLRW spacetime with +flat three-dimensional metric. Consider a free Dirac spin-1/2 field, described by the four-component +spinor ψ. In our conventions the Dirac action in curved spacetime is given by +Sψ(x) = − +ˆ +d4x√−g +�1 +2i +� ¯ψγµ∇µψ − +� +∇µ ¯ψ +� +γµψ +� ++ mψ ¯ψψ +� +. +(29) +In the above expression mψ denotes the mass of the Dirac field and ¯ψ ≡ ψ†γ0 the adjoint spinor. +Since we are in a curved background, the partial derivative of a spinor ∂µψ has been replaced +with the corresponding covariant derivative ∇µψ, which is defined below. Moreover, gamma ma- +trices in curved spacetime are also needed, they are sometimes indicated (as above) with an un- +derline to distinguish them from the Minkowski space gamma matrices, i.e. +γµ(x) (which are +generally functions of the coordinates) versus the constant matrices γα in flat spacetime. +As +it is well-known, to obtain a representation for the curved spacetime gamma matrices in terms +of the Minkowskian gamma matrices we need to introduce the local tetrad or vierbein field (in +4-dimensional spacetime) e µ +α . It is defined in each tangent space of the spacetime manifold and re- +lates the curved spacetime metric with the Minkowskian one as follows: gµν(x) = eµ +α(x)eν +β(x)ηαβ, +where ηαβ is the Lorentz metric in the local inertial frame associated to normal coordinates at +the given spacetime point. +The general relation between the two sorts of gamma matrices is +γµ(x) = eµ +α(x)γα. Specifically, in a spatially flat FLRW spacetime the vierbein in conformal co- +ordinates is eµ +α = diag (1/a(τ), 1/a(τ), 1/a(τ), 1/a(τ)) where a(τ) is the scale factor as a function +of the conformal time. Whence the gamma matrices in this background are time-dependent and +related to the constant flat spacetime ones as follows: γµ(τ) = γµ/a(τ). This relation insures that +they satisfy the following anti-commutation relations: +� +γµ, γν� += −2gµνI4 , +(30) +provided, of course, the (constant) flat space gamma matrices satisfy +� +γα, γβ� += −2ηαβI4. In +order to obtain the equation of motion, i.e. the covariant Dirac equation in curved spacetime, one +has to vary the covariant action (29) with respect to the spinor field, giving +iγµ∇µψ + mψψ = ieµ +αγα∇µψ + mψψ = i1 +a (γα∂α − γαΓα) ψ + mψψ = 0 . +(31) +The covariant derivative is defined through the spin connection, ∇µ ≡ ∂µ−Γµ. The spinorial affine +connection Γµ satisfies the equation [87] +� +Γν, γµ(x) +� += ∂γµ(x) +∂xν ++ Γµ +νργρ(x) , +(32) +where Γµ +νρ are the Christoffel symbols. The above equation is tantamount to require the vanish- +ing of the covariant derivative of the curved space gamma matrices: ∇νγµ(x) = 0 [2], i.e. the +curved-space gamma matrices are defined to be covariantly constant over the spacetime manifold. +Using the Christoffel symbols in the conformally flat FLRW metric as given in Appendix A, the +15 + +explicit solution of Eq. (32) can be found, with the following result: Γ0 = 0, Γj = − (H/2) γjγ0 = +− (a′/2a) γjγ0. Therefore, γαΓα = 3(a′/2a)γ0 = −3(a′/2a)γ0. This expression can then be inserted +in Eq. (31). +In this way we have obtained an explicit form for the Dirac equation in FLRW spacetime with +spatially flat metric. We are now in position to attempt a solution by expanding the quantized +fermion field in mode functions as follows: +ψ(x) = +ˆ +d3k +� +λ=±1 +(B⃗k,λu⃗k,λ(x) + D† +⃗k,λv⃗k,λ(x)). +(33) +Here B⃗k,λ and D† +⃗k,λ are creation and annihilation operators which satisfy the standard anticom- +mutation relations, +� +D⃗k,λ, D† +⃗q,λ′ +� += +� +B⃗k,λ, B† +⃗q,λ′ +� += δλ,λ′δ(3) � +⃗k − ⃗q +� +, +� +D⃗k,λ, D⃗q,λ′ +� += +� +D† +⃗k,λ, D† +⃗q,λ′ +� += +� +B⃗k,λ, B⃗q,λ′ +� += +� +B† +⃗k,λ, B† +⃗q,λ′ +� += 0. +(34) +The momentum expansion of the mode functions u⃗k,λ and their charge conjugates v⃗k,λ can be +conveniently written in terms of two 2-component spinors ξλ(⃗k) and corresponding spinor modes +hI +k and hII +k : +u⃗k,λ(τ, x) = +ei⃗k·⃗x +� +(2πa)3 +� +hII +k (τ)ξλ(⃗k) +hI +k(τ)⃗σ.⃗k +k ξλ(⃗k) +� +, +v⃗k,λ(τ, x) = +e−i⃗k·⃗x +� +(2πa)3 +� +−hI∗ +k (τ)⃗σ.⃗k +k ξ−λ(⃗k) +−hII∗ +k (τ)ξ−λ(⃗k) +� +, +(35) +with +⃗σ · ⃗k +k ξλ(⃗k) = λξλ(⃗k), +λ = ±1 , +ξ† +λ(⃗k)ξλ(⃗k) = 1 . +(36) +Using this representation, Eq. (31) splits into two coupled first order equations for each of the two +types of spinor modes hI +k and hII +k : +hII +k = ia +k (1 +a∂τ + imψ)hI +k(τ), +hI +k = ia +k (1 +a∂τ − imψ)hII +k (τ). +(37) +After straightforward calculation, these equations can be rewritten as two second order decoupled +equations: +� +∂2 +τ + imψa′ + a2m2 +ψ + k2 +� +hI +k(τ) = 0 → +� +∂2 +τ + Ω2 +k(τ) +� +hI +k(τ) = 0, +� +∂2 +τ − imψa′ + a2m2 +ψ + k2 +� +hII +k (τ) = 0 → +� +∂2 +τ + (Ω2 +k(τ))∗� +hII +k (τ) = 0, +(38) +where +Ω2 +k ≡ ω2 +k + a2∆2 + iσ(τ) , +(39) +with +ωk(M) ≡ +� +k2 + M2a2, +σ ≡ mψa′ = +� +M2 + ∆2 a′. +(40) +16 + +The fact that (38) only depends on the modulus of the momentum, k, justifies the notation used for +the modes hI +k, hII +k , with no arrows. Following the same prescription as in the case of scalar fields (cf. +Sec. 2), we have introduced an off-shell scale M, which again will take the role of renormalization +scale. Correspondingly, we have defined ∆2 ≡ m2 +ψ −M2 and once more assigned adiabaticity order +2 to it. We did not change the notation ∆ as compared to the scalar case since the final formulas +do not depend on ∆ but on M and the respective physical masses. The argument of ωk will be +omitted from now on, unless it takes a different value from M. The normalization conditions for +the mode functions involved in ψ is performed through the Dirac scalar product as follows: +(u⃗k,λ, u⃗k′,λ′) = +ˆ +d3x a3u† +⃗k,λu⃗k′,λ′ = δλλ′δ3(⃗k − ⃗k′) +(41) +and similarly for (v⃗k,λ, v⃗k′,λ′) = δλλ′δ3(⃗k − ⃗k′). It follows that +|hI +k|2 + |hII +k |2 = 1. +(42) +As mentioned in the previous section, the number of time derivatives of the cosmological scale +factor a(τ) that appear in a term of the expansion is called adiabatic order of the term. +In order to solve the differential equations (38) we may follow a recursive process which preserves +the adiabatic hierarchy, just as we did with the scalar fields. Let us first redefine hI +k and the time +variable as follow +hI +k,1 ≡ +� +ΩkhI +k +dτ1 = Ωkdτ. +(43) +Substituting these relations into the equation for hI +k in (38) we find +d2 +dτ 2 +1 +hI +k,1 + Ω2 +k,1hI +k,1 = 0, +Ω2 +k,1 ≡ 1 + ǫ2, +ǫ2 ≡ −Ω−1/2 +k +d2 +dτ 2 +1 +Ω1/2 +k . +(44) +Since ǫ2 includes two derivatives, it contains terms of second and higher adiabatic order. We can +ignore it to find the leading order solution +hI +k,1 ≈ e−iτ1, +(45) +so that we get a first approximation +hI +k ≈ e−i ´ τ Ωkd˜τ +√Ωk +. +(46) +Notice that hI +k,1 formally satisfies a differential equation with the same form as (38) for hI +k. So +that, we can repeat the process: +hI +k,2 ≡ +� +Ωk,1hI +k,1, +dτ2 ≡ Ωk,1dτ1. +(47) +The corresponding differential equation for hI +k,2 is +� ∂2 +∂τ 2 +2 ++ Ω2 +k,2 +� +hI +k,2 = 0, +Ω2 +k,2 ≡ 1 + ǫ4, +ǫ4 ≡ −Ω−1/2 +k,1 +d2 +dτ 2 +2 +Ω1/2 +k,1 . +(48) +Once again, ǫ4 consists of terms of adiabatic order 4 and higher. We can approximate a solution +of (48) by neglecting ǫ4: +hI +k,2 ≈ e−iτ2 , +(49) +17 + +whereby the approximation to hI +k can be further improved: +hI +k ≈ e−i ´ τ ΩkΩk,1 d˜τ +�ΩkΩk,1 +. +(50) +By iterating the procedure, we can obtain a better and better approximation to hI +k, and after ℓ > 1 +steps we find +hI +k ≈ e−i +´ τ Ωk···Ωk,ℓ−1 d˜τ +� +Ωk · · · Ωk,ℓ−1 +, +(51) +where, for ℓ ≥ 1, +Ω2 +k,ℓ ≡ 1 + ǫ2ℓ, +dτℓ ≡ Ωk,ℓ−1dτℓ−1, +ǫ2ℓ ≡ −Ω−1/2 +k,ℓ−1 +d2 +dτ 2 +ℓ +Ω1/2 +k,ℓ−1. +(52) +Now that the general method has been set up, let’s find the 0th order solution for hI +k. From (46), +the most generic solution for hI +k is +hI +k(τ) ≈ f (0) +k +√Ωk +e−i +´ τ Ωk d˜τ = +f (0) +k +(ω2 +k + a2∆2 + iσ)1/4 e−i +´ τ √ +ω2 +k+a2∆2+iσ d˜τ, +(53) +where the time independent function f (0) +k +(of adiabatic order 0) accounts for the integration ‘con- +stant’ (strictly speaking, a function of the momentum but not of conformal time) in the exponential. +As for hII +k , by comparing both lines of (38) it is clear that it is possible to proceed in an analogous +manner. So we obtain +hII +k (τ) ≈ g(0) +k +�Ω∗ +k +e−i ´ τ Ω∗ +k d˜τ = +g(0) +k +(ω2 +k + a2∆2 − iσ)1/4 e−i ´ τ √ +ω2 +k+a2∆2−iσ d˜τ , +(54) +where g(0) +k +has the same paper as f (0) +k . To find the zeroth adiabatic order it is just enough to +expand this solution and keep zero order terms. However, some extra caution is needed when +dealing with the integrand in the exponential of (53), which may be expanded up to 1st order as +Ω(0−1) +k += ωk + ω(1) +k +, +(55) +where +ω(1) +k +≡ iaM +2ωk +a′ +a . +(56) +The reason is that the integration of the second term in the exponential factor is: +e−i +´ +ω(1) +k dτ = +�ωk + aM +k +�1/2 += +�ωk + aM +ωk − aM +�1/4 +, +(57) +so it yields a real term of adiabatic order zero, meaning that the expansion of Ωk up to 1st order +in the integral was mandatory. We have not included an explicit multiplicative factor related with +the constant of integration4 since it is already represented by f (0) +k . We choose f (0) +k +such that the +4The same situation happens with indefinite integrals of higher order terms in the imaginary exponential of +Eq. (51). They are written in an appropriate manner, contributing at bigger adiabatic orders. The final results, +though, just depend on f (0) +k +and not on the other higher order integrations constants, as dictated by the normalization +condition (42). See Appendix B for more details. +18 + +above solution can be compatible with mode functions in Minkowskian spacetime, so we can write +f (0) +k += +� +k +2, +hI(0) +k +(τ) = +� +ωk + aM +2ωk +e−i +´ τ ωk d˜τ, +g(0) +k += +� +k +2, +hII(0) +k +(τ) = +� +ωk − aM +2ωk +e−i +´ τ ωk d˜τ. +(58) +Next we move on to the solution at 1st adiabatic order. As we have mentioned, the quantity ǫ2 +defined in (44), contains terms of adiabatic order two and higher, so it is not necessary to find the +first order solution. It is enough to find the first order term from the denominator of (53). So, +hI(0−1) +k +≈ +� +1 +√ωk +� +f (0) +k ++ f (1) +k +� � +1 − iMa′ +4ω2 +k +� +e +´ τ Ma′ +2ωk d˜τ +� +e−i +´ τ ωk d˜τ += +� +ωk + aM +2ωk +� +1 − iMa′ +4ω2 +k ++ +� +2 +kf (1) +k +� +e−i ´ τ ωk d˜τ. +(59) +Similarly for the second spinor mode hII +k : +hII(0−1) +k +≈ +� +1 +√ωk +� +g(0) +k ++ g(1) +k +� � +1 + iMa′ +4ω2 +k +� +e +− ´ τ Ma′ +2ωk d˜τ +� +e−i ´ τ ωk d˜τ += +� +ωk − aM +2ωk +� +1 + iMa′ +4ω2 +k ++ +� +2 +kg(1) +k +� +e−i +´ τ ωk d˜τ , +(60) +where f (1) +k +and g(1) +k +come from integration constants, as mentioned in the footnote of the previous +page. By imposing the normalization condition (42), which has to be satisfied at each adiabatic +order, it is possible to see that these constants are purely imaginary, that is +Re f (1) +k += Re g(1) +k += 0. +(61) +To continue, we deal with the 2nd adiabatic order of the mode functions, i.e. hI,II(2) +k +. At this time, +we have to include Ω2 +k,1 = 1 + ǫ2 in our considerations (this term contains 2nd order adiabatic +terms and beyond). Starting from Eq. (50), we have +hI +k ≈ f (0) +k ++ f (1) +k ++ f (2) +k +� +Ωk(1 + ǫ2)1/2 e−i ´ τ Ωk(1+ǫ2)1/2 d˜τ , +(62) +where ǫ2 can be computed to be +ǫ2 = +5 +16Ω6 +k +� +2aa′m2 +ψ + imψa′′�2 − +1 +4Ω4 +k +� +2a′2m2 +ψ + 2aa′′m2 +ψ + imψa′′′� +. +(63) +With this result, it is immediate to obtain an approximation for Ωk,1 valid up to third adiabatic +order: +Ωk,1 = (1 + ǫ2)1/2 = 1 − a2M2 +4ω4 +k +a′′ +a − a2M2 +4ω4 +k +�a′ +a +�2 ++ 5 +8 +a4M4 +ω6 +k +�a′ +a +�2 +− iaM +8ω4 +k +a′′′ +a + ia3M3 +2ω6 +k +�a′ +a +�3 +− 15ia5M5 +8ω8 +k +�a′ +a +�3 ++ 9ia3M3 +8ω6 +k +a′ +a +a′′ +a + . . . +(64) +19 + +On the other hand, an expansion of the product ΩkΩk,1 is necessary to improve the approxima- +tion of hI,II +k , as one can see from equation (50). As earlier, if we wish to present a second order +approximation of the modes we have to expand the mentioned product up to 3rd adiabatic order +in the exponential. The expansion can be presented as follows: +ΩkΩk,1 = Ωk (1 + ǫ2)1/2 = ωk + ω(1) +k ++ ω(2) +k ++ ω(3) +k ++ . . . +(65) +where the dots represent the contributions of adiabatic order bigger than 3, and the indicated +terms in the expansion read +ω(1) +k +≡ iaM +2ωk +a′ +a , +ω(2) +k +≡ −a2M2 +8ω3 +k +�a′ +a +�2 +− a2M2 +4ω3 +k +a′′ +a + 5a4M4 +8ω5 +k +�a′ +a +�2 ++ a2∆2 +2ωk +, +ω(3) +k +≡ 5ia3M3 +16ω5 +k +�a′ +a +�3 ++ ia3M3 +ω5 +k +a′ +a +a′′ +a − iaM +8ω3 +k +a′′′ +a + ia∆2 +4Mωk +a′ +a − 25ia5M5 +16ω7 +k +�a′ +a +�3 +− ia3M∆2 +4ω3 +k +a′ +a . +(66) +As noted before, ω(1) +k +and ω(3) +k +are purely imaginary, while ωk and ω(2) +k +are real. Again, when +integrated inside the exponential of equation (50) the former two give a real contribution, whereas +the latter two become part of the phase of the mode and play the role of an effective frequency: +exp +� +−i +ˆ τ +Ωk (1 + ǫ2)1/2 d˜τ +� +≈ exp +� +−i +ˆ τ � +ω(1) +k ++ ω(3) +k +� +d˜τ +� +exp +� +−i +ˆ τ � +ωk + ω(2) +k +� +d˜τ +� += +�ωk + aM +ωk − aM +�1/4 +exp +� +5a3M3 +16ω5 +k +�a′ +a +�2 +− aM +8ω3 +k +a′′ +a + a∆2 +4Mωk +� +exp +� +−i +ˆ τ � +ωk + ω(2) +k +� +d˜τ +� +≈ +�ωk + aM +ωk − aM +�1/4 � +1 + 5a3M3 +16ω5 +k +�a′ +a +�2 +− aM +8ω3 +k +a′′ +a + a∆2 +4Mωk +� +exp +� +−i +ˆ τ � +ωk + ω(2) +k +� +d˜τ +� +(67) +The last result holds good up to an arbitrary function of momentum (constant in conformal time) +multiplying the whole result. We account for this arbitrary constant by introducing the functions +f (0) +k , f (1) +k , f (2) +k , . . . at each order. +An efficient strategy to compute the integrals involved in the above calculation (and many +other ones of a similar sort, see Appendix B for a sample of them) is to set up an ansatz which +respects the adiabaticity order of the calculation. The ansatz consists of a finite number of terms +(in fact, a linear combination of them) taken each at the given adiabatic order and with coefficients +(or ‘form factors’) which must be determined. The terms of the ansatz are constructed out of the +derivatives of the scale factor and the parameter ∆2 (which we recall is of second adiabatic order). +For instance, in order to compute the integral of ω(3) +k +in Eq. (66), we know that the result must be +of second adiabatic order. Hence as a suitable ansatz we use a linear combination of second order +adiabatic terms as follows: +− i +ˆ τ +w(3) +k d˜τ = Q1 (a, ωk) +�a′ +a +�2 ++ Q2 (a, ωk) a′′ +a + Q3 (a, ωk) ∆2 + const. +(68) +where again the term ‘const.’ at the end means that it does not depend on the integration variable, +˜τ. By taking derivatives with respect to (conformal) time of the last expression and comparing +with ω(3) +k +one can identify the form factors Q1 = 5a3M3 +16ω5 +k , Q2 = − aM +8ω3 +k and Q3 = +a +4Mωk . +20 + +Using (67) together with (65) and (62), the expansion of hI +k up to 2nd order is +hI(0−2) +k += +�ωk + aM +2ωk +�1/2 � +1 − ia′M +4ω2 +k ++ +� +2 +kf (1) +k +� +1 − ia′M +4ω2 +k +� ++ +� +2 +kf (2) +k +− Ma′′ +8ω3 +k ++ 5M3a′2a +16ω5 +k +− 5a2a′2M4 +16ω6 +k +− a′2M2 +32ω4 +k ++ aa′′M2 +8ω4 +k ++ a∆2 +4Mωk +− a2∆2 +4ω2 +k +� +e−i ´ τ � +ωk+ω(2) +k +� +d˜τ . +(69) +In a similar way, +hII(0−2) +k += +�ωk − aM +2ωk +�1/2 � +1 + ia′M +4ω2 +k ++ +� +2 +kg(1) +k +� +1 + ia′M +4ω2 +k +� ++ +� +2 +kg(2) +k ++ Ma′′ +8ω3 +k +− 5M3a′2a +16ω5 +k +− 5a2a′2M4 +16ω6 +k +− a′2M2 +32ω4 +k ++ aa′′M2 +8ω4 +k +− a∆2 +4Mωk +− a2∆2 +4ω2 +k +� +e−i ´ τ� +ωk+ω(2) +k +� +d˜τ . +(70) +The normalization condition fixes the following relations: +���f (1) +k +��� +2 += − +√ +2k Re f (2) +k , +���g(1) +k +��� +2 += − +√ +2k Re g(2) +k . +(71) +So far, the expansion for the modes hI +k and hII +k up to 2nd order has been presented. One can +continue with the procedure formerly described to reach higher orders, although of course the +calculation becomes more and more involved. We should keep in mind, though, that the adiabatic +expansion is an asymptotic expansion. While for renormalization purposes it is enough to stop +the expansion at 4th adiabatic order (in 4-dimensional spacetime), it is nonetheless necessary to +reach up to 6th order to meet the finite terms ∼ H6 that are dominant in the early universe and +capable of triggering inflation in this framework (cf. Sect. 5.3)5. We shall refrain from presenting +these cumbersome formulas in the main text, see Appendix B. +It is worth to mention that there is some residual freedom in the previous calculations since, +we can not determine entirely the set of integration constants that appear during the calculations +f (1) +k , g(1) +k , f (2) +k , g(2) +k , . . . Because of the normalization condition (42) of the mode functions, some +restrictions such as (61) and (71) apply. Fortunately, as commented in more detail in Appendix B, +the satisfaction of these restrictions is enough for the observables to be independent from this +residual freedom. So that, is enough to set all of them to 0 to get, for instance, the desired values +of the energy density and pressure. +4 +ZPE and VED for a spin-1/2 field in FLRW spacetime +The computation of the Fourier modes for a quantized fermion field through adiabatic expansion +as explained in the previous section is just the first step to compute the vacuum energy density +(VED). The next step towards the VED is to obtain the ZPE associated to Dirac fermions in curved +spacetime. As it well known, traditional computations of ZPE suffer from the well-known headache +of carrying highly unacceptable contributions proportional to the quartic powers of the masses, +5As explained in [14], owing to the renormalization prescription of the EMT – see e.g. Eq. (19) for the scalar +case and its fermionic counterpart, Eq. (79) below – the explicit 4th order powers H4 just cancel out. As a result, +the 6th order is the first non-vanishing contribution on-shell. +21 + +∼ m4. This is so both for scalar and fermion fields, and it is already the case in flat, Minkowskian, +spacetime, see e.g. [30, 31] for a detailed discussion and more references. +In curved spacetime +we have the same situation, in principle, but in addition we encounter subleading, curvature +dependent, contributions which do not exist in the flat case, as we shall see in a moment. To +handle this issue, an appropriate renormalization prescription is called for. +The calculation of the ZPE performed here for spin-1/2 fermions is closely related with the +one previously put forward for scalar fields in [14, 15] and summarized in Sec. 2. Once more the +computation will be done through adiabatic expansion of the field modes and will be carried out +up to 6th adiabatic order, since this is the first non-vanishing order on-shell, i.e. +when fixing +the renormalization scale M to the value of the mass of the fermion mψ. However, the off-shell +computation at 4th order is already very useful as a means to determine the RG running of the +VED as a function of the scale M. This is actually one of the main new features of the ARP +method proposed in [14, 15], which leads to the cosmic evolution of the VED. Next we consider +the actual calculation for spinor fields. +To find out the ZPE, we start from the definition of EMT in Eq. (4). In this case we have to +evaluate the functional derivative +T ψ +µν = − +2 +√−g +δSψ +δgµν , +(72) +applied to the fermion action (29). Upon a straightforward calculation we arrive at the following +symmetric expression: +T ψ +µν = i +4 +¯ψ +� +γµ∇ν + γν∇µ +� +ψ − i +4 +�� +∇µ ¯ψ +� +γν + +� +∇ν ¯ψ +� +γµ +� +ψ , +(73) +in which the equation of motion (31) and its hermitian conjugate have been used. We treat this +spinor field as a field operator and upon using its expansion in Fourier modes and utilizing the +anticommuting algebra of the creation and annihilation operators, Eq. (34), we can compute the +VEV of the various components, which reflect the contribution from the vacuum fluctuations of +the quantized fermion fields. We find that the VEV of the 00th component of the EMT can be +cast as follows: +� +T δψ +00 +� += +1 +2π2a +ˆ +dkk2ρk, +(74) +where ρk is a function of the previously defined mode functions: +ρk = i +a +� +hI +kh′I∗ +k + hII +k h′II∗ +k +− hI∗ +k h′I +k − hII∗ +k h′II +k +� +. +(75) +The explicit form of the adiabatic expansion of ρk is rather cumbersome; the reader may find the +final result of ⟨T δψ +00 ⟩ in the Appendix B. Let us note that for off-shell renormalization at a point M +it suffices to adiabatically expand the solution up to 4th order (see Eq. (19) and Eq. (79) below) but +we provide the result up to 6th order so as to be sensitive to the on-shell result (when M = mψ) +and also because it is important for the inflationary mechanism in the early universe (cf. Sec. 5.3). +Renormalization is indeed necessary since the VEV of the EMT is formally divergent. The UV- +divergent contributions appear up to 4th adiabatic order (in n = 4 spacetime dimensions), so that +one has to subtract terms at least up to this order to obtain a finite result. +4.1 +Divergence balance between bosons and fermions in vacuum +The VEV can be split in two different parts, divergent (in the UV sense) and non-divergent. +Explicit calculation using the formulas of AppendixC) shows that the divergent part is +� +T δψ +00 +� += +1 +2π2a2 +ˆ ∞ +0 +dkk2 +� +−2ωk − a2∆2 +ωk ++ a4∆4 +4ω3 +k +� ++ +1 +2π2 +ˆ ∞ +0 +dkk2 +� M2 +4ω3 +k ++ ∆2 +4ω3 +k +� +H2 . +(76) +22 + +As it is easy to see, there are terms diverging quartically, quadratically and logarithmically. The +non-divergent part contains the remaining terms, all of them being finite. The above ZPE is an +unrenormalized result at this point. However, before we proceed to renormalize that expression, +it may be instructive to check if there is a chance for the cancellation between UV-divergent +terms between fermions and bosons in the supersymmetric (SUSY) limit, at least for the leading +divergences. In the on-shell case (M = m and hence ∆2 = 0) the above equation (76) simplifies to +⟨T δψ +00 ⟩ +��� +(M=m) = − +1 +π2a2 +ˆ +dkk2ωk(m) + +1 +8π2 +ˆ ∞ +0 +dkk2 +m2 +ω3 +k(m)H2 . +(77) +It coincides with the Minkowskian result for a = 1 (since H = 0). Now in a SUSY theory, in which +the number of boson and fermion degrees of freedom (d.o.f.) is perfectly balanced, we should expect +that the leading (quartic) divergences cancel among the fermionic and bosonic contributions in the +vacuum state [92,93] since in such case the scalar and fermionic fields have the same mass m. Thus +the quartically divergent contribution from the first term of (77) should be minus four times the +corresponding result for one real scalar field. Indeed it is so, for in the on-shell limit and projecting +the UV-divergent terms of the first two adiabatic orders only, we find that the contribution from +one real scalar field in FLRW spacetime with spatially flat metric is [15] +⟨T δφ +00 ⟩(0−2)��� +(M=m) = +1 +4π2a2 +ˆ +dkk2ωk(m) − 3 +� +ξ − 1 +6 +� +4π2a2 +ˆ +dkk2 +� +H2 +ωk(m) + a2m2H2 +ω3 +k(m) +� +. +(78) +We confirm that the first term (the quartically divergent one) of this expression is of opposite sign +to the first one in(77) and is a factor of 4 smaller, as we indicated. So, in a SUSY theory, where we +would have 4 real scalar d.o.f. for each Dirac fermion, there would be an exact cancellation of the +leading UV-divergent terms. In addition, we can see at once from (78) that both the quadratic and +logarithmic divergences of bosons are associated to effects of the spacetime curvature since they are +proportional to H2. These terms, therefore, vanish in Minkowski spacetime but are unavoidably +present in the FLRW background. On the other hand from the second term on the r.h.s. of Eq. +(77) it is clear that for fermions we only have subleading divergences of logarithmic type, which are +also associated to curvature effects since they are again proportional to H2 and would also vanish in +Minkowski space. Hence there is no possible cancellation of these subleading divergences between +bosonic and fermionic d.o.f., in FLRW spacetime, even in the exact SUSY limit. Of course, our +framework is not placed in the context of supersymmetry, but it serves as a consistency check of +our calculations. See also the discussion in [94,95]. +Although it is possible to introduce a cutoff for a preliminary treatment of the subleading +diverges (and maybe to speculate on its possible meaning) it is not really necessary. One simply +has to implement appropriate renormalization since renormalization is anyway necessary to deal +meaningfully with the VED, as there is no way to cure the divergences from the combined contri- +butions from bosons and fermions and it is not useful to be left with a “physical” cutoff. Dealing +with a cutoff is always ambiguous as it is generally not a covariant quantity. Renormalization gets +rid of cutoffs and one can preserve covariance, which is safer for a physical interpretation of the +results. The adiabatic renormalization is ideal in this sense since the adiabatic expansion generates +automatically a covariant result. +It is well-known that the renormalization program in QFT requires the presence of a renor- +malization point, as well as a renormalization prescription. The renormalization point is a floating +scale characteristic of the RG. As in the ordinary adiabatic procedure, to implement the renormal- +ization of the EMT in 4 spacetime dimensions we perform a subtraction of the first four adiabatic +orders, which are the only ones that can be UV-divergent [1–3]. However, in contrast to the usual +recipe, in which the subtraction is performed on the mass shell value m of the quantum field, we +23 + +perform it at an arbitrary scale M since this enables us to explore the RG evolution of the VED +and ultimately connect it with its cosmic evolution. This is the specific feature of the adiabatic +renormalization procedure (ARP) for the VED that was proposed in [14,15] – see also [31] for ad- +ditional details and a comparison with other renormalization schemes. The resulting renormalized +VED ensuing from this procedure is free from the usual troubles associated to the quartic powers +of the masses and the associated fine tuning problems. +Finally, let us note that dealing with the CCP in Minkowski spacetime using, for instance, +the MS scheme and assigning some value to the ’t Hooft’s mass unit µ in DR (as discussed so +many times in the literature), is entirely meaningless. It is not only devoid of meaning in that +a non-vanishing cosmological constant cannot be defined in Minkowski space without manifestly +violating Einstein’s equations; it is meaningless also on account of the fact that there is no sense in +associating the scale µ to a cosmological variable, say H, since, if Einstein’s equations are invoked, +the Λ term as such in these equations cannot exist in Minkowski spacetime unless the VED is +exactly ρΛ + ZPE = 0. So there is no cosmology whatsoever to do in flat spacetime, despite some +stubborn attempts in the literature. Persisting in this attitude leads to the nonsense of having to +cope with ∼ m4 effects which must then be fine tuned among all the particles involved. This point +has been driven home repeatedly e.g. in [30] and also recently in [31], see also [96]. A realistic +approach to the VED within QFT in curved spacetime must get rid of Minkowski space pseudo- +argumentations. The approach that we present here is fully formulated in curved spacetime and +the vacuum energy density just evolves with the curvature effects (powers of H) rather than with +powers of the masses, i.e. we pursue the successful renormalization program of [14,15]. Therefore, +when the background curvature vanishes, we consistently predict that the non-trivial effects which +are responsible for the value of the vacuum energy density and the cosmological constant disappear +(and hence we are left with no Λ nor VED in the universe). Such is, of course, the situation in +Minkowski space. +In practice, however, we cannot reach that flat spacetime situation in our +universe since there exists four-dimensional curvature at all times during the indefinite process of +expansion. But by the same token such an impossibility evinces the fact that the VED and its +dynamical nature is a direct consequence of the expansion process (and of the spacetime curvature +inherent to it). The expected size of the VED and of Λ in our framework is indeed provided by +the magnitude of the spacetime curvature, which is of the typical value of the measured Λ. It +is therefore not caused by the quartic power of the masses of the fields (which is the very root +of the CC problem in most approaches). These powers do not affect the running of the VED in +our framework. To put it in a nutshell: the renormalized VED in our framework is like a small +quantum ‘ripple’ imprinted on the existing (classical) background curvature owing to the vacuum +fluctuations of the quantized matter fields. In the absence of the background curvature, the ripple +would disappear too since it is proportional to it through the coefficicient νeff, which encodes the +quantum effects from the quantized matter fields. +Following the same approach as for scalar fields, in the next section we compute the quantum +effects contributing to the VED from the quantized spin-1/2 fields and express them in renormalized +form using the same substraction scheme devised in [14,15]. +4.2 +Renormalized ZPE for fermions +Thus, following the same prescription (19) as in the case of the scalar field, the renormalized form +of the fermionic EMT is defined in our case as follows: +� +T δψ +µν +� +ren (M) ≡ +� +T δψ +µν +� +(mψ) − +� +T δψ +µν +�(0−4) +(M) . +(79) +24 + +In particular, this can be alternatively written as 6 +� +T δψ +00 +� +ren (M) = +� +T δψ +00 +� +Div (mψ) − +� +T δψ +00 +� +Div (M) + +� +T δψ +00 +�(0−4) +Non−Div (mψ) − +� +T δψ +00 +�(0−4) +Non−Div (M) ++ +� +T δψ +00 +�(6) +(mψ) + . . . += +1 +2π2a +ˆ ∞ +0 +dkk2 +� +−2ωk(mψ) +a ++ 2ωk(M) +a ++ +a∆2 +ωk(M) − +a3∆4 +4ω3 +k(M) +� ++ +1 +2π2a +ˆ ∞ +0 +dkk2 +� +am2 +ψ +4ω3 +k(mψ) − +aM2 +4ω3 +k(M) − +a∆2 +4ω3 +k(M) +� �a′ +a +�2 ++ +� +T δψ +00 +�(0−4) +Non−Div (mψ) − +� +T δψ +00 +�(0−4) +Non−Div (M) + +� +T δψ +00 +�(6) +(mψ) + . . . +(80) +where we have used the calculational results for the unrenormalized components of the vacuum +EMT recorded in Appendix C and we have introduced the notation ωk(M) ≡ +√ +k2 + a2M2 and +ωk(mψ) ≡ +� +k2 + a2m2 +ψ. The last line of (80) contains all the non-divergent terms, which consti- +tute a perfectly finite contribution and is made of finite parts from the 4th order expansion and of +the entire 6th order term, which is fully finite but rather cumbersome. On the other hand, the first +two lines in the last equality are a collection of terms that are individually divergent, but whose +combination makes the integral convergent. In fact, by making use of simple algebraic manipula- +tions at the level of the integrand one can show that explicitly. For instance, the rearrangement +in the integrand +dkk2 +� +ω(mψ) − ω(M) − a2∆2 +2ω(M) + +a4∆4 +8ω3(M) +� += dkk2a6∆6 +ω(mψ) + 3ω(M) +8ω3(M)(ω(mψ) + ω(M))3 +(81) +shows that terms seemingly diverging as ∼ k4 organize themselves to eventually converge as ∼ 1/k2. +Needless to say, this is the consequence of the subtraction that has been operated. Similarly with +the second integral in (80), whose individual terms are logarithmically divergent, but overall the +integral is once more convergent thanks to the involved subtraction. +The above renormalized result (80) would, of course, vanish for M = mψ if we were to stop the +calculation at 4th order, so in case that one wishes to obtain the renormalized on-shell result one +has to either compute the exact unrenormalized EMT on-shell before subtracting the divergent +adiabatic orders – which is possible but only in simpler cases such as in de Sitter space [73,74] – or +one has to face the calculation of the adiabatic expansion up to 6th-order at least. In the last case +the term ⟨T δψ +00 ⟩(6)(mψ) indicated at the end of Eq. (80) must be computed. This is what we have +done here since an exact solution in the FLRW case is not possible. The amount of calculation to +reach up to 6th adiabatic order is significant. The un-renormalized components of the EMT up to +the desired order are explicitly collected in Appendix C. To subsequently obtain the renormalized +EMT one has to implement the subtraction (79) and compute all the involved integrals. The final +result up to the mentioned order can nevertheless be presented through a rather compact formula, +as follows:7. +6The subscript ’Div’ refers to the part of the EMT calculation comprising divergent integrals. These appear only +up to the 4th adiabatic order. The subscript ’Non-Div’, on the other hand, refers, of course, to the part of the EMT +calculation involving finite integrals only. +7We refer the reader to Appendix A.2 of [15] for the computation/regularization of the involved integrals (depend- +ing if they are convergent or divergent) from the master DR formula indicated there. Use of DR can be convenient +since in certain cases the needed rearrangement of terms in the integrand to verify that the overall integral is actu- +25 + +� +T δψ +00 +�(0−6) +ren +(M, H) = +a2 +32π2 +� +3m4 +ψ − 4m2 +ψM2 + M4 − 2m4 +ψ ln +m2 +ψ +M2 +� ++ +1 +16π2 +� +m2 +ψ − M2 − m2 +ψ ln +m2 +ψ +M2 +� +H2 ++ +1 +20160π2a4m2 +ψ +� +204H4H′ + 26 +� +H′�3 − 30H3H′′ + 9 +� +H′′�2 + 9H2 � +3 +� +H′�2 − 8H′′′� +− 18H′H′′′ + H(−78H′H′′ + 18H′′′′) +� += +a2 +32π2 +� +3m4 +ψ − 4m2 +ψM2 + M4 − 2m4 +ψ ln +m2 +ψ +M2 +� ++ a2H2 +16π2 +� +m2 +ψ − M2 − m2 +ψ ln +m2 +ψ +M2 +� ++ +a2 +20160π2m2 +ψ +� +− 31H6 − 108H4 ˙H − 46 ˙H3 + 126H3 ¨H + 9 ¨H2 − 18 ˙H +... +H ++ 27H2 � +7 ˙H2 + 4 +... +H +� ++ 6H(23 ˙H ¨H + 3 +.... +H ) +� +. +(82) +The final equality corresponds to the expression in terms of the cosmic time (d()/dt ≡ ˙()) with +H ≡ ˙a/a. We point out that there is an explicit dependency on the Hubble function (and its +derivatives) coming from Gµν. This justifies the notation ⟨T δψ +00 ⟩ren(M, H), with two arguments, +where the dependence on the time derivatives of H is omitted for simplicity. We note that in the +fermionic case there are no terms of O(H4) in the evolution of the ZPE (and the VED, see next +section), in stark contrast to the situation with scalars, see the last line of Eq. (20), where we can +recognize terms of the form H2 ˙H, H ¨H and ˙H2 all of them of O(H4). We also remark what has +been previously anticipated: for M = mψ (on-shell point) only the 6th-order terms remain, which +are the ones in the last two lines of Eq. (82). These terms are relevant for the RVM mechanism of +inflation in the very early universe (cf. Sec. 5.3). However, for the study of the renormalized theory +at the point M (generally different from the on-shell mass point mψ) it is enough to consider the +terms up to 4th adiabatic order, those in the first line of Eq. (82), see the next section. +So far, we have been able to provide the desired formula for the renormalized ZPE at the energy +scale M up to 6th adiabatic order, as expressed by Eq. (82). This is, however, not the end of the +story, since a proper expression for the VED needs to take into account also the renormalized +parameter ρΛ in the Einstein-Hilbert action (2), as this parameter is part of the unrenormalized +vacuum action and after renormalization it also runs with the scale M, i.e. ρΛ(M). Both the +renormalized ZPE and ρΛ(M) run with the scale and this will be crucial to study the properties +of the renormalized VED. The running of the ZPE part between two different scales M and M0 +can be illustrated by considering the difference of the respective ZPE values at these scales. From +(82) we find +� +T δψ +00 +� +ren (M, H) − +� +T δψ +00 +� +ren (M0, H) = +a2 +32π2 +� +M4 − M4 +0 − 4m2 +ψ(M2 − M2 +0 ) + 2m4 +ψ ln M2 +M2 +0 +� ++ a2H2 +16π2 +� +−M2 + M2 +0 + m2 +ψ ln M2 +M2 +0 +� +. +(83) +ally convergent can be complicated. Let us emphasize, however, that DR is only used as an auxiliary regularization +tool for intermediate steps. The final result has no memory of this intermediate step, see e.g. Appendix B of [14] +for an explicit nontrivial example. To be sure, no MS prescription is used for renormalization at any point of our +calculation. The crucial difference between the ARP and the MS-like schemes is that the subtraction (79) involves +not just the UV-divergences but also the finite parts. +26 + +The finite parts, and in particular the 6th order terms cancel of course in the above difference, but +the latter will be essential in the on-shell case since the result would be zero without these higher +order effects 8. We should notice that, in contradistinction to the case with scalar fields, there are +no contributions of O(H4) such as H2 ˙H, H ¨H or ˙H2 in the expression for the ZPE, as can be +seen on comparing equations (20) and (82). For this reason it is unnecessary to use the higher +derivative (HD) tensor (1)Hµν (cf. AppendixA) as part of the renormalized Einstein’s equations +in the case of the fermion fields, again in contrast to the situation with the scalar fields – see [15] +for details. Therefore, for fermions the subtraction at the two scales of the renormalized form of +Einstein’s equations can be done using the ordinary form of Einstein equations (3) without higher +order curvature terms, and we find +� +T δψ +µν +� +ren (M, H) − +� +T δψ +µν +� +ren (M0, H) = (ρΛ(M) − ρΛ(M0)) gµν + +� +1 +8πG(M) − +1 +8πG(M0) +� +Gµν. +(84) +By comparison equations (83) and (84), and taking into account the tensorial structure of (84) +and the explicit form of Gµν in FLRW spacetime (cf. Appendix A) , we can perform the following +identifications: +ρΛ(M) − ρΛ(M0) = − +1 +32π2 +� +M4 − M4 +0 − 4m2 +ψ(M2 − M2 +0 ) + 2m4 +ψ ln M2 +M2 +0 +� +, +1 +8πG(M) − +1 +8πG(M0) = +1 +48π2 +� +−M2 + M2 +0 + m2 +ψ ln M2 +M2 +0 +� +. +(85) +4.3 +Renormalized VED +Once the renormalized ZPE has been obtained, the same consideration as for the scalar field case +(see (22) and (23) ) can be repeated intact here, thus leading to the expression for the renormalized +VED of the fermionic field: +ρδψ +vac(M, H) = ⟨T vac +00 ⟩ (M, H) +a2 += ρΛ(M) + +� +T δψ +00 +� +ren (M, H) +a2 +. +(86) +Now if the subtraction of scales is done, +ρδψ +vac(M, H) − ρδψ +vac(M0, H) = ⟨T vac +00 ⟩ (M, H) − ⟨T vac +00 ⟩ (M0, H) +a2 += ρΛ(M) − ρΛ(M0) + +� +T δψ +00 +� +ren (M, H) − +� +T δψ +00 +� +ren (M0, H) +a2 += ρΛ(M) − ρΛ(M0) − (ρΛ(M) − ρΛ(M0)) + 3H2 +� +1 +8πG(M) − +1 +8πG(M0) +� += H2 +16π2 +� +−M2 + M2 +0 + m2 +ψ ln M2 +M2 +0 +� +, +(87) +where in the last equality (85) was used. As expected, when written in terms of the ordinary +Hubble function H in cosmic time, the evolution of the VED does not depend explicitly on the +8Let us remark that the difference (83) is an exact result, in the sense that it does not depend on the adiabaticity +order we are working. This is obvious from the renormalization prescription (79), as all higher orders beyond the 4th +one (not only the 6th) cancel out in the subtraction, the reason being that these adiabatic orders are independent +of the renormalization point M. The latter is involved in the calculation of the EMT up to 4th order only (as these +are the only adiabatic orders that are UV-divergent). +27 + +scale factor. +For the sake of emphasizing the point, in the above equation we have explicitly +indicated the cancellation of the terms carrying along the quartic powers of the masses, see the +third equality in the above equation. As we can see, it is essential that the structure of the VED +is obtained from the sum “VED = ρΛ +ZPE”, i.e. as in Eq. (23), since the mentioned cancellation +occurs between the renormalized expressions of ρΛ and ZPE upon being subtracted at the two +arbitrary scales M and M0. This means that the two values of the VED at these scales are related +in a very smooth manner: in fact, they differ only by a term proportional to H2, as it is obvious +from (87). +Even though Eq. (87) is formally correct, our job is not finished in the physical arena yet. +Despite of the fact that such an equation describes the precise mathematical evolution of the +VED with the renormalization scale, M, it is necessary to associate the latter with a suitable +physical scale in order to extract useful phenomenological information out of it, exactly as in +the companion studies of the VED for scalar fields previously presented in [14, 15]. As pointed +out in these references and also in Sec. 2.2 regarding the contribution from the scalar fields, the +Hubble rate H is a characteristic energy scale (in natural units) of the expanding universe in the +FLRW metric, and hence proves to be a natural candidate for a representative physical scale in +this context. Whereby by following the same prescription used in the aforementioned references, +we set the renormalization energy scale to M = H(t) (at the end of our calculations) in order +to track the physical evolution of the VED. In other words, this prescription should allow us to +explore the VED at different expansion history times H(t) in a physically meaningful way. In this +way we obtain a well behaved evolution of the VED, which means that, given its value at one scale +all other values at nearby scales are very close to it. The dynamics of the VED is slow and can be +encoded into an effective contribution to the νeff parameter, as we did for bosons in Eq. (26). The +combined contribution from bosons and fermions to this parameter will be given in Sec. 5. Let +us finally clarify the sense of this scale setting, Namely, the full effective action does not depend +on M, of course, but the renormalized VED indeed does since the effective action of vacuum is +only a part of the full effective action. The scale dependence on M from the other terms of the +action, for example the terms carrying the running couplings of the RG-improved classical action, +compensates for the M-dependence of the vacuum action. Put another way, only the full effective +action (involving the classical part plus the nontrivial quantum vacuum effects) is scale- (i.e. RG-) +independent. This is of course the standard lore of the renormalization group (RG), see also [31] for +an expanded discussion. The choice of a particular scale helps of course in enhancing the physical +significance of particular sectors of the full effective action. The procedure is of course akin to +the usage of the RG in conventional gauge theories of strong and electroweak interactions, except +that here one has to pick out an appropriate cosmological energy scale which is most suitable for +the description of the universe’s expansion. The distinguished scale H appears to be the natural +choice if the universe where we live is indeed appropriately described by the FLRW metric. In the +next section we apply this approach to derive the important RG equation of the VED itself. +4.4 +Renormalization group equation for the vacuum energy density +One can also compute the β function of the running vacuum associated to fermionic quantum +fluctuations. Only the adiabatic terms below 4th order carry M-dependence by definition since +the higher orders are finite and hence are not subtracted in the renormalization procedure. As it +was already noted before, in contrast to the scalar case the terms of 4th adiabatic order are not +present for fermions. The computation follows the same strategy as for scalars [15]. In this case +28 + +we make use of equations (83) and (86), and we find +βδψ +ρvac =M ∂ρδψ +vac(M) +∂M += βδψ +ρΛ + +1 +8π2 +� +M2 − m2 +ψ +�2 − +1 +8π2 H2 � +M2 − m2 +ψ +� += − 1 +8π2 H2 � +M2 − m2 +ψ +� +. +(88) +The second equality holds immediately after computing the β-function of the parameter ρΛ. From +the first equation (85) we find that +βδψ +ρΛ = M ∂ρΛ(M) +∂M += − 1 +8π2 +� +M2 − m2 +ψ +�2 . +(89) +and hence contains a term proportional to the quartic power of the particle mass; what’s more, +there is an exact cancellation between the terms of the ZPE containing quartic powers of M and +mψ and the expression of βρΛ. The result (88) can also be consistently obtained directly from +Eq. (87). Notice that neither the parameter ρΛ nor the ZPE have physical meaning in an isolated +way, only the sum makes physical sense and defines the VED in the present context. +Let us +compare the above results with those following from the contribution of one real scalar field φ [15]: +βδφ +ρvac = +� +ξ − 1 +6 +� 3H2 +8π2 +� +M2 − m2 +φ +� ++ O(H4) +(90) +and +βδφ +ρΛ(M) = +1 +2(4π)2 (M2 − m2 +φ)2 . +(91) +where we omit the O(H4) terms in the scalar case (not present in the fermionic case) since it is +enough to check the comparison at low energies. We can see that in both cases the β-function of +the VED is proportional to βρvac ∝ H2 � +M2 − m2� +, where m = mφ or mψ, and therefore has a +very smooth behavior thanks to the factor H2. In contrast, the β-function for the parameter ρΛ +in the gravitational action (which is often incorrectly identified as the VED in some explicit QFT +calculations of the vacuum energy in the literature) behaves in both cases as βρΛ ∝ +� +M2 − m2�2 +and hence leads to undesired quartic contributions ∼ m4 to the running. These are the problematic +terms leading to fine tuning problems, but as can be seen these terms exactly cancel in βρvac for the +vacuum energy both for fermions and bosons in our renormalization scheme. Notice that there is +a factor of 4 between equations (89) and (91) and have opposite sign. In a SUSY context, Eq. (91) +should by multiplied by 4 to equalize bosonic and fermionic d.o.f in a given matter supermultiplet, +all of whose members possess the same mass. Then βδφ +ρΛ → 4βδφ +ρΛ ≡ βδφ (SUSY) +ρΛ +, and the sum of the +two coefficients will indeed vanish in a supersymmetric context: +βδψ (SUSY) +ρΛ ++ βδφ (SUSY) +ρΛ += 0 . +(92) +But this is, of course, not a cancellation of the β-function coefficients for the VED of bosons +and fermions in the SUSY limit, but only the cancellation of the contributions to the β-function +coefficient for the formal parameter ρΛ in the EH action (2). This property is obviously connected +with the discussion in Sec. 4.1 about the balance of UV-divergences between fermions and bosons. +In a SUSY theory the quartic divergences cancel prior to any renormalization process, as we +have noticed, and the resulting β-function for the parameter ρΛ is zero. By the same token the +running of the VED is freed from ∼ m4 effects, which cancel among fermions and bosons in a +SUSY context. The quartic powers are independent of the curvature of spacetime. However, the +subleading divergences do depend on the background curvature and do not cancel at all, even in +29 + +the exact SUSY limit 9. The “residual” (finite) parts left in the renormalization process do not +cancel either; they are actually proportional to the curvature of the FLRW background, R ∼ H2. +This fact translates into a correction to the physical vacuum energy density of order ∼ m2H2 both +for bosons and fermions, which is far smaller than m4. So the finite, curvature dependent, terms +that remain after ARP renormalization are de facto the most important ones for our purposes +since they lead to the RVM form of the VED! The renormalization of the formal parameter ρΛ, +in contrast, has no physical imprint in the final result for the VED, except that the unwanted m4 +terms cancel against those involved in the ZPE, thus rendering the renormalized V ED = ρΛ +ZPE +free from quartic mass dependencies. +From the above RG equations we may write down the total contribution to the β-function of +the VED from the matter fields. Consider Nf species of fermion fields with masses mψ,ℓ for each +species ℓ ∈ {1, 2, . . . , Nf}, and similarly let Ns be the number of scalar field species with masses +mφ,j, j ∈ {1, 2, . . . , Ns}. Some of these species may have the same mass, but this aspect is not +relevant here, our formulas will include a summation over all contributions irrespective if some +of them may be equal. The total β-function of the VED from an arbitrary number of quantized +matter fields can now be cast as follows: +βquant.matt. +ρvac +≡ +Ns +� +j=1 +βδφj +ρvac+ +Nf +� +ℓ=1 +βδψℓ +ρvac = 3H2 +8π2 + + +Ns +� +j=1 +� +ξj − 1 +6 +� +(M2 − m2 +φj) − 1 +3 +Nf +� +ℓ=1 +(M2 − m2 +ψℓ) + ++O(H4) . +(93) +The net outcome, therefore, is that the β-function of the vacuum energy density is free from +undesirable contributions proportional to quartic mass powers of the quantized fields, ∼ m4, and +hence these contributions do not appear in the renormalized theory. This is of course an extremely +welcome feature of our renormalization framework, which is, on inspection of the above equation, +fully shared by both scalar and fermion fields. Indeed, up to numerical factors fermions and scalar +fields provide the same kind of leading contribution to the time evolution of the cosmological +vacuum energy. Overall we find that the running of ρvac depends only on quadratic terms in the +fermion mass, namely ∼ m2 +ψℓH2, which are of the same type as in the case of bosons, namely +∼ m2 +φjH2, as discussed in Sec. 2.2 and previously demonstrated in great detail in [14,15]. These +terms are actually very smooth owing to the presence of the H2 factor. +Integrating the RG +equation associated to the β-function (93) one finds the expression for the evolution of the VED +as a function of the renormalization scale M in the presence of any number of matter fields, see +Sec. 5. In particular, integrating (88) for the case of one single fermion it is easy to verify that it +leads exactly to (87). +The kind of much tempered behavior of the VED evolution that we have found here within +our ARP renormalization program is of the sort that was expected on the basis of semi-qualitative +RG arguments and constitutes the characteristic running law of the so-called Running Vacuum +Models (RVM), see [30, 31] and references therein. +Thus, there is no need for fine-tuning in +this scenario, since in such a renormalization procedure we have already gotten rid of the ugly +contributions associated to the quartic powers of the masses. In other words, the ‘problem’ with +the quartic powers of the masses does not appear in the physically renormalized theory. While the +running of the formal parameter ρΛ with M indeed carries ∼ m4 contributions, as it is obvious +from the formulas above, this fact has no physical implication since ρΛ is not itself a physical +9Needless to say, the SUSY considerations made here in passing only intend to clarify that in curved spacetime, +irrespective of whether the quantized matter fields belong to a supersymmetric theory or not, the renormalization +program is in any case mandatory to finally get rid of all the UV divergences. The calculations in this work, however, +do not presume any SUSY context at all, not even a SUSY-broken theory. Our treatment of scalar and fermion +fields is indeed completely general, in the sense that we are dealing with an arbitrary number of matter fields of both +species without enforcing any balance between bosonic and fermionic d.o.f. – see Sec. 5 for more details. +30 + +parameter (if taken in isolation) and the unwanted terms carried by it exactly cancel out in the +β-function for the VED, as we have just proven. As a result, the running of the VED is much +softer, the ‘slope’ is ∼ m2H2 rather than ∼ m4. +At variance with this result, in the context +of the MS renormalization approach, in which ρΛ runs with the unphysical mass unit µ coming +from dimensional regularization, one is enforced to fine tune ρΛ(µ) against the large contribution +proportional to ∼ m4 terms [31]. +4.5 +Renormalization of the fermionic vacuum pressure +Taking into account the perfect fluid form of the EMT associated to the vacuum, the corresponding +pressure is defined through the iith-components. Any of them can be utilized owing to the assumed +homogeneity and isotropy. So, it is just enough to compute the VEV of the 11th-component10: +Pvac(M) = ⟨T vac +11 ⟩ +a2 += −ρΛ(M) + +� +T δψ +11 +�ren +(M) +a2 +. +(94) +From (73) and using once more the expansion of the spin-1/2 fermion fields in Fourier modes (cf. +AppendixB and AppendixC) the result can be expressed, after considerable work, as follows: +� +T δψ +11 +� += +1 +2π2a +ˆ ∞ +0 +dkk2Pk, +(95) +with +Pk ≡ −2k +3a +� +hI +khII∗ +k ++ hI∗ +k hII +k +� +(96) +and where the explicit expressions (in WKB-expanded form) for the fermion modes hI +k and hII +k can +be found in the aforementioned appendices. Notice that there is a relation between ρk and Pk +Pk = − ρ′ +k +3H , +(97) +which follows from (75) using the mode equations (37). This relation can be used as an alternative +way to calculate ⟨T δψ +11 ⟩ from ⟨T δψ +00 ⟩: +� +T δψ +11 +� += − 1 +3H +�� +T δψ +00 +�′ ++ H +� +T δψ +00 +�� +(98) +For the sake of simplicity, the remaining discussions of this section will be restric to the case +of one single neutral scalar field and one single Dirac fermion. We shall retake the multifield case +in Sec. 5. Following the same steps and considerations made in the previous sections for the 00th +component of the EMT, we reach the following expression for the renormalized value of the VEV +of the 11th-component of the EMT up to 6th adiabatic order: +� +T δψ +11 +�(0−6) +ren +(M) = − a2 +32π2 +� +M4 + 3m4 +ψ − 4m2 +ψM2 − 2m4 +ψ ln +m2 +ψ +M2 +� ++ +1 +48π2 +� +M2 − m2 +ψ + m2 +ψ ln +m2 +ψ +M2 +� +H2 + +1 +24π2 +� +M2 − m2 +ψ + m2 +ψ ln +m2 +ψ +M2 +� +H′ ++ +1 +20160π2a4m2 +ψ +� +−245H2 � +H′�2 + 8 +� +H′�3 − 98H3H′′ + 35 +� +H′′�2 − 62H2H′′′ ++204H4H′ − 66HH′H′′ + 56H′H′′′ + 42HH′′′′ − 6H′′′′′� +10One can either compute the VEV of the T11 component, as we do here, or use the formula (24), which allows +to compute the vacuum pressure from the 00th component and the trace of the EMT. The result is the same, of +course, owing to the isotropy of vacuum. In Ref. [15], for instance, we presented the computation of the pressure for +the scalar fields using this second method. +31 + += − a2 +32π2 +� +M4 + 3m4 +ψ − 4m2 +ψM2 − 2m4 +ψ ln +m2 +ψ +M2 +� ++ +a2 +16π2 +� +M2 − m2 +ψ + m2 +ψ ln +m2 +ψ +M2 +� +H2 + +a2 +24π2 +� +M2 − m2 +ψ + m2 +ψ ln +m2 +ψ +M2 +� +˙H ++ +a2 +20160π2m2 +ψ +� +31H6 + 170H4 ˙H − 45H2 ˙H2 − 80 ˙H3 − 90H3 ¨H − 55 ¨H2 +−150H2... +H − 100 ˙H +... +H − 6H(65 ˙H ¨H + 9 +.... +H ) − 6 +..... +H +� +. +(99) +We may now proceed to compute the vacuum EoS for the fermion fields up to the sixth adiabatic +order. The best strategy is to compute first the pressure through Eq. (99), which can be inserted +into the relation (94). Using next the VED expression (86) for fermions – with ⟨T δψ +00 ⟩ given by +(82) – the vacuum pressure can be seen to be equal to minus the VED plus some additional terms: +Pvac(M) = −ρvac(M) + +1 +24π2 +� +M2 − m2 +ψ + m2 +ψ ln +m2 +ψ +M2 +� +˙H ++ +1 +20160π2m2 +ψ +� +62H4 ˙H + 144H2 ˙H2 − 126 ˙H3 + 36H3 ¨H − 46 ¨H2 − 42H2... +H +− 118 ˙H +... +H − 6H(42 ˙H ¨H + 4 +.... +H ) − 6 +..... +H +� ++ . . . +(100) +The additional terms represent a small (but worth noticing) deviation from the classical vacuum +EoS relation Pvac = −ρvac. The dominant vacuum EoS is still the classical one up to a leading +correction of O( ˙H) (the second term on the r.h.s of the above equation) and several sorts of higher +order corrections of O(H6) indicated in the last two lines. The ∼ ˙H correction in the first line +of Eq. (100) can obviously be relevant for the present universe, and in particular it can modify +the equation of state of the vacuum it to depart from −1 at present (cf. Sec. 5.4). The higher +order terms in the last two lines, in contrast, might be relevant only for the very early universe. +in principle. However, these terms involve time derivatives and hence vanish for H =const. This +fact will have implications for our discussion of RVM-inflation in Sec. 5.3, since inflation can be +shown to exist in this framework for H =const., cf. Sec. 5.3. So at the end of the day, the higher +order terms in the last two lines of Eq. (100) become irrelevant both at low and high energies in +this framework. The remarkable consequence is that the EoS of the quantum vacuum is very close +to −1 during inflation, in contrast to the vacuum EoS in subsequent eras of the cosmic evolution +(cf. Sec. 5.4). +4.6 +Trace Anomaly +It is a very well known result that if a field theories has a classical action which is conformally +invariant, then the trace of the classical EMT vanishes exactly. For this it is necessary to work +with a massless field, otherwise the presence of a mass breaks the symmetry since it introduces a +fixed length scale. For instance, for a massless scalar field, +lim +ξ→1/6 lim +mφ→0 TCl. (φ) = 0. +(101) +This follows immediately from (6) and (7). However, it is also true that this simple result does not +hold when one considers the pure quantum contribution associated to the quantum fluctuations +of the scalar field and constitutes a conformal anomaly, [1]. This follows after a careful study of +the diverging part of the vacuum effective action, W Div +eff , in which Weff was defined in Eq. (17). +32 + +The part W Div +eff +is not conformally invariant for an arbitrary number of spacetime dimensions n +(although Weff is so in the massless limit), except for the case n = 4. As a consequence, W Div +eff +receives a finite payoff for n → 4 owing to the existing pole 1/(n − 4) in it. Correspondingly, the +VEV of the on-shell EMT receives a nontrivial contribution in the massless limit coming from the +divergent part of the effective action, even in the case ξ = 1/6: +lim +mφ→0 lim +ξ→1/6 +� +T δφ� += − lim +mφ→0 m2 +φ +� +δφ2� +. +(102) +The term +� +δφ2� +contains some elements of 4th adiabatic order proportional to 1/m2 +φ, so that the +corresponding limit results in a finite contribution. The same idea applies in the fermionic case, +lim +mψ→0 +� +T δψ� += − lim +mψ→0 mψ +� ¯ψψ +� +. +(103) +Here the term +� ¯ψψ +� +contains 4th adiabatic order terms that are proportional to 1/mψ which +make the limit non-trivial. Technically speaking (102) and (103) are not yet what we call the +trace anomaly or conformal anomaly . This is due to the fact that the total effective action is +conformally invariant and the associated EMT is traceless, so the part of the trace associated to +the finite and divergent parts should be equal but with opposite sign in the conformal limit [1]. +The anomaly is associated to the finite part, so its actual value for the scalar field case is +� +T δφ�Anomaly += − lim +mφ→0 +� +T δφ� += +1 +480π2a4 +� +4H2H′ − H′′′� += +1 +2880π2 +� +RµνRµν − 1 +3R2 + □R +� +, +(104) +where the conversion of the anomaly result into an invariant expression in the last step can be +performed using the formulae of Appendix A. This result was explicitly verified in the calculation +of [15]. +We remark that for an arbitrary curved background the expression for the conformal +anomaly is more involved [1]. However, since the spatially flat FLRW spacetime is conformally +flat (i.e. conformal to Minkowski space) the contribution from the Weyl tensor vanishes identically +and hence also its square (entering the anomaly). Additional terms beyond 4th adiabatic order +decouple when mφ → ∞, satisfying the Appelquist-Carazzone decoupling theorem [97]. These +terms are not finite in the massless limit, and hence do not take part of the anomaly. +In practice we have derived the anomaly (104) from the unrenormalized trace of the vacuum +EMT for scalar fields, +� +T δφ� +, which is given in full detail in [15]. The corresponding conformal +anomaly for fermions can be similarly extracted from the unrenormalized +� +T δψ� +and it is a bit +cumbersome as well, so we shall spare details here. We limit ourselves to provide the final result. +Once more we can recognize the expression of the anomaly as a linear combination of finite terms +of adiabatic order 4 which are independent of the mass scale. We find +� +T δψ�Anomaly += − lim +mψ→0 +� +T δψ� += +1 +240π2a4 +� +7H′H2 − 3H′′′� += +1 +2880π2 +� +11RµνRµν − 11 +3 R2 + 6□R +� +. +(105) +One natural question is related with the physical consequences of the conformal anomaly. It is +well-known that it is a valuable theoretical concept encoding essential information on the VEV +of the renormalized EMT [1], although it need not be itself part of the observable quantities of +the renormalized theory. There are some attempts in the literature to remove the anomaly by +particular prescriptions or definitions of the renormalized EMT [98]. This is also the case of the +renormalization procedure employed in this work, as defined in (19) and (79), where the anomaly +has no physical effects. The reason is that the on-mass-shell value of the vacuum EMT is subtracted +33 + +at an arbitrary scale, M, up to 4th adiabatic order. Since the anomaly is of 4th adiabatic order and +it is independent from the mass of the fields and, of course, also from the arbitrary renormalization +point, it gets cancelled exactly in our ARP renormalization procedure. Alternatively, one can think +in terms of the effective action. Indeed, in [15], we defined the renormalized effective lagrangian +density off-shell at an arbitrary scale M, +LRen +W (M) ≡ LW(m) − LDiv +W (M) +(106) +and it was shown by expanding it through the DeWitt-Schwinger series that it eventually leads +exactly to the same renormalized EMT defined by (19). This result was obtained explicitly for a +scalar field φ and can be repeated for fermions, although we shall not provide details here. Now +the anomaly is related with the divergent part of the effective Lagrangian, corresponding to the +lowest adiabatic orders (up to 4h order). As a consequence it gets once more exactly cancelled in +(106) analogously to the subtraction of the EMT. +As previously indicated, the anomaly part of the trace is contained in the un-renormalized trace +of the EMT (even though the anomaly itself is a finite part of it). In our framework, however, the +anomaly cancels since the anomaly is independent of the mass scale and our renormalized EMT is +defined through a subtraction of the vacuum EMT evaluated at two different scales, see equations +(19) and (79). Thus the conformal anomaly is not involved in the renormalized expressions for the +vacuum energy density and pressure in our framework. Despite it not having physical consequences +in our approach, the explicit calculation of the anomaly is certainly useful as a nontrivial cross- +check of our intermediate results. +5 +Combined fermionic and bosonic contributions +Let us now determine the combined vacuum contributions from a multiplicity of non-interacting +fermionic and bosonic degrees of freedom. As defined before (cf. Sec. 4.4 ), we consider Nf species +of fermion fields with masses mψ,ℓ (ℓ ∈ {1, 2, . . . , Nf}), and Ns scalar field species with masses mφ,j +(j ∈ {1, 2, . . . , Ns}). +5.1 +Running vacuum from an arbitrary number of quantized matter fields +The renormalized expression of the vacuum energy density is, in that case, +ρvac(M, H) = ρΛ(M) + +�Ns +j=1 +� +T δφj +00 +� +(M, H) + �Nf +ℓ=1 +� +T δψℓ +00 +� +(M, H) +a2 +. +(107) +As in Sec. 4.2, considering two different energy scales M and M0 is a necessary step to obtain a +meaningful result. Einstein’s equations, after subtraction, can be written in the following manner: +Ns +� +j=1 +�� +T δφj +00 +� +(M, H) − +� +T δφj +00 +� +(M0, H) +� ++ +Nf +� +ℓ=1 +�� +T δψℓ +00 +� +(M, H) − +� +T δψℓ +00 +� +(M0, H) +� += +Ns +� +j=1 +� +a2 +128π2 +� +−M4 + M4 +0 + 4m2 +φj +� +M2 − M2 +0 +� +− 2m4 +φj ln M2 +M2 +0 +� ++ 3 +� +ξj − 1 +6 +� +a2H2 +16π2 +� +M2 − M2 +0 − m2 +φj ln M2 +M2 +0 +� ++ 9 +� +ξj − 1 +6 +�2 a2 +16π2 +� +˙H2 − 2 ¨HH − 6H2 ˙H +� +ln M2 +M2 +0 +� +34 + ++ +Nf +� +ℓ=1 +� +a2 +32π2 +� +M4 − M4 +0 − 4m2 +ψℓ +� +M2 − M2 +0 +� ++ 2m4 +ψℓ ln M2 +M2 +0 +� ++ a2H2 +16π2 +� +−M2 + M2 +0 + m2 +ψℓ ln M2 +M2 +0 +� � += (ρΛ(M) − ρΛ(M0)) g00 + +� +1 +8πG(M) +− +1 +8πG(M0) +� +G00 + (a1(M) − a1(M0)) (1)H00. +(108) +Notice the appearance of the 00th component of (1)Hµν, which is a HD tensor of O(H4), hence of +adiabatic order 4 [1]. Its presence in the generalized Einstein’s GR equations is indispensable for +renormalization and constitutes a UV completion of the field equations. No additional HD tensors +are needed for conformally flat spacetimes. In our case, (1)Hµν is necessary for the renormalization +of the short-distance effects produced by the quantum fluctuations of the scalar fields, as these +involve O(H4) corrections. However, as previously indicated in Sec. 4.2, the renormalized EMT +for fermions does not contain O(H4) terms. By using the expression of (1)H00 in AppendixA, we +can recognize the tensorial structure of the various terms and find the running of couplings: +ρΛ(M) − ρΛ(M0) = +1 +128π2 (−4Nf + Ns) +� +M4 − M4 +0 +� ++ +1 +32π2 + +4 +Nf +� +ℓ=1 +m2 +ψℓ − +Ns +� +j=1 +m2 +φj + + � +M2 − M2 +0 +� ++ +1 +64π2 + +−4 +Nf +� +ℓ=1 +m4 +ψℓ + +Ns +� +j=1 +m4 +φj + + ln M2 +M2 +0 +, +(109) +1 +8πG(M) − +1 +8πG(M0) = +1 +48π2 + +−Nf + 3 +Ns +� +j=1 +� +ξj − 1 +6 +� + � +M2 − M2 +0 +� ++ +1 +48π2 + + +Nf +� +ℓ=1 +m2 +ψℓ − 3 +Ns +� +j=1 +� +ξj − 1 +6 +� +m2 +φj + + ln M2 +M2 +0 +, +(110) +a1(M) − a1(M0) = − +1 +32π2 +Ns +� +j=1 +� +ξj − 1 +6 +�2 +ln M2 +M2 +0 +. +(111) +From the above formulas we can now use Eq. (107) to find out the difference between the values +of the VED at two different scales: +ρvac(M, H) − ρvac(M0, H0) = +3 +16π2 H2 +Ns +� +j=1 +� +ξj − 1 +6 +� � +M2 − m2 +φj + m2 +φj ln +m2 +φj +M2 +� +− +3 +16π2 H2 +0 +Ns +� +j=1 +� +ξj − 1 +6 +� � +M2 +0 − m2 +φj + m2 +φj ln +m2 +φj +M2 +0 +� ++ +1 +16π2 H2 +Nf +� +ℓ=1 +� +−M2 + m2 +ψℓ − m2 +ψℓ ln +m2 +ψℓ +M2 +� +− +1 +16π2 H2 +0 +Nf +� +ℓ=1 +� +−M2 +0 + m2 +ψℓ − m2 +ψℓ ln +m2 +ψℓ +M2 +0 +� +35 + ++ +9 +16π2 +� +2H ¨H + 6H2 ˙H − ˙H2� Ns +� +j=1 +� +ξj − 1 +6 +�2 +ln +m2 +φj +M2 +− +9 +16π2 +� +2H0 ¨H0 + 6H2 +0 ˙H0 − ˙H2 +0 +� Ns +� +j=1 +� +ξj − 1 +6 +�2 +ln +m2 +φj +M2 +0 ++ +�Ns +j=1 +� +T δφj +00 +�(6) +ren (M, H) + �Nf +ℓ=1 +� +T ψℓ +00 +�(6) +ren (M, H) +a2 +− +�Ns +j=1 +� +T δφj +00 +�(6) +ren (M0, H0) + �Nf +ℓ=1 +� +T ψℓ +00 +�(6) +ren (M0, H0) +a2 ++ . . . +(112) +In the last line, the dots collectively represent all the terms of adiabatic 8 or beyond, which are +not considered in our analysis. Notice that in the previous expression we have used the important +relation (109), which is essential to cancel the quartic mass contributions from the matter fields. +Following the same prescription that we used before to derive equations (25) and (26) for +a single scalar field, we implement now the scale settings M = H and M = H0 in order to +compare the evolution of the VED between these two points, in the present case involving the +full contributions from all the matter fields. For simplicity, let us call ρvac(H) ≡ ρvac(H, H) and +ρvac(H0) ≡ ρvac(H0, H0) when using the above expression (112) The expansion history times H +and H0 can be arbitrary, of course, but for obvious reasons we choose H0 = H(t0) to be the value +of the Hubble function at the present time, t0, and H = H(t) a value at a point in our past (t < t0). +Therefore, the running of the VED between these two points can be expressed as follows: +ρvac(H) − ρvac(H0) = +3 +16π2 H2 +Ns +� +j=1 +� +ξj − 1 +6 +� � +H2 − m2 +φj + m2 +φj ln +m2 +φj +H2 +� +− +3 +16π2 H2 +0 +Ns +� +j=1 +� +ξj − 1 +6 +� � +H2 +0 − m2 +φj + m2 +φj ln +m2 +φj +H2 +0 +� ++ +1 +16π2 H2 +Nf +� +ℓ=1 +� +−H2 + m2 +ψℓ − m2 +ψℓ ln +m2 +ψℓ +H2 +� +− +1 +16π2 H2 +0 +Nf +� +ℓ=1 +� +−H2 +0 + m2 +ψℓ − m2 +ψℓ ln +m2 +ψℓ +H2 +0 +� ++ +9 +16π2 +� +2H ¨H + 6H2 ˙H − ˙H2� Ns +� +j=1 +� +ξj − 1 +6 +�2 +ln +m2 +φj +H2 +− +9 +16π2 +� +2H0 ¨H0 + 6H2 +0 ˙H0 − ˙H2 +0 +� Ns +� +j=1 +� +ξj − 1 +6 +�2 +ln +m2 +φj +H2 +0 ++ +�Ns +j=1 +� +T δφj +00 +�(6) +ren (H, H) + �Nf +ℓ=1 +� +T ψℓ +00 +�(6) +ren (H, H) +a2 +− +�Ns +j=1 +� +T δφj +00 +�(6) +ren (H0, H0) + �Nf +ℓ=1 +� +T ψℓ +00 +�(6) +ren (H0, H0) +a2 +(113) +Obviously if the point H is in the nearby past we can neglect all the O(H4) terms generated in the +above expression since they are much smaller than the O(H2) contributions. We will do this in +36 + +the next section, where we study in more detail the low energy regime, in particular the late time +universe where we live. Let us however clarify that the O(H2) terms are dominant not only for the +late time universe around our time, but in actual fact for the entire post-inflationary regime, which +is when the higher order powers of H become activated and are actually dominant (see Sec. 5.3 for +the study of the inflationary epoch). +Finally, we can extract the running of the gravitational constant from eq. (110), with the +following result: +G(M) = +G(M0) +1 + G(M0) +2π +� +Ns +� +j=1 +� +ξj − 1 +6 +� +− Nf +3 +� +(M2 − M2 +0 ) + G(M0) +2π +� +Nf +� +ℓ=1 +m2 +ψℓ +3 +− +Ns +� +j=1 +� +ξj − 1 +6 +� +m2 +φj +� +ln M2 +M2 +0 +. +(114) +5.2 +The low energy regime: evolution of ρvac and G in the present universe +Of paramount importance is the evolution of the VED and of the gravitational coupling G in the +low energy regime, especially around our time. Therefore, following our prescription, we evaluate +(113) for the late universe, when the dominant powers of H are the H2 ones. Such an expression +then boils down to +ρvac(H) = ρvac(H0) + 3νeff(H) +8π +m2 +Pl +� +H2 − H2 +0 +� ++ O(H4) . +(115) +The function νeff(H) is defined as +νeff(H) = 1 +2π +� Ns +� +j=1 +� +ξj − 1 +6 +� m2 +φj +m2 +Pl +� +ln +m2 +φj +H2 +0 +− 1 +� +− 1 +3 +Nf +� +ℓ=1 +m2 +ψℓ +m2 +Pl +� +ln +m2 +ψℓ +H2 +0 +− 1 +� ++ +H2 +H2 − H2 +0 +ln H2 +H2 +0 + +1 +3 +Nf +� +ℓ=1 +m2 +ψℓ +m2 +Pl +− +Ns +� +j=1 +� +ξj − 1 +6 +� m2 +φj +m2 +Pl + + +� +. +(116) +It is indeed a function evolving with the Hubble rate, but is almost constant since the dependence +on H is very mild, as we shall make manifest in a moment. Let us emphasize that the O(H4) terms +correcting the r.h.s. of Eq. (115) are completely irrelevant for the current universe, and in point +of fact they can be safely ignored for the FLRW regime, i.e. during the entire period following the +inflationary stage (cf. next section). Therefore, equation (115) should actually be relevant for the +full cosmological evolution that is accessible (directly or indirectly) to our physical measurements +and observations. +It is convenient to define the parameter +ǫ ≡ 1 +2π + + +Ns +� +j=1 +� +ξj − 1 +6 +� m2 +φj +m2 +Pl +− 1 +3 +Nf +� +ℓ=1 +m2 +ψℓ +m2 +Pl + + +(117) +This parameter is connected to the β-function (93) at low energies. Indeed, when we consider +M = H in the low energy regime, it is obvious that H2 ≪ m2 for any particle mass, and hence +Eq. (93) reduces to +βquant.matt. +ρvac += − 3 +4π ǫ m2 +Pl H2 . +(118) +Therefore ǫ plays the role of coefficient of the low-energy β-function of the VED. However, the +eventual coefficient that effectively controls the final evolution of the VED is actually enhanced +37 + +with respect to ǫ by a big logarithmic factor. To see this, let us take the current limit (H → H0) +of the function (116): +ν0 +eff ≡ +lim +H→H0 νeff(H) = 1 +2π + + +Ns +� +j=1 +� +ξj − 1 +6 +� m2 +φj +m2 +Pl +� +ln +m2 +φj +H2 +0 +− 2 +� +− 1 +3 +Nf +� +ℓ=1 +m2 +ψℓ +m2 +Pl +� +ln +m2 +ψℓ +H2 +0 +− 2 +� + . +(119) +A simple rearrangement now shows that we can rephrase (116) in terms of ǫ and ν0 +eff as follows: +νeff(H) = ν0 +eff + +� +1 − +H2 +H2 − H2 +0 +ln H2 +H2 +0 +� +ǫ . +(120) +This formulas is exact, but in practice some simplifications are perfectly possible. For example, +take the big logarithms ln m2 +i /H2 +0 (with mi any particle mass, boson or fermion) involved in νeff +but not in ǫ. For any known massive particle, we have ln m2 +i /H2 +0 ≫ 1, this being true even for +the lightest neutrinos (recall that H0 ∼ 10−42 GeV= 10−30 meV). Typically ln m2 +i /H2 +0 = O(100) +in all cases. But as a matter of fact the only relevant contributions to νeff(H) come from the +heavy massive particles that belong to a GUT at a characteristic scale MX ∼ 1016GeV. For these +particles (whether bosons or fermions, with masses mi ∼ MX) we have m2 +i /m2 +Pl ∼ M2 +X/m2 +Pl and +this number is not so small since it may thrust the value of νeff up to νeff ∼ 10−3, if one takes into +account the large multiplicities of heavy fields existing in a typical GUT. This was first estimated +long ago in [99]. Thus, it is natural to expect +��ν0 +eff +�� ≫ |ǫ|. It follows that we can safely neglect +the term proportional to ǫ in (120). This means that in practice we can neglect the very mild +time-dependence of νeff(H) and replace it with the constant coefficient ν0 +eff in which H is evaluated +at the current time, H = H0. By the same token we can also ignore the numerical term of 2 +accompanying the big logarithms in (119). Whence, to a very good approximation the evolution +of the vacuum energy density can be described through the formula +ρvac(H) = ρvac(H0) + 3νeff +8π m2 +Pl(H2 − H2 +0) . +(121) +with an effective νeff ≃ ν0 +eff given by +νeff = 1 +2π + + +Ns +� +j=1 +� +ξj − 1 +6 +� m2 +φj +m2 +Pl +ln +m2 +φj +H2 +0 +− 1 +3 +Nf +� +ℓ=1 +m2 +ψℓ +m2 +Pl +ln +m2 +ψℓ +H2 +0 + + . +(122) +That is to say, it all amounts to replace νeff(H) → νeff in (115) and it still retains a great degree +of accuracy in the result. As it is manifest from the previous two equations, coefficient νeff plays +the role of β-function for the running vacuum directly as a function of H. If we compare (122) +with (117) we can see that νeff and ǫ are roughly ‘proportional’ through a big log as follows: +νeff ∼ ǫ ln m2 +i +H2 +0 +∼ O(100) ǫ . +(123) +Despite there being a summation over different masses, and hence such a proportionality not being +strict, the above relation is nevertheless true in order of magnitude. The presence of the big log +factor in νeff makes the running of the VED faster than the tiny value of ǫ might convey. On the +other hand, as we shall see below, it is ǫ alone that controls the (much smaller) running of the +gravitational coupling G, which does not receive any enhancement from log factors. +38 + +Equations (121) and (122) suffice to study the behavior of the VED near our time11. These +simplified formulas have been previously used de facto to fit the value of νeff from the latest +cosmological data, see e.g. +[50] and references therein. Here, however, we provide for the first +time the full theoretical structure behind this parameter in the QFT context from the quantum +effects induced by an arbitrary number of quantized matter fields. Remarkably, the typical fitting +value obtained in the mentioned reference is νeff ∼ 10−3 and positive, which is well within the +aforementioned expectations. +This phenomenological determination picks up of course the net +outcome from the various quantum matter fields involved in (122), which at this point cannot be +discriminated in an individual way. +Finally, insofar as the running gravitational constant is concerned, it can be written using the +same renormalization scale as follows: +G(H) = +GN +1 + 1 +2π +� +Ns +� +j=1 +� +ξj − 1 +6 +� +− Nf +3 +� +H2−H2 +0 +m2 +Pl +− ǫ ln H2 +H2 +0 +. +(124) +The former expression can be derived straightforwardly from eq. (114) by setting M = H and +M0 = H0, with H0 being the current value of the Hubble function, and we have defined GN ≡ +G(M0) (the current value of the gravitational coupling). We follow exactly the same recipe as for +the VED. In the low energy regime, where H2 ≪ m2 +Pl, we can approximate with high accuracy the +former expression by just +G(H) = +GN +1 − ǫ ln H2 +H2 +0 +, +(125) +where ǫ contains contributions from all the matter fields, see Eq. (117). Recall from (123) that +|ǫ| ≪ |νeff|, and also that the running of G(H) is logarithmic, in contrast to the running of ρvac(H) +which is quadratic in H2 at low energies. Therefore, the running of G is much smaller and also +much slower than the running of the VED. It may, however, be interesting to note that when H +approaches mPl the term ∼ H2/m2 +Pl in the denominator of the more accurate formula Eq. (124) +could be dominant over the logarithmic one. If the multiplicity of matter fields is large enough, +such a term could make the gravitational coupling to evolve asymptotically free at very large +energies when we approach the Planck scale. In the absence of that term, G increases at high +energies for ǫ > 0. +As an additional cross-check, we can see that the low energy regime of the vacuum energy +density and the running gravitational constant are compatible through the Bianchi identity. This +can be translated into a local energy exchange between the vacuum fluid and the background +gravitational field due to the quantum fluctuations. A deeper insight on the local (covariant) energy +conservation and the Bianchi identity can be found in the previous works [15,16], where the reader +may find a detailed derivation of the logarithmic evolution law, that is to say, equation (125), in +the simpler scenario of one real scalar field. In fact, one finds that the β-function (90) for the VED +running is crucially involved also in the local conservation law of the VED, which can be writen +in two alternative ways [15]: +˙ρvac + 3H (ρvac + Pvac) = +˙M +M βρvac = − +˙G +G +3H2 +8πG . +(126) +The first equality expresses the fact that the non-conservation of the VED is due to both the +running of ρvac with M (i.e. the fact that βρvac ̸= 0) and also to the cosmic time dependence of M +11It is easy to check that for one single neutral scalar field and no fermion field the above expressions reduce to +the formulas (25) and (26), as should be expected. +39 + +(viz. +˙M ̸= 0), whereas the second equality is a direct reflex of the Bianchi identity in Einstein’s +equations with variable ρvac and G, and hence provides a link between the time variation of the VED +and that of the gravitational coupling G. The former equation does not depend on the number or +nature of the fields involved, and holds as long as the matter components are covariantly conserved +on their own. The non-conservation of ρvac, however, preserves the Bianchi identity thanks to the +corresponding running of the gravitational coupling. This does not preclude, however, that one can +still formulate scenarios where matter can exchange energy with ρvac, but we do not address this +situation here. Taking the leading terms of βρvac from the r.h.s. of (93) for the present universe, +and setting M = H, we readily obtain the following differential equation +1 +G2 +dG +dt = 2ǫ +GN +˙H +H , +(127) +where ǫ is the full expression (117) involving the contributions from all the matter fields.dd Its +solution is precisely Eq. (125), as can be readily checked. +It is also interesting to note from the above formulas that this framework predicts a (mild) +cosmic time variation of the “fundamental constants”, such as the gravitational coupling G and +ρvac, with H(t) and hence an evolution of these ‘constants’ the cosmological expansion. +The +possibility for such a variation has long been discussed in the literature [100] and is still a matter +of debate and intensive test, see e.g. [101] and the ample bibliography provided in it. Specific +theoretical models accounting for such a possible variation are manifold, and in some cases they +imply a time-dependence of the running couplings and masses in the particle and nuclear physics +world, see e.g. [102,103]. While most of the proposals are based on strict particle physics scenarios, +in particular on GUT’s, testing the evolution of the VED in curved spacetime is a novel feature +suggested in our framework, which was actually put forward on more phenomenological grounds +sometime ago in [104]. +The QFT calculations presented in the current work provide indeed a +solid theoretical support to these same ideas but from first principles. +Recall that when one +“fundamental constant” varies, then all of them vary! +The formulas discussed above concern important epochs of the cosmological expansion such +as the radiation-dominated epoch, matter-dominated epoch and the current epoch in which the +vacuum energy resurfaces and became finally dominant over matter. During the entire FLRW +regime the dominant power of the Hubble rate in the VED is H2 or ˙H (which are of the same +adiabatic order). +The terms with powers of H (or of equal adiabatic order) higher than H2 +(indicated by O(H4) in (115)) acquire real relevance much early on in the expansion history since +only during the early universe we encounter a truly high energy scenario. As previously anticipated, +an interesting feature regarding these higher powers is that they can provide us with a possible +mechanism for inflation. Namely, if the early cosmic era possesses a short period where H remains +approximately constant and very large (typically near a GUT scale), the universe may go through +a phase of exponential expansion in which the VED starts from a huge value which then quickly +decays into radiation and triggers the ordinary FLRW regime. This situation is possible also in +the RVM framework, and is called ‘RVM-inflation’ [15]. We reassess it in the next section in the +extended context of the present considerations, where we now have both scalar and fermion fields. +5.3 +Inflation from running vacuum +It was noted in [15] that the quantum effects computed from the adiabatic expansion lead to +higher powers of the Hubble rate and its derivatives which are irrelevant for the current universe +but capable to bring about inflation in the very early universe. They are characterized by a short +period where H=const., provided this constant value takes, of course, a large value which we +expect to lie around a characteristic GUT scale. The regime H=const. is totally unrelated to the +40 + +ground state of a scalar field potential and therefore this new mechanism does not require any ad +hoc inflaton field. It is called ‘RVM-inflation’. Here we consider the contribution from the fermions +fields and provide a formula for the dominant term of the energy density receiving contributions +from an arbitrary number of non-minimally coupled scalar fields and also an arbitrary number of +fermions fields. The payoff from the latter stems from setting H =const in Eq. (82), where we +can see that all the time derivatives of the Hubble rate vanish except for a single term which is +proportional to H6/m2 +ψ. The contribution from a non-minimal scalar was computed in [15] and +here we just combine it with that of fermions assuming an arbitrary number of families of both +species. Overall, we find that the total VED involving the contributions from bosons and fermions +at very high energies (hence relevant for triggering RVM-inflation in the very early universe) reads +as follows: +ρinf +vac =CinfH6 + F( ˙H, ¨H, +... +H...), +(128) +where +Cinf ≡ +1 +80π2 + + + +Ns +� +j +1 +m2 +φj +�� +ξj − 1 +6 +� +− 2 +63 − 360 +� +ξj − 1 +6 +�3� +− 31 +252 +Nf +� +ℓ +1 +m2 +ψℓ + + + . +(129) +The terms collected in the function F( ˙H, ¨H, +... +H...) depend on different combinations of powers of +H involving derivatives of H in all cases, and hence they all vanish for H =const. That is to say, +F = 0 for H =const. Overall we see that the dominant contribution is of the form ρinf +vac ∝ H6 +with a complicated coefficient Cinf which depends on the number of scalar and fermions fields, +their masses and multiplicities and also on the non-minimal couplings of the different scalars. In +the case of fermions this contribution is seen to be negative-definite, whereas in the case of the +scalars it can be positive. Let us note that during the inflationary period the EoS of the quantum +vacuum is essentially −1, with very tiny deviations caused by terms which depend on the various +time derivatives of the Hubble rate. To the extent that the condition H=const. is fulfilled these +deviations are extremely small, see Eq. (100). In the next section we shall see that, in contrast, the +EoS of the quantum vacuum in the present time can deviate from −1 by an small amount which +is not as negligible as in the very early universe and therefore could be detected and even mimic +quintessence behavior. +The solution of the cosmological equations proceed along the same lines as in [15], except that +now the fermionic contribution is also included but it only modifies the specific coefficient of H6. +Therefore, one finds again that a short period of inflation is produced with H ≈ const. and the +vacuum decays quickly into radiation [15]: +H(a) = +HI +(1 + ˆa8), +(130) +ρr(ˆa) = ρI ˆa8 � +1 + ˆa8�− 3 +2 , +ρvac(ˆa) = ρI +� +1 + ˆa8�− 3 +2 . +(131) +in which ˆa ≡ a/a∗, with a∗ is the transition point from the regime of vacuum dominance into that +of radiation dominance, which can be estimated to be around a∗ ∼ 10−30 (see Eq. (137) below), HI +and ρI are the value of H and ρvac, respectively, at the beginning of inflation, with ρI = CinfH6 +I . +Applying the Friedmann equation, we find +HI = +� +3 +8πGICinf +�1/4 +, +(132) +41 + +ρI = +3 +8πGI +H2 +I = +3 +8πGI +� +3 +8πGICinf +�1/2 += C−1/2 +inf +� +3 +8πGI +�3/2 +, +(133) +where GI ≡ G(HI) is the gravitational coupling at H = HI, the latter being value of the Hubble +parameter at the primeval (inflationary) era. Needless to say, the difference between GI and the +usual GN is not very important here since the running of G is logarithmic, and hence the effect +is very small as compared to the fast evolution of the H6 term, so in practice we can neglect the +running of G for these considerations. To trigger inflation in an effective way, we must have a +positive coefficient Cinf > 0. In the light of Eq. (129) we can see that this is perfectly possible +since the couplings ξj and masses of the fields can take a variety of values that make this possible, +as can be shown in a devoted study that will be presented elsewhere [105]. Moreover the masses +of the relevant fields involved must be very large, say around a typical GUT scale, MX ∼ 1016 +GeV. This may not be obvious at first sight. A naive interpretation of the higher order terms +of the VED, which are related with the 6th adiabatic order of the ZPE (see equation (82)), may +give the impression that the relevant masses are to be the lightest possible ones, but this is not +so at all since in such a case the adiabatic expansion would break down. On the other hand, the +analysis trough the Friedmann equations reveals the correct dependency of the VED and of the +Hubble function on the masses during the inflationary regime. From equation (129) it is obvious +that C−1/2 +inf +∝ mφ,ψ, where the notation stands for a linear combination of the typical masses of the +matter fields. The inflationary parameters (132)-(133), therefore, depend on a positive power of +mφ,ψ, and as a result the RVM-inflation is actually dominated by the heavier masses, in contrast +to naive expectations; namely, masses mφ,ψ ∼ MX ∼ 1016GeV of order of a typical GUT, as +mentioned above. It then follows that the same heavy masses which may generate a mild (but +non-negligible) quadratic running ∼ H2 of the VED (with a coefficient νeff ∼ 10−3) may also drive +fast inflation early on at the beginning of the cosmological expansion. To see this feature more +explicitly, let us recall that the differential equation driving the Hubble function in the presence +of a high power H6 in the VED (128) reads [106–108] +˙H + 3 +2 (1 + ωm) H2 � +1 − H4 +H4 +I +� += 0 , +(134) +where ωm = 1/3 is the EoS of matter in the relativistic epoch, and HI is given in (132). We have +neglected the influence of the term H2 and also of the CC term in the very early universe. It is +obvious from (134) that there is a constant solution H = HI to that equation, which is precisely +the one which triggers the inflationary period. From this observation one can then solve equation +(134) exactly to find Eq. (130). The latter shows clearly the departure of H from HI when ˆa > 1 +(i.e. a > a∗). The inflationary phase actually occurs during the short period when the departure +remains small, namely when H remains approximately constant, H ≃ HI. During such period the +F-term on the r.h.s. of Eq. (128) just vanishes, F( ˙H, ¨H, +... +H...) = 0, since all dependence on H is +through time derivatives. From equations (132) and (133), it is clear that the order of magnitude +of the physical scales involved in RVM-inflation reads as follows: +HI ∼ (MX mPl)1/2 ∼ 1017 GeV , +ρI ∼ MX m3 +Pl ∼ +� +1018GeV +�4 . +(135) +up to numerical coefficients and multiplicity factors, of course. However, it is clear that if the masses +of the relevant matter fields lie in the expected range for a GUT, the right order of magnitude for +the relevant physical parameters can be obtained. In fact, it should be recalled at this point that +the mechanism of RVM-inflation can also be motivated in ‘stringy’ scenarios [45–49], see [109] for +a recent review. Therefore, it should be natural to expect RVM-infation in the range between the +GUT scale and the Planck scale. This is exactly what the above estimates suggest in order of +magnitude. +42 + +One more remarkable observation is in order. One can easily check from (131) that for ˆa ≫ 1 +(i.e. a ≫ a∗) we retrieve the standard decaying behavior of radiation, ρr(a) ∼ a−4. This condition +enforces the following relation between ρ0 +r, accounting for the current value of radiation energy +density, and ρI: +ρI ≈ ρ0 +ra−4 +∗ , +(136) +Following the line of the previous estimations, it yields an equality time between vacuum energy +and radiation of +a∗ = +� +Ω0 +r +ρ0 +c +ρI +� 1 +4 +≃ +� +10−4 10−47 +1072 +�1/4 +∼ 10−30 . +(137) +where ρ0 +c ∼ 10−47 GeV4 is the current critical density. In the meantime the vacuum energy becomes +negligible and does not disturb primordial BBN, see [15,16] for more details. See also [61,106–108] +for interesting phenomenological applications prior to the QFT treatment of RVM-inflation which +was first presented in [15]. +Finally, we should mention that RVM-inflation is genuinely different from e.g. Starobinsky’s +inflation, as explained in detail in [45,61]. While it may be natural to conceive that a consistent +inflationary model of the very early Universe should be a good candidate for an effective theory of +quantum gravity, at least at energies much less than the Planck scale, RVM-inflation reveals itself +as one such possible candidate, all the more if we take into account that a a low-energy ‘stringy’ +version of RVM-inflation has been also identified and sharing most of the virtues of the current +QFT formulation [45,46]. +5.4 +Equation of state of the quantum vacuum +The quantum effects of the fields have an imprint on the vacuum equation of state, which is not +exactly the traditional one Pvac = −ρvac. From the expressions of the renormalized energy density +and pressure of the vacuum that have been obtained in the previous sections and considering their +generalization to an arbitrary number of fermions and scalars, we arrive at the following expression +for the EoS of the quantum vacuum: +ωvac(H) = Pvac(H) +ρvac(H) += −1 + +1 +8π2ρvac(H) +� Nf +� +ℓ=1 +� ˙H +3 +� +H2 − m2 +ψℓ + m2 +ψℓ ln +m2 +ψℓ +H2 +� � ++ +Ns +� +j=1 +� � +ξj − 1 +6 +� +˙H +� +m2 +φj − H2 − m2 +φj ln +m2 +φj +H2 +� +− 3 +� +ξj − 1 +6 +�2 � +6 ˙H2 + 3H ¨H + +... +H +� +ln +m2 +φj +H2 +�� ++ O +�H6 +m2 +� += −1 + +1 +8π2ρvac(H) +�� +1 +3 +Nf +� +ℓ=1 +� +H2 − m2 +ψℓ + m2 +ψℓ ln +m2 +ψℓ +H2 +� ++ +Ns +� +j=1 +� +ξj − 1 +6 +� � +m2 +φj − H2 − m2 +φj ln +m2 +φj +H2 +� � +˙H +− 3 + + +Ns +� +j=1 +� +ξj − 1 +6 +�2 +ln +m2 +φj +H2 + + +� +6 ˙H2 + 3H ¨H + +... +H +� � ++ O +�H6 +m2 +� +(138) +43 + +The term O +� +H6/m2� +represents the higher order adiabatic terms, containing 6 derivatives of the +cosmic time, such as the own H6/m2, ¨H2/m2, . . . coming from ⟨T δψℓ +00 ⟩(6) +ren(H, H), ⟨T δφℓ +00 ⟩(6) +ren(H, H), +⟨T δψℓ +11 ⟩(6) +ren(H, H) and ⟨T δφℓ +11 ⟩(6) +ren(H, H). Here, m = mψ, mφ. They do not give a significant contri- +bution during the postinflationary era, so that we can perfectly avoid them in what follows. This +relation shows that the quantum vacuum is of dynamical nature. As we can see there is a devia- +tion with respect −1, the traditional EoS associated to the Cosmological Constant in the ΛCDM +framework. The correction terms due to both bosonic and fermionic fields are small in the present +era, in comparison with the constant term, but need not be negligible since the particle masses +involved can be from a typical GUT, and hence one can estimate that the effective parameter νeff +could reach up to 10−3 [99]. Furthermore, if we focus only on the O(H2) terms relevant for the +current universe and the radiation epoch we may neglect also the higher order adiabatic terms of +O(H4) in the last lines of Eq. (138). Finally, if we consider a linear approximation in νeff – the +parameter defined in Eq. (122) – the EoS can be written in a rather compact form as a function +of the cosmological redshift as follows: +wvac = −1 + +� +νeff − ǫ +� +−1 + ln E2(z) +�� � +Ω0 +m(1 + z)3 + 4Ω0 +r +3 (1 + z)4� +Ω0vac + νeff (−1 + E2(z)) − ǫ (1 − E2(z) + E2(z) ln E2(z)) + O +� +ν2 +eff +� +, +(139) +where νeff contains the combined effects from fermions and bosons, see Eq. (119), and we have +defined the normalized Hubble rate with respect to the present time (H0): +E2(z) ≡ H(z) +H0 += Ω0 +vac + Ω0 +m (1 + z)3 + Ω0 +r (1 + z)4 . +(140) +Here Ω0 +vac = ρ0 +vac/ρ0 +c ≈ 0.7, Ω0 +m = ρ0 +m/ρ0 +c ≈ 0.3 and Ω0 +r = ρ0 +r/ρ0 +c ≈ 10−4 are the current fractions +of vacuum energy, dust-like matter and radiation, respectively. The EoS formula may be further +simplified if we neglect the effect of the small coefficient ǫ in Eq. (120). This is justified since +��νeff +�� ≫ +��ǫ +�� owing to the logarithmic extra terms ln m2/H2 +0 contained in ν0 +eff, which can typically +be of O(100), see (123). Thus, to within very good approximation, we can write +wvac ≃ −1 + νeff +Ω0 +m(1 + z)3 + 4Ω0 +r +3 (1 + z)4 +Ω0vac + νeff (−1 + E2(z)) , +(141) +Notice that the term proportional to νeff in the denominator cannot be neglected at large z since +it becomes dominant. In this case, the EoS takes on the form +wvac ≈ −1 + Ω0 +m(1 + z)3 + 4Ω0 +r +3 (1 + z)4 +E2(z) +(z ≫ 1) , +(142) +where νeff has cancelled. For example, for z large enough but within the matter-dominated epoch +the dominant term in equation (140) is the ∼ (1 + z)3 one, and we can see from (142) that the +vacuum EoS then mimics matter since wvac ≃ 0. Similarly, at much larger values of z already in +the radiation-dominated epoch, where ∼ (1 + z)4 is the dominant term, then wvac ≃ 1/3 and the +vacuum imitates radiation. Such a ‘chameleonic’ behavior of the quantum vacuum was first noticed +in [16], and in fact the formula of the vacuum EoS that we have found here is a generalization +for an arbitrary number of fermion and boson fields of the expression previously found in [16]. +Last but not least, the behavior of the vacuum EoS in the late universe is no less remarkable and +striking. From (141) we find +wvac(z) ≃ −1 + νeff +Ω0 +m +Ω0vac +(1 + z)3 +(z ≲ 5) . +(143) +44 + +and in this approximation we recover once more the form (28), but in this case with νeff involving +the contributions from all the quantized matter fields. Taking into account that the last fits of the +RVM to the overall cosmological data favor a value of νeff > 0 and of order 10−3 [50] (see also +the previous phenomenological studies on the RVM reported in recent years [51–55]), we learn +that the deviation of the EoS from −1 in the present universe is not completely negligible and it +imitates quintessence since wvac ≳ −1. +To summarize, the running vacuum mimics the EoS of the dominant component at a given time +of the cosmic evolution [16]. Namely, wvac is close to 1/3 in the radiation dominated epoch, 0 in the +matter dominated epoch and asymptotes to −1 in the future. Furthermore, at present it mimics +quintessence since νeff is found to be positive in the phenomenological tests of the RVM mentioned +before and hence wvac is slightly above −1, the traditional value of the classic vacuum. Therefore, +we can have at present an effective quintessence behavior of the quantum vacuum without need of +invoking ad hoc scalar fields. Finally, we should also mention that the EoS of the quantum vacuum +in the very early universe (the period when inflation took place, see Sec. 5.3) is very close to −1 +(viz. the traditional vacuum value), exactly as in the remote future since the universe asimptotes +towards a new inflationary period. +6 +Conclusions +In this paper, we have evaluated the contributions to the vacuum energy density (VED) from the +quantized matter fields in a semiclassical gravity context. More specifically, by using a regulariza- +tion technique of Quantum Field Theory (QFT) in curved spacetime called adiabatic regularization +and making use of a subtraction prescription amply tested in previous works [14–16], we have been +able to calculate the mode functions and the renormalized zero-point energy (ZPE) from spin-1/2 +quantum fields in a FLRW background up to sixth adiabatic order. Combining with the contribu- +tion from the ρΛ term in the Einstein-Hilbert action, we have obtained the properly renormalized +VED. Since the corresponding computation for scalar fields had already been accounted for in +the aforementioned works, we have put forward here the combined contribution to the VED from +an arbitrary number of quantized matter fields. We do not consider interactions among them, +however, as the free field calculation in curved spacetime is already rather cumbersome in itself. +One interesting difference between the expression of the ZPE of these two types of fields is that +in the fermionic case the terms of fourth adiabatic order (viz. involving four time derivatives) are +not present. The final result is that the overall VED of the quantized matter fields upon adiabatic +renormalization appears to be a soft dynamical quantity with the cosmological evolution. This is a +most remarkable outcome of the present study. More specifically, the VED shows up in the form of +an expansion in powers of the Hubble rate H and its time derivatives, all these powers being of even +adiabatic order, a property which is fully consistent (and expected) from the general covariance of +the theory. Noteworthy, too, is the fact that the expression for the renormalized VED emerging +from our calculation appears to take the form of the running vacuum model (RVM) , see [30, 31] +and references therein. Therefore, as it is characteristic in the RVM, the leading quantum effects +obtained for the late universe are of second adiabatic order, e.g. ∼ H2 and ∼ ˙H. Obviously, this +may have consequences for the present universe, and these consequences have been tested in pre- +vious phenomenological works. In particular, these quantum effects turn out to impact positively +on a possible solution to the ΛCDM tensions, see e.g. [50–55]. +In this study we have also discussed some of the theoretical difficulties in trying to renormalize +the cosmological term, Λ, and its relation with the VED. To start with, it should be emphasized +that these are two different concepts that can only be properly related in non-flat spacetime. If Λ +is taken to be the physically measured value of the cosmological term at present, then its relation +with the current VED is ρ0 +vac = Λ/(8πGN). However, at a formal QFT level these quantities have +45 + +to be derived from a gravitational action in curved spacetime and a lot more of care needs to be +exercised. Leaving for the moment quantum gravity considerations for a better future (viz. for +when, hopefully, the quantum treatment of the gravitational field becomes possible), the more +pedestrian renormalization of ρvac in QFT in curved spacetime proves to be already quite helpful +at present [14,15]. It shows, for example – and the present works attests once more for this fact +– that the renormalized VED in the FLRW background is a mild dynamical quantity evolving +with the cosmic expansion, and hence ρ0 +vac is just its value at present. There is no such thing +as a rigid cosmological constant in the context of QFT in the FLRW background. In general, +ρvac = ρvac(H) is a function of the Hubble rate and its time derivatives. The longstanding and +widespread confusion in the literature about cosmological constant, Λ, and vacuum energy density, +ρvac, has prevented to achieve a proper treatment of the renormalization of these quantities in +cosmological spacetime. In particular, the attempts to relate these concepts in the context of flat +spacetime calculations are meaningless and their repeated iteration has been counterproducing [31]. +In the simplified scenario considered here, where only interactions with the gravitational back- +ground are allowed, the VED is the sum of two contributions, a parameter in the effective action, +ρΛ, and the ZPE of the fields. After renormalization, the VED depends on a scale M, and the +setting M = H at the end of the calculation allows us to compare the VED at different epochs +of the cosmic history, in a manner similar to the standard association made of the renormal- +ization point with a characteristic energy scale of a given process in ordinary gauge theories. +Thus the difference between the VED values at any two points of the cosmological expansion, +say H(t1) and H(t2), provides a smooth running of the VED. Remarkably, such an evolution +turns out to be free from the undesirable ∼ m4 contributions associated to the quantized matter +fields. As a result there is no fine tuning involved in the evolution of the VED in the present +context. The VED, in fact, adopts the standard form of the RVM, which in the late universe +reads ρvac(H2) ≈ ρvac(H1) + 3νeff/(8π)m2 +Pl(H2 +2 − H2 +1), where H1 and H2 can be, for example, the +current value, H0, and another value H near our past. Finally, νeff is a small parameter related to +the β-function of the renormalization group running of the VED whose value has been explicitly +computed in this work from the fluctuations of the quantized matter fields. Depending on the sign +of νeff, the VED can mimic a quintessence or phantom-like behavior. +Much earlier in the cosmic history, the higher powers of H (larger than H2) take their turn and +can be relevant for the inflationary regime, in the sense that they can indeed trigger inflation in +what has been called ‘RVM-inflation’ [14–16]. While the scalar field contribution to this inflationary +mechanism had been computed in the previous references, in this work we have accounted for the +spin-1/2 fermionic contribution and combined both types of effects for an arbitrary matter content. +In both cases (scalar and fermion fields) the sixth order adiabatic terms had to be computed. +Finally, the renormalized vacuum fluid’s pressure, Pvac, has been determined using the same QFT +techniques as for ρvac. Equipped with these nontrivial results the equation of state (EoS) of the +quantum vacuum can be computed from first principles. We find that the EoS function Pvac/ρvac +deviates from the traditional result -1, which is noticeable. This is true in most of the cosmological +history, especially after inflation (which is the only period in our past where the vacuum EoS stayed +very close to −1). It is no less remarkable, as previously noted, that in the late universe, and most +particularly near our time, the vacuum EoS behaves as quintessence for νeff > 0, the latter being +the sign preferred by the existing phenomenological fits to the overall cosmological data – see [50], +for example. For higher and higher redshifts during the FLRW regime, the vacuum EoS mimics +the equation of state of the dominant matter component (relativistic or non-relativistic) at the +corresponding epoch. Such a peculiar behavior of the running vacuum energy density was referred +to as “chameleonic” in [16]. The tracking of the EoS of matter ceases in the late universe, where +the DE epoch breaks through, and then it behaves as effective quintessence, the reason being that +the EoS is then in the process to asymptote towards −1 in the remote future, exactly as it was in +46 + +the primeval inflationary time. In fact, the inflationary process in the late universe is eventually +resumed, but very slowly. +Overall, by combining the results from an arbitrary number of quantized matter fields we find +that the main cosmic running of ρvac depends on the quadratic terms in the boson and fermion +masses times the square of the Hubble function, i.e. ∼ m2 +ψH2 and ∼ m2 +φH2. These effects are +obviously much softer than the naively expected (hard) contributions ∼ m4. As noted, the soft +terms have been amply tested in phenomenological studies of the RVM existing in the literature, +see [30, 31] and associated references. The QFT effects that we have computed here and in the +preceding studies [14–16] provide a solid theoretical underpinning of the RVM phenomenology. +They even bring to light new relevant features such as the dynamical character of the EoS of +the quantum vacuum, which is unprecedented in the literature to the best of our knowledge. +In particular, they suggest that if in future cosmological observations clear signs are found that +the EoS of the dark energy departs from −1, such a feature could be explained by the running +vacuum, which is no longer a state with EoS exactly equal to −1 in QFT, and therefore it opens +the possibility that such observations may be accounted for from fundamental properties of QFT +ultimately attributable to the fluctuations of the quantized matter fields in curved spacetime. This +could be an extremely interesting signature of this approach, which does not rely at all on ad hoc +quintessence fields and the like to explain a possible dynamical evolution of the DE and its EoS +in today’s universe. The EoS dynamics is prompted here by the virtual quantum effects produced +by quantized fermion and boson fields, the same kind of effects which trigger a smooth and very +mild evolution of the vacuum energy density in cosmological spacetime. The physical outcome, +at a pure cosmological/observational level, is a remarkable and qualitatively new feature, to wit: +the non-constancy of the ‘cosmological constant’, Λ, in Einstein’s equations. Theoretically, this +conclusion emerges from explicit QFT calculations in the FLRW background and may point to +a possible explanation for a variety of problems that have been dealt with phenomenologically +in the past in terms of ad hoc quintessence or phantom fields. Therefore, in our approach, the +measured cosmological ‘constant’ is neither mimicked nor supplanted by any ersatz entity from +the already crammed black box of the dark energy. The physical Λ here is, instead, a quantity +directly connected with the vacuum energy density in QFT in curved spacetime, and as such is +running genuinely with the quantum renormalization effects. As we have also emphasized, this +same QFT framework helps also in relieving the current tensions in the ΛCDM and, ultimately, +it might provide an explanation for the cosmic acceleration observed in our Universe from first +principles. +Acknowledgements +Two of us (CMP and JSP) are funded by projects PID2019-105614GB-C21 and FPA2016-76005- +C2-1-P (MINECO, Spain), 2017-SGR-929 (Generalitat de Catalunya) and CEX2019-000918-M +(ICCUB). CMP is also partially supported by the fellowship 2019 FI−B 00351. +The work of +JSP is also partially supported by the COST Association Action CA18108 “Quantum Gravity +Phenomenology in the Multimessenger Approach (QG-MM)”. CMP and JSP acknowledge as well +the participation in the COST Action CA21136 “Addressing observational tensions in cosmology +with systematics and fundamental physics” (CosmoVerse). SC is supported by the Transilvania +Fellowship Program, 2022. CMP and JSP are very grateful to A. G´omez-Valent and J. de Cruz +P´erez for the fruitful collaboration over the years in the task of understanding the RVM and its +manyfold phenomenological implications. +47 + +A +Appendix: Conventions and Useful Formulas +In this work, natural units are used. That is ℏ = c = 1 and GN = 1/m2 +Pl ≈ 1.22 × 1019 GeV. +Our framework is a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) background with null spatial +curvature. The conventions are (+, +, +) in the Misner-Thon-Wheeler notation [110]: +• gµν = a2(τ)diag(−, +, +, +), with τ denoting conformal time. +• Rλ +µνσ = ∂νΓλ +µσ + Γρ +µσΓλ +ρν + . . . , with Rµν = Rλ +µλν and R = Rµνgµν. +• The Einstein field equation can be cast as Gµν + Λgµν = 8πGTµν, with Gµν ≡ Rµν − 1 +2Rgµν. +For derivatives, the notations ()′ = d/dτ and ˙() = d/dt are used. +In particular, H = ˙a/a, +H ≡ a′/a = aH, hence a′ = aH = a2H, a′′ = a3(2H2 + ˙H) etc. (see Appendix A.1 of [15]). +In this case, the non-vanishing Christoffel symbols in the conformal frame are +Γ0 +00 = H, +Γ0 +ij = Hδij, +Γj +i0 = Hδj +i . +(144) +On the other hand, the Ricci tensor is +R = 6a′′ +a = 6 +a2 +� +H′ + H2� += 6(2H2 + ˙H), +(145) +and the 00th components of the Ricci and Einstein tensors are +R00 = −3a2 � +H2 + ˙H +� +, +G00 = 3a2H2. +(146) +The renormalization program requires taking into account higher derivative (HD) terms in Ein- +stein’s equations [1]. In the particular case of FLRW spacetime, it is enough to consider just one +of the characteristic higher order tensors, (1)Hµν, given by the metric functional derivative of R2 +in the effective action (we refer once more to Appendix A.1 of [15] for more details): +(1)Hµν = +1 +√−g +δ +δgµν +ˆ +d4x√−g R2 = −2∇µ∇νR + 2gµν□R − 1 +2gµνR2 + 2RRµν. +(147) +Its 00th component is +(1)H00 = −18a2 � +˙H2 − 2H ¨H − 6H2 ˙H +� +(148) +and its 11th component is +(1)H11 = −a2 � +108H2 ˙H + 54 ˙H2 + 72 ˙H ¨H + 12 +... +H +� +. +(149) +We will also need the invariant relations +RµνRµν = 12 +a4 +� +H′2 + H′H2 + H4� +, +□R = − 6 +a4 +� +H′′′ − 6H′H2� +, +(150) +which hold good for flat three-dimensional FLRW spacetime. +For gamma matrices (in flat spacetime), the standard Dirac basis is chosen for our calculations +with spin-1/2 fermions: +γ0 = +�I +0 +0 +−I +� +γk = +� 0 +σk +−σk +0 +� +, +(151) +where σk (k = 1, 2, 3) are the usual Pauli matrices. In terms of the above γα, the curved spacetime +γ-matrices read γµ(x) = eµ +α(x)γα, where eµ +α(x) is the vierbein (cf. Sec.3). +48 + +B +Appendix: Adiabatic expansion of the spin-1/2 field modes +In the main text (cf. Sec. 3) we have presented an iterative procedure which allows us to determine +the two types of field modes hI +k and hII +k which are necessary to construct the 2-component spinor +fields. They are functions of both momentum (k) and conformal time (τ) and have the following +structure: +hI +k(τ) = +� +ωk + aM +2ωk +F(τ) e−i +´ τ Wk(˜τ)d˜τ, +hII +k (τ) = +� +ωk − aM +2ωk +G(τ) e−i +´ τ Wk(˜τ)d˜τ , +(152) +where +F ≡ 1 + F (1) + F (2) + F (3) + · · · , +(153) +G ≡ 1 + G(1) + G(2) + G(3) + · · · , +(154) +Wk ≡ ωk + ω(2) +k ++ ω(4) +k ++ ω(6) +k ++ · · · +(155) +Here Wk is a real function playing an analogous role to the effective frequency introduced (with +the same notation) in the scalar field case, Eq. (12). The superscript (n = 1, 2, 3, ...) indicates that +the corresponding quantity is of adiabatic order n. The modes (152) are constrained to satisfy the +normalization condition +|hI +k|2 + |hII +k |2 = 1. +(156) +Some of the notation is similar to that of [73–75], although we use conformal metric and different +conventions, and moreover we deal with FLRW spacetime rather than de Sitter (where the EMT +takes a simpler form). In addition, as explained in the main text, we perform the renormalization +at the arbitrary scale M (not at the on-shell point). This is important in order to test the scaling +dependence of the renormalized VED, which is the main aim of our calculation. +In what follows we use the notation ωk ≡ +√ +k2 + M2a2. Recall that unless it is explicitly noted +the mass scale involved is the off-shell point M. When the subtraction (79) is implemented within +our renormalization procedure, one just sets M = mψ in the subtracted part. The mass mψ can +be conveniently expanded in even adiabatic orders as +mψ = +� +M2 + ∆2 = M + ∆2 +2M − ∆4 +8M3 + +∆6 +16M5 + . . . +(157) +After completing ℓ ≥ 1 steps in the process described in Sec. 3, we end up with Eq. (51) in the +main text, which depends on the following expression: +ΩkΩk,1 · · · Ωk,ℓ−1 = ωk + ω(1) +k ++ ω(2) +k ++ · · · + ω(2ℓ−1) +k ++ . . . +(158) +where ω(j) +k +represents a function of adiabatic order j. Functions Wk(τ, M), F(τ, M) and G(τ, M) in +the ansatz (152) can be estimated with the help of this product. However the following clarifications +may be necessary to better understand this process, together with the explanations already given +in the main text, see Sec. 3: +• For ℓ = 1 we only have performed one iterative step, and at this point we need to deal with +the square root of +Ω2 +k = ω2 +k + iσ + a2∆2 = ω2 +k + iMa′ + a2∆2 + ia′∆2 +2M ++ . . . +(159) +49 + +as defined in (40). The dots “...” in (159) account for terms of adiabatic order 4 and beyond. +The square root of the previous result yields +Ωk = ωk + ω(1) +k ++ a2ωk +8 +� +M2 +ω4 +k +�a′ +a +�2 ++ 4∆2 +ω2 +k +� ++ iaM +16ωk +a′ +a +� +4∆2 +M2 − 4a2∆2 +ω2 +k +− a2M2 +ω4 +k +�a′ +a +�2� ++ . . . +(160) +with +ω(1) +k +≡ iMa +2ωk +a′ +a . +(161) +From the r.h.s of (160), the first two terms, ωk and ω(1) +k , are used in the first order approxi- +mation of the modes (see equation (59) in the main text). +Now suppose that we proceed with a further step in the iterative process, ℓ = 2. We have to +deal with the square root of +Ω2 +kΩ2 +k,1 = +� +ω2 +k + iMa′ + a2∆2 + ia∆2 +2M +a′ +a + . . . +� +(1 + ǫ2) . +(162) +The introduction of +ǫ2 = ǫ(2) +2 ++ ǫ(3) +2 ++ . . . , +(163) +whose expression can be seen in (63), does not alter neither the 0th nor the 1st orders of +(159) and (160) since ǫ2 is a sum of terms of adiabatic order 2 and higher: +ΩkΩk,1 =ωk + ω(1) +k ++ ωk +2 ǫ(2) +2 ++ a2∆2 +2ωk ++ a2M2 +8ω3 +k +�a′ +a +�2 ++ ωk +2 ǫ(3) +2 ++ iaM +4ωk +ǫ(2) +2 +a′ +a ++ ia∆2 +4Mωk +a′ +a − ia3M∆2 +4ω3 +k +a′ +a − ia3M3 +16ω5 +k +�a′ +a +�3 ++ . . . +(164) +Nonetheless, the 2nd, 3rd, . . . adiabatic orders of (164) do not coincide with the ones of (160). +Similarly, by going to the next iterative step, ℓ = 3, implies working with the square root of +the product +Ω2 +kΩ2 +k,1Ω2 +k,2 = +� +ω2 +k + iMa′ + a2∆2 + . . . +� +(1 + ǫ2) (1 + ǫ4) +(165) +with +ǫ4 = ǫ(4) +4 ++ ǫ(5) +4 ++ . . . +(166) +The introduction of ǫ4 does not alter neither the 0th, 1st, 2nd nor the 3rd adiabatic orders +of (162) or (164), since ǫ4 is a sum of terms of adiabatic order 4 and higher. By the same +token, the 4th and 5th adiabatic orders and beyond in (162) do not coincide with the ones +in (165). Similar considerations apply to the square root of these quantities. +We can sum up this by saying that for each iterative step we can compute two consecutive +adiabatic orders of (158) that will not be altered by the subsequent steps. Then, after ℓ steps, +the 0, 1, . . . , 2ℓ − 1 adiabatic orders of the product (158) are trustworthy for the estimation +of the modes. +• The RHS of (158) has a pure imaginary part conformed by the odd orders, precisely those +which do no take part in the computation of Wk: +− i +ˆ τ � +ωk + ω(1) +k ++ ω(2) +k ++ · · · + ω(2ℓ−1) +k +� +d˜τ += −i +ˆ τ � +W (0−2(ℓ−1)) +k +� +d˜τ − i +ˆ τ � +ω(1) +k ++ ω(3) +k ++ · · · + ω(2ℓ−1) +k +� +d˜τ. +(167) +50 + +However, because of the factor −i in front of the integral, the imaginary terms in the integrand +become real and are then necessary for the computation of F and G in (152). +• We did not specify the limits of integration in (51) for the following reasons. On the one +hand, even terms take part in the pure imaginary exponential of (152). Now because the +imaginary exponential does not appear in the final result of the relevant quantities that we +compute in the main text (since they cancel in the products of a function times its complex +conjugate) one might wrongly be led to conclude that Wk does not influence the calculation +of the EMT. This would, however, be incorrect since the derivatives of the modes hI,II +k +are +present in these calculations. On the other hand, after integrating the odd terms without +specifying the limits in the integral, there exists some residual freedom in the form of a +set of functions of the momentum only (i.e. +not depending on time). +These are called +f (0) +k , g(0) +k , f (1) +k , g(1) +k , . . . where the superscript indicates the adiabatic order. If our goal is to +compute an adiabatic expansion of the modes up to 6th order we need 7 arbitrary constants +for hI, namely f (0) +k , . . . , f (6) +k . Similarly for hII, which we denote g(0) +k , . . . , g(6) +k . +• From Eqs. (37) and (38), it is clear that hI +k(τ, M) = hII +k (τ, −M) so F(τ, M) = G(τ, −M) +and f (n) +k +(M) = g(n) +k (−M). +With this considerations in mind let’s us put forward the adiabatic expansion of Wk explicitly. +As said, Wk can be specified though the even terms of (158). We are interested to compute at +least up to 6th adiabatic order (that means, at least, ℓ = 4 steps). Therefore we find: +W (0−6) +k +(τ) = ωk + ω(2) +k ++ ω(4) +k ++ ω(6) +k , +(168) +with +ω(0) +k += ωk , +(169) +ω(2) +k += a2∆2 +2ωk +− a2M2 +8ω3 +k +�a′ +a +�2 ++ 5a4M4 +8ω5 +k +�a′ +a +�2 +− a2M2 +4ω3 +k +a′′ +a , +(170) +ω(4) +k += −a4∆4 +8ω3 +k ++ +� +−25a6M4 +16ω7 +k ++ 23a4M2 +16ω5 +k +− a2 +8ω3 +k +� +∆2 +�a′ +a +�2 ++ +�3a4M2 +8ω5 +k +− a2 +4ω3 +k +� +∆2 a′′ +a +− +�1105a8M8 +128ω11 +k +− 267a6M6 +64ω9 +k ++ 21a4M4 +128ω7 +k +� �a′ +a +�4 ++ +�221a6M6 +32ω9 +k +− 57a4M4 +32ω7 +k +� �a′ +a +�2 a′′ +a +− +�19a4M4 +32ω7 +k +− a2M2 +32ω5 +k +� �a′′ +a +�2 +− +�7a4M4 +8ω7 +k +− a2M2 +16ω5 +k +� a′ +a +a′′′ +a + a2M2 +16ω5 +k +a′′′′ +a , +(171) +ω(6) +k += a6∆6 +16ω5 +k ++ +�175a8M4 +64ω9 +k +− 215a6M2 +64ω7 +k ++ 13a4 +16ω5 +k +� +∆4 +�a′ +a +�2 +− +�15a6M2 +32ω7 +k +− 3a4 +8ω5 +k +� +∆4 a′′ +a ++ +�12155a10M8 +256ω13 +k +− 6823a8M6 +128ω11 +k ++ 3351a6M4 +256ω9 +k +− 21a4M2 +64ω7 +k +� +∆2 +�a′ +a +�4 ++ +�133a6M4 +64ω9 +k +− 81a4M2 +64ω7 +k ++ +a2 +32ω5 +k +� +∆2 +�a′′ +a +�2 +51 + +− +�1989a8M6 +64ω11 +k +− 1725a6M4 +64ω9 +k ++ 57a4M2 +16ω7 +k +� +∆2 +�a′ +a +�2 a′′ +a ++ +�49a6M4 +16ω9 +k +− 61a4M2 +32ω7 +k ++ +a2 +16ω5 +k +� +∆2 a′ +a +a′′′ +a − +�5a4M2 +32ω7 +k +− +a2 +16ω5 +k +� +∆2 a′′′′ +a ++ +�414125a12M12 +1024ω17 +k +− 338935a10M10 +1024ω15 +k ++ 56271a8M8 +1024ω13 +k +− 869a6M6 +1024ω11 +k +� �a′ +a +�6 +− +�248475a10M10 +512ω15 +k +− 73457a8M8 +256ω13 +k ++ 12087a6M6 +512ω11 +k +� �a′ +a +�4 a′′ +a ++ +�34503a8M8 +256ω13 +k +− 3225a6M6 +64ω11 +k ++ 249a4M4 +256ω9 +k +� �a′ +a +�2 �a′′ +a +�2 +− +�631a6M6 +128ω11 +k +− 109a4M4 +128ω9 +k +� �a′′ +a +�3 ++ +�1055a8M8 +16ω13 +k +− 3171a6M6 +128ω11 +k ++ 69a4M4 +128ω9 +k +� �a′ +a +�3 a′′′ +a +− +�1391a6M6 +64ω11 +k +− 245a4M4 +64ω9 +k +� a′ +a +a′′ +a +a′′′ +a + +�69a4M4 +128ω9 +k +− a2M2 +128ω7 +k +� �a′′′ +a +�2 +− +�815a6M6 +128ω11 +k +− 149a4M4 +128ω9 +k +� �a′ +a +�2 a′′′′ +a + +�55a4M4 +64ω9 +k +− a2M2 +64ω7 +k +� a′′ +a +a′′′′ +a ++ +�27a4M4 +64ω9 +k +− a2M2 +64ω7 +k +� a′ +a +a′′′′′ +a +− a2M2 +64ω7 +k +a′′′′′′ +a . +(172) +The odd terms in the expansion (158) yield a real exponential contribution in the integrals involved +in (152) and hence do not contribute to the expansion of Wk in (155), but are nevertheless necessary +to compute the amplitude of the modes. Notice that after computing the integral, the adiabatic +order decreases by one unit, so in order to estimate the amplitude up to 6th order is mandatory +to compute up to ω(7) +k . The corresponding integrals for these terms are listed below: +− i +ˆ +ω(1) +k dτ = −i +ˆ τ �iMa′ +2ωk +� +d˜τ = log +�ωk + aM +ωk − aM +�1/4 +, +(173) +−i +ˆ +ω(3) +k dτ = − i∆2 +ˆ τ � +−ia2a′M +4ω3 +k ++ +ia′ +4Mωk +� +d˜τ +− i +ˆ τ � +−25ia2M5a′3 +16ω7 +k ++ 5iM3a′3 +16ω5 +k ++ iaM3a′a′′ +ω5 +k +− iMa′′′ +8ω3 +k +� +d˜τ += +a∆2 +4Mωk +− aM +8ω3 +k +a′′ +a + 5a3M3 +16ω5 +k +�a′ +a +�2 +, +(174) +−i +ˆ τ +ω(5) +k d˜τ = −i∆4 +ˆ τ �3ia4Ma′ +16ω5 +k +− ia2a′ +8Mω3 +k +− +ia′ +16M3ωk +� +d˜τ +− i∆2 +ˆ τ �175ia4M5a′3 +32ω9 +k +− 75ia2M3a′3 +16ω7 +k ++ 15iMa′3 +32ω5 +k +− 5ia3M3a′a′′ +2ω7 +k ++ 3iaMa′a′′ +2ω5 +k ++ 3ia2Ma′′′ +16ω5 +k +− +ia′′′ +16Mω3 +k +� +d˜τ +52 + +− i +ˆ τ � +12155ia4M9a′5 +256ω13 +k +− 3453ia2M7a′5 +128ω11 +k ++ 399iM5a′5 +256ω9 +k +− 1547ia3M7a′3a′′ +32ω11 +k ++ 543iaM5a′3a′′ +32ω9 +k ++ 575ia2M5a′a′′2 +64ω9 +k +− 89iM3a′a′′2 +64ω7 +k ++ 417ia2M5a′2a′′′ +64ω9 +k +− 63iM3a′2a′′′ +64ω7 +k +− 33iaM3a′′a′′′ +32ω7 +k +− 19iaM3a′a′′′′ +32ω7 +k ++ iMa′′′′′ +32ω5 +k +� +d˜τ += +� +− +a3 +16Mω3 +k +− +a +16M3ωk +� +∆4 + +�3a3M +16ω5 +k +− +a +16Mω3 +k +� +∆2 a′′ +a ++ +� +−25a5M3 +32ω7 +k ++ 15a3M +32ω5 +k +� +∆2 +�a′ +a +�2 ++ +�221a5M5 +64ω9 +k +− 35a3M3 +64ω7 +k +� a′′ +a +�a′ +a +�2 +− +�1105a7M7 +256ω11 +k +− 399a5M5 +256ω9 +k +� �a′ +a +�4 +− 19a3M3 +64ω7 +k +�a′′ +a +�2 +− 7M3 +16ω7 +k +a′ +a +a′′′ +a + aM +32ω5 +k +a′′′′ +a , +(175) +−i +ˆ τ +ω(7) +k d˜τ = − i∆6 +ˆ τ � +−5ia6Ma′ +32ω7 +k ++ 3ia4a′ +32Mω5 +k ++ +ia2a′ +32M3ω3 +k ++ +ia′ +32M5ωk +� +d˜τ +− i∆4 +ˆ τ � +− 1575ia6M5a′3 +128ω11 +k ++ 1925ia4M3a′3 +128ω9 +k +− 525ia2Ma′3 +128ω7 +k ++ +15ia′3 +128Mω5 +k ++ 35ia5M3a′a′′ +8ω9 +k +− 15ia3Ma′a′′ +4ω7 +k ++ 3iaa′a′′ +8Mω5 +k +− 15ia4Ma′′′ +64ω7 +k ++ 3ia2a′′′ +32Mω5 +k ++ +ia′′′ +64M3ω3 +k +� +d˜τ +− i∆2 +ˆ τ � +− 158015ia6M9a′5 +512ω15 +k ++ 185361ia4M7a′5 +512ω13 +k +− 51933ia2M5a′5 +512ω11 +k ++ 1995iM3a′5 +512ω9 +k ++ 17017ia5M7a′3a′′ +64ω13 +k +− 3929ia3M5a′3a′′ +16ω11 +k ++ 2715iaM3a′3a′′ +64ω9 +k +− 5175ia4M5a′a′′2 +128ω11 +k ++ 1749ia2M3a′a′′2 +64ω9 +k +− 267iMa′a′′2 +128ω7 +k +− 3753ia4M5a′2a′′′ +128ω11 +k ++ 1263ia2M3a′2a′′′ +64ω9 +k +− 189iMa′2a′′′ +128ω7 +k ++ 231ia3M3a′′a′′′ +64ω9 +k +− 99iaMa′′a′′′ +64ω7 +k ++ 133ia3M3a′a′′′′ +64ω9 +k +− 57iaMa′a′′′′ +64ω7 +k +− 5ia2Ma′′′′′ +64ω7 +k ++ +ia′′′′′ +64Mω5 +k +� +d˜τ +− i +ˆ τ � +− 7040125ia6M13a′7 +2048ω19 +k ++ 6664175ia4M11a′7 +2048ω17 +k +− 1429935ia2M9a′7 +2048ω15 +k ++ 39325iM7a′7 +2048ω13 +k ++ 1242375ia5M11a′5a′′ +256ω17 +k +− 449881ia3M9a′5a′′ +128ω15 +k ++ 112779iaM7a′5a′′ +256ω13 +k +− 945489ia4M9a′3a′′2 +512ω15 +k ++ 30273ia2M7a′3a′′2 +32ω13 +k +53 + +− 24435iM5a′3a′′2 +512ω11 +k ++ 10361ia3M7a′a′′3 +64ω13 +k +− 1639iaM5a′a′′3 +32ω11 +k ++ 90425ia3M7a′2a′′a′′′ +256ω13 +k +− 687335ia4M9a′4a′′′ +1024ω15 +k ++ 173583ia2M7a′4a′′′ +512ω13 +k +− 17259iM5a′4a′′′ +1024ω11 +k +− 27923iaM5a′2a′′a′′′ +256ω11 +k +− 4675ia2M5a′′2a′′′ +256ω11 +k ++ 649iM3a′′2a′′′ +256ω9 +k +− 3403ia2M5a′a′′′2 +256ω11 +k ++ 447iM3a′a′′′2 +256ω9 +k ++ 17405ia3M7a′3a′′′′ +256ω13 +k +− 5215iaM5a′3a′′′′ +256ω11 +k +− 2701ia2M5a′a′′a′′′′ +128ω11 +k ++ 349iM3a′a′′a′′′′ +128ω9 +k ++ 31iaM3a′′′a′′′′ +32ω9 +k +− 1301ia2M5a′2a′′′′′ +256ω11 +k ++ 159iM3a′2a′′′′′ +256ω9 +k ++ 41iaM3a′′a′′′′′ +64ω9 +k ++ 17iaM3a′a′′′′′′ +64ω9 +k +− iMa′′′′′′′ +128ω7 +k +� +d˜τ += +� +a5 +32Mω5 +k ++ +a3 +48M3ω3 +k ++ +a +32M5ωk +� +∆6 ++ +�175a7M3 +128ω9 +k +− 75a5M +64ω7 +k ++ +15a3 +128Mω5 +k +� +∆4 +�a′ +a +�2 +− +�15a5M +64ω7 +k +− +3a3 +32Mω5 +k +− +a +64M3ω3 +k +� +∆4 a′′ +a + +�133a5M3 +128ω9 +k +− 57a3M +128ω7 +k +� +∆2 +�a′′ +a +�2 ++ +�12155a9M7 +512ω13 +k +− 11326a7M5 +512ω11 +k ++ 1995a5M3 +512ω9 +k +� +∆2 +�a′ +a +�4 +− +�1989a7M5 +128ω11 +k +− 1350a5M3 +128ω9 +k ++ 105a3M +128ω7 +k +� +∆2 a′′ +a +�a′ +a +�2 ++ +�49a5M3 +32ω9 +k +− 21a3M +32ω7 +k +� +∆2 a′′′ +a +a′ +a − +�5a3M +64ω7 +k +− +a +64Mω5 +k +� +∆2 a′′′′ +a ++ +�414125a11M11 +2048ω17 +k +− 459355a9M9 +3072ω15 +k ++ 39325a7M7 +2048ω13 +k +� �a′ +a +�6 ++ +�34503a7M7 +512ω13 +k +− 11037a5M5 +512ω11 +k +� �a′′ +a +�2 �a′ +a +�2 ++ 55a3M3 +128ω9 +k +a′′′′ +a +a′′ +a ++ +�1055a7M7 +32ω13 +k +− 330a5M5 +32ω11 +k +� a′′′ +a +�a′ +a +�3 ++ 27a3M3 +128ω9 +k +a′′′′′ +a +a′ +a +− +�248475a9M9 +1024ω15 +k +− 64863a7M7 +512ω13 +k ++ 6699a5M5 +1024ω11 +k +� a′′ +a +�a′ +a +�4 ++ 69aM3 +256ω9 +k +�a′′′ +a +�2 +− +�631a5M5 +256ω11 +k +− 271a3M3 +768ω9 +k +� �a′′ +a +�3 +− +�1391a5M5 +128ω11 +k +− 189a3M3 +128ω9 +k +� a′′′ +a +a′′ +a +a′ +a +− +�815a5M5 +256ω11 +k +− 105a3M3 +256ω9 +k +� a′′′′ +a +�a′ +a +�2 +− +aM +128ω7 +k +a′′′′′′ +a +(176) +Use of Mathematica [81] has been helpful to work out the above integrals. The computational +strategy consists in using a sufficiently general ansatz for the structure of the result that is com- +54 + +patible with the adiabaticity order of the calculation, and then solve for the coefficients (form +factors) of the ansatz from pure algebraic manipulations. The procedure has been illustrated with +a specific example in Sec. 3. +Finally, by applying the normalization condition (156) for the modes at each order, there +exists a constraint to fix the residual freedom mentioned earlier, which is parametrized by the time +independent factors fk and gk (only depending on the momentum k): +Ref (1) +k += 0, +� +Imf (1) +k +�2 ++ +√ +2kRef (2) +k += 0, +2Imf (2) +k Imf (1) +k ++ +√ +2kRef (3) +k += 0, +���f (2) +k +��� +2 ++ 2Imf (1) +k Imf (3) +k ++ +√ +2kRef (4) +k += 0, +2Imf (1) +k Imf (4) +k ++ 2Imf (2) +k Imf (3) +k ++ 2Ref (2) +k Ref (3) +k ++ +√ +2kRef (5) +k += 0, +2Imf (1) +k Imf (5) +k ++ 2Imf (2) +k Imf (4) +k ++ +���f (3) +k +��� +2 ++ 2Ref (2) +k Ref (4) +k ++ +√ +2kRef (6) +k += 0. +(177) +Notice that, imposing the former conditions is equivalent to claim that +�����1 + +� +2 +k +� +f (1) +k ++ f (2) +k ++ f (3) +k ++ f (4) +k ++ f (5) +k ++ f (6) +k +� ����� +2 +≈ 1, +(178) +where the departure from 1 are just terms of adiabatic order 7 or bigger. Similarly for the functions +gk. It can be shown that the observables made out of quadratic terms in the modes hI +k, hII +k (such +as e.g. the EMT), depend on the particular values of fk in the form (178). It follows that they are +not actual degrees of freedom if they satisfy the conditions (177). It is then safe to set particular +values for the functions as long as quantities are computed up to 6th adiabatic order. The simplest +solution satisfying the normalization conditions (177) is f (1) +k += f (2) +k += f (3) +k += f (4) +k += f (5) +k += f (6) +k += 0 +and it is the chosen option for the formulas shown in the rest of this Appendix. +Equipped with these results we are able to calculate the different orders of F (τ, M) up to 6th +order. A comparison between the general equation (51) and the ansatz (152), the different orders +of F are conformed by the rightful combinations of terms of the denominator �ΩkΩk,1Ωk,2Ωk,3 +and the real factors of the exponential exp +� +−i +´ τ ΩkΩk,1Ωk,2Ωk,3d˜τ +� +. Now the different orders of +F are +F (1)(M) = −iaM +4ω2 +k +a′ +a , +(179) +F (2)(M) = +� +− a2 +4ω2 +k ++ +a +4Mωk +� +∆2 + +� +−5a4M4 +16ω6 +k ++ 5a3M3 +16ω5 +k +− M2a2 +32ω4 +k +� �a′ +a +�2 ++ +�a2M2 +8ω4 +k +− aM +8ω3 +k +� a′′ +a , +(180) +F (3)(M) = i +�5Ma3 +16ω4 +k +− +a2 +16ω3 +k +− +a +8Mω2 +k +� a′ +a ∆2 + i +�65M5a5 +64ω8 +k +− 5M4a4 +64ω7 +k +− 21M3a3 +128ω6 +k +� �a′ +a +�3 ++ i +� +−19M3a3 +32ω6 +k ++ M2a2 +32ω5 +k +� a′a′′ +a2 + i aM +16ω4 +k +a′′′ +a , +(181) +55 + +F (4)(M) = +� 5a4 +32ω4 +k +− +a3 +8Mω3 +k ++ +a2 +32M2ω2 +k +− +a +16M3ωk +� +∆4 ++ +�65a6M4 +64ω8 +k +− 15a5M3 +16ω7 +k +− 61a4M2 +128ω6 +k ++ 59a3M +128ω5 +k +− +a2 +32ω4 +k +� �a′ +a +�2 +∆2 ++ +� +−9a4M2 +32ω6 +k ++ a3M +4ω5 +k ++ 3a2 +32ω4 +k +− +a +16Mω3 +k +� a′′ +a ∆2 ++ +�2285a8M8 +512ω12 +k +− 565a7M7 +128ω11 +k +− 349a6M6 +256ω10 +k ++ 793a5M5 +512ω9 +k +− 85a4M4 +2048ω8 +k +� �a′ +a +�4 ++ +� +−457a6M6 +128ω10 +k ++ 113a5M5 +32ω9 +k ++ 113a4M4 +256ω8 +k +− 139a3M3 +256ω7 +k +� �a′ +a +�2 a′′ +a ++ +�41a4M4 +128ω8 +k +− 5a3M3 +16ω7 +k +− a2M2 +128ω6 +k +� �a′′ +a +�2 ++ +�7a4M4 +16ω8 +k +− 7a3M3 +16ω7 +k ++ a2M2 +64ω6 +k +� a′ +a +a′′′ +a ++ +� +−a2M2 +32ω6 +k ++ aM +32ω5 +k +� a′′′′ +a , +(182) +F (5)(M) = i +� +−45a5M +128ω6 +k ++ 3a4 +32ω5 +k ++ +19a3 +128Mω4 +k +− +a2 +64M2ω3 +k ++ +a +32M3ω2 +k +� a′ +a ∆4 ++ i +� +−1105a7M5 +256ω10 +k ++ 35a6M4 +64ω9 +k ++ 1563a5M3 +512ω8 +k +− 101a4M2 +512ω7 +k +− 63a3M +256ω6 +k +� �a′ +a +�3 +∆2 ++ i +�247a5M3 +128ω8 +k +− 15a4M2 +64ω7 +k +− 113a3M +128ω6 +k ++ +a2 +32ω5 +k +� a′ +a +a′′ +a ∆2 ++ i +� +−9a3M +64ω6 +k ++ +a2 +64ω5 +k ++ +a +32Mω4 +k +� a′′′ +a ∆2 ++ +� +−57125a9M9 +2048ω14 +k ++ 715a8M8 +512ω13 +k ++ 7657a7M7 +512ω12 +k +− 903a6M6 +2048ω11 +k +− 6511a5M5 +8192ω10 +k +� �a′ +a +�5 ++ i +�14167a7M7 +512ω12 +k +− 301a6M6 +256ω11 +k +− 9273a5M5 +1024ω10 +k ++ 161a4M4 +1024ω9 +k +� a′′ +a +�a′ +a +�3 ++ i +� +−2525a5M5 +512ω10 +k ++ 19a4M4 +128ω9 +k ++ 361a3M3 +512ω8 +k +� a′ +a +�a′′ +a +�2 ++ i +� +−933a5M5 +256ω10 +k ++ 33a4M4 +256ω9 +k ++ 255a3M3 +512ω8 +k +� �a′ +a +�2 a′′′ +a ++ i +�69a3M3 +128ω8 +k +− M2a2 +128ω7 +k +� a′′ +a +a′′′ +a + i +�41a3M3 +128ω8 +k +− M2a2 +128ω7 +k +� a′ +a +a′′′′ +a − i aM +64ω6 +k +a′′′′′ +a +, +(183) +56 + +F (6)(M) = +� +− 15a6 +128ω6 +k ++ +11a5 +128Mω5 +k +− +3a4 +128M2ω4 +k ++ +5a3 +128M3ω3 +k +− +a2 +64M4ω2 +k ++ +a +32M5ωk +� +∆6 ++ +� +−1105a8M4 +512ω10 +k ++ 965a7M3 +512ω9 +k ++ 1713a4M2 +1024ω8 +k +− 715a5M +512ω7 +k +− 149a4 +1024ω6 +k ++ +57a3 +512Mω5 +k +� �a′ +a +�2 +∆4 ++ +�117a6M2 +256ω8 +k +− 97a5M +256ω7 +k +− 55a4 +256ω6 +k ++ +33a3 +256Mω5 +k +− +a2 +128M2ω4 +k ++ +a +64M3ω3 +k +� a′′ +a ∆4 ++ +� +−57125a10M8 +2048ω14 +k ++ 54265a9M7 +2048ω13 +k ++ 24479a8M6 +1024ω12 +k +− 47405a7M5 +2048ω11 +k +− 28887a6M4 +8192ω10 +k ++ 31635a5M3 +8192ω9 +k +− 85a4M2 +1024ω8 +k +� �a′ +a +�4 +∆2 ++ +�9597a8M6 +512ω12 +k +− 9045a7M5 +512ω11 +k +− 11985a6M4 +1024ω10 +k ++ 5619a5M3 +512ω9 +k ++ 765a4M2 +1024ω8 +k +−417a3M +512ω7 +k +� �a′ +a +�2 a′′ +a ∆2 + +�13a4M2 +128ω8 +k +− 3a3M +32ω7 +k +− +3a2 +128ω6 +k ++ +a +64Mω5 +k +� a′′′′ +a ∆2 ++ +� +−697a6M4 +512ω10 +k ++ 641a5M3 +512ω9 +k ++ 301a4M2 +512ω8 +k +− 241a3M +512ω7 +k +− +a2 +128ω6 +k +� �a′′ +a +�2 +∆2 ++ +� +−119a6M4 +64ω10 +k ++ 7a5M3 +4ω9 +k ++ 183a4M2 +256ω8 +k +− 167a3M +256ω7 +k ++ +a2 +64ω6 +k +� a′ +a +a′′′ +a ∆2 ++ +� +−1690275a12M12 +8192ω18 +k ++ 1678975a11M11 +8192ω17 +k ++ 2377685a10M10 +16384ω16 +k +− 1231405a9M9 +8192ω15 +k +−519009a8M8 +32768ω14 +k ++ 627179a7M7 +32768ω13 +k +− 13989a6M6 +65536ω12 +k +� �a′ +a +�6 ++ +�1014165a10M10 +4096ω16 +k +− 1007385a9M9 +4096ω15 +k +− 250133a8M8 +2048ω14 +k ++ 521273a7M7 +4096ω13 +k ++74799a6M6 +16384ω12 +k +− 106819a5M5 +16384ω11 +k +� a′′ +a +�a′ +a +�4 ++ +� +−141309a8M8 +2048ω14 +k ++ 140205a7M7 +2048ω13 +k ++ 85737a6M6 +4096ω12 +k +− 44397a5M5 +2048ω11 +k ++ 441a4M4 +4096ω10 +k +� �a′ +a +�2 �a′′ +a +�2 ++ +�2643a6M6 +1024ω12 +k +− 2603a5M5 +1024ω11 +k +− 403a4M4 +1024ω10 +k ++ 363a3M3 +1024ω9 +k +� �a′′ +a +�3 ++ +� +−8545a8M8 +256ω14 +k ++ 4255a7M7 +128ω13 +k ++ 9721a6M6 +1024ω12 +k +− 10541a5M5 +1024ω11 +k ++ 277a4M4 +2048ω10 +k +� a′′′ +a +�a′ +a +�3 ++ +�353a6M6 +32ω12 +k +− 1405a5M5 +128ω11 +k +− 697a4M4 +512ω10 +k ++ 755a3M3 +512ω9 +k +� a′ +a +a′′ +a +a′′′ +a ++ +� +−69a4M4 +256ω10 +k ++ 69a3M3 +256ω9 +k +− a2M2 +512ω8 +k +� �a′′′ +a +�2 +57 + ++ +� +−113a4M4 +256ω10 +k ++ 7a3M3 +16ω9 +k ++ a2M2 +256ω8 +k +� a′′′′ +a +a′′ +a ++ +�1645a6M6 +512ω12 +k +− 205a5M5 +64ω11 +k +− 349a4M4 +1024ω10 +k ++ 419a3M3 +1024ω9 +k +� a′′′′ +a +�a′ +a +�2 ++ +� +−27a4M4 +128ω10 +k ++ 27a3M3 +128ω9 +k +− a2M2 +256ω8 +k +� a′ +a +a′′′′′ +a ++ +� a2M2 +128ω8 +k +− +aM +128ω7 +k +� a′′′′′′ +a +. +(184) +For hII +k , one can make use of the relation G(n)(M) = F (n)(−M). +C +Appendix: Adiabatic expansion of ⟨Tµν⟩ for spin-1/2 fields +The unrenormalized components of the vacuum EMT, ⟨Tµν⟩, for spin-1/2 fermions can be obtained +through the adiabatic expansion, which we compute up to 6th order. 12 For the 00 component we +have +� +T δψ +00 +� += +� +T δψ +00 +�(0) ++ +� +T δψ +00 +�(2) ++ +� +T δψ +00 +�(4) ++ +� +T δψ +00 +�(6) ++ · · · +(185) +The various terms in the expansion (185) read as follows: +� +T δψ +00 +�(0) += +1 +2π2a +ˆ ∞ +0 +dkk2 +� +−2ωk +a +� +, +(186) +� +T δψ +00 +�(2) += +1 +2π2a +ˆ ∞ +0 +dkk2 +� +−a∆2 +ωk +− a3M4 +4ω5 +k +�a′ +a +�2 ++ aM2 +4ω3 +k +�a′ +a +�2� +, +(187) +� +T δψ +00 +�(4) += +1 +2π2a +ˆ ∞ +0 +dkk2 +� +a3∆4 +4ω3 +k ++ +�5a5M4∆2 +8ω7 +k +− 7a3M2∆2 +8ω5 +k ++ a∆2 +4ω3 +k +� �a′ +a +�2 ++ +�105a7M8 +64ω11 +k +− 63a5M6 +32ω9 +k ++ 21a3M4 +64ω7 +k +� �a′ +a +�4 ++ +� +−7a5M6 +8ω9 +k ++ 7M4a3 +8ω7 +k +� a′′ +a +�a′ +a +�2 ++ +� +−a3M4 +16ω7 +k ++ aM2 +16ω5 +k +� �a′′ +a +�2 ++ +�a3M4 +8ω7 +k +− aM2 +8ω5 +k +� a′ +a +a′′′ +a +� +, +(188) +12Details of the corresponding computation for scalar fields were provided in [14,15]. A summary of these calcu- +lations is presented in Sec. 2 of the current work. +58 + +� +T δψ +00 +�(6) += +1 +2π2a +ˆ ∞ +0 +dkk2 +� +− a5∆6 +8ω5 +k ++ +� +−35a7∆4M4 +32ω9 +k ++ 55a5M2 +32ω7 +k +− 5a3 +8ω5 +k +� �a′ +a +�2 +∆4 ++ +� +−1155a9M8 +128ω13 +k ++ 987a7M6 +64ω11 +k +− 903a5M4 +128ω9 +k ++ 21a3M2 +32ω7 +k +� �a′ +a +�4 +∆2 ++ +�63a7M6 +16ω11 +k +− 91a5M4 +16ω9 +k ++ 7a3M2 +4ω7 +k +� a′′ +a +�a′ +a +�2 +∆2 ++ +�7a5M4 +32ω9 +k +− 9a3M2 +32ω7 +k ++ +a +16ω5 +k +� �a′′ +a +�2 +∆2 + +� +−7a5M4 +16ω9 +k ++ 9a3M2 +16ω7 +k +− +a +8ω5 +k +� a′ +a +a′′′ +a ∆2 ++ +� +− 25025a11M12 +512ω17 +k ++ 39039a9M10 +512ω15 +k +− 14883a7M8 +512ω13 +k ++ 869a5M6 +512ω11 +k +� �a′ +a +�6 ++ +�3003a9M10 +64ω15 +k +− 2013a7M8 +32ω13 +k ++ 1023a5M6 +64ω11 +k +� a′′ +a +�a′ +a +�4 ++ +� +−5a5M6 +16ω11 +k ++ 5a3M4 +16ω9 +k +� �a′′ +a +�3 ++ +� +−891a7M8 +128ω13 +k ++ 501a5M6 +64ω11 +k +− 111a3M4 +128ω9 +k +� �a′′ +a +�2 �a′ +a +�2 ++ +� +−429a7M8 +64ω13 +k ++ 249a5M6 +32ω11 +k +− 69a3M4 +64ω9 +k +� �a′ +a +�3 a′′′ +a + +�15a5M6 +16ω11 +k +− 15a3M4 +16ω9 +k +� a′ +a +a′′ +a +a′′′ +a ++ +� +−a3M4 +64ω9 +k ++ aM2 +64ω7 +k +� �a′′′ +a +�2 ++ +�9a5M6 +16ω11 +k +− 9a3M4 +16ω9 +k +� a′′′′ +a +�a′ +a +�2 ++ +�a3M4 +32ω9 +k +− aM2 +32ω7 +k +� a′′ +a +a′′′′ +a + +� +−a3M4 +32ω9 +k ++ aM2 +32ω7 +k +� a′ +a +a′′′′′ +a +� +. +(189) +To obtain the component ⟨T δψ +11 ⟩ a similar expansion as in (185) holds. However, it is possible to +use the following relation with the previously computed ⟨T00⟩ component: +� +T δψ +11 +� += +� +T δψ +00 +� +− +� +T δψ +00 +�′ +3H +. +(190) +As a result we find: +� +T δψ +11 +�(0) += +1 +2π2a +ˆ ∞ +0 +dkk2 +�2aM2 +3ωk +− 2ωk +3a +� +, +(191) +� +T δψ +11 +�(2) += +1 +2π2a +ˆ ∞ +0 +dkk2 +� � +−a3M2 +3ω3 +k ++ +a +3ωk +� +∆2 + +� +−5a5M6 +12ω7 +k ++ a3M4 +3ω5 +k ++ aM2 +12ω3 +k +� �a′ +a +�2 ++ +�a3M4 +6ω5 +k +− aM2 +6ω3 +k +� a′′ +a +� +, +(192) +59 + +� +T δψ +11 +�(4) += +1 +2π2a +ˆ ∞ +0 +dkk2 +� �a5M2 +4ω5 +k +− a3 +4ω3 +k +� +∆4 + +� +−5a5M4 +12ω7 +k ++ 7a3M2 +12ω5 +k +− +a +6ω3 +k +� a′′ +a ∆2 ++ +�35a7M6 +24ω9 +k +− 25a5M4 +12ω7 +k ++ 13a3M2 +24ω5 +k ++ +a +12ω3 +k +� �a′ +a +�2 +∆2 ++ +�385a9M10 +64ω13 +k +− 483a7M8 +64ω11 +k ++ 91a5M6 +64ω9 +k ++ 7a3M4 +64ω7 +k +� �a′ +a +�4 ++ +� +−77a7M8 +16ω11 +k ++ 21a5M6 +4ω9 +k +− 7a3M4 +16ω7 +k +� a′′ +a +�a′ +a +�2 ++ +�7a5M6 +16ω9 +k +− 11a3M4 +24ω7 +k ++ aM2 +48ω5 +k +� �a′′ +a +�2 ++ +�7a5M6 +12ω9 +k +− 13a3M4 +24ω7 +k +− aM2 +24ω5 +k +� a′ +a +a′′′ +a + +� +−a3M4 +24ω7 +k ++ aM2 +24ω5 +k +� a′′′′ +a +� +, +(193) +� +T δψ +11 +�(6) += +1 +2π2a +ˆ ∞ +0 +dkk2 +� � +−5a7M2 +24ω7 +k ++ 5a5 +24ω5 +k +� +∆6 + +�35a7M4 +48ω9 +k +− 55a5M2 +48ω7 +k ++ 5a3 +12ω5 +k +� a′′ +a ∆4 ++ +� +−105a9M6 +32ω11 +k ++ 35a7M4 +6ω9 +k +− 265a5M2 +96ω7 +k ++ 5a3 +24ω5 +k +� �a′ +a +�2 +∆4 ++ +� +−5005a11M10 +128ω15 +k ++ 9163a9M8 +128ω13 +k +− 4683a7M6 +128ω11 +k ++ 497a5M4 +128ω9 +k ++ 7a3M2 +32ω7 +k +� �a′ +a +�4 +∆2 ++ +�847a9M8 +32ω13 +k +− 343a7M6 +8ω11 +k ++ 553a5M4 +32ω9 +k +− 7a3M2 +8ω7 +k +� a′′ +a +�a′ +a +�2 +∆2 ++ +� +−63a7M6 +32ω11 +k ++ 35a5M4 +12ω9 +k +− 31a3M2 +32ω7 +k ++ +a +48ω5 +k +� �a′′ +a +�2 +∆2 ++ +� +−21a7M6 +8ω11 +k ++ 175a5M4 +48ω9 +k +− 47a3M2 +48ω7 +k +− +a +24ω5 +k +� a′ +a +a′′′ +a ∆2 ++ +�7a5M4 +48ω9 +k +− 3a3M2 +16ω7 +k ++ +a +24ω5 +k +� a′′′′ +a ∆2 ++ +� +−425425a13M14 +1536ω19 +k ++ 355355a11M12 +768ω17 +k +− 25883a9M10 +128ω15 +k ++ 12221a7M8 +768ω13 +k ++ 869a5M6 +1536ω11 +k +� �a′ +a +�6 ++ +�85085a11M12 +256ω17 +k +− 124839a9M10 +256ω15 +k ++ 40623a7M8 +256ω13 +k +− 869a5M6 +256ω11 +k +� a′′ +a +�a′ +a +�4 ++ +� +−11869a9M10 +128ω15 +k ++ 15301a7M8 +128ω13 +k +− 3395a5M6 +128ω11 +k +− 37a3M4 +128ω9 +k +� �a′ +a +�2 �a′′ +a +�2 ++ +�671a7M8 +192ω13 +k +− 391a5M6 +96ω11 +k ++ 37a3M4 +64ω9 +k +� �a′′ +a +�3 ++ +� +−715a9M10 +16ω15 +k ++ 3597a7M8 +64ω13 +k +− 357a5M6 +32ω11 +k +− 23a3M4 +64ω9 +k +� a′′′ +a +�a′ +a +�3 +60 + ++ +�473a7M8 +32ω13 +k +− 263a5M6 +16ω11 +k ++ 53a3M4 +32ω9 +k +� a′′′ +a +a′′ +a +a′ +a ++ +� +−23a5M6 +64ω11 +k ++ 17a3M4 +48ω9 +k ++ aM2 +192ω7 +k +� �a′′′ +a +�2 ++ +�275a7M8 +64ω13 +k +− 149a5M6 +32ω11 +k ++ 23a3M4 +64ω9 +k +� �a′ +a +�2 a′′′′ +a ++ +� +−19a5M6 +32ω11 +k ++ 29a3M4 +48ω9 +k +− aM2 +96ω7 +k +� a′′ +a +a′′′′ +a ++ +� +−9a5M6 +32ω11 +k ++ 13a3M4 +48ω9 +k ++ aM2 +96ω7 +k +� a′ +a +a′′′′′ +a ++ +�a3M4 +96ω9 +k +− aM2 +96ω7 +k +� a′′′′′′ +a +� +. +(194) +We refer the reader to the master formula in Appendix A.2 of [15] for the explicit computation +of the integrals in the above results. 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Freeman, +1973. +67 + diff --git a/gtE4T4oBgHgl3EQfrA2b/content/tmp_files/load_file.txt b/gtE4T4oBgHgl3EQfrA2b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4dd53422cb50452d0b967ec6a880303126bb558 --- /dev/null +++ b/gtE4T4oBgHgl3EQfrA2b/content/tmp_files/load_file.txt @@ -0,0 +1,5279 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf,len=5278 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='05205v1 [gr-qc] 12 Jan 2023 Running vacuum in FLRW spacetime: The dynamics of ρvac(H) from the quantized matter fields Cristian Moreno-Pulido, Joan Sol`a Peracaula Departament de F´ısica Qu`antica i Astrof´ısica, and Institute of Cosmos Sciences, Universitat de Barcelona, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Diagonal 647, E-08028 Barcelona, Catalonia, Spain Samira Cheraghchi Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 50, 500091 Brasov, Romania E-mails: cristian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='moreno@fqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='edu, sola@fqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='edu, samira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='cheraghchi@unitbv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ro Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Phenomenological work in the last few years has provided significant support to the idea that the vacuum energy density (VED) is a running quantity with the cos- mological evolution and that this running helps to alleviate the cosmological tensions afflicting the ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the theoretical side, recent devoted studies have shown that the properly renormalized ρvac in FLRW spacetime adopts the ‘running vacuum model’ (RVM) form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While in three previous studies by two of us (CMP and JSP) such compu- tations focused solely on scalar fields non-minimally coupled to gravity, in the present work we compute the spin-1/2 fermionic contributions and combine them both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The calculation is performed using a new version of the adiabatic renormalization procedure based on subtracting the UV divergences at an off-shell renormalization point M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The quantum scaling of ρvac with M turns into cosmic evolution with the Hubble rate, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As a result the ‘cosmological constant’ Λ appears in our framework as the nearly sustained value of 8πG(H)ρvac(H) around (any) given epoch H, where G(H) is the gravitational coupling, which is also running, although very mildly (logarithmically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We find that the VED evolution at present reads δρvac(H) ∼ νeffm2 Pl � H2 − H2 0 � (|νeff| ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The coefficient νeff receives contributions from all the quantized fields, bosons and fermions, which we compute here for an arbitrary number of matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Remarkably, there also exist higher powers O(H6) which can trigger inflation in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Fi- nally, the equation of state (EoS) of the vacuum receives also quantum corrections from bosons and fermion fields, shifting its value from -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The remarkable consequence is that the EoS of the quantum vacuum may nowadays effectively look like quintessence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 1 Contents 1 Introduction 3 2 Vacuum energy of a non-minimally coupled scalar field 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 Zero-point energy and adiabatic expansion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 11 3 Quantization of a spin-1/2 fermion field in curved spacetime 14 4 ZPE and VED for a spin-1/2 field in FLRW spacetime 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 Divergence balance between bosons and fermions in vacuum .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': 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universe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 37 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 43 6 Conclusions 45 A Appendix: Conventions and Useful Formulas 48 B Appendix: Adiabatic expansion of the spin-1/2 field modes 49 C Appendix: Adiabatic expansion of ⟨Tµν⟩ for spin-1/2 fields 58 2 1 Introduction Despite having coexisted for many decades, a completely successful theory of gravity that combines Quantum Field Theory (QFT) and General Relativity (GR) does not exist yet, unfortunately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, a variety of different approaches and techniques are available in the literature which allow one to study the subject of quantum fields in the gravitational context, and more specifically the physics of the expanding universe and its current speeding up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Our aim is to understand such an acceleration on fundamental grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To be precise, in this work we will concentrate on the well-known semiclassical approach which goes under the name of QFT in curved spacetime [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This means that gravity is still a classical external (background) field, whereas the matter fields are quantum field operators obeying suitable commutation or anticommutation relations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' A further step in the path of understanding gravity in the QFT context is quantum gravity (QG), in which spacetime itself (the metric) is quantized and hence functional integration over metrics is mandatory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [5–7], and the review [8] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' At the same time a lot of exciting QG phenomenology is being investigated in the current multi-messenger era, characterized by an outburst of experimental data that are being obtained from the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the theoretical side, effective field theory methods and the possibility of quantum gravitational effects leading to quantum hair may provide useful hints of QG which have been explored recently [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, while the QG option has, of course, to be kept in mind since it can be very important when QG can be (hopefully) formulated in a fully consistent way [8], QFT in curved spacetime may still be of great help to further describe the role of quantum fields in a gravitational context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this work, we will continue dwelling upon these lines and shall focus exclusively on the, more modest, but effective, semiclassical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It goes without saying that the latter has had also its own problems and successes over the years, and still have [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, new perspectives have recently been explored in this context concerning the vacuum energy and the cosmological constant (CC) [14–16] which may be of significance, and for this reason we wish to further pursue this line of approach here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The agent responsible for the accelerated cosmic expansion is generically called Dark Energy (DE), an entity which constitutes a key piece in the cosmological puzzle, but whose fundamental nature is still undisclosed [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Typically, it is associated with the CC in Einstein’s equations, Λ, as done routinely in the standard or concordance model of cosmology, aka ΛCDM [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The model has been a rather successful paradigm for the phenomenological description of the universe for about three decades,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' but it became consolidated only in the mid nineties [21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='22] and especially after consistent measurements of Λ made in the last twenty years using independent cosmological sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' in particular including distant type Ia supernovae (SnIa),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' baryon-acoustic oscillations (BAO),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the data on large-scale structure formation and of course the anisotropies of the cosmic microwave background (CMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' All in all they have put the very experimental basis for the concordance ΛCDM model of cosmology [23–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The situation is far from being satisfactory, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The problems with the ΛCDM are both of theoretical and observational nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As for the theoretical problems, recall that the value of Λ is traditionally associated to a parameter called the vacuum energy density (VED) in the universe, although in the context of the ΛCDM is nothing but a name for the following quantity with dimensions of energy density: ρvac = Λ/(8πGN) (GN being Newton’s constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Its theoretical significance is not explained at all in the context of the standard cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' If, however, we take quantum theory seriously, the most universal contribution to this vacuum energy density is the zero-point energy (ZPE) of the massive quantum fields in the standard model of particle physics, and in fact in any QFT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, it is well- known that a naive calculation of this quantity leads to very large contributions proportional to the quartic power of the mass of the particles, ρZPE ∼ m4, which is in blatant discordance with the 3 order of magnitude obtained for this quantity from cosmological observations: ρobs vac ∼ 10−47GeV4 (expressed in natural units, with ℏ = c = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Even taking for instance the electron field, one finds a mismatch of 34 orders of magnitude: ρobs vac/ρZPE ∼ 10−34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The huge discrepancy between a typical Standard Model ZPE and the measured value of VED constitutes the so-called Cosmological Constant Problem (CCP) [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' See also [30, 31] for a recent account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Despite the enormous discrepancies between usual theory predictions and factual measurements, estimates on the value of Λ within the right order of magnitude have been attempted under certain assumptions in the context of QG in different approaches, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [32–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While the aforementioned measurements of ρvac indicate that the vacuum can gravitate within an energy density order of magnitude of ∼ 10−47GeV4, what is difficult to understand theoretically is why the vacuum can only gravitate in that tiny range, given the fact that any typical quantum effect rockets its contribution to much larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is of course a rephrasing of the same puzzle associated to the CCP, expressed in the QFT context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, new avenues for a possible solution have been suggested recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The renormalization approach presented in the present work and in the preceding studies [14–16] offers some hope to eschew part of these difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' First and foremost, the renormalized quantum effects found here endow the VED with a mild dynamical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The latter thus appears as a slowly varying function of the cosmic expansion, specifically of the Hubble rate H, see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Second, the renormalized VED as reported here proves well behaved and can perfectly accommodate the measured value of Λ from observational cosmology without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Technically, this is because the “running” of Λ is proportional to the tiny values of the β-function coefficients for bosons and fermions, which are responsible for the renormalization group evolution of the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As a result, at any given epoch of the late universe Λ appears essentially as constant, but it is not strictly so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, a third crucial ingredient of our approach is that, in the very early universe, the VED becomes, in contrast, very large and fast evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' There it can take the capital role of bringing about inflation, as we shall see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As previously mentioned, in addition to the traditional theoretical problems, other issues of more practical and mundane nature have been perturbing cosmologists in the last few years, which put the ΛCDM against the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The practical problems are the presently irreconcilable obser- vational differences between the standard picture and a number of different kinds of observations involving structure formation data (the so-called σ8 tension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' at a moderate level of ∼ 3σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' where σ8 is the root mean square of fluctuations in density perturbations at the 8h−1 Mpc scale),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' and above all the notorious disagreement between the local value of the Hubble parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' obtained from the traditional distant ladder techniques,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' and the value extracted from the early universe using CMB data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' see [37–40] and references therein for extensive reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The discrepancy in the latter case is at the level of ∼ 5σ, hence a tension whose notable severity (and persistence over time) may well be passing the point of being attributable to a fluke [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So the prevailing model of cosmology may well be facing a crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Science, however, thrives on crisis since new ideas are then stimulated which could help to overcome the crisis and maybe refine some aspects of the underlying paradigm or even originate a new one subsuming the old paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Many proposals have indeed been made to alleviate these tensions, which include different forms of DE as well, even though many of them are essentially ad hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As indicated above, clues to eventually substantiate the nature of the DE may come from a variety of cosmic and even astrophysical messengers [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For instance, the possibility of measuring the bending of light in the Solar System scale has been proposed [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' But whatever the nature of the DE might be, we must provide an explanation for the role played by the vacuum energy in QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Indeed, in the absence of a correct understanding of the VED from first principles many DE proposals may look as an escape forward rather than a real alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This work, in contrast, intends to dwell further on the methods of QFT in curved spacetime so as to shed some useful light on these difficult problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Thus we take the Λ term seriously, not just as a mere fitting parameter but as a formal quantity in the gravitational action 4 from which one can determine the VED, a fundamental concept in QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, we report here on progress made along the lines of the preceding comprehensive works [14–16], where a detailed account was made of the virtual contribution to the VED from quantized scalar matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We found that these effects, when appropriately renormalized, translate into a (mild) dynamical evolution of the VED with the cosmic expansion, ρvac = ρvac(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is not excluded by the cosmological principle, as it permits a homogeneous and isotropic dependence in time of physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' More specifically, it was shown by explicit calculation and by appropriate renormal- ization that the VED behaves in the characteristic manner of the running vacuum model (RVM), see [31] for a recent comprehensive review (for a shorter summary, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us also note that a similar RVM interpretation of the vacuum energy can be achieved in the string context, what has been called ‘stringy-RVM’ [45–49] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The fact that a QFT calcu- lation and an effective string theory approach can lead to the same kind of RVM solution seems to indicate that the VED as a rigid concept is not natural and that a dynamical evolution of the vacuum energy should be more plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In actual fact, it has been recently shown that this fact can help significantly to improve the description of the overall cosmological data and in particular opens a viable solution to the well-known tensions afflicting the ΛCDM, see particularly the last phenomenological analysis [50], which was preceded by several other works, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [51–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is also interesting to remark that the RVM structure of the vacuum energy has been successfully tested against competing models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ghost models and holographic models of the DE) using cos- mographical methods, which are essentially model-independent – see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [56–58] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The model has indeed passed a battery of different tests [59,60] and the outcome is that the quality fit provided by the overall cosmological data is comparable, actually better, than that of the ΛCDM, if we attend to the verdict of the standard information criteria [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this paper we continue the task of computing the dynamics of ρvac(H) induced by the quantum effects of the quantized matter fields in Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetime, which two of us (CMP and JSP) initiated in previous works [14–16], see also [31] for a comprehensive review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The result of the present, more complete, calculation (involving for the first time the fermionic contributions) reconfirms that the combined dynamics of the vacuum adopts the RVM form indicated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In these works, the adiabatic regularization prescription (ARP) was used to compute the renormalized VED, ρvac, for a non-minimally coupled real scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The method is based on a series expansion in the number of derivatives of the scale factor which introduces a hierarchy in some physical quantities evolving in a dynamical background [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Not only it was shown that the running of the VED was free from dangerous large contributions proportional to m4 (quartic powers of the mass of the particles), but ρvac was shown to be mildly evolving with the Hubble rate and hence with the cosmic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, if t1 and t2 are two particular values of cosmic time, both close to the present, the corresponding values of ρvac were shown to be related as follows: ρvac(H2) − ρvac(H1) ≈ νeff m2 Pl � H2 2 − H2 1 � , (1) where H1 ≡ H(t1) and H2 ≡ H(t2) are the values of the Hubble function at times t1 and t2, respectively and |νeff| ≪ 1 is a small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While the above relation is relevant for the (very mild) evolution of the VED in the current universe, the corresponding analysis of the early universe leads to a new mechanism of inflation called ‘RVM-inflation’, which relies on the existence of quantum effects of 6th adiabatic order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' up to terms ∼ O(H6) which have been first accounted for scalar fields in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While the family of Running Vacuum Models (RVM) has been in the literature for quite some time (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [30,31,61] and references therein), a full-fledged account based on QFT principles is much more recent [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This work is tightly related to the preceding studies, in which the adiabatic regularization was applied to the ‘simple’ case of one real scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It was natural to perform the next step and 5 check if spin-1/2 fermions do preserve the main conclusions derived from scalars, above all to verify if the corresponding vacuum fluctuations induce also a running of the VED independent of the quartic powers of their masses and hence remain also free from the traditional fine-tuning illness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So the main goal of this paper is to extend the computations done for the scalar field, by considering the quantization of spin-1/2 fermions in the FLRW background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The extension proves rather non- trivial since we are in curved spacetime and the computations with fermions are no less involved owing to the Fermi-Dirac statistics and the formal peculiarities associated with spinor calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is however reassuring to find that the many new technicalities involved in the calculation do not alter the main conclusions derived from the calculation with scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The RVM form (1) at low energies is once more attained, but the contribution to the coefficient νeff is, of course, different and involves non-trivial computational details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Similarly, we compute the fermionic contribution to the ∼ O(H6) terms which are involved in the RVM mechanism of inflation occurring in the very early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The final results concerning the renormalized VED can be obtained by considering the contributions from an arbitrary number of quantized scalar and fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We combine the two types of effects and present a final formula which we refer to as the bosonic and fermionic contributions to the VED, with the understanding that additional effects from gauge fields and their interactions with matter would be necessary in our calculation in a more complete approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is nevertheless not necessary in our study since our fields are free, except for the non-minimal coupling of the quantized scalar fields with the external (non-quantized) gravitational field and the necessary spinorial affine connection of the fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' All that said, the computation of the free field contribution from bosons and fermions in curved spacetime is already a formidable task, so for the sake of a stepwise and clearer presentation we will address the fermionic contributions here on equal footing to the presentation of the scalar part in the references [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This work is structured as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2, we consider a quantized scalar field φ non- minimally coupled to curvature and review the computation of its energy-momentum tensor (EMT) and corresponding vacuum expectation value (VEV) induced by the vacuum fluctuations of that field [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The 00th component of its VEV constitutes the ZPE of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We define also the VED and the vacuum pressure, Pvac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' All these quantities are unrenormalized at this point and hence UV-divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the same section we review the off-shell adiabatic renormalization method used in the previous references, which involves the distinctive feature of our adiabatic renormalization of the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 3, we review the quantization of a Dirac fermion in a curved background, the corresponding Dirac equation and its spinor solutions obtained from adiabatic expansion of the field modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The computation of the EMT and of its VEV for the case of a free quantized fermionic field in a spatially flat FLRW background is performed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The off-shell adiabatic renormalization of the EMT for spin-1/2 fermions is addressed and we derive the corresponding renormalized ZPE, VED and Pvac in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Additionally, some remarks on the trace anomaly and its role in our approach are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5 contains the combined results from all the quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Specifically, we display the renormalized VED for a system made of an arbitrary number of quantized scalar fields non-minimally coupled to curvature (with different masses and non-minimal couplings) and an arbitrary number of quantized spin-1/2 free fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the same section we compute the corresponding running of the gravitational coupling G = G(H), which is associated with the running of ρvac(H) in order to preserve the Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We also discuss the mechanism of ‘RVM-inflation’ with the combined contribution from all these fields, and compute the equation of state (EoS) of the quantum vacuum for that system of quantized bosons and fermions fluctuating in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Remarkably, the vacuum EoS is no longer equal to wvac = −1, the reason being that the vacuum pressure and the VED are not exactly related in the usual way Pvac = −ρvac since Pvac and ρvac are independent functions of the Hubble rate H and its time derivatives owing to the quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the current universe, there is still some remnant of these quantum effects which induce a small (but potentially measurable) departure 6 making the quantum vacuum mimic quintessence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The conclusions are delivered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 6 together with some additional discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, three appendices are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In AppendixA, we define our conventions and some useful expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The last two appendices, B and C, are rather bulky since they collect a number of cumbersome formulas related to the adiabatic expansion of the EMT and the Fourier modes of the fermionic field (computed up to 6th order for the first time in the literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2 Vacuum energy of a non-minimally coupled scalar field In this section we summarize the results for the VED from quantized scalar fields in FLRW spacetime obtained in [14,15] and in passing we introduce some notation which will be useful also for the fermionic calculation that will be subsequently reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The Einstein-Hilbert (EH) action for gravity plus matter reads SEH+m = SEH + Sm = ˆ d4x√−g � 1 16πG R − ρΛ � + Sm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (2) The term ρΛ has dimension of energy density and sometimes is called the vacuum energy density, but this is inaccurate in the formal QFT context since renormalization is necessary and the physical vacuum energy density, ρvac, is not just that term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, ρΛ is at this point just a bare parameter of the action, as the gravitational coupling G itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Varying the action with respect to the metric provides Einstein’s equations 1 8πGGµν = −ρΛgµν + T m µν , (3) with Gµν = Rµν − (1/2)gµνR the usual Einstein tensor and T m µν the energy-momentum tensor (EMT) of matter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=': T m µν = − 2 √−g δSm δgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (4) The matter action Sm may contain a variety of contributions, including those from incoherent matter, but it will be enough to focus on fundamental effects from quantized scalar and fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Here we shall compute the fermionic contribution to the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' But let us summarize first the situation with the scalar field part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The latter was dealt with in great detail in the two previous studies [14,15], in which the VED calculation was addressed under the assumption that no effective potential was present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, we admitted a non-minimal coupling of the scalar field to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' That calculation in curved spacetime is already sufficiently demanding and in addition it furnishes the universal part of the VED through the zero-point energy (ZPE) effects in the curved background, see next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The classical action associated to a non-minimally coupled real scalar field is the following: Sφ = − ˆ d4x√−g �1 2gµν∂µφ∂νφ + 1 2 � m2 φ + ξR � φ2 � , (5) where ξ is the non-minimal coupling with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is well known that this action enjoys (local) conformal symmetry in the massless case with ξ = 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, the value of ξ is not fixed in our computation and in general we do not assume the presence of such a symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Varying the above action with respect to the scalar field leads to the Klein-Gordon (KG) equation with non-minimal coupling: � □ − m2 φ − ξR � φ2 = 0, (6) 1Our geometric conventions and other formulas of interest for this calculation are collected in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 7 where □φ = gµν∇µ∇νφ = (−g)−1/2∂µ (√−g gµν∂νφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The corresponding EMT associated to φ follows from the metric variation of the action (5) according to the recipe (4), and yields Tµν(φ) = − 2 √−g δSφ δgµν = (1 − 2ξ) ∂µφ∂νφ + � 2ξ − 1 2 � gµν∂σφ∂σφ − 2ξφ∇µ∇νφ + 2ξgµνφ + ξGµνφ2 − 1 2m2 φgµνφ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (7) As indicated, we perform the calculation in cosmological (FLRW) spacetime with flat three- dimensional metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For convenience we use the conformal frame ds2 = a2(τ)ηµνdxµdxν, with ηµν = diag(−1, +1, +1, +1) the Minkowski metric in our conventions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' AppendixA), a(τ) is the scale factor and τ the conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Differentiation with respect to τ will be denoted with a prime, so for example H ≡ a′/a is the corresponding Hubble function in conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We will perform the explicit calculations using the conformal metric but our final results will eventually be rendered in terms of the usual Hubble function H(t) = ˙a/a in cosmic time t (where a dot denotes differentiation with respect to t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Recall that dτ = dt/a and hence H = aH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' If we switch on the quantum fluctuations of the φ field it is natural to considering the following decomposition: φ (⃗x, τ) = φb(τ) + δφ (⃗x, τ) , (8) in which the background φb and the fluctuating part δφ are understood to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In particular, this is also the case for the corresponding Fourier decomposition in frequency modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The decomposition of the fluctuating part reads δφ(τ, x) = 1 (2π)3/2a ˆ d3k � Akeik·xhk(τ) + A† ke−ik·xh∗ k(τ) � , (9) with the usual commutation relations for the creation and annihilation operators, Ak and A† k: [Ak, A′† k] = δ(k − k′), [Ak, A′ k] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (10) By using these relations, the KG-equation (6) in terms of the frequency modes can be put as h′′ k + Ω2 k(τ)hk = 0, (11) where Ω2 k ≡ k2 + a2m2 φ + (ξ − 1/6) R and we recall that ()′ ≡ d/dτ ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The above differential equation does not possess a close analytic solution for the entire cosmological evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, it can be solved by means of what is called an adiabatic series expansion, which is essentially WKB-type solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' First of all, it is necessary to introduce the following ansatz for the mode functions: hk(τ) = 1 � 2Wk(τ) exp � −i ˆ τ Wk(˜τ)d˜τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (12) Notice that the modes are normalized through the Wronskian condition hkh∗′ k − h′ kh∗ k = i , (13) which is essential to preserve the canonical commutation relations for the quantized field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' By introducing the above ansatz into (11), the function Wk (effective frequency) is the solution of the (WKB-type) non-linear differential equation W 2 k = Ω2 k − 1 2 W ′′ k Wk + 3 4 �W ′ k Wk �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (14) 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 Zero-point energy and adiabatic expansion For a slowly varying effective frequency Ωk(τ) one can proceed to solve this equation perturbatively with the help of an asymptotic series which can be organized through adiabatic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This approach constitutes the basis for the aforementioned ARP [62–69], see also [70–75] and [76–80] for more recent applications and extensions, and the textbooks [1–3] for a more systematic presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The quantities k2 and a are taken to be of adiabatic order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Of adiabatic order 1 are: a′ and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The quantities a′′, a′2, H′ and H2 and linear combinations are taken of adiabatic order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It can be summarized by saying that each time derivative increases one unit the adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So, the ansatz solution for Wk can be written as an adiabatic series expansion: Wk = W (0) k + W (2) k + W (4) k + W (6) k + · · · , (15) in which the superscript indicates the adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We note that only even orders are allowed, which is justified from the general covariance of the result since only tensors of even adiabatic order can be present in the effective action and the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Explicit calculation indeed corroborates the absence of the odd adiabatic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The “seed” to initiate the adiabatic series (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the zeroth order contribution to Wk) is W (0) k ≡ ωk = � k2 + a2M2, (16) where M is an arbitrary off-shell scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Nothing enforces us to take the mass of the particle at this point, we only need to preserve the adiabaticity of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The floating quantity M will play the role of renormalization scale, as it will be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, as it was shown in [15], this parameter can also be used as the renormalization scale in the DeWitt-Schwinger expansion [4] of the vacuum effective action Weff [1], to wit: the effective action obtained from integrating out the vacuum fluctuations of the quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From the explicit expression of Weff one can also derive the VEV of the EMT – denoted ⟨Tµν⟩ and often referred to in this paper as the ‘vacuum EMT’ – by computing its metric functional derivative as follows: ⟨Tµν⟩ = − 2 √−g δWeff δgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (17) This formula is of course similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (4), but for the vacuum effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This alternative method provides exactly the same result as the WKB expansion of the field modes, as outlined below, and it was illustrated in great detail in [15] for the case of the quantized scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Such a cross-check in the determination of the vacuum EMT involves a significant amount of calculations and provides a nontrivial validation of the entire renormalization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The same holds good for the case of fermions but we shall not present the details of the DeWitt-Schwinger method for fermions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Next we summarize the mode expansion for scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' By introducing W (0) k given above in the RHS of (14), the terms of adiabatic order 2 can be collected to find the next term in the series W (2) k , with the result W (2) k = a2∆2 2ωk + a2R 2ωk � ξ − 1 6 � − ω′′ k 4ω2 k + 3 (ω′ k)2 8ω3 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (18) Here ∆2 ≡ m2−M2 is the difference between the quadratic mass of the field and that of the off-shell scale, and is of adiabatic order 2 because it is necessary for renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Loosely speaking, since M2 and ∆2 appear together in the expansion they need to be of different adiabatic order so as 9 to obtain a consistent adiabatic expansion exploring the off-shell regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Since M2 is of adiabatic order 0, the next-to-leading order for ∆2 to be compatible with general covariance is precisely order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This fact is reconfirmed on using the aforementioned DeWitt-Schwinger expansion [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Introducing W (2) k on the RHS of (14) we can obtain W (4) k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The higher order adiabatic terms become progressively more and more cumbersome since the number of terms involved in the expansion becomes larger and larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In our case we reach up to order 6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' we compute the series up to W (6) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This was done for the first time in the literature in [15] for scalar fields, and we will also be done here to order W (6) k for the first time for fermions, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notice that attaining the order W (6) k is indispensable in order to study RVM-inflation in the early universe [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Extensive use of Mathematica [81] has been made to handle these bulky calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Once obtained the expansion of Wk we can compute the mode functions hk and other physical quantities such as the EMT to the suitable order, in particular the EMT trace, which can be used to compute the vacuum pressure (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The technical details for the scalars are not going to be repeated here2, but it is important to remark that these quantities present divergent terms up to 4th adiabatic order (in 4-dimensional spacetime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The approach we adopt here is the same as that which was proposed and amply tested in [14,15], namely we define the renormalized vacuum expectation value (VEV) of the EMT (or renormalized vacuum EMT) by taking the on-shell value (at an arbitrary adiabatic order) and subtracting from it the divergent orders at an arbitrary scale, which we denote M: � T δφ µν � ren (M) = � T δφ µν � (mφ) − � T δφ µν �(0−4) (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (19) Here the superscript (0 − 4) refers to the UV-divergent subtracted orders, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' from 0th up to 4th adiabatic order, all of them being UV-divergent (the higher adiabatic orders being all finite in n = 4 spacetime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notice that for M = mφ the above definition provides the natural generalization of the subtraction of divergent constants performed to obtain finite results on trivial backgrounds (such as Minkowski spacetime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, in curved backgrounds the mode by mode subtraction implied in the above prescription is not just a constant term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' and moreover for arbitrary M the corresponding renormalized result allows us to test the evolution of the VED with the scale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As previous indicated, this feature obviously offers a floating scale which is characteristic of the renormalization group (RG) analysis in cosmology [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us clarify, however, that we distinguish M from the ’t Hooft’s mass unit µ in dimensional regularization (DR), which will not be used in this work at any point, although it can be invoked as an intermediate regularization procedure (not at all as renormalization though) if one likes [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The parameter µ is unphysical and is used in the minimal subtraction scheme (MS) with DR to define the renormalization point [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We should emphasize that we do not use MS at all in the present work, although one could make (optional) use of DR in intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In these cases, the quantity µ always cancels out and the final renormalized expressions depend on M only, as it is the case e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' of the effective action of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' But the full effective action (which involves the classical and quantum parts) is of course independent of M as well, as the running of the couplings exactly compensates for the explicit M-dependence of the quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is, of course, the standard lore of the RG program, see [15] for detailed considerations along these lines and making use of the effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We refrain from writing out the unrenormalized expression for the EMT in the case of scalar fields, see [14,15] for full details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us however quote the renormalized result emerging from the ARP prescription (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Expressing the final result in terms of the cosmic time and the correspond- ing Hubble function H = ˙a/a, we find for the 00-component of the vacuum EMT the following 2They are provided in [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Analogous computations will be reported in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 3 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4 for the case of the fermion field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 10 result: � T δφ 00 � ren (M) = a2 128π2 � −M4 + 4m2 φM2 − 3m4 φ + 2m4 φ ln m2 φ M2 � − � ξ − 1 6 � 3a2H2 16π2 � m2 φ − M2 − m2 φ ln m2 φ M2 � + � ξ − 1 6 �2 9a2 16π2 � 6H2 ˙H + 2H ¨H − ˙H2� ln m2 φ M2 + O � H6 m2 φ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (20) and similarly for the VEV of its trace � T δφ� ren (M) = 1 32π2 � 3m4 φ − 4m2 φM2 + M4 − 2m2 φ ln m2 φ M2 � + 3 � ξ − 1 6 � 8π2 � 2H2 + ˙H � � m2 φ − M2 − m2 φ ln m2 φ M2 � − 9 8π2 � ξ − 1 6 �2 � 12H2 ˙H + 4 ˙H2 + 7H ¨H + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H � ln m2 φ M2 + O � H6 m2 φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (21) We used the notation O(H6/m2 φ) to collectively refer to the terms of adiabatic order 6 (consisting of 6 time derivatives of the scale factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It may include terms such as H6/m2 φ, but also many other combinations such as ˙H3/m2 φ, H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H/m2 φ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These terms are rather lengthy and have been computed and reported explicitly in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We refrain from writing them down again here and invite the reader to check the aforementioned paper for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We will report explicitly on the 6th-order adiabatic terms only in the case of fermions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4) since these are computed for the first time in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2 Renormalized vacuum energy and vacuum pressure We are now ready to compute the vacuum EMT, ⟨Tµν⟩, which will lead us to the VED, ρvac, and vacuum’s pressure, Pvac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As in [15], we write the vacuum EMT as the sum of the renormalized parameter ρΛ and the renormalized ZPE, which embodies the finite form of the adiabatically renormalized vacuum fluctuations: � T vac µν � = −ρΛ(M)gµν + � T δφ µν � ren (M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (22) Since the vacuum is expected to be a most symmetric state free of any new parameter, this expression must take on the form of a perfect fluid: ⟨T vac µν ⟩ = Pvacgµν + (ρvac + Pvac) uµuν, where uµ is the 4-velocity (uµuµ = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In conformal coordinates in the comoving (FLRW) frame, uµ = (−a, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Taking the 00th and iith-component (any i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 3 is good owing to isotropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' so we take i = 1) one finds the precise form of the vacuum energy density and pressure [15]: ρvac(M) = ⟨T vac 00 ⟩ a2 = ρΛ(M) + � T δφ 00 �ren (M) a2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (23) 11 Pvac(M) = ⟨T vac 11 ⟩ a2 = −ρΛ(M) + � T δφ 11 �ren (M) a2 = −ρΛ(M) + 1 3 \uf8eb \uf8ed � T δφ�ren (M) + � T δφ 00 �ren (M) a2 \uf8f6 \uf8f8 = −ρvac(M) + 1 3 \uf8eb \uf8ed � T δφ�ren (M) + 4 � T δφ 00 �ren (M) a2 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (24) where isotropy allows to express the result in terms of the trace T δφ of the fluctuating part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notice that ρΛ(M) in the above expressions is the renormalized form of the corresponding bare parameter appearing in the EH action (2) and it has units of energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The VED, however, is not just this renormalized parameter but the renormalized sum (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Although is tantalizing to call ρΛ(M) “the CC density”, and in fact this has been common in the literature (especially when the discussion is strictly classical without considering quantum effects), this is not strictly correct since the physical CC is not simply 8πGρΛ but 8πGρvac, that is, the physical vacuum energy density is connected with the physical Λ through ρvac = Λ/(8πG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The parameter Λ which is measured in the observations is indeed defined through this expression, which is precisely computable in QFT from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We shall show once more for fermions (as we did for scalar fields in the previous works [14, 15]), that the adiabatically renormalized form of the running VED is free from the huge ∼ m4 contributions that are usually attributed to the VED in other (inappropriate) renormalization schemes, and therefore the renormalized expression that we will obtain can be perfectly consistent with the measured Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To be sure, it is not our aim to predict this value but rather to show that the theoretical formula points naturally to a value as small (in natural units) as measured by the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' A simple way to condense these ideas is to say that the VED is related with the ZPE and ρΛ as follows: “VED = ρΛ + ZPE”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Parameter ρΛ is initially just a bare coupling in the effective action and it has no direct phenomenological interpretation, not even after renor- malization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the other hand, the ZPE embodies the quantum fluctuations of the massive fields and calls also for renormalization since it is originally UV-divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The physical VED in this context is then the renormalized sum of these two contributions, and it can not be split apart since the separate terms make no sense in isolated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Observations are sensitive only to the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Furthermore, as we shall see explicitly for the fermionic case, there is a crucial cancellation of the quartic mass terms when we consider the evolution of the sum ρΛ + ZPE as a function of the renormalization point, which does not occur if the two terms are dealt with separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This was already pinpointed for the case of scalar fields in [14–16] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' With these provisos, the expression for the VED associated to the scalar field can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notwithstanding, the final renormalized result still requires a physical interpretation since it de- pends on the renormalization scale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, recall that the result depends on both the values of M and H (and corresponding time derivatives), which are independent arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The scale M can not be left arbitrary at this point since we wish to provide an estimate of the VED at a given expansion epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As previously indicated, the vacuum effective action Weff is explicitly dependent on M despite the full effective action is of course RG-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Thus, following , [14,15] an ap- propriate choice of the renormalization point M is to select it equal to the value of H at the epoch under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This corresponds to choose the RG scale around the characteristic energy scale of FLRW spacetime at any given moment, and hence it should have physical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In actual fact this is in analogy with the standard practice in ordinary gauge theories, where the choice of the renormalization group scale is made near the typical energy of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In what follows we derive the ‘low energy’ form of the VED along these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is actually the form that applies for the current universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Subsequently we will focus on the running gravitational coupling G(M) and its relation with the running ρvac(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 12 We should also point out that the reach of our considerations concern the calculation of the evolution (or ‘running’) of the VED only, rather than predicting its current value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Given that value, however, we can predict how it evolves with H around our epoch, or any other epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Now in the absence of an observational input at some expansion epoch H(t) we cannot compute ρvac(H) at other values of H (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' at other epochs of the cosmic evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To compute the value of the VED at present is out of the scope of the renormalization program since the latter is based on the RG flow, which requires a boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is exactly the same situation as in any renormalization calculation, we need the input values of the parameters at one scale to predict some observable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' a cross-section) at another scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The truly relevant feature of our calculational approach, as it should be clear at this point, is that the RG-flow is completely smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It only depends on the evolution of H and is completely free from spurious effects associated to the large contributions from the quartic masses of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These quartic terms are the traditional kind of undesirable effects which spoil the physical interpretation of the renormalization program concerning the CC and the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' They are typically found in calculations of the VED whose renormalization is based on, say, the MS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Most existing approaches to the CC problem in the literature exhibit this unwanted feature, which is already at the basis of the Minkowskian calculation and is of course unacceptable in curved spacetime [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' A similar situation is found in Schwarzschild and de Sitter backgrounds, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [83,84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Bearing in mind the above considerations, the final result for the running VED at low energies (specifically the part triggered by the quantized scalar fields), can be best written in terms of the evolution between two expansion history times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is natural that we choose the current epoch (characterized by the value H0 of the Hubble parameter) and relate it with the value of the VED at a nearby epoch H of the cosmic evolution3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The approximate final result can be rendered in a very compact form as follows: ρvac(H) = ρ0 vac + 3νeff 8π � H2 − H2 0 � m2 Pl, (25) where νeff ≡ 1 2π � ξ − 1 6 � m2 φ m2 Pl ln m2 φ H2 0 (26) and ρ0 vac ≡ ρvac(H0), is the current value of the vacuum energy (accessible by observations) with H0 the current value of the Hubble function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is necessary to remark that νeff is an effective parameter expected to be small due to its proportionality to m2 φ/m2 Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Remarkably, the above dynamical form of the VED turns out to adopt the RVM form, see [31] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Phenomenological studies based on fitting the above RVM formula to the overall cosmological data indeed provide an estimate for νeff at the level of νeff ∼ 10−3 and positive [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The order of magnitude is reasonable if we take into account that the masses involved here pertain of course to the scale of a typical Grand Unified Theory (GUT) where, in addition, a large factor must be included to account for the large multiplicity of heavy particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5 we provide a more general formula where an arbitrary number of species of bosons and fermion fields are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is worth noticing that the order of magnitude of νeff picked out in the mentioned study is perfectly compatible with the result recently obtained from the Big Bang nucleosynthesis (BBN) bound in [85], although in the latter case the bound was not sensitive to the sign of νeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the computation of the pressure in an analogous way (we refrain from providing more details on the scalar field contribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' see once more [15] and [16] for a full- fledged account) enables us to write an explicit expression for the equation of state (EoS) of the 3In this context,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' a nearby epoch does not necessarily mean that it is very close to the current epoch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' it rather refers to any cosmic span for which the VED running is still driven by the ∼ H2 terms and not by higher powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The higher powers are only relevant for the very early universe, namely during the inflationary time, and hence the low-energy formula applies virtually to any (post-inflationary) epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For more details, see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 13 vacuum [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The leading expression for the current universe is the following: wvac = Pvac(H) ρvac(H) ≈ −1 − νeff ˙Hm2 Pl 4πρ0vac .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (27) For very low redshift z and in terms of the current cosmological parameters Ω0 i = ρ0 i /ρ0 c = 8πGNρ0 i /(3H2 0) the above expression reduces to wvac(z) ≈ −1 + νeff Ω0 vac Ω0m (1 + z)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (28) This result is especially remarkable since it predicts a small departure from -1 which could be measured around the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Recall that the traditional value associated to a Cosmological Constant is just −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This means that the EoS for the quantum vacuum receives small quantum effects which trigger a departure from −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For instance, if we adopt the positive sign for νeff, as obtained from the latest fitting analysis to a large set of different kinds of observational data [50], then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (28) predicts that the vacuum energy behaves as quintessence around the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As noted, the EoS formula (28) is valid only for small values of the redshift z, but one can show that the departure is even bigger in the past, adopting a kind of chameleonic behaviour by which the EoS of the quantum vacuum tracks the EoS of matter at high redshifts, see [16] and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' All in all, these surprising results have been predicted from first principles, nearly from explicit QFT calculations in the FLRW background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In particular, the fact that the quantum vacuum may currently mimic quintessence is truly remarkable since the result does not rely on ad-hoc fields or on any other phenomenological ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 3 Quantization of a spin-1/2 fermion field in curved spacetime As indicated in the introduction, the main goal of this work is to extend the QFT results for the VED obtained for quantized scalar fields, which we have summarized in the previous section, to the case of quantized spin-1/2 Dirac fermion fields and then combine the two types of contributions in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The calculation of the renormalized VED for fermions is again nontrivial and requires a devoted study, which we present here in detail (see also the appendices provided at the end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While the QFT treatment is analogous to the case of scalars the technicalities are quite different and no less intricate, but fortunately the final result proves to be in consonance with the one derived for the scalars, so it is perfectly possible to furnish a combined contribution to the VED which involves an arbitrary number of scalar and fermion fields, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The study of the solutions of the Dirac equation in curved spacetime goes back to the works from many decades ago by Fock, Tetrode, Schr¨odinger, McVittie, Bargmann, Wheeler and others: see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [86–90], where the relevant historical references are given and different aspects of spin-1/2 fermions in curved spacetime are studied, including a detailed account for the solutions in FLRW spacetime – see also the review [91], with a rather complete list of references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the other hand, the subject of adiabatic regularization for fermions has been previously treated in the literature in different applications, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [67] as well as the more recent papers [72–75] where emphasis is made on exact solutions e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' in de Sitter spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The calculation of the renormalized VED in FLRW spacetime, however, can be appropriately performed using an off-shell variant of the ARP framework [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' which leads to the RVM behavior of the vacuum energy [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The RVM framework has proven successful to mitigate the cosmological tensions [39,40], as shown in different phenomenological analyses [50–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the theoretical side, attempts at computing the VED with other procedures has led to the traditional calamity with the quartic powers of the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Here we will show that using the ARP to tackle the VED contribution from fermions generates a result 14 which is free from these difficulties and fully along the lines of what has been obtained for the scalar fields in the previous sections and originally in [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, the combined contribution from fermions and scalar fields to the VED is compatible with a smooth running of the cosmological vacuum energy and is consistent with the aforementioned phenomenological analysis of the RVM as a possible solution to the cosmological tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Since we will use the formalism employed in some of the aforementioned papers to treat fermions within the adiabatic approach, it is convenient to summarize first the necessary aspects of that formalism before we put forward our main results concerning the VED for fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This will be useful also to fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Once more we perform the calculations in FLRW spacetime with flat three-dimensional metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Consider a free Dirac spin-1/2 field, described by the four-component spinor ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In our conventions the Dirac action in curved spacetime is given by Sψ(x) = − ˆ d4x√−g �1 2i � ¯ψγµ∇µψ − � ∇µ ¯ψ � γµψ � + mψ ¯ψψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (29) In the above expression mψ denotes the mass of the Dirac field and ¯ψ ≡ ψ†γ0 the adjoint spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Since we are in a curved background, the partial derivative of a spinor ∂µψ has been replaced with the corresponding covariant derivative ∇µψ, which is defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Moreover, gamma ma- trices in curved spacetime are also needed, they are sometimes indicated (as above) with an un- derline to distinguish them from the Minkowski space gamma matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' γµ(x) (which are generally functions of the coordinates) versus the constant matrices γα in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As it is well-known, to obtain a representation for the curved spacetime gamma matrices in terms of the Minkowskian gamma matrices we need to introduce the local tetrad or vierbein field (in 4-dimensional spacetime) e µ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is defined in each tangent space of the spacetime manifold and re- lates the curved spacetime metric with the Minkowskian one as follows: gµν(x) = eµ α(x)eν β(x)ηαβ, where ηαβ is the Lorentz metric in the local inertial frame associated to normal coordinates at the given spacetime point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The general relation between the two sorts of gamma matrices is γµ(x) = eµ α(x)γα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Specifically, in a spatially flat FLRW spacetime the vierbein in conformal co- ordinates is eµ α = diag (1/a(τ), 1/a(τ), 1/a(τ), 1/a(τ)) where a(τ) is the scale factor as a function of the conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Whence the gamma matrices in this background are time-dependent and related to the constant flat spacetime ones as follows: γµ(τ) = γµ/a(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This relation insures that they satisfy the following anti-commutation relations: � γµ, γν� = −2gµνI4 , (30) provided, of course, the (constant) flat space gamma matrices satisfy � γα, γβ� = −2ηαβI4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In order to obtain the equation of motion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the covariant Dirac equation in curved spacetime, one has to vary the covariant action (29) with respect to the spinor field, giving iγµ∇µψ + mψψ = ieµ αγα∇µψ + mψψ = i1 a (γα∂α − γαΓα) ψ + mψψ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (31) The covariant derivative is defined through the spin connection, ∇µ ≡ ∂µ−Γµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The spinorial affine connection Γµ satisfies the equation [87] � Γν, γµ(x) � = ∂γµ(x) ∂xν + Γµ νργρ(x) , (32) where Γµ νρ are the Christoffel symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The above equation is tantamount to require the vanish- ing of the covariant derivative of the curved space gamma matrices: ∇νγµ(x) = 0 [2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the curved-space gamma matrices are defined to be covariantly constant over the spacetime manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Using the Christoffel symbols in the conformally flat FLRW metric as given in Appendix A, the 15 explicit solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (32) can be found, with the following result: Γ0 = 0, Γj = − (H/2) γjγ0 = − (a′/2a) γjγ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, γαΓα = 3(a′/2a)γ0 = −3(a′/2a)γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This expression can then be inserted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this way we have obtained an explicit form for the Dirac equation in FLRW spacetime with spatially flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We are now in position to attempt a solution by expanding the quantized fermion field in mode functions as follows: ψ(x) = ˆ d3k � λ=±1 (B⃗k,λu⃗k,λ(x) + D† ⃗k,λv⃗k,λ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (33) Here B⃗k,λ and D† ⃗k,λ are creation and annihilation operators which satisfy the standard anticom- mutation relations, � D⃗k,λ, D† ⃗q,λ′ � = � B⃗k,λ, B† ⃗q,λ′ � = δλ,λ′δ(3) � ⃗k − ⃗q � , � D⃗k,λ, D⃗q,λ′ � = � D† ⃗k,λ, D† ⃗q,λ′ � = � B⃗k,λ, B⃗q,λ′ � = � B† ⃗k,λ, B† ⃗q,λ′ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (34) The momentum expansion of the mode functions u⃗k,λ and their charge conjugates v⃗k,λ can be conveniently written in terms of two 2-component spinors ξλ(⃗k) and corresponding spinor modes hI k and hII k : u⃗k,λ(τ, x) = ei⃗k·⃗x � (2πa)3 � hII k (τ)ξλ(⃗k) hI k(τ)⃗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='⃗k k ξλ(⃗k) � , v⃗k,λ(τ, x) = e−i⃗k·⃗x � (2πa)3 � −hI∗ k (τ)⃗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='⃗k k ξ−λ(⃗k) −hII∗ k (τ)ξ−λ(⃗k) � , (35) with ⃗σ · ⃗k k ξλ(⃗k) = λξλ(⃗k), λ = ±1 , ξ† λ(⃗k)ξλ(⃗k) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (36) Using this representation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (31) splits into two coupled first order equations for each of the two types of spinor modes hI k and hII k : hII k = ia k (1 a∂τ + imψ)hI k(τ), hI k = ia k (1 a∂τ − imψ)hII k (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (37) After straightforward calculation, these equations can be rewritten as two second order decoupled equations: � ∂2 τ + imψa′ + a2m2 ψ + k2 � hI k(τ) = 0 → � ∂2 τ + Ω2 k(τ) � hI k(τ) = 0, � ∂2 τ − imψa′ + a2m2 ψ + k2 � hII k (τ) = 0 → � ∂2 τ + (Ω2 k(τ))∗� hII k (τ) = 0, (38) where Ω2 k ≡ ω2 k + a2∆2 + iσ(τ) , (39) with ωk(M) ≡ � k2 + M2a2, σ ≡ mψa′ = � M2 + ∆2 a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (40) 16 The fact that (38) only depends on the modulus of the momentum, k, justifies the notation used for the modes hI k, hII k , with no arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Following the same prescription as in the case of scalar fields (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2), we have introduced an off-shell scale M, which again will take the role of renormalization scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Correspondingly, we have defined ∆2 ≡ m2 ψ −M2 and once more assigned adiabaticity order 2 to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We did not change the notation ∆ as compared to the scalar case since the final formulas do not depend on ∆ but on M and the respective physical masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The argument of ωk will be omitted from now on, unless it takes a different value from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The normalization conditions for the mode functions involved in ψ is performed through the Dirac scalar product as follows: (u⃗k,λ, u⃗k′,λ′) = ˆ d3x a3u† ⃗k,λu⃗k′,λ′ = δλλ′δ3(⃗k − ⃗k′) (41) and similarly for (v⃗k,λ, v⃗k′,λ′) = δλλ′δ3(⃗k − ⃗k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It follows that |hI k|2 + |hII k |2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (42) As mentioned in the previous section, the number of time derivatives of the cosmological scale factor a(τ) that appear in a term of the expansion is called adiabatic order of the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In order to solve the differential equations (38) we may follow a recursive process which preserves the adiabatic hierarchy, just as we did with the scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us first redefine hI k and the time variable as follow hI k,1 ≡ � ΩkhI k dτ1 = Ωkdτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (43) Substituting these relations into the equation for hI k in (38) we find d2 dτ 2 1 hI k,1 + Ω2 k,1hI k,1 = 0, Ω2 k,1 ≡ 1 + ǫ2, ǫ2 ≡ −Ω−1/2 k d2 dτ 2 1 Ω1/2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (44) Since ǫ2 includes two derivatives, it contains terms of second and higher adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We can ignore it to find the leading order solution hI k,1 ≈ e−iτ1, (45) so that we get a first approximation hI k ≈ e−i ´ τ Ωkd˜τ √Ωk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (46) Notice that hI k,1 formally satisfies a differential equation with the same form as (38) for hI k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So that, we can repeat the process: hI k,2 ≡ � Ωk,1hI k,1, dτ2 ≡ Ωk,1dτ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (47) The corresponding differential equation for hI k,2 is � ∂2 ∂τ 2 2 + Ω2 k,2 � hI k,2 = 0, Ω2 k,2 ≡ 1 + ǫ4, ǫ4 ≡ −Ω−1/2 k,1 d2 dτ 2 2 Ω1/2 k,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (48) Once again, ǫ4 consists of terms of adiabatic order 4 and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We can approximate a solution of (48) by neglecting ǫ4: hI k,2 ≈ e−iτ2 , (49) 17 whereby the approximation to hI k can be further improved: hI k ≈ e−i ´ τ ΩkΩk,1 d˜τ �ΩkΩk,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (50) By iterating the procedure, we can obtain a better and better approximation to hI k, and after ℓ > 1 steps we find hI k ≈ e−i ´ τ Ωk···Ωk,ℓ−1 d˜τ � Ωk · · · Ωk,ℓ−1 , (51) where, for ℓ ≥ 1, Ω2 k,ℓ ≡ 1 + ǫ2ℓ, dτℓ ≡ Ωk,ℓ−1dτℓ−1, ǫ2ℓ ≡ −Ω−1/2 k,ℓ−1 d2 dτ 2 ℓ Ω1/2 k,ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (52) Now that the general method has been set up, let’s find the 0th order solution for hI k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From (46), the most generic solution for hI k is hI k(τ) ≈ f (0) k √Ωk e−i ´ τ Ωk d˜τ = f (0) k (ω2 k + a2∆2 + iσ)1/4 e−i ´ τ √ ω2 k+a2∆2+iσ d˜τ, (53) where the time independent function f (0) k (of adiabatic order 0) accounts for the integration ‘con- stant’ (strictly speaking, a function of the momentum but not of conformal time) in the exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As for hII k , by comparing both lines of (38) it is clear that it is possible to proceed in an analogous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So we obtain hII k (τ) ≈ g(0) k �Ω∗ k e−i ´ τ Ω∗ k d˜τ = g(0) k (ω2 k + a2∆2 − iσ)1/4 e−i ´ τ √ ω2 k+a2∆2−iσ d˜τ , (54) where g(0) k has the same paper as f (0) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To find the zeroth adiabatic order it is just enough to expand this solution and keep zero order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, some extra caution is needed when dealing with the integrand in the exponential of (53), which may be expanded up to 1st order as Ω(0−1) k = ωk + ω(1) k , (55) where ω(1) k ≡ iaM 2ωk a′ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (56) The reason is that the integration of the second term in the exponential factor is: e−i ´ ω(1) k dτ = �ωk + aM k �1/2 = �ωk + aM ωk − aM �1/4 , (57) so it yields a real term of adiabatic order zero, meaning that the expansion of Ωk up to 1st order in the integral was mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We have not included an explicit multiplicative factor related with the constant of integration4 since it is already represented by f (0) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We choose f (0) k such that the 4The same situation happens with indefinite integrals of higher order terms in the imaginary exponential of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' They are written in an appropriate manner, contributing at bigger adiabatic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The final results, though, just depend on f (0) k and not on the other higher order integrations constants, as dictated by the normalization condition (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' See Appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 18 above solution can be compatible with mode functions in Minkowskian spacetime, so we can write f (0) k = � k 2, hI(0) k (τ) = � ωk + aM 2ωk e−i ´ τ ωk d˜τ, g(0) k = � k 2, hII(0) k (τ) = � ωk − aM 2ωk e−i ´ τ ωk d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (58) Next we move on to the solution at 1st adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As we have mentioned, the quantity ǫ2 defined in (44), contains terms of adiabatic order two and higher, so it is not necessary to find the first order solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is enough to find the first order term from the denominator of (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So, hI(0−1) k ≈ � 1 √ωk � f (0) k + f (1) k � � 1 − iMa′ 4ω2 k � e ´ τ Ma′ 2ωk d˜τ � e−i ´ τ ωk d˜τ = � ωk + aM 2ωk � 1 − iMa′ 4ω2 k + � 2 kf (1) k � e−i ´ τ ωk d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (59) Similarly for the second spinor mode hII k : hII(0−1) k ≈ � 1 √ωk � g(0) k + g(1) k � � 1 + iMa′ 4ω2 k � e − ´ τ Ma′ 2ωk d˜τ � e−i ´ τ ωk d˜τ = � ωk − aM 2ωk � 1 + iMa′ 4ω2 k + � 2 kg(1) k � e−i ´ τ ωk d˜τ , (60) where f (1) k and g(1) k come from integration constants, as mentioned in the footnote of the previous page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' By imposing the normalization condition (42), which has to be satisfied at each adiabatic order, it is possible to see that these constants are purely imaginary, that is Re f (1) k = Re g(1) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (61) To continue, we deal with the 2nd adiabatic order of the mode functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' hI,II(2) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' At this time, we have to include Ω2 k,1 = 1 + ǫ2 in our considerations (this term contains 2nd order adiabatic terms and beyond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (50), we have hI k ≈ f (0) k + f (1) k + f (2) k � Ωk(1 + ǫ2)1/2 e−i ´ τ Ωk(1+ǫ2)1/2 d˜τ , (62) where ǫ2 can be computed to be ǫ2 = 5 16Ω6 k � 2aa′m2 ψ + imψa′′�2 − 1 4Ω4 k � 2a′2m2 ψ + 2aa′′m2 ψ + imψa′′′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (63) With this result, it is immediate to obtain an approximation for Ωk,1 valid up to third adiabatic order: Ωk,1 = (1 + ǫ2)1/2 = 1 − a2M2 4ω4 k a′′ a − a2M2 4ω4 k �a′ a �2 + 5 8 a4M4 ω6 k �a′ a �2 − iaM 8ω4 k a′′′ a + ia3M3 2ω6 k �a′ a �3 − 15ia5M5 8ω8 k �a′ a �3 + 9ia3M3 8ω6 k a′ a a′′ a + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (64) 19 On the other hand, an expansion of the product ΩkΩk,1 is necessary to improve the approxima- tion of hI,II k , as one can see from equation (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As earlier, if we wish to present a second order approximation of the modes we have to expand the mentioned product up to 3rd adiabatic order in the exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The expansion can be presented as follows: ΩkΩk,1 = Ωk (1 + ǫ2)1/2 = ωk + ω(1) k + ω(2) k + ω(3) k + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (65) where the dots represent the contributions of adiabatic order bigger than 3, and the indicated terms in the expansion read ω(1) k ≡ iaM 2ωk a′ a , ω(2) k ≡ −a2M2 8ω3 k �a′ a �2 − a2M2 4ω3 k a′′ a + 5a4M4 8ω5 k �a′ a �2 + a2∆2 2ωk , ω(3) k ≡ 5ia3M3 16ω5 k �a′ a �3 + ia3M3 ω5 k a′ a a′′ a − iaM 8ω3 k a′′′ a + ia∆2 4Mωk a′ a − 25ia5M5 16ω7 k �a′ a �3 − ia3M∆2 4ω3 k a′ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (66) As noted before, ω(1) k and ω(3) k are purely imaginary, while ωk and ω(2) k are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' when integrated inside the exponential of equation (50) the former two give a real contribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' whereas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='the latter two become part of the phase of the mode and play the role of an effective frequency: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Ωk (1 + ǫ2)1/2 d˜τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='≈ exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ω(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ω(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='d˜τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ωk + ω(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='d˜τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�ωk + aM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ωk − aM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�1/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='5a3M3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16ω5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�a′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− aM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='8ω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a + a∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='4Mωk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ωk + ω(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='d˜τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='≈ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�ωk + aM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ωk − aM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�1/4 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 + 5a3M3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16ω5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�a′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− aM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='8ω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a + a∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='4Mωk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ωk + ω(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='d˜τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='(67) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='The last result holds good up to an arbitrary function of momentum (constant in conformal time) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='multiplying the whole result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We account for this arbitrary constant by introducing the functions f (0) k , f (1) k , f (2) k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' at each order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' An efficient strategy to compute the integrals involved in the above calculation (and many other ones of a similar sort, see Appendix B for a sample of them) is to set up an ansatz which respects the adiabaticity order of the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The ansatz consists of a finite number of terms (in fact, a linear combination of them) taken each at the given adiabatic order and with coefficients (or ‘form factors’) which must be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The terms of the ansatz are constructed out of the derivatives of the scale factor and the parameter ∆2 (which we recall is of second adiabatic order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For instance, in order to compute the integral of ω(3) k in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (66), we know that the result must be of second adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Hence as a suitable ansatz we use a linear combination of second order adiabatic terms as follows: − i ˆ τ w(3) k d˜τ = Q1 (a, ωk) �a′ a �2 + Q2 (a, ωk) a′′ a + Q3 (a, ωk) ∆2 + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (68) where again the term ‘const.’ at the end means that it does not depend on the integration variable, ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' By taking derivatives with respect to (conformal) time of the last expression and comparing with ω(3) k one can identify the form factors Q1 = 5a3M3 16ω5 k , Q2 = − aM 8ω3 k and Q3 = a 4Mωk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 20 Using (67) together with (65) and (62), the expansion of hI k up to 2nd order is hI(0−2) k = �ωk + aM 2ωk �1/2 � 1 − ia′M 4ω2 k + � 2 kf (1) k � 1 − ia′M 4ω2 k � + � 2 kf (2) k − Ma′′ 8ω3 k + 5M3a′2a 16ω5 k − 5a2a′2M4 16ω6 k − a′2M2 32ω4 k + aa′′M2 8ω4 k + a∆2 4Mωk − a2∆2 4ω2 k � e−i ´ τ � ωk+ω(2) k � d˜τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (69) In a similar way, hII(0−2) k = �ωk − aM 2ωk �1/2 � 1 + ia′M 4ω2 k + � 2 kg(1) k � 1 + ia′M 4ω2 k � + � 2 kg(2) k + Ma′′ 8ω3 k − 5M3a′2a 16ω5 k − 5a2a′2M4 16ω6 k − a′2M2 32ω4 k + aa′′M2 8ω4 k − a∆2 4Mωk − a2∆2 4ω2 k � e−i ´ τ� ωk+ω(2) k � d˜τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (70) The normalization condition fixes the following relations: ���f (1) k ��� 2 = − √ 2k Re f (2) k , ���g(1) k ��� 2 = − √ 2k Re g(2) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (71) So far, the expansion for the modes hI k and hII k up to 2nd order has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' One can continue with the procedure formerly described to reach higher orders, although of course the calculation becomes more and more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We should keep in mind, though, that the adiabatic expansion is an asymptotic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While for renormalization purposes it is enough to stop the expansion at 4th adiabatic order (in 4-dimensional spacetime), it is nonetheless necessary to reach up to 6th order to meet the finite terms ∼ H6 that are dominant in the early universe and capable of triggering inflation in this framework (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We shall refrain from presenting these cumbersome formulas in the main text, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is worth to mention that there is some residual freedom in the previous calculations since, we can not determine entirely the set of integration constants that appear during the calculations f (1) k , g(1) k , f (2) k , g(2) k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Because of the normalization condition (42) of the mode functions, some restrictions such as (61) and (71) apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Fortunately, as commented in more detail in Appendix B, the satisfaction of these restrictions is enough for the observables to be independent from this residual freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So that, is enough to set all of them to 0 to get, for instance, the desired values of the energy density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4 ZPE and VED for a spin-1/2 field in FLRW spacetime The computation of the Fourier modes for a quantized fermion field through adiabatic expansion as explained in the previous section is just the first step to compute the vacuum energy density (VED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The next step towards the VED is to obtain the ZPE associated to Dirac fermions in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As it well known, traditional computations of ZPE suffer from the well-known headache of carrying highly unacceptable contributions proportional to the quartic powers of the masses, 5As explained in [14], owing to the renormalization prescription of the EMT – see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (19) for the scalar case and its fermionic counterpart, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (79) below – the explicit 4th order powers H4 just cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As a result, the 6th order is the first non-vanishing contribution on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 21 ∼ m4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is so both for scalar and fermion fields, and it is already the case in flat, Minkowskian, spacetime, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [30, 31] for a detailed discussion and more references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In curved spacetime we have the same situation, in principle, but in addition we encounter subleading, curvature dependent, contributions which do not exist in the flat case, as we shall see in a moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To handle this issue, an appropriate renormalization prescription is called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The calculation of the ZPE performed here for spin-1/2 fermions is closely related with the one previously put forward for scalar fields in [14, 15] and summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Once more the computation will be done through adiabatic expansion of the field modes and will be carried out up to 6th adiabatic order, since this is the first non-vanishing order on-shell, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' when fixing the renormalization scale M to the value of the mass of the fermion mψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, the off-shell computation at 4th order is already very useful as a means to determine the RG running of the VED as a function of the scale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is actually one of the main new features of the ARP method proposed in [14, 15], which leads to the cosmic evolution of the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Next we consider the actual calculation for spinor fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To find out the ZPE, we start from the definition of EMT in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this case we have to evaluate the functional derivative T ψ µν = − 2 √−g δSψ δgµν , (72) applied to the fermion action (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Upon a straightforward calculation we arrive at the following symmetric expression: T ψ µν = i 4 ¯ψ � γµ∇ν + γν∇µ � ψ − i 4 �� ∇µ ¯ψ � γν + � ∇ν ¯ψ � γµ � ψ , (73) in which the equation of motion (31) and its hermitian conjugate have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We treat this spinor field as a field operator and upon using its expansion in Fourier modes and utilizing the anticommuting algebra of the creation and annihilation operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (34), we can compute the VEV of the various components, which reflect the contribution from the vacuum fluctuations of the quantized fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We find that the VEV of the 00th component of the EMT can be cast as follows: � T δψ 00 � = 1 2π2a ˆ dkk2ρk, (74) where ρk is a function of the previously defined mode functions: ρk = i a � hI kh′I∗ k + hII k h′II∗ k − hI∗ k h′I k − hII∗ k h′II k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (75) The explicit form of the adiabatic expansion of ρk is rather cumbersome;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the reader may find the final result of ⟨T δψ 00 ⟩ in the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us note that for off-shell renormalization at a point M it suffices to adiabatically expand the solution up to 4th order (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (19) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (79) below) but we provide the result up to 6th order so as to be sensitive to the on-shell result (when M = mψ) and also because it is important for the inflationary mechanism in the early universe (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Renormalization is indeed necessary since the VEV of the EMT is formally divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The UV- divergent contributions appear up to 4th adiabatic order (in n = 4 spacetime dimensions), so that one has to subtract terms at least up to this order to obtain a finite result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 Divergence balance between bosons and fermions in vacuum The VEV can be split in two different parts, divergent (in the UV sense) and non-divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Explicit calculation using the formulas of AppendixC) shows that the divergent part is � T δψ 00 � = 1 2π2a2 ˆ ∞ 0 dkk2 � −2ωk − a2∆2 ωk + a4∆4 4ω3 k � + 1 2π2 ˆ ∞ 0 dkk2 � M2 4ω3 k + ∆2 4ω3 k � H2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (76) 22 As it is easy to see, there are terms diverging quartically, quadratically and logarithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The non-divergent part contains the remaining terms, all of them being finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The above ZPE is an unrenormalized result at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, before we proceed to renormalize that expression, it may be instructive to check if there is a chance for the cancellation between UV-divergent terms between fermions and bosons in the supersymmetric (SUSY) limit, at least for the leading divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the on-shell case (M = m and hence ∆2 = 0) the above equation (76) simplifies to ⟨T δψ 00 ⟩ ��� (M=m) = − 1 π2a2 ˆ dkk2ωk(m) + 1 8π2 ˆ ∞ 0 dkk2 m2 ω3 k(m)H2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (77) It coincides with the Minkowskian result for a = 1 (since H = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Now in a SUSY theory, in which the number of boson and fermion degrees of freedom (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=') is perfectly balanced, we should expect that the leading (quartic) divergences cancel among the fermionic and bosonic contributions in the vacuum state [92,93] since in such case the scalar and fermionic fields have the same mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Thus the quartically divergent contribution from the first term of (77) should be minus four times the corresponding result for one real scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Indeed it is so, for in the on-shell limit and projecting the UV-divergent terms of the first two adiabatic orders only, we find that the contribution from one real scalar field in FLRW spacetime with spatially flat metric is [15] ⟨T δφ 00 ⟩(0−2)��� (M=m) = 1 4π2a2 ˆ dkk2ωk(m) − 3 � ξ − 1 6 � 4π2a2 ˆ dkk2 � H2 ωk(m) + a2m2H2 ω3 k(m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (78) We confirm that the first term (the quartically divergent one) of this expression is of opposite sign to the first one in(77) and is a factor of 4 smaller, as we indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So, in a SUSY theory, where we would have 4 real scalar d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' for each Dirac fermion, there would be an exact cancellation of the leading UV-divergent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In addition, we can see at once from (78) that both the quadratic and logarithmic divergences of bosons are associated to effects of the spacetime curvature since they are proportional to H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These terms, therefore, vanish in Minkowski spacetime but are unavoidably present in the FLRW background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the other hand from the second term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (77) it is clear that for fermions we only have subleading divergences of logarithmic type, which are also associated to curvature effects since they are again proportional to H2 and would also vanish in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Hence there is no possible cancellation of these subleading divergences between bosonic and fermionic d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=', in FLRW spacetime, even in the exact SUSY limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Of course, our framework is not placed in the context of supersymmetry, but it serves as a consistency check of our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' See also the discussion in [94,95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Although it is possible to introduce a cutoff for a preliminary treatment of the subleading diverges (and maybe to speculate on its possible meaning) it is not really necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' One simply has to implement appropriate renormalization since renormalization is anyway necessary to deal meaningfully with the VED, as there is no way to cure the divergences from the combined contri- butions from bosons and fermions and it is not useful to be left with a “physical” cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Dealing with a cutoff is always ambiguous as it is generally not a covariant quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Renormalization gets rid of cutoffs and one can preserve covariance, which is safer for a physical interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The adiabatic renormalization is ideal in this sense since the adiabatic expansion generates automatically a covariant result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is well-known that the renormalization program in QFT requires the presence of a renor- malization point, as well as a renormalization prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The renormalization point is a floating scale characteristic of the RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As in the ordinary adiabatic procedure, to implement the renormal- ization of the EMT in 4 spacetime dimensions we perform a subtraction of the first four adiabatic orders, which are the only ones that can be UV-divergent [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, in contrast to the usual recipe, in which the subtraction is performed on the mass shell value m of the quantum field, we 23 perform it at an arbitrary scale M since this enables us to explore the RG evolution of the VED and ultimately connect it with its cosmic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is the specific feature of the adiabatic renormalization procedure (ARP) for the VED that was proposed in [14,15] – see also [31] for ad- ditional details and a comparison with other renormalization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The resulting renormalized VED ensuing from this procedure is free from the usual troubles associated to the quartic powers of the masses and the associated fine tuning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, let us note that dealing with the CCP in Minkowski spacetime using, for instance, the MS scheme and assigning some value to the ’t Hooft’s mass unit µ in DR (as discussed so many times in the literature), is entirely meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is not only devoid of meaning in that a non-vanishing cosmological constant cannot be defined in Minkowski space without manifestly violating Einstein’s equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' it is meaningless also on account of the fact that there is no sense in associating the scale µ to a cosmological variable, say H, since, if Einstein’s equations are invoked, the Λ term as such in these equations cannot exist in Minkowski spacetime unless the VED is exactly ρΛ + ZPE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So there is no cosmology whatsoever to do in flat spacetime, despite some stubborn attempts in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Persisting in this attitude leads to the nonsense of having to cope with ∼ m4 effects which must then be fine tuned among all the particles involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This point has been driven home repeatedly e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' in [30] and also recently in [31], see also [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' A realistic approach to the VED within QFT in curved spacetime must get rid of Minkowski space pseudo- argumentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The approach that we present here is fully formulated in curved spacetime and the vacuum energy density just evolves with the curvature effects (powers of H) rather than with powers of the masses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' we pursue the successful renormalization program of [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, when the background curvature vanishes, we consistently predict that the non-trivial effects which are responsible for the value of the vacuum energy density and the cosmological constant disappear (and hence we are left with no Λ nor VED in the universe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Such is, of course, the situation in Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In practice, however, we cannot reach that flat spacetime situation in our universe since there exists four-dimensional curvature at all times during the indefinite process of expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' But by the same token such an impossibility evinces the fact that the VED and its dynamical nature is a direct consequence of the expansion process (and of the spacetime curvature inherent to it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The expected size of the VED and of Λ in our framework is indeed provided by the magnitude of the spacetime curvature, which is of the typical value of the measured Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is therefore not caused by the quartic power of the masses of the fields (which is the very root of the CC problem in most approaches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These powers do not affect the running of the VED in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To put it in a nutshell: the renormalized VED in our framework is like a small quantum ‘ripple’ imprinted on the existing (classical) background curvature owing to the vacuum fluctuations of the quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the absence of the background curvature, the ripple would disappear too since it is proportional to it through the coefficicient νeff, which encodes the quantum effects from the quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Following the same approach as for scalar fields, in the next section we compute the quantum effects contributing to the VED from the quantized spin-1/2 fields and express them in renormalized form using the same substraction scheme devised in [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2 Renormalized ZPE for fermions Thus, following the same prescription (19) as in the case of the scalar field, the renormalized form of the fermionic EMT is defined in our case as follows: � T δψ µν � ren (M) ≡ � T δψ µν � (mψ) − � T δψ µν �(0−4) (M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (79) 24 In particular, this can be alternatively written as 6 � T δψ 00 � ren (M) = � T δψ 00 � Div (mψ) − � T δψ 00 � Div (M) + � T δψ 00 �(0−4) Non−Div (mψ) − � T δψ 00 �(0−4) Non−Div (M) + � T δψ 00 �(6) (mψ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' = 1 2π2a ˆ ∞ 0 dkk2 � −2ωk(mψ) a + 2ωk(M) a + a∆2 ωk(M) − a3∆4 4ω3 k(M) � + 1 2π2a ˆ ∞ 0 dkk2 � am2 ψ 4ω3 k(mψ) − aM2 4ω3 k(M) − a∆2 4ω3 k(M) � �a′ a �2 + � T δψ 00 �(0−4) Non−Div (mψ) − � T δψ 00 �(0−4) Non−Div (M) + � T δψ 00 �(6) (mψ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (80) where we have used the calculational results for the unrenormalized components of the vacuum EMT recorded in Appendix C and we have introduced the notation ωk(M) ≡ √ k2 + a2M2 and ωk(mψ) ≡ � k2 + a2m2 ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The last line of (80) contains all the non-divergent terms, which consti- tute a perfectly finite contribution and is made of finite parts from the 4th order expansion and of the entire 6th order term, which is fully finite but rather cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the other hand, the first two lines in the last equality are a collection of terms that are individually divergent, but whose combination makes the integral convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, by making use of simple algebraic manipula- tions at the level of the integrand one can show that explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For instance, the rearrangement in the integrand dkk2 � ω(mψ) − ω(M) − a2∆2 2ω(M) + a4∆4 8ω3(M) � = dkk2a6∆6 ω(mψ) + 3ω(M) 8ω3(M)(ω(mψ) + ω(M))3 (81) shows that terms seemingly diverging as ∼ k4 organize themselves to eventually converge as ∼ 1/k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Needless to say, this is the consequence of the subtraction that has been operated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Similarly with the second integral in (80), whose individual terms are logarithmically divergent, but overall the integral is once more convergent thanks to the involved subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The above renormalized result (80) would, of course, vanish for M = mψ if we were to stop the calculation at 4th order, so in case that one wishes to obtain the renormalized on-shell result one has to either compute the exact unrenormalized EMT on-shell before subtracting the divergent adiabatic orders – which is possible but only in simpler cases such as in de Sitter space [73,74] – or one has to face the calculation of the adiabatic expansion up to 6th-order at least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the last case the term ⟨T δψ 00 ⟩(6)(mψ) indicated at the end of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (80) must be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is what we have done here since an exact solution in the FLRW case is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The amount of calculation to reach up to 6th adiabatic order is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The un-renormalized components of the EMT up to the desired order are explicitly collected in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To subsequently obtain the renormalized EMT one has to implement the subtraction (79) and compute all the involved integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The final result up to the mentioned order can nevertheless be presented through a rather compact formula, as follows:7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 6The subscript ’Div’ refers to the part of the EMT calculation comprising divergent integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These appear only up to the 4th adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The subscript ’Non-Div’, on the other hand, refers, of course, to the part of the EMT calculation involving finite integrals only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 7We refer the reader to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2 of [15] for the computation/regularization of the involved integrals (depend- ing if they are convergent or divergent) from the master DR formula indicated there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Use of DR can be convenient since in certain cases the needed rearrangement of terms in the integrand to verify that the overall integral is actu- 25 � T δψ 00 �(0−6) ren (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) = a2 32π2 � 3m4 ψ − 4m2 ψM2 + M4 − 2m4 ψ ln m2 ψ M2 � + 1 16π2 � m2 ψ − M2 − m2 ψ ln m2 ψ M2 � H2 + 1 20160π2a4m2 ψ � 204H4H′ + 26 � H′�3 − 30H3H′′ + 9 � H′′�2 + 9H2 � 3 � H′�2 − 8H′′′� − 18H′H′′′ + H(−78H′H′′ + 18H′′′′) � = a2 32π2 � 3m4 ψ − 4m2 ψM2 + M4 − 2m4 ψ ln m2 ψ M2 � + a2H2 16π2 � m2 ψ − M2 − m2 ψ ln m2 ψ M2 � + a2 20160π2m2 ψ � − 31H6 − 108H4 ˙H − 46 ˙H3 + 126H3 ¨H + 9 ¨H2 − 18 ˙H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H + 27H2 � 7 ˙H2 + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H � + 6H(23 ˙H ¨H + 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='. H ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (82) The final equality corresponds to the expression in terms of the cosmic time (d()/dt ≡ ˙()) with H ≡ ˙a/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We point out that there is an explicit dependency on the Hubble function (and its derivatives) coming from Gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This justifies the notation ⟨T δψ 00 ⟩ren(M, H), with two arguments, where the dependence on the time derivatives of H is omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We note that in the fermionic case there are no terms of O(H4) in the evolution of the ZPE (and the VED, see next section), in stark contrast to the situation with scalars, see the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (20), where we can recognize terms of the form H2 ˙H, H ¨H and ˙H2 all of them of O(H4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We also remark what has been previously anticipated: for M = mψ (on-shell point) only the 6th-order terms remain, which are the ones in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These terms are relevant for the RVM mechanism of inflation in the very early universe (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, for the study of the renormalized theory at the point M (generally different from the on-shell mass point mψ) it is enough to consider the terms up to 4th adiabatic order, those in the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (82), see the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So far, we have been able to provide the desired formula for the renormalized ZPE at the energy scale M up to 6th adiabatic order, as expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is, however, not the end of the story, since a proper expression for the VED needs to take into account also the renormalized parameter ρΛ in the Einstein-Hilbert action (2), as this parameter is part of the unrenormalized vacuum action and after renormalization it also runs with the scale M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ρΛ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Both the renormalized ZPE and ρΛ(M) run with the scale and this will be crucial to study the properties of the renormalized VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The running of the ZPE part between two different scales M and M0 can be illustrated by considering the difference of the respective ZPE values at these scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From (82) we find � T δψ 00 � ren (M, H) − � T δψ 00 � ren (M0, H) = a2 32π2 � M4 − M4 0 − 4m2 ψ(M2 − M2 0 ) + 2m4 ψ ln M2 M2 0 � + a2H2 16π2 � −M2 + M2 0 + m2 ψ ln M2 M2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (83) ally convergent can be complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us emphasize, however, that DR is only used as an auxiliary regularization tool for intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The final result has no memory of this intermediate step, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Appendix B of [14] for an explicit nontrivial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To be sure, no MS prescription is used for renormalization at any point of our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The crucial difference between the ARP and the MS-like schemes is that the subtraction (79) involves not just the UV-divergences but also the finite parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 26 The finite parts, and in particular the 6th order terms cancel of course in the above difference, but the latter will be essential in the on-shell case since the result would be zero without these higher order effects 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We should notice that, in contradistinction to the case with scalar fields, there are no contributions of O(H4) such as H2 ˙H, H ¨H or ˙H2 in the expression for the ZPE, as can be seen on comparing equations (20) and (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For this reason it is unnecessary to use the higher derivative (HD) tensor (1)Hµν (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' AppendixA) as part of the renormalized Einstein’s equations in the case of the fermion fields, again in contrast to the situation with the scalar fields – see [15] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, for fermions the subtraction at the two scales of the renormalized form of Einstein’s equations can be done using the ordinary form of Einstein equations (3) without higher order curvature terms, and we find � T δψ µν � ren (M, H) − � T δψ µν � ren (M0, H) = (ρΛ(M) − ρΛ(M0)) gµν + � 1 8πG(M) − 1 8πG(M0) � Gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (84) By comparison equations (83) and (84), and taking into account the tensorial structure of (84) and the explicit form of Gµν in FLRW spacetime (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Appendix A) , we can perform the following identifications: ρΛ(M) − ρΛ(M0) = − 1 32π2 � M4 − M4 0 − 4m2 ψ(M2 − M2 0 ) + 2m4 ψ ln M2 M2 0 � , 1 8πG(M) − 1 8πG(M0) = 1 48π2 � −M2 + M2 0 + m2 ψ ln M2 M2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (85) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3 Renormalized VED Once the renormalized ZPE has been obtained, the same consideration as for the scalar field case (see (22) and (23) ) can be repeated intact here, thus leading to the expression for the renormalized VED of the fermionic field: ρδψ vac(M, H) = ⟨T vac 00 ⟩ (M, H) a2 = ρΛ(M) + � T δψ 00 � ren (M, H) a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (86) Now if the subtraction of scales is done, ρδψ vac(M, H) − ρδψ vac(M0, H) = ⟨T vac 00 ⟩ (M, H) − ⟨T vac 00 ⟩ (M0, H) a2 = ρΛ(M) − ρΛ(M0) + � T δψ 00 � ren (M, H) − � T δψ 00 � ren (M0, H) a2 = ρΛ(M) − ρΛ(M0) − (ρΛ(M) − ρΛ(M0)) + 3H2 � 1 8πG(M) − 1 8πG(M0) � = H2 16π2 � −M2 + M2 0 + m2 ψ ln M2 M2 0 � , (87) where in the last equality (85) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As expected, when written in terms of the ordinary Hubble function H in cosmic time, the evolution of the VED does not depend explicitly on the 8Let us remark that the difference (83) is an exact result, in the sense that it does not depend on the adiabaticity order we are working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is obvious from the renormalization prescription (79), as all higher orders beyond the 4th one (not only the 6th) cancel out in the subtraction, the reason being that these adiabatic orders are independent of the renormalization point M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The latter is involved in the calculation of the EMT up to 4th order only (as these are the only adiabatic orders that are UV-divergent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 27 scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For the sake of emphasizing the point, in the above equation we have explicitly indicated the cancellation of the terms carrying along the quartic powers of the masses, see the third equality in the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As we can see, it is essential that the structure of the VED is obtained from the sum “VED = ρΛ +ZPE”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (23), since the mentioned cancellation occurs between the renormalized expressions of ρΛ and ZPE upon being subtracted at the two arbitrary scales M and M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This means that the two values of the VED at these scales are related in a very smooth manner: in fact, they differ only by a term proportional to H2, as it is obvious from (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Even though Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (87) is formally correct, our job is not finished in the physical arena yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Despite of the fact that such an equation describes the precise mathematical evolution of the VED with the renormalization scale, M, it is necessary to associate the latter with a suitable physical scale in order to extract useful phenomenological information out of it, exactly as in the companion studies of the VED for scalar fields previously presented in [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As pointed out in these references and also in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2 regarding the contribution from the scalar fields, the Hubble rate H is a characteristic energy scale (in natural units) of the expanding universe in the FLRW metric, and hence proves to be a natural candidate for a representative physical scale in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Whereby by following the same prescription used in the aforementioned references, we set the renormalization energy scale to M = H(t) (at the end of our calculations) in order to track the physical evolution of the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In other words, this prescription should allow us to explore the VED at different expansion history times H(t) in a physically meaningful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this way we obtain a well behaved evolution of the VED, which means that, given its value at one scale all other values at nearby scales are very close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The dynamics of the VED is slow and can be encoded into an effective contribution to the νeff parameter, as we did for bosons in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The combined contribution from bosons and fermions to this parameter will be given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us finally clarify the sense of this scale setting, Namely, the full effective action does not depend on M, of course, but the renormalized VED indeed does since the effective action of vacuum is only a part of the full effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The scale dependence on M from the other terms of the action, for example the terms carrying the running couplings of the RG-improved classical action, compensates for the M-dependence of the vacuum action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Put another way, only the full effective action (involving the classical part plus the nontrivial quantum vacuum effects) is scale- (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' RG-) independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is of course the standard lore of the renormalization group (RG), see also [31] for an expanded discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The choice of a particular scale helps of course in enhancing the physical significance of particular sectors of the full effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The procedure is of course akin to the usage of the RG in conventional gauge theories of strong and electroweak interactions, except that here one has to pick out an appropriate cosmological energy scale which is most suitable for the description of the universe’s expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The distinguished scale H appears to be the natural choice if the universe where we live is indeed appropriately described by the FLRW metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the next section we apply this approach to derive the important RG equation of the VED itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='4 Renormalization group equation for the vacuum energy density One can also compute the β function of the running vacuum associated to fermionic quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Only the adiabatic terms below 4th order carry M-dependence by definition since the higher orders are finite and hence are not subtracted in the renormalization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As it was already noted before, in contrast to the scalar case the terms of 4th adiabatic order are not present for fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The computation follows the same strategy as for scalars [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this case 28 we make use of equations (83) and (86), and we find βδψ ρvac =M ∂ρδψ vac(M) ∂M = βδψ ρΛ + 1 8π2 � M2 − m2 ψ �2 − 1 8π2 H2 � M2 − m2 ψ � = − 1 8π2 H2 � M2 − m2 ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (88) The second equality holds immediately after computing the β-function of the parameter ρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From the first equation (85) we find that βδψ ρΛ = M ∂ρΛ(M) ∂M = − 1 8π2 � M2 − m2 ψ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (89) and hence contains a term proportional to the quartic power of the particle mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' what’s more, there is an exact cancellation between the terms of the ZPE containing quartic powers of M and mψ and the expression of βρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The result (88) can also be consistently obtained directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notice that neither the parameter ρΛ nor the ZPE have physical meaning in an isolated way, only the sum makes physical sense and defines the VED in the present context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us compare the above results with those following from the contribution of one real scalar field φ [15]: βδφ ρvac = � ξ − 1 6 � 3H2 8π2 � M2 − m2 φ � + O(H4) (90) and βδφ ρΛ(M) = 1 2(4π)2 (M2 − m2 φ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (91) where we omit the O(H4) terms in the scalar case (not present in the fermionic case) since it is enough to check the comparison at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We can see that in both cases the β-function of the VED is proportional to βρvac ∝ H2 � M2 − m2� , where m = mφ or mψ, and therefore has a very smooth behavior thanks to the factor H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In contrast, the β-function for the parameter ρΛ in the gravitational action (which is often incorrectly identified as the VED in some explicit QFT calculations of the vacuum energy in the literature) behaves in both cases as βρΛ ∝ � M2 − m2�2 and hence leads to undesired quartic contributions ∼ m4 to the running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These are the problematic terms leading to fine tuning problems, but as can be seen these terms exactly cancel in βρvac for the vacuum energy both for fermions and bosons in our renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notice that there is a factor of 4 between equations (89) and (91) and have opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In a SUSY context, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (91) should by multiplied by 4 to equalize bosonic and fermionic d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='f in a given matter supermultiplet, all of whose members possess the same mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Then βδφ ρΛ → 4βδφ ρΛ ≡ βδφ (SUSY) ρΛ , and the sum of the two coefficients will indeed vanish in a supersymmetric context: βδψ (SUSY) ρΛ + βδφ (SUSY) ρΛ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (92) But this is, of course, not a cancellation of the β-function coefficients for the VED of bosons and fermions in the SUSY limit, but only the cancellation of the contributions to the β-function coefficient for the formal parameter ρΛ in the EH action (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This property is obviously connected with the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 about the balance of UV-divergences between fermions and bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In a SUSY theory the quartic divergences cancel prior to any renormalization process, as we have noticed, and the resulting β-function for the parameter ρΛ is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' By the same token the running of the VED is freed from ∼ m4 effects, which cancel among fermions and bosons in a SUSY context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The quartic powers are independent of the curvature of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, the subleading divergences do depend on the background curvature and do not cancel at all, even in 29 the exact SUSY limit 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The “residual” (finite) parts left in the renormalization process do not cancel either;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' they are actually proportional to the curvature of the FLRW background, R ∼ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This fact translates into a correction to the physical vacuum energy density of order ∼ m2H2 both for bosons and fermions, which is far smaller than m4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So the finite, curvature dependent, terms that remain after ARP renormalization are de facto the most important ones for our purposes since they lead to the RVM form of the VED!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The renormalization of the formal parameter ρΛ, in contrast, has no physical imprint in the final result for the VED, except that the unwanted m4 terms cancel against those involved in the ZPE, thus rendering the renormalized V ED = ρΛ +ZPE free from quartic mass dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From the above RG equations we may write down the total contribution to the β-function of the VED from the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Consider Nf species of fermion fields with masses mψ,ℓ for each species ℓ ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , Nf}, and similarly let Ns be the number of scalar field species with masses mφ,j, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , Ns}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Some of these species may have the same mass, but this aspect is not relevant here, our formulas will include a summation over all contributions irrespective if some of them may be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The total β-function of the VED from an arbitrary number of quantized matter fields can now be cast as follows: βquant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ρvac ≡ Ns � j=1 βδφj ρvac+ Nf � ℓ=1 βδψℓ ρvac = 3H2 8π2 \uf8ee \uf8f0 Ns � j=1 � ξj − 1 6 � (M2 − m2 φj) − 1 3 Nf � ℓ=1 (M2 − m2 ψℓ) \uf8f9 \uf8fb+O(H4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (93) The net outcome, therefore, is that the β-function of the vacuum energy density is free from undesirable contributions proportional to quartic mass powers of the quantized fields, ∼ m4, and hence these contributions do not appear in the renormalized theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is of course an extremely welcome feature of our renormalization framework, which is, on inspection of the above equation, fully shared by both scalar and fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Indeed, up to numerical factors fermions and scalar fields provide the same kind of leading contribution to the time evolution of the cosmological vacuum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Overall we find that the running of ρvac depends only on quadratic terms in the fermion mass, namely ∼ m2 ψℓH2, which are of the same type as in the case of bosons, namely ∼ m2 φjH2, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2 and previously demonstrated in great detail in [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These terms are actually very smooth owing to the presence of the H2 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Integrating the RG equation associated to the β-function (93) one finds the expression for the evolution of the VED as a function of the renormalization scale M in the presence of any number of matter fields, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In particular, integrating (88) for the case of one single fermion it is easy to verify that it leads exactly to (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The kind of much tempered behavior of the VED evolution that we have found here within our ARP renormalization program is of the sort that was expected on the basis of semi-qualitative RG arguments and constitutes the characteristic running law of the so-called Running Vacuum Models (RVM), see [30, 31] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Thus, there is no need for fine-tuning in this scenario, since in such a renormalization procedure we have already gotten rid of the ugly contributions associated to the quartic powers of the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In other words, the ‘problem’ with the quartic powers of the masses does not appear in the physically renormalized theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While the running of the formal parameter ρΛ with M indeed carries ∼ m4 contributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' as it is obvious from the formulas above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' this fact has no physical implication since ρΛ is not itself a physical 9Needless to say,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the SUSY considerations made here in passing only intend to clarify that in curved spacetime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' irrespective of whether the quantized matter fields belong to a supersymmetric theory or not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the renormalization program is in any case mandatory to finally get rid of all the UV divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The calculations in this work, however, do not presume any SUSY context at all, not even a SUSY-broken theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Our treatment of scalar and fermion fields is indeed completely general, in the sense that we are dealing with an arbitrary number of matter fields of both species without enforcing any balance between bosonic and fermionic d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' – see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 30 parameter (if taken in isolation) and the unwanted terms carried by it exactly cancel out in the β-function for the VED, as we have just proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As a result, the running of the VED is much softer, the ‘slope’ is ∼ m2H2 rather than ∼ m4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' At variance with this result, in the context of the MS renormalization approach, in which ρΛ runs with the unphysical mass unit µ coming from dimensional regularization, one is enforced to fine tune ρΛ(µ) against the large contribution proportional to ∼ m4 terms [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='5 Renormalization of the fermionic vacuum pressure Taking into account the perfect fluid form of the EMT associated to the vacuum, the corresponding pressure is defined through the iith-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Any of them can be utilized owing to the assumed homogeneity and isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So, it is just enough to compute the VEV of the 11th-component10: Pvac(M) = ⟨T vac 11 ⟩ a2 = −ρΛ(M) + � T δψ 11 �ren (M) a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (94) From (73) and using once more the expansion of the spin-1/2 fermion fields in Fourier modes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' AppendixB and AppendixC) the result can be expressed, after considerable work, as follows: � T δψ 11 � = 1 2π2a ˆ ∞ 0 dkk2Pk, (95) with Pk ≡ −2k 3a � hI khII∗ k + hI∗ k hII k � (96) and where the explicit expressions (in WKB-expanded form) for the fermion modes hI k and hII k can be found in the aforementioned appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notice that there is a relation between ρk and Pk Pk = − ρ′ k 3H , (97) which follows from (75) using the mode equations (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This relation can be used as an alternative way to calculate ⟨T δψ 11 ⟩ from ⟨T δψ 00 ⟩: � T δψ 11 � = − 1 3H �� T δψ 00 �′ + H � T δψ 00 �� (98) For the sake of simplicity, the remaining discussions of this section will be restric to the case of one single neutral scalar field and one single Dirac fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We shall retake the multifield case in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Following the same steps and considerations made in the previous sections for the 00th component of the EMT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' we reach the following expression for the renormalized value of the VEV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='of the 11th-component of the EMT up to 6th adiabatic order: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='T δψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�(0−6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ren ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='(M) = − a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='32π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M4 + 3m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ − 4m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψM2 − 2m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='48π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='24π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='20160π2a4m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−245H2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H′�2 + 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H′�3 − 98H3H′′ + 35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H′′�2 − 62H2H′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+204H4H′ − 66HH′H′′ + 56H′H′′′ + 42HH′′′′ − 6H′′′′′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='10One can either compute the VEV of the T11 component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' as we do here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' or use the formula (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' which allows to compute the vacuum pressure from the 00th component and the trace of the EMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The result is the same, of course, owing to the isotropy of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [15], for instance, we presented the computation of the pressure for the scalar fields using this second method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 31 = − a2 32π2 � M4 + 3m4 ψ − 4m2 ψM2 − 2m4 ψ ln m2 ψ M2 � + a2 16π2 � M2 − m2 ψ + m2 ψ ln m2 ψ M2 � H2 + a2 24π2 � M2 − m2 ψ + m2 ψ ln m2 ψ M2 � ˙H + a2 20160π2m2 ψ � 31H6 + 170H4 ˙H − 45H2 ˙H2 − 80 ˙H3 − 90H3 ¨H − 55 ¨H2 −150H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H − 100 ˙H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H − 6H(65 ˙H ¨H + 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='. H ) − 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (99) We may now proceed to compute the vacuum EoS for the fermion fields up to the sixth adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The best strategy is to compute first the pressure through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (99), which can be inserted into the relation (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Using next the VED expression (86) for fermions – with ⟨T δψ 00 ⟩ given by (82) – the vacuum pressure can be seen to be equal to minus the VED plus some additional terms: Pvac(M) = −ρvac(M) + 1 24π2 � M2 − m2 ψ + m2 ψ ln m2 ψ M2 � ˙H + 1 20160π2m2 ψ � 62H4 ˙H + 144H2 ˙H2 − 126 ˙H3 + 36H3 ¨H − 46 ¨H2 − 42H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H − 118 ˙H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H − 6H(42 ˙H ¨H + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='. H ) − 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (100) The additional terms represent a small (but worth noticing) deviation from the classical vacuum EoS relation Pvac = −ρvac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The dominant vacuum EoS is still the classical one up to a leading correction of O( ˙H) (the second term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='s of the above equation) and several sorts of higher order corrections of O(H6) indicated in the last two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The ∼ ˙H correction in the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (100) can obviously be relevant for the present universe, and in particular it can modify the equation of state of the vacuum it to depart from −1 at present (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The higher order terms in the last two lines, in contrast, might be relevant only for the very early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, these terms involve time derivatives and hence vanish for H =const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This fact will have implications for our discussion of RVM-inflation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3, since inflation can be shown to exist in this framework for H =const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=', cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' So at the end of the day, the higher order terms in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (100) become irrelevant both at low and high energies in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The remarkable consequence is that the EoS of the quantum vacuum is very close to −1 during inflation, in contrast to the vacuum EoS in subsequent eras of the cosmic evolution (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 Trace Anomaly It is a very well known result that if a field theories has a classical action which is conformally invariant, then the trace of the classical EMT vanishes exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For this it is necessary to work with a massless field, otherwise the presence of a mass breaks the symmetry since it introduces a fixed length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For instance, for a massless scalar field, lim ξ→1/6 lim mφ→0 TCl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (101) This follows immediately from (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, it is also true that this simple result does not hold when one considers the pure quantum contribution associated to the quantum fluctuations of the scalar field and constitutes a conformal anomaly, [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This follows after a careful study of the diverging part of the vacuum effective action, W Div eff , in which Weff was defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 32 The part W Div eff is not conformally invariant for an arbitrary number of spacetime dimensions n (although Weff is so in the massless limit), except for the case n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As a consequence, W Div eff receives a finite payoff for n → 4 owing to the existing pole 1/(n − 4) in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Correspondingly, the VEV of the on-shell EMT receives a nontrivial contribution in the massless limit coming from the divergent part of the effective action, even in the case ξ = 1/6: lim mφ→0 lim ξ→1/6 � T δφ� = − lim mφ→0 m2 φ � δφ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (102) The term � δφ2� contains some elements of 4th adiabatic order proportional to 1/m2 φ, so that the corresponding limit results in a finite contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The same idea applies in the fermionic case, lim mψ→0 � T δψ� = − lim mψ→0 mψ � ¯ψψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (103) Here the term � ¯ψψ � contains 4th adiabatic order terms that are proportional to 1/mψ which make the limit non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Technically speaking (102) and (103) are not yet what we call the trace anomaly or conformal anomaly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is due to the fact that the total effective action is conformally invariant and the associated EMT is traceless, so the part of the trace associated to the finite and divergent parts should be equal but with opposite sign in the conformal limit [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The anomaly is associated to the finite part, so its actual value for the scalar field case is � T δφ�Anomaly = − lim mφ→0 � T δφ� = 1 480π2a4 � 4H2H′ − H′′′� = 1 2880π2 � RµνRµν − 1 3R2 + □R � , (104) where the conversion of the anomaly result into an invariant expression in the last step can be performed using the formulae of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This result was explicitly verified in the calculation of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We remark that for an arbitrary curved background the expression for the conformal anomaly is more involved [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, since the spatially flat FLRW spacetime is conformally flat (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' conformal to Minkowski space) the contribution from the Weyl tensor vanishes identically and hence also its square (entering the anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Additional terms beyond 4th adiabatic order decouple when mφ → ∞, satisfying the Appelquist-Carazzone decoupling theorem [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These terms are not finite in the massless limit, and hence do not take part of the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In practice we have derived the anomaly (104) from the unrenormalized trace of the vacuum EMT for scalar fields, � T δφ� , which is given in full detail in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The corresponding conformal anomaly for fermions can be similarly extracted from the unrenormalized � T δψ� and it is a bit cumbersome as well, so we shall spare details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We limit ourselves to provide the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Once more we can recognize the expression of the anomaly as a linear combination of finite terms of adiabatic order 4 which are independent of the mass scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We find � T δψ�Anomaly = − lim mψ→0 � T δψ� = 1 240π2a4 � 7H′H2 − 3H′′′� = 1 2880π2 � 11RµνRµν − 11 3 R2 + 6□R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (105) One natural question is related with the physical consequences of the conformal anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is well-known that it is a valuable theoretical concept encoding essential information on the VEV of the renormalized EMT [1], although it need not be itself part of the observable quantities of the renormalized theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' There are some attempts in the literature to remove the anomaly by particular prescriptions or definitions of the renormalized EMT [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is also the case of the renormalization procedure employed in this work, as defined in (19) and (79), where the anomaly has no physical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The reason is that the on-mass-shell value of the vacuum EMT is subtracted 33 at an arbitrary scale, M, up to 4th adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Since the anomaly is of 4th adiabatic order and it is independent from the mass of the fields and, of course, also from the arbitrary renormalization point, it gets cancelled exactly in our ARP renormalization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Alternatively, one can think in terms of the effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Indeed, in [15], we defined the renormalized effective lagrangian density off-shell at an arbitrary scale M, LRen W (M) ≡ LW(m) − LDiv W (M) (106) and it was shown by expanding it through the DeWitt-Schwinger series that it eventually leads exactly to the same renormalized EMT defined by (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This result was obtained explicitly for a scalar field φ and can be repeated for fermions, although we shall not provide details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Now the anomaly is related with the divergent part of the effective Lagrangian, corresponding to the lowest adiabatic orders (up to 4h order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As a consequence it gets once more exactly cancelled in (106) analogously to the subtraction of the EMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As previously indicated, the anomaly part of the trace is contained in the un-renormalized trace of the EMT (even though the anomaly itself is a finite part of it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In our framework, however, the anomaly cancels since the anomaly is independent of the mass scale and our renormalized EMT is defined through a subtraction of the vacuum EMT evaluated at two different scales, see equations (19) and (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Thus the conformal anomaly is not involved in the renormalized expressions for the vacuum energy density and pressure in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Despite it not having physical consequences in our approach, the explicit calculation of the anomaly is certainly useful as a nontrivial cross- check of our intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5 Combined fermionic and bosonic contributions Let us now determine the combined vacuum contributions from a multiplicity of non-interacting fermionic and bosonic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As defined before (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='4 ), we consider Nf species of fermion fields with masses mψ,ℓ (ℓ ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , Nf}), and Ns scalar field species with masses mφ,j (j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , Ns}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 Running vacuum from an arbitrary number of quantized matter fields The renormalized expression of the vacuum energy density is, in that case, ρvac(M, H) = ρΛ(M) + �Ns j=1 � T δφj 00 � (M, H) + �Nf ℓ=1 � T δψℓ 00 � (M, H) a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (107) As in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2, considering two different energy scales M and M0 is a necessary step to obtain a meaningful result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Einstein’s equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' after subtraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' can be written in the following manner: Ns � j=1 �� T δφj 00 � (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) − � T δφj 00 � (M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) � + Nf � ℓ=1 �� T δψℓ 00 � (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) − � T δψℓ 00 � (M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='128π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−M4 + M4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 + 4m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 − M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− 2m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ln M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a2H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 − M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ln M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�2 a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='˙H2 − 2 ¨HH − 6H2 ˙H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ln M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='32π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M4 − M4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 − 4m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 − M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ 2m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ln M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ a2H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−M2 + M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ln M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='= (ρΛ(M) − ρΛ(M0)) g00 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='8πG(M) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='8πG(M0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='G00 + (a1(M) − a1(M0)) (1)H00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (108) Notice the appearance of the 00th component of (1)Hµν, which is a HD tensor of O(H4), hence of adiabatic order 4 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Its presence in the generalized Einstein’s GR equations is indispensable for renormalization and constitutes a UV completion of the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' No additional HD tensors are needed for conformally flat spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In our case, (1)Hµν is necessary for the renormalization of the short-distance effects produced by the quantum fluctuations of the scalar fields, as these involve O(H4) corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, as previously indicated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2, the renormalized EMT for fermions does not contain O(H4) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' By using the expression of (1)H00 in AppendixA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' we can recognize the tensorial structure of the various terms and find the running of couplings: ρΛ(M) − ρΛ(M0) = 1 128π2 (−4Nf + Ns) � M4 − M4 0 � + 1 32π2 \uf8eb \uf8ed4 Nf � ℓ=1 m2 ψℓ − Ns � j=1 m2 φj \uf8f6 \uf8f8 � M2 − M2 0 � + 1 64π2 \uf8eb \uf8ed−4 Nf � ℓ=1 m4 ψℓ + Ns � j=1 m4 φj \uf8f6 \uf8f8 ln M2 M2 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (109) 1 8πG(M) − 1 8πG(M0) = 1 48π2 \uf8eb \uf8ed−Nf + 3 Ns � j=1 � ξj − 1 6 �\uf8f6 \uf8f8 � M2 − M2 0 � + 1 48π2 \uf8eb \uf8ed Nf � ℓ=1 m2 ψℓ − 3 Ns � j=1 � ξj − 1 6 � m2 φj \uf8f6 \uf8f8 ln M2 M2 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (110) a1(M) − a1(M0) = − 1 32π2 Ns � j=1 � ξj − 1 6 �2 ln M2 M2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (111) From the above formulas we can now use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (107) to find out the difference between the values of the VED at two different scales: ρvac(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) − ρvac(M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H0) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−M2 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2H ¨H + 6H2 ˙H − ˙H2� Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2H0 ¨H0 + 6H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ˙H0 − ˙H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='T δφj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ren (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) + �Nf ℓ=1 � T ψℓ 00 �(6) ren (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) a2 − �Ns j=1 � T δφj 00 �(6) ren (M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H0) + �Nf ℓ=1 � T ψℓ 00 �(6) ren (M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H0) a2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (112) In the last line, the dots collectively represent all the terms of adiabatic 8 or beyond, which are not considered in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notice that in the previous expression we have used the important relation (109), which is essential to cancel the quartic mass contributions from the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Following the same prescription that we used before to derive equations (25) and (26) for a single scalar field, we implement now the scale settings M = H and M = H0 in order to compare the evolution of the VED between these two points, in the present case involving the full contributions from all the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For simplicity, let us call ρvac(H) ≡ ρvac(H, H) and ρvac(H0) ≡ ρvac(H0, H0) when using the above expression (112) The expansion history times H and H0 can be arbitrary, of course, but for obvious reasons we choose H0 = H(t0) to be the value of the Hubble function at the present time, t0, and H = H(t) a value at a point in our past (t < t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the running of the VED between these two points can be expressed as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ρvac(H) − ρvac(H0) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−H2 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ψℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2H ¨H + 6H2 ˙H − ˙H2� Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2H0 ¨H0 + 6H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ˙H0 − ˙H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ξj − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='φj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�Ns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='T δφj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ren (H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) + �Nf ℓ=1 � T ψℓ 00 �(6) ren (H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H) a2 − �Ns j=1 � T δφj 00 �(6) ren (H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H0) + �Nf ℓ=1 � T ψℓ 00 �(6) ren (H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H0) a2 (113) Obviously if the point H is in the nearby past we can neglect all the O(H4) terms generated in the above expression since they are much smaller than the O(H2) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We will do this in 36 the next section, where we study in more detail the low energy regime, in particular the late time universe where we live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us however clarify that the O(H2) terms are dominant not only for the late time universe around our time, but in actual fact for the entire post-inflationary regime, which is when the higher order powers of H become activated and are actually dominant (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3 for the study of the inflationary epoch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, we can extract the running of the gravitational constant from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (110), with the following result: G(M) = G(M0) 1 + G(M0) 2π � Ns � j=1 � ξj − 1 6 � − Nf 3 � (M2 − M2 0 ) + G(M0) 2π � Nf � ℓ=1 m2 ψℓ 3 − Ns � j=1 � ξj − 1 6 � m2 φj � ln M2 M2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (114) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2 The low energy regime: evolution of ρvac and G in the present universe Of paramount importance is the evolution of the VED and of the gravitational coupling G in the low energy regime, especially around our time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, following our prescription, we evaluate (113) for the late universe, when the dominant powers of H are the H2 ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Such an expression then boils down to ρvac(H) = ρvac(H0) + 3νeff(H) 8π m2 Pl � H2 − H2 0 � + O(H4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (115) The function νeff(H) is defined as νeff(H) = 1 2π � Ns � j=1 � ξj − 1 6 � m2 φj m2 Pl � ln m2 φj H2 0 − 1 � − 1 3 Nf � ℓ=1 m2 ψℓ m2 Pl � ln m2 ψℓ H2 0 − 1 � + H2 H2 − H2 0 ln H2 H2 0 \uf8eb \uf8ed1 3 Nf � ℓ=1 m2 ψℓ m2 Pl − Ns � j=1 � ξj − 1 6 � m2 φj m2 Pl \uf8f6 \uf8f8 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (116) It is indeed a function evolving with the Hubble rate, but is almost constant since the dependence on H is very mild, as we shall make manifest in a moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us emphasize that the O(H4) terms correcting the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (115) are completely irrelevant for the current universe, and in point of fact they can be safely ignored for the FLRW regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' during the entire period following the inflationary stage (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, equation (115) should actually be relevant for the full cosmological evolution that is accessible (directly or indirectly) to our physical measurements and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is convenient to define the parameter ǫ ≡ 1 2π \uf8eb \uf8ed Ns � j=1 � ξj − 1 6 � m2 φj m2 Pl − 1 3 Nf � ℓ=1 m2 ψℓ m2 Pl \uf8f6 \uf8f8 (117) This parameter is connected to the β-function (93) at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Indeed, when we consider M = H in the low energy regime, it is obvious that H2 ≪ m2 for any particle mass, and hence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (93) reduces to βquant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ρvac = − 3 4π ǫ m2 Pl H2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (118) Therefore ǫ plays the role of coefficient of the low-energy β-function of the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, the eventual coefficient that effectively controls the final evolution of the VED is actually enhanced 37 with respect to ǫ by a big logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To see this, let us take the current limit (H → H0) of the function (116): ν0 eff ≡ lim H→H0 νeff(H) = 1 2π \uf8ee \uf8f0 Ns � j=1 � ξj − 1 6 � m2 φj m2 Pl � ln m2 φj H2 0 − 2 � − 1 3 Nf � ℓ=1 m2 ψℓ m2 Pl � ln m2 ψℓ H2 0 − 2 �\uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (119) A simple rearrangement now shows that we can rephrase (116) in terms of ǫ and ν0 eff as follows: νeff(H) = ν0 eff + � 1 − H2 H2 − H2 0 ln H2 H2 0 � ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (120) This formulas is exact, but in practice some simplifications are perfectly possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For example, take the big logarithms ln m2 i /H2 0 (with mi any particle mass, boson or fermion) involved in νeff but not in ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For any known massive particle, we have ln m2 i /H2 0 ≫ 1, this being true even for the lightest neutrinos (recall that H0 ∼ 10−42 GeV= 10−30 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Typically ln m2 i /H2 0 = O(100) in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' But as a matter of fact the only relevant contributions to νeff(H) come from the heavy massive particles that belong to a GUT at a characteristic scale MX ∼ 1016GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For these particles (whether bosons or fermions, with masses mi ∼ MX) we have m2 i /m2 Pl ∼ M2 X/m2 Pl and this number is not so small since it may thrust the value of νeff up to νeff ∼ 10−3, if one takes into account the large multiplicities of heavy fields existing in a typical GUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This was first estimated long ago in [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Thus, it is natural to expect ��ν0 eff �� ≫ |ǫ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It follows that we can safely neglect the term proportional to ǫ in (120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This means that in practice we can neglect the very mild time-dependence of νeff(H) and replace it with the constant coefficient ν0 eff in which H is evaluated at the current time, H = H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' By the same token we can also ignore the numerical term of 2 accompanying the big logarithms in (119).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Whence, to a very good approximation the evolution of the vacuum energy density can be described through the formula ρvac(H) = ρvac(H0) + 3νeff 8π m2 Pl(H2 − H2 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (121) with an effective νeff ≃ ν0 eff given by νeff = 1 2π \uf8ee \uf8f0 Ns � j=1 � ξj − 1 6 � m2 φj m2 Pl ln m2 φj H2 0 − 1 3 Nf � ℓ=1 m2 ψℓ m2 Pl ln m2 ψℓ H2 0 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (122) That is to say, it all amounts to replace νeff(H) → νeff in (115) and it still retains a great degree of accuracy in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As it is manifest from the previous two equations, coefficient νeff plays the role of β-function for the running vacuum directly as a function of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' If we compare (122) with (117) we can see that νeff and ǫ are roughly ‘proportional’ through a big log as follows: νeff ∼ ǫ ln m2 i H2 0 ∼ O(100) ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (123) Despite there being a summation over different masses, and hence such a proportionality not being strict, the above relation is nevertheless true in order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The presence of the big log factor in νeff makes the running of the VED faster than the tiny value of ǫ might convey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the other hand, as we shall see below, it is ǫ alone that controls the (much smaller) running of the gravitational coupling G, which does not receive any enhancement from log factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 38 Equations (121) and (122) suffice to study the behavior of the VED near our time11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These simplified formulas have been previously used de facto to fit the value of νeff from the latest cosmological data, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [50] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Here, however, we provide for the first time the full theoretical structure behind this parameter in the QFT context from the quantum effects induced by an arbitrary number of quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Remarkably, the typical fitting value obtained in the mentioned reference is νeff ∼ 10−3 and positive, which is well within the aforementioned expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This phenomenological determination picks up of course the net outcome from the various quantum matter fields involved in (122), which at this point cannot be discriminated in an individual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, insofar as the running gravitational constant is concerned, it can be written using the same renormalization scale as follows: G(H) = GN 1 + 1 2π � Ns � j=1 � ξj − 1 6 � − Nf 3 � H2−H2 0 m2 Pl − ǫ ln H2 H2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (124) The former expression can be derived straightforwardly from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (114) by setting M = H and M0 = H0, with H0 being the current value of the Hubble function, and we have defined GN ≡ G(M0) (the current value of the gravitational coupling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We follow exactly the same recipe as for the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the low energy regime, where H2 ≪ m2 Pl, we can approximate with high accuracy the former expression by just G(H) = GN 1 − ǫ ln H2 H2 0 , (125) where ǫ contains contributions from all the matter fields, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (117).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Recall from (123) that |ǫ| ≪ |νeff|, and also that the running of G(H) is logarithmic, in contrast to the running of ρvac(H) which is quadratic in H2 at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, the running of G is much smaller and also much slower than the running of the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It may, however, be interesting to note that when H approaches mPl the term ∼ H2/m2 Pl in the denominator of the more accurate formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (124) could be dominant over the logarithmic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' If the multiplicity of matter fields is large enough, such a term could make the gravitational coupling to evolve asymptotically free at very large energies when we approach the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the absence of that term, G increases at high energies for ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As an additional cross-check, we can see that the low energy regime of the vacuum energy density and the running gravitational constant are compatible through the Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This can be translated into a local energy exchange between the vacuum fluid and the background gravitational field due to the quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' A deeper insight on the local (covariant) energy conservation and the Bianchi identity can be found in the previous works [15,16], where the reader may find a detailed derivation of the logarithmic evolution law, that is to say, equation (125), in the simpler scenario of one real scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, one finds that the β-function (90) for the VED running is crucially involved also in the local conservation law of the VED, which can be writen in two alternative ways [15]: ˙ρvac + 3H (ρvac + Pvac) = ˙M M βρvac = − ˙G G 3H2 8πG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (126) The first equality expresses the fact that the non-conservation of the VED is due to both the running of ρvac with M (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the fact that βρvac ̸= 0) and also to the cosmic time dependence of M 11It is easy to check that for one single neutral scalar field and no fermion field the above expressions reduce to the formulas (25) and (26), as should be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 39 (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ˙M ̸= 0), whereas the second equality is a direct reflex of the Bianchi identity in Einstein’s equations with variable ρvac and G, and hence provides a link between the time variation of the VED and that of the gravitational coupling G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The former equation does not depend on the number or nature of the fields involved, and holds as long as the matter components are covariantly conserved on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The non-conservation of ρvac, however, preserves the Bianchi identity thanks to the corresponding running of the gravitational coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This does not preclude, however, that one can still formulate scenarios where matter can exchange energy with ρvac, but we do not address this situation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Taking the leading terms of βρvac from the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' of (93) for the present universe, and setting M = H, we readily obtain the following differential equation 1 G2 dG dt = 2ǫ GN ˙H H , (127) where ǫ is the full expression (117) involving the contributions from all the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='dd Its solution is precisely Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (125), as can be readily checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is also interesting to note from the above formulas that this framework predicts a (mild) cosmic time variation of the “fundamental constants”, such as the gravitational coupling G and ρvac, with H(t) and hence an evolution of these ‘constants’ the cosmological expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The possibility for such a variation has long been discussed in the literature [100] and is still a matter of debate and intensive test, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [101] and the ample bibliography provided in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Specific theoretical models accounting for such a possible variation are manifold, and in some cases they imply a time-dependence of the running couplings and masses in the particle and nuclear physics world, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [102,103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While most of the proposals are based on strict particle physics scenarios, in particular on GUT’s, testing the evolution of the VED in curved spacetime is a novel feature suggested in our framework, which was actually put forward on more phenomenological grounds sometime ago in [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The QFT calculations presented in the current work provide indeed a solid theoretical support to these same ideas but from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Recall that when one “fundamental constant” varies, then all of them vary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The formulas discussed above concern important epochs of the cosmological expansion such as the radiation-dominated epoch, matter-dominated epoch and the current epoch in which the vacuum energy resurfaces and became finally dominant over matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' During the entire FLRW regime the dominant power of the Hubble rate in the VED is H2 or ˙H (which are of the same adiabatic order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The terms with powers of H (or of equal adiabatic order) higher than H2 (indicated by O(H4) in (115)) acquire real relevance much early on in the expansion history since only during the early universe we encounter a truly high energy scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As previously anticipated, an interesting feature regarding these higher powers is that they can provide us with a possible mechanism for inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Namely, if the early cosmic era possesses a short period where H remains approximately constant and very large (typically near a GUT scale), the universe may go through a phase of exponential expansion in which the VED starts from a huge value which then quickly decays into radiation and triggers the ordinary FLRW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This situation is possible also in the RVM framework, and is called ‘RVM-inflation’ [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We reassess it in the next section in the extended context of the present considerations, where we now have both scalar and fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3 Inflation from running vacuum It was noted in [15] that the quantum effects computed from the adiabatic expansion lead to higher powers of the Hubble rate and its derivatives which are irrelevant for the current universe but capable to bring about inflation in the very early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' They are characterized by a short period where H=const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=', provided this constant value takes, of course, a large value which we expect to lie around a characteristic GUT scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The regime H=const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' is totally unrelated to the 40 ground state of a scalar field potential and therefore this new mechanism does not require any ad hoc inflaton field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is called ‘RVM-inflation’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Here we consider the contribution from the fermions fields and provide a formula for the dominant term of the energy density receiving contributions from an arbitrary number of non-minimally coupled scalar fields and also an arbitrary number of fermions fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The payoff from the latter stems from setting H =const in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (82), where we can see that all the time derivatives of the Hubble rate vanish except for a single term which is proportional to H6/m2 ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The contribution from a non-minimal scalar was computed in [15] and here we just combine it with that of fermions assuming an arbitrary number of families of both species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Overall, we find that the total VED involving the contributions from bosons and fermions at very high energies (hence relevant for triggering RVM-inflation in the very early universe) reads as follows: ρinf vac =CinfH6 + F( ˙H, ¨H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='), (128) where Cinf ≡ 1 80π2 \uf8f1 \uf8f2 \uf8f3 Ns � j 1 m2 φj �� ξj − 1 6 � − 2 63 − 360 � ξj − 1 6 �3� − 31 252 Nf � ℓ 1 m2 ψℓ \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (129) The terms collected in the function F( ˙H, ¨H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=') depend on different combinations of powers of H involving derivatives of H in all cases, and hence they all vanish for H =const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' That is to say, F = 0 for H =const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Overall we see that the dominant contribution is of the form ρinf vac ∝ H6 with a complicated coefficient Cinf which depends on the number of scalar and fermions fields, their masses and multiplicities and also on the non-minimal couplings of the different scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the case of fermions this contribution is seen to be negative-definite, whereas in the case of the scalars it can be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Let us note that during the inflationary period the EoS of the quantum vacuum is essentially −1, with very tiny deviations caused by terms which depend on the various time derivatives of the Hubble rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To the extent that the condition H=const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' is fulfilled these deviations are extremely small, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the next section we shall see that, in contrast, the EoS of the quantum vacuum in the present time can deviate from −1 by an small amount which is not as negligible as in the very early universe and therefore could be detected and even mimic quintessence behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The solution of the cosmological equations proceed along the same lines as in [15], except that now the fermionic contribution is also included but it only modifies the specific coefficient of H6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, one finds again that a short period of inflation is produced with H ≈ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' and the vacuum decays quickly into radiation [15]: H(a) = HI (1 + ˆa8), (130) ρr(ˆa) = ρI ˆa8 � 1 + ˆa8�− 3 2 , ρvac(ˆa) = ρI � 1 + ˆa8�− 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (131) in which ˆa ≡ a/a∗, with a∗ is the transition point from the regime of vacuum dominance into that of radiation dominance, which can be estimated to be around a∗ ∼ 10−30 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (137) below), HI and ρI are the value of H and ρvac, respectively, at the beginning of inflation, with ρI = CinfH6 I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Applying the Friedmann equation, we find HI = � 3 8πGICinf �1/4 , (132) 41 ρI = 3 8πGI H2 I = 3 8πGI � 3 8πGICinf �1/2 = C−1/2 inf � 3 8πGI �3/2 , (133) where GI ≡ G(HI) is the gravitational coupling at H = HI, the latter being value of the Hubble parameter at the primeval (inflationary) era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Needless to say, the difference between GI and the usual GN is not very important here since the running of G is logarithmic, and hence the effect is very small as compared to the fast evolution of the H6 term, so in practice we can neglect the running of G for these considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To trigger inflation in an effective way, we must have a positive coefficient Cinf > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the light of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (129) we can see that this is perfectly possible since the couplings ξj and masses of the fields can take a variety of values that make this possible, as can be shown in a devoted study that will be presented elsewhere [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Moreover the masses of the relevant fields involved must be very large, say around a typical GUT scale, MX ∼ 1016 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This may not be obvious at first sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' A naive interpretation of the higher order terms of the VED, which are related with the 6th adiabatic order of the ZPE (see equation (82)), may give the impression that the relevant masses are to be the lightest possible ones, but this is not so at all since in such a case the adiabatic expansion would break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the other hand, the analysis trough the Friedmann equations reveals the correct dependency of the VED and of the Hubble function on the masses during the inflationary regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From equation (129) it is obvious that C−1/2 inf ∝ mφ,ψ, where the notation stands for a linear combination of the typical masses of the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The inflationary parameters (132)-(133), therefore, depend on a positive power of mφ,ψ, and as a result the RVM-inflation is actually dominated by the heavier masses, in contrast to naive expectations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' namely, masses mφ,ψ ∼ MX ∼ 1016GeV of order of a typical GUT, as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It then follows that the same heavy masses which may generate a mild (but non-negligible) quadratic running ∼ H2 of the VED (with a coefficient νeff ∼ 10−3) may also drive fast inflation early on at the beginning of the cosmological expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To see this feature more explicitly, let us recall that the differential equation driving the Hubble function in the presence of a high power H6 in the VED (128) reads [106–108] ˙H + 3 2 (1 + ωm) H2 � 1 − H4 H4 I � = 0 , (134) where ωm = 1/3 is the EoS of matter in the relativistic epoch, and HI is given in (132).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We have neglected the influence of the term H2 and also of the CC term in the very early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is obvious from (134) that there is a constant solution H = HI to that equation, which is precisely the one which triggers the inflationary period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From this observation one can then solve equation (134) exactly to find Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (130).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The latter shows clearly the departure of H from HI when ˆa > 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' a > a∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The inflationary phase actually occurs during the short period when the departure remains small, namely when H remains approximately constant, H ≃ HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' During such period the F-term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (128) just vanishes, F( ˙H, ¨H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=') = 0, since all dependence on H is through time derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From equations (132) and (133), it is clear that the order of magnitude of the physical scales involved in RVM-inflation reads as follows: HI ∼ (MX mPl)1/2 ∼ 1017 GeV , ρI ∼ MX m3 Pl ∼ � 1018GeV �4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (135) up to numerical coefficients and multiplicity factors, of course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, it is clear that if the masses of the relevant matter fields lie in the expected range for a GUT, the right order of magnitude for the relevant physical parameters can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, it should be recalled at this point that the mechanism of RVM-inflation can also be motivated in ‘stringy’ scenarios [45–49], see [109] for a recent review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, it should be natural to expect RVM-infation in the range between the GUT scale and the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is exactly what the above estimates suggest in order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 42 One more remarkable observation is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' One can easily check from (131) that for ˆa ≫ 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' a ≫ a∗) we retrieve the standard decaying behavior of radiation, ρr(a) ∼ a−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This condition enforces the following relation between ρ0 r, accounting for the current value of radiation energy density, and ρI: ρI ≈ ρ0 ra−4 ∗ , (136) Following the line of the previous estimations, it yields an equality time between vacuum energy and radiation of a∗ = � Ω0 r ρ0 c ρI � 1 4 ≃ � 10−4 10−47 1072 �1/4 ∼ 10−30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (137) where ρ0 c ∼ 10−47 GeV4 is the current critical density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the meantime the vacuum energy becomes negligible and does not disturb primordial BBN, see [15,16] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' See also [61,106–108] for interesting phenomenological applications prior to the QFT treatment of RVM-inflation which was first presented in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, we should mention that RVM-inflation is genuinely different from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Starobinsky’s inflation, as explained in detail in [45,61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While it may be natural to conceive that a consistent inflationary model of the very early Universe should be a good candidate for an effective theory of quantum gravity, at least at energies much less than the Planck scale, RVM-inflation reveals itself as one such possible candidate, all the more if we take into account that a a low-energy ‘stringy’ version of RVM-inflation has been also identified and sharing most of the virtues of the current QFT formulation [45,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='4 Equation of state of the quantum vacuum The quantum effects of the fields have an imprint on the vacuum equation of state, which is not exactly the traditional one Pvac = −ρvac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From the expressions of the renormalized energy density and pressure of the vacuum that have been obtained in the previous sections and considering their generalization to an arbitrary number of fermions and scalars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' we arrive at the following expression for the EoS of the quantum vacuum: ωvac(H) = Pvac(H) ρvac(H) = −1 + 1 8π2ρvac(H) � Nf � ℓ=1 � ˙H 3 � H2 − m2 ψℓ + m2 ψℓ ln m2 ψℓ H2 � � + Ns � j=1 � � ξj − 1 6 � ˙H � m2 φj − H2 − m2 φj ln m2 φj H2 � − 3 � ξj − 1 6 �2 � 6 ˙H2 + 3H ¨H + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H � ln m2 φj H2 �� + O �H6 m2 � = −1 + 1 8π2ρvac(H) �� 1 3 Nf � ℓ=1 � H2 − m2 ψℓ + m2 ψℓ ln m2 ψℓ H2 � + Ns � j=1 � ξj − 1 6 � � m2 φj − H2 − m2 φj ln m2 φj H2 � � ˙H − 3 \uf8ee \uf8f0 Ns � j=1 � ξj − 1 6 �2 ln m2 φj H2 \uf8f9 \uf8fb � 6 ˙H2 + 3H ¨H + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H � � + O �H6 m2 � (138) 43 The term O � H6/m2� represents the higher order adiabatic terms, containing 6 derivatives of the cosmic time, such as the own H6/m2, ¨H2/m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' coming from ⟨T δψℓ 00 ⟩(6) ren(H, H), ⟨T δφℓ 00 ⟩(6) ren(H, H), ⟨T δψℓ 11 ⟩(6) ren(H, H) and ⟨T δφℓ 11 ⟩(6) ren(H, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Here, m = mψ, mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' They do not give a significant contri- bution during the postinflationary era, so that we can perfectly avoid them in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This relation shows that the quantum vacuum is of dynamical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As we can see there is a devia- tion with respect −1, the traditional EoS associated to the Cosmological Constant in the ΛCDM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The correction terms due to both bosonic and fermionic fields are small in the present era, in comparison with the constant term, but need not be negligible since the particle masses involved can be from a typical GUT, and hence one can estimate that the effective parameter νeff could reach up to 10−3 [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Furthermore, if we focus only on the O(H2) terms relevant for the current universe and the radiation epoch we may neglect also the higher order adiabatic terms of O(H4) in the last lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (138).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, if we consider a linear approximation in νeff – the parameter defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (122) – the EoS can be written in a rather compact form as a function of the cosmological redshift as follows: wvac = −1 + � νeff − ǫ � −1 + ln E2(z) �� � Ω0 m(1 + z)3 + 4Ω0 r 3 (1 + z)4� Ω0vac + νeff (−1 + E2(z)) − ǫ (1 − E2(z) + E2(z) ln E2(z)) + O � ν2 eff � , (139) where νeff contains the combined effects from fermions and bosons, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (119), and we have defined the normalized Hubble rate with respect to the present time (H0): E2(z) ≡ H(z) H0 = Ω0 vac + Ω0 m (1 + z)3 + Ω0 r (1 + z)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (140) Here Ω0 vac = ρ0 vac/ρ0 c ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='7, Ω0 m = ρ0 m/ρ0 c ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3 and Ω0 r = ρ0 r/ρ0 c ≈ 10−4 are the current fractions of vacuum energy, dust-like matter and radiation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The EoS formula may be further simplified if we neglect the effect of the small coefficient ǫ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is justified since ��νeff �� ≫ ��ǫ �� owing to the logarithmic extra terms ln m2/H2 0 contained in ν0 eff, which can typically be of O(100), see (123).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Thus, to within very good approximation, we can write wvac ≃ −1 + νeff Ω0 m(1 + z)3 + 4Ω0 r 3 (1 + z)4 Ω0vac + νeff (−1 + E2(z)) , (141) Notice that the term proportional to νeff in the denominator cannot be neglected at large z since it becomes dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this case, the EoS takes on the form wvac ≈ −1 + Ω0 m(1 + z)3 + 4Ω0 r 3 (1 + z)4 E2(z) (z ≫ 1) , (142) where νeff has cancelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For example, for z large enough but within the matter-dominated epoch the dominant term in equation (140) is the ∼ (1 + z)3 one, and we can see from (142) that the vacuum EoS then mimics matter since wvac ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Similarly, at much larger values of z already in the radiation-dominated epoch, where ∼ (1 + z)4 is the dominant term, then wvac ≃ 1/3 and the vacuum imitates radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Such a ‘chameleonic’ behavior of the quantum vacuum was first noticed in [16], and in fact the formula of the vacuum EoS that we have found here is a generalization for an arbitrary number of fermion and boson fields of the expression previously found in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Last but not least, the behavior of the vacuum EoS in the late universe is no less remarkable and striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From (141) we find wvac(z) ≃ −1 + νeff Ω0 m Ω0vac (1 + z)3 (z ≲ 5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (143) 44 and in this approximation we recover once more the form (28), but in this case with νeff involving the contributions from all the quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Taking into account that the last fits of the RVM to the overall cosmological data favor a value of νeff > 0 and of order 10−3 [50] (see also the previous phenomenological studies on the RVM reported in recent years [51–55]), we learn that the deviation of the EoS from −1 in the present universe is not completely negligible and it imitates quintessence since wvac ≳ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To summarize, the running vacuum mimics the EoS of the dominant component at a given time of the cosmic evolution [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Namely, wvac is close to 1/3 in the radiation dominated epoch, 0 in the matter dominated epoch and asymptotes to −1 in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Furthermore, at present it mimics quintessence since νeff is found to be positive in the phenomenological tests of the RVM mentioned before and hence wvac is slightly above −1, the traditional value of the classic vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, we can have at present an effective quintessence behavior of the quantum vacuum without need of invoking ad hoc scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, we should also mention that the EoS of the quantum vacuum in the very early universe (the period when inflation took place, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3) is very close to −1 (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the traditional vacuum value), exactly as in the remote future since the universe asimptotes towards a new inflationary period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 6 Conclusions In this paper, we have evaluated the contributions to the vacuum energy density (VED) from the quantized matter fields in a semiclassical gravity context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' More specifically, by using a regulariza- tion technique of Quantum Field Theory (QFT) in curved spacetime called adiabatic regularization and making use of a subtraction prescription amply tested in previous works [14–16], we have been able to calculate the mode functions and the renormalized zero-point energy (ZPE) from spin-1/2 quantum fields in a FLRW background up to sixth adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Combining with the contribu- tion from the ρΛ term in the Einstein-Hilbert action, we have obtained the properly renormalized VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Since the corresponding computation for scalar fields had already been accounted for in the aforementioned works, we have put forward here the combined contribution to the VED from an arbitrary number of quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We do not consider interactions among them, however, as the free field calculation in curved spacetime is already rather cumbersome in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' One interesting difference between the expression of the ZPE of these two types of fields is that in the fermionic case the terms of fourth adiabatic order (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' involving four time derivatives) are not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The final result is that the overall VED of the quantized matter fields upon adiabatic renormalization appears to be a soft dynamical quantity with the cosmological evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is a most remarkable outcome of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' More specifically, the VED shows up in the form of an expansion in powers of the Hubble rate H and its time derivatives, all these powers being of even adiabatic order, a property which is fully consistent (and expected) from the general covariance of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Noteworthy, too, is the fact that the expression for the renormalized VED emerging from our calculation appears to take the form of the running vacuum model (RVM) , see [30, 31] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, as it is characteristic in the RVM, the leading quantum effects obtained for the late universe are of second adiabatic order, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ∼ H2 and ∼ ˙H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Obviously, this may have consequences for the present universe, and these consequences have been tested in pre- vious phenomenological works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In particular, these quantum effects turn out to impact positively on a possible solution to the ΛCDM tensions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' [50–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this study we have also discussed some of the theoretical difficulties in trying to renormalize the cosmological term, Λ, and its relation with the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' To start with, it should be emphasized that these are two different concepts that can only be properly related in non-flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' If Λ is taken to be the physically measured value of the cosmological term at present, then its relation with the current VED is ρ0 vac = Λ/(8πGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, at a formal QFT level these quantities have 45 to be derived from a gravitational action in curved spacetime and a lot more of care needs to be exercised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Leaving for the moment quantum gravity considerations for a better future (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' for when, hopefully, the quantum treatment of the gravitational field becomes possible), the more pedestrian renormalization of ρvac in QFT in curved spacetime proves to be already quite helpful at present [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It shows, for example – and the present works attests once more for this fact – that the renormalized VED in the FLRW background is a mild dynamical quantity evolving with the cosmic expansion, and hence ρ0 vac is just its value at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' There is no such thing as a rigid cosmological constant in the context of QFT in the FLRW background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In general, ρvac = ρvac(H) is a function of the Hubble rate and its time derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The longstanding and widespread confusion in the literature about cosmological constant, Λ, and vacuum energy density, ρvac, has prevented to achieve a proper treatment of the renormalization of these quantities in cosmological spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In particular, the attempts to relate these concepts in the context of flat spacetime calculations are meaningless and their repeated iteration has been counterproducing [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the simplified scenario considered here, where only interactions with the gravitational back- ground are allowed, the VED is the sum of two contributions, a parameter in the effective action, ρΛ, and the ZPE of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' After renormalization, the VED depends on a scale M, and the setting M = H at the end of the calculation allows us to compare the VED at different epochs of the cosmic history, in a manner similar to the standard association made of the renormal- ization point with a characteristic energy scale of a given process in ordinary gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Thus the difference between the VED values at any two points of the cosmological expansion, say H(t1) and H(t2), provides a smooth running of the VED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Remarkably, such an evolution turns out to be free from the undesirable ∼ m4 contributions associated to the quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As a result there is no fine tuning involved in the evolution of the VED in the present context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The VED, in fact, adopts the standard form of the RVM, which in the late universe reads ρvac(H2) ≈ ρvac(H1) + 3νeff/(8π)m2 Pl(H2 2 − H2 1), where H1 and H2 can be, for example, the current value, H0, and another value H near our past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, νeff is a small parameter related to the β-function of the renormalization group running of the VED whose value has been explicitly computed in this work from the fluctuations of the quantized matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Depending on the sign of νeff, the VED can mimic a quintessence or phantom-like behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Much earlier in the cosmic history, the higher powers of H (larger than H2) take their turn and can be relevant for the inflationary regime, in the sense that they can indeed trigger inflation in what has been called ‘RVM-inflation’ [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' While the scalar field contribution to this inflationary mechanism had been computed in the previous references, in this work we have accounted for the spin-1/2 fermionic contribution and combined both types of effects for an arbitrary matter content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In both cases (scalar and fermion fields) the sixth order adiabatic terms had to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally, the renormalized vacuum fluid’s pressure, Pvac, has been determined using the same QFT techniques as for ρvac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Equipped with these nontrivial results the equation of state (EoS) of the quantum vacuum can be computed from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We find that the EoS function Pvac/ρvac deviates from the traditional result -1, which is noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is true in most of the cosmological history, especially after inflation (which is the only period in our past where the vacuum EoS stayed very close to −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is no less remarkable, as previously noted, that in the late universe, and most particularly near our time, the vacuum EoS behaves as quintessence for νeff > 0, the latter being the sign preferred by the existing phenomenological fits to the overall cosmological data – see [50], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For higher and higher redshifts during the FLRW regime, the vacuum EoS mimics the equation of state of the dominant matter component (relativistic or non-relativistic) at the corresponding epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Such a peculiar behavior of the running vacuum energy density was referred to as “chameleonic” in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The tracking of the EoS of matter ceases in the late universe, where the DE epoch breaks through, and then it behaves as effective quintessence, the reason being that the EoS is then in the process to asymptote towards −1 in the remote future, exactly as it was in 46 the primeval inflationary time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In fact, the inflationary process in the late universe is eventually resumed, but very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Overall, by combining the results from an arbitrary number of quantized matter fields we find that the main cosmic running of ρvac depends on the quadratic terms in the boson and fermion masses times the square of the Hubble function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ∼ m2 ψH2 and ∼ m2 φH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These effects are obviously much softer than the naively expected (hard) contributions ∼ m4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As noted, the soft terms have been amply tested in phenomenological studies of the RVM existing in the literature, see [30, 31] and associated references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The QFT effects that we have computed here and in the preceding studies [14–16] provide a solid theoretical underpinning of the RVM phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' They even bring to light new relevant features such as the dynamical character of the EoS of the quantum vacuum, which is unprecedented in the literature to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In particular, they suggest that if in future cosmological observations clear signs are found that the EoS of the dark energy departs from −1, such a feature could be explained by the running vacuum, which is no longer a state with EoS exactly equal to −1 in QFT, and therefore it opens the possibility that such observations may be accounted for from fundamental properties of QFT ultimately attributable to the fluctuations of the quantized matter fields in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This could be an extremely interesting signature of this approach, which does not rely at all on ad hoc quintessence fields and the like to explain a possible dynamical evolution of the DE and its EoS in today’s universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The EoS dynamics is prompted here by the virtual quantum effects produced by quantized fermion and boson fields, the same kind of effects which trigger a smooth and very mild evolution of the vacuum energy density in cosmological spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The physical outcome, at a pure cosmological/observational level, is a remarkable and qualitatively new feature, to wit: the non-constancy of the ‘cosmological constant’, Λ, in Einstein’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Theoretically, this conclusion emerges from explicit QFT calculations in the FLRW background and may point to a possible explanation for a variety of problems that have been dealt with phenomenologically in the past in terms of ad hoc quintessence or phantom fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore, in our approach, the measured cosmological ‘constant’ is neither mimicked nor supplanted by any ersatz entity from the already crammed black box of the dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The physical Λ here is, instead, a quantity directly connected with the vacuum energy density in QFT in curved spacetime, and as such is running genuinely with the quantum renormalization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As we have also emphasized, this same QFT framework helps also in relieving the current tensions in the ΛCDM and, ultimately, it might provide an explanation for the cosmic acceleration observed in our Universe from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Acknowledgements Two of us (CMP and JSP) are funded by projects PID2019-105614GB-C21 and FPA2016-76005- C2-1-P (MINECO, Spain), 2017-SGR-929 (Generalitat de Catalunya) and CEX2019-000918-M (ICCUB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' CMP is also partially supported by the fellowship 2019 FI−B 00351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The work of JSP is also partially supported by the COST Association Action CA18108 “Quantum Gravity Phenomenology in the Multimessenger Approach (QG-MM)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' CMP and JSP acknowledge as well the participation in the COST Action CA21136 “Addressing observational tensions in cosmology with systematics and fundamental physics” (CosmoVerse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' SC is supported by the Transilvania Fellowship Program, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' CMP and JSP are very grateful to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' G´omez-Valent and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' de Cruz P´erez for the fruitful collaboration over the years in the task of understanding the RVM and its manyfold phenomenological implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 47 A Appendix: Conventions and Useful Formulas In this work, natural units are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' That is ℏ = c = 1 and GN = 1/m2 Pl ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='22 × 1019 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Our framework is a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) background with null spatial curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The conventions are (+, +, +) in the Misner-Thon-Wheeler notation [110]: gµν = a2(τ)diag(−, +, +, +), with τ denoting conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Rλ µνσ = ∂νΓλ µσ + Γρ µσΓλ ρν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , with Rµν = Rλ µλν and R = Rµνgµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The Einstein field equation can be cast as Gµν + Λgµν = 8πGTµν, with Gµν ≡ Rµν − 1 2Rgµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For derivatives, the notations ()′ = d/dτ and ˙() = d/dt are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In particular, H = ˙a/a, H ≡ a′/a = aH, hence a′ = aH = a2H, a′′ = a3(2H2 + ˙H) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 of [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In this case, the non-vanishing Christoffel symbols in the conformal frame are Γ0 00 = H, Γ0 ij = Hδij, Γj i0 = Hδj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (144) On the other hand, the Ricci tensor is R = 6a′′ a = 6 a2 � H′ + H2� = 6(2H2 + ˙H), (145) and the 00th components of the Ricci and Einstein tensors are R00 = −3a2 � H2 + ˙H � , G00 = 3a2H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (146) The renormalization program requires taking into account higher derivative (HD) terms in Ein- stein’s equations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In the particular case of FLRW spacetime, it is enough to consider just one of the characteristic higher order tensors, (1)Hµν, given by the metric functional derivative of R2 in the effective action (we refer once more to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 of [15] for more details): (1)Hµν = 1 √−g δ δgµν ˆ d4x√−g R2 = −2∇µ∇νR + 2gµν□R − 1 2gµνR2 + 2RRµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (147) Its 00th component is (1)H00 = −18a2 � ˙H2 − 2H ¨H − 6H2 ˙H � (148) and its 11th component is (1)H11 = −a2 � 108H2 ˙H + 54 ˙H2 + 72 ˙H ¨H + 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (149) We will also need the invariant relations RµνRµν = 12 a4 � H′2 + H′H2 + H4� , □R = − 6 a4 � H′′′ − 6H′H2� , (150) which hold good for flat three-dimensional FLRW spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' For gamma matrices (in flat spacetime), the standard Dirac basis is chosen for our calculations with spin-1/2 fermions: γ0 = �I 0 0 −I � γk = � 0 σk −σk 0 � , (151) where σk (k = 1, 2, 3) are the usual Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In terms of the above γα, the curved spacetime γ-matrices read γµ(x) = eµ α(x)γα, where eµ α(x) is the vierbein (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 48 B Appendix: Adiabatic expansion of the spin-1/2 field modes In the main text (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 3) we have presented an iterative procedure which allows us to determine the two types of field modes hI k and hII k which are necessary to construct the 2-component spinor fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' They are functions of both momentum (k) and conformal time (τ) and have the following structure: hI k(τ) = � ωk + aM 2ωk F(τ) e−i ´ τ Wk(˜τ)d˜τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' hII k (τ) = � ωk − aM 2ωk G(τ) e−i ´ τ Wk(˜τ)d˜τ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (152) where F ≡ 1 + F (1) + F (2) + F (3) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (153) G ≡ 1 + G(1) + G(2) + G(3) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (154) Wk ≡ ωk + ω(2) k + ω(4) k + ω(6) k + · · · (155) Here Wk is a real function playing an analogous role to the effective frequency introduced (with the same notation) in the scalar field case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The superscript (n = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=') indicates that the corresponding quantity is of adiabatic order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The modes (152) are constrained to satisfy the normalization condition |hI k|2 + |hII k |2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (156) Some of the notation is similar to that of [73–75], although we use conformal metric and different conventions, and moreover we deal with FLRW spacetime rather than de Sitter (where the EMT takes a simpler form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In addition, as explained in the main text, we perform the renormalization at the arbitrary scale M (not at the on-shell point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This is important in order to test the scaling dependence of the renormalized VED, which is the main aim of our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' In what follows we use the notation ωk ≡ √ k2 + M2a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Recall that unless it is explicitly noted the mass scale involved is the off-shell point M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' When the subtraction (79) is implemented within our renormalization procedure, one just sets M = mψ in the subtracted part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The mass mψ can be conveniently expanded in even adiabatic orders as mψ = � M2 + ∆2 = M + ∆2 2M − ∆4 8M3 + ∆6 16M5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (157) After completing ℓ ≥ 1 steps in the process described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 3, we end up with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (51) in the main text, which depends on the following expression: ΩkΩk,1 · · · Ωk,ℓ−1 = ωk + ω(1) k + ω(2) k + · · · + ω(2ℓ−1) k + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (158) where ω(j) k represents a function of adiabatic order j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Functions Wk(τ, M), F(τ, M) and G(τ, M) in the ansatz (152) can be estimated with the help of this product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However the following clarifications may be necessary to better understand this process, together with the explanations already given in the main text, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 3: For ℓ = 1 we only have performed one iterative step, and at this point we need to deal with the square root of Ω2 k = ω2 k + iσ + a2∆2 = ω2 k + iMa′ + a2∆2 + ia′∆2 2M + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (159) 49 as defined in (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The dots “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='..” in (159) account for terms of adiabatic order 4 and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The square root of the previous result yields Ωk = ωk + ω(1) k + a2ωk 8 � M2 ω4 k �a′ a �2 + 4∆2 ω2 k � + iaM 16ωk a′ a � 4∆2 M2 − 4a2∆2 ω2 k − a2M2 ω4 k �a′ a �2� + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (160) with ω(1) k ≡ iMa 2ωk a′ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (161) From the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='s of (160), the first two terms, ωk and ω(1) k , are used in the first order approxi- mation of the modes (see equation (59) in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Now suppose that we proceed with a further step in the iterative process, ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We have to deal with the square root of Ω2 kΩ2 k,1 = � ω2 k + iMa′ + a2∆2 + ia∆2 2M a′ a + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' � (1 + ǫ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (162) The introduction of ǫ2 = ǫ(2) 2 + ǫ(3) 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , (163) whose expression can be seen in (63), does not alter neither the 0th nor the 1st orders of (159) and (160) since ǫ2 is a sum of terms of adiabatic order 2 and higher: ΩkΩk,1 =ωk + ω(1) k + ωk 2 ǫ(2) 2 + a2∆2 2ωk + a2M2 8ω3 k �a′ a �2 + ωk 2 ǫ(3) 2 + iaM 4ωk ǫ(2) 2 a′ a + ia∆2 4Mωk a′ a − ia3M∆2 4ω3 k a′ a − ia3M3 16ω5 k �a′ a �3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (164) Nonetheless, the 2nd, 3rd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' adiabatic orders of (164) do not coincide with the ones of (160).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Similarly, by going to the next iterative step, ℓ = 3, implies working with the square root of the product Ω2 kΩ2 k,1Ω2 k,2 = � ω2 k + iMa′ + a2∆2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' � (1 + ǫ2) (1 + ǫ4) (165) with ǫ4 = ǫ(4) 4 + ǫ(5) 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (166) The introduction of ǫ4 does not alter neither the 0th, 1st, 2nd nor the 3rd adiabatic orders of (162) or (164), since ǫ4 is a sum of terms of adiabatic order 4 and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' By the same token, the 4th and 5th adiabatic orders and beyond in (162) do not coincide with the ones in (165).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Similar considerations apply to the square root of these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We can sum up this by saying that for each iterative step we can compute two consecutive adiabatic orders of (158) that will not be altered by the subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Then, after ℓ steps, the 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , 2ℓ − 1 adiabatic orders of the product (158) are trustworthy for the estimation of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The RHS of (158) has a pure imaginary part conformed by the odd orders, precisely those which do no take part in the computation of Wk: − i ˆ τ � ωk + ω(1) k + ω(2) k + · · · + ω(2ℓ−1) k � d˜τ = −i ˆ τ � W (0−2(ℓ−1)) k � d˜τ − i ˆ τ � ω(1) k + ω(3) k + · · · + ω(2ℓ−1) k � d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (167) 50 However, because of the factor −i in front of the integral, the imaginary terms in the integrand become real and are then necessary for the computation of F and G in (152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We did not specify the limits of integration in (51) for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the one hand, even terms take part in the pure imaginary exponential of (152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Now because the imaginary exponential does not appear in the final result of the relevant quantities that we compute in the main text (since they cancel in the products of a function times its complex conjugate) one might wrongly be led to conclude that Wk does not influence the calculation of the EMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' This would, however, be incorrect since the derivatives of the modes hI,II k are present in these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' On the other hand, after integrating the odd terms without specifying the limits in the integral, there exists some residual freedom in the form of a set of functions of the momentum only (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' not depending on time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' These are called f (0) k , g(0) k , f (1) k , g(1) k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' where the superscript indicates the adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' If our goal is to compute an adiabatic expansion of the modes up to 6th order we need 7 arbitrary constants for hI, namely f (0) k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , f (6) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Similarly for hII, which we denote g(0) k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' , g(6) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (37) and (38), it is clear that hI k(τ, M) = hII k (τ, −M) so F(τ, M) = G(τ, −M) and f (n) k (M) = g(n) k (−M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' With this considerations in mind let’s us put forward the adiabatic expansion of Wk explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' As said, Wk can be specified though the even terms of (158).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We are interested to compute at least up to 6th adiabatic order (that means, at least, ℓ = 4 steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Therefore we find: W (0−6) k (τ) = ωk + ω(2) k + ω(4) k + ω(6) k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (168) with ω(0) k = ωk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (169) ω(2) k = a2∆2 2ωk − a2M2 8ω3 k �a′ a �2 + 5a4M4 8ω5 k �a′ a �2 − a2M2 4ω3 k a′′ a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (170) ω(4) k = −a4∆4 8ω3 k + � −25a6M4 16ω7 k + 23a4M2 16ω5 k − a2 8ω3 k � ∆2 �a′ a �2 + �3a4M2 8ω5 k − a2 4ω3 k � ∆2 a′′ a − �1105a8M8 128ω11 k − 267a6M6 64ω9 k + 21a4M4 128ω7 k � �a′ a �4 + �221a6M6 32ω9 k − 57a4M4 32ω7 k � �a′ a �2 a′′ a − �19a4M4 32ω7 k − a2M2 32ω5 k � �a′′ a �2 − �7a4M4 8ω7 k − a2M2 16ω5 k � a′ a a′′′ a + a2M2 16ω5 k a′′′′ a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='(171) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ω(6) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a′′′′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (172) The odd terms in the expansion (158) yield a real exponential contribution in the integrals involved in (152) and hence do not contribute to the expansion of Wk in (155), but are nevertheless necessary to compute the amplitude of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Notice that after computing the integral, the adiabatic order decreases by one unit, so in order to estimate the amplitude up to 6th order is mandatory to compute up to ω(7) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The corresponding integrals for these terms are listed below: − i ˆ ω(1) k dτ = −i ˆ τ �iMa′ 2ωk � d˜τ = log �ωk + aM ωk − aM �1/4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (173) −i ˆ ω(3) k dτ = − i∆2 ˆ τ � −ia2a′M 4ω3 k + ia′ 4Mωk � d˜τ − i ˆ τ � −25ia2M5a′3 16ω7 k + 5iM3a′3 16ω5 k + iaM3a′a′′ ω5 k − iMa′′′ 8ω3 k � d˜τ = a∆2 4Mωk − aM 8ω3 k a′′ a + 5a3M3 16ω5 k �a′ a �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='(174) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ω(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k d˜τ = −i∆4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ �3ia4Ma′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16ω5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ia2a′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='8Mω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ia′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16M3ωk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='d˜τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− i∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ �175ia4M5a′3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='32ω9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− 75ia2M3a′3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16ω7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ 15iMa′3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='32ω5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− 5ia3M3a′a′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2ω7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ 3iaMa′a′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2ω5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ 3ia2Ma′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16ω5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ia′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16Mω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='d˜τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='12155ia4M9a′5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='256ω13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− 3453ia2M7a′5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='128ω11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− 33iaM3a′′a′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='32ω7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− 19iaM3a′a′′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='32ω7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ iMa′′′′′ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='(175) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='−i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ω(7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k d˜τ = − i∆6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ τ � ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='(176) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='Use of Mathematica [81] has been helpful to work out the above integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The computational strategy consists in using a sufficiently general ansatz for the structure of the result that is com- 54 patible with the adiabaticity order of the calculation, and then solve for the coefficients (form factors) of the ansatz from pure algebraic manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The procedure has been illustrated with a specific example in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' by applying the normalization condition (156) for the modes at each order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' there exists a constraint to fix the residual freedom mentioned earlier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' which is parametrized by the time independent factors fk and gk (only depending on the momentum k): Ref (1) k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' � Imf (1) k �2 + √ 2kRef (2) k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2Imf (2) k Imf (1) k + √ 2kRef (3) k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ���f (2) k ��� 2 + 2Imf (1) k Imf (3) k + √ 2kRef (4) k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2Imf (1) k Imf (4) k + 2Imf (2) k Imf (3) k + 2Ref (2) k Ref (3) k + √ 2kRef (5) k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2Imf (1) k Imf (5) k + 2Imf (2) k Imf (4) k + ���f (3) k ��� 2 + 2Ref (2) k Ref (4) k + √ 2kRef (6) k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (177) Notice that, imposing the former conditions is equivalent to claim that �����1 + � 2 k � f (1) k + f (2) k + f (3) k + f (4) k + f (5) k + f (6) k � ����� 2 ≈ 1, (178) where the departure from 1 are just terms of adiabatic order 7 or bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Similarly for the functions gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It can be shown that the observables made out of quadratic terms in the modes hI k, hII k (such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' the EMT), depend on the particular values of fk in the form (178).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It follows that they are not actual degrees of freedom if they satisfy the conditions (177).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' It is then safe to set particular values for the functions as long as quantities are computed up to 6th adiabatic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' The simplest solution satisfying the normalization conditions (177) is f (1) k = f (2) k = f (3) k = f (4) k = f (5) k = f (6) k = 0 and it is the chosen option for the formulas shown in the rest of this Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Equipped with these results we are able to calculate the different orders of F (τ, M) up to 6th order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' A comparison between the general equation (51) and the ansatz (152), the different orders of F are conformed by the rightful combinations of terms of the denominator �ΩkΩk,1Ωk,2Ωk,3 and the real factors of the exponential exp � −i ´ τ ΩkΩk,1Ωk,2Ωk,3d˜τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' Now the different orders of F are F (1)(M) = −iaM 4ω2 k a′ a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (179) F (2)(M) = � − a2 4ω2 k + a 4Mωk � ∆2 + � −5a4M4 16ω6 k + 5a3M3 16ω5 k − M2a2 32ω4 k � �a′ a �2 + �a2M2 8ω4 k − aM 8ω3 k � a′′ a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (180) F (3)(M) = i �5Ma3 16ω4 k − a2 16ω3 k − a 8Mω2 k � a′ a ∆2 + i �65M5a5 64ω8 k − 5M4a4 64ω7 k − 21M3a3 128ω6 k � �a′ a �3 + i � −19M3a3 32ω6 k + M2a2 32ω5 k � a′a′′ a2 + i aM 16ω4 k a′′′ a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='(181) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='F (4)(M) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� 5a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='32ω4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='8Mω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='32M2ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='16M3ωk ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='aM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='128ω7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� a′′′′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (184) For hII k , one can make use of the relation G(n)(M) = F (n)(−M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' C Appendix: Adiabatic expansion of ⟨Tµν⟩ for spin-1/2 fields The unrenormalized components of the vacuum EMT, ⟨Tµν⟩, for spin-1/2 fermions can be obtained through the adiabatic expansion, which we compute up to 6th order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 12 For the 00 component we have � T δψ 00 � = � T δψ 00 �(0) + � T δψ 00 �(2) + � T δψ 00 �(4) + � T δψ 00 �(6) + · · · (185) The various terms in the expansion (185) read as follows: � T δψ 00 �(0) = 1 2π2a ˆ ∞ 0 dkk2 � −2ωk a � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (186) � T δψ 00 �(2) = 1 2π2a ˆ ∞ 0 dkk2 � −a∆2 ωk − a3M4 4ω5 k �a′ a �2 + aM2 4ω3 k �a′ a �2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (187) � T δψ 00 �(4) = 1 2π2a ˆ ∞ 0 dkk2 � a3∆4 4ω3 k + �5a5M4∆2 8ω7 k − 7a3M2∆2 8ω5 k + a∆2 4ω3 k � �a′ a �2 + �105a7M8 64ω11 k − 63a5M6 32ω9 k + 21a3M4 64ω7 k � �a′ a �4 + � −7a5M6 8ω9 k + 7M4a3 8ω7 k � a′′ a �a′ a �2 + � −a3M4 16ω7 k + aM2 16ω5 k � �a′′ a �2 + �a3M4 8ω7 k − aM2 8ω5 k � a′ a a′′′ a � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (188) 12Details of the corresponding computation for scalar fields were provided in [14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' A summary of these calcu- lations is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 2 of the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='T δψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2π2a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='ˆ ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='dkk2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a′′′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (189) To obtain the component ⟨T δψ 11 ⟩ a similar expansion as in (185) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' However, it is possible to use the following relation with the previously computed ⟨T00⟩ component: � T δψ 11 � = � T δψ 00 � − � T δψ 00 �′ 3H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (190) As a result we find: � T δψ 11 �(0) = 1 2π2a ˆ ∞ 0 dkk2 �2aM2 3ωk − 2ωk 3a � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (191) � T δψ 11 �(2) = 1 2π2a ˆ ∞ 0 dkk2 � � −a3M2 3ω3 k + a 3ωk � ∆2 + � −5a5M6 12ω7 k + a3M4 3ω5 k + aM2 12ω3 k � �a′ a �2 + �a3M4 6ω5 k − aM2 6ω3 k � a′′ a � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='(192) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='T δψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='�(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' (194) We refer the reader to the master formula in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content='2 of [15] for the explicit computation of the integrals in the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' We refrain from providing additional details here, as the remaining calculations can be handled straightforwardly with the help of the mentioned master formula.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} +page_content=' 67' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE4T4oBgHgl3EQfrA2b/content/2301.05205v1.pdf'} diff --git a/idFMT4oBgHgl3EQf4zEN/content/tmp_files/2301.12453v1.pdf.txt b/idFMT4oBgHgl3EQf4zEN/content/tmp_files/2301.12453v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b328822f8f28289d7730ef88fa925a96ed098c50 --- /dev/null +++ b/idFMT4oBgHgl3EQf4zEN/content/tmp_files/2301.12453v1.pdf.txt @@ -0,0 +1,2616 @@ +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +1 +Boosting Automated Patch Correctness +Prediction via Pre-trained Language Model +Quanjun Zhang, Chunrong Fang*, Weisong Sun, Yan Liu, Tieke He*, Xiaodong Hao, Zhenyu Chen +Abstract—Automated program repair (APR) aims to fix software bugs automatically without human debugging efforts and plays a +crucial role in software development and maintenance. Despite the recent significant progress in the number of fixed bugs, APR is still +challenged by a long-standing overfitting problem (i.e., the generated patch is plausible but overfitting). Various techniques have thus +been proposed to address the overfitting problem. Among them, leveraging deep learning approaches to predict patch correctness +automatically is emerging along with the available large-scale patch benchmarks recently. However, existing learning-based techniques +mainly rely on manually-designed code features, which can be extremely costly and challenging to construct in practice. In this paper, +we propose APPT, a pre-trained model-based automated patch correctness assessment technique, which treats the source code as a +sequence of tokens without extra overhead to design a mass of features from different perspectives. In particular, APPT adopts a +pre-trained model as the encoder stack, followed by an LSTM stack and a deep learning classifier. Although our idea is general and +can be built on various existing pre-trained models, we have implemented APPT based on the BERT model. We conduct an extensive +experiment on 1,183 Defects4J patches and the experimental results show that APPT achieves prediction accuracy of 79.0% and +recall of 81.3%, outperforming the state-of-the-art technique CACHE by 3.6% and 4.8%. Our additional investigation on 49,694 +real-world patches shows that APPT achieves the optimum performance (exceeding 99% in five common metrics for assessing patch +classification techniques) compared with existing representation learning techniques. We also prove that adopting advanced +pre-trained models can further provide substantial advancement (e.g., GraphCodeBERT-based APPT improves BERT-based APPT by +3.0% and 2.6% in precision and recall, respectively), highlighting the generalizability of APPT. +Index Terms—Automated Program Repair, Patch Correctness, Pre-trained Model +! +1 +INTRODUCTION +S +Oftware bugs are inevitable in modern software systems +and result in fatal consequences, such as costing trillions +of dollars in financial loss and affecting billions of people +around the world [1], [2]. It is incredibly time-consuming +and labor-intensive for developers to fix such bugs due +to the increasing size and complexity of modern software +systems [3]. Automated program repair (APR) aims to fix +revealed software bugs without human intervention auto- +matically and has attracted massive attention from both +academia and industry in the past decades [4], [5]. Despite +an emerging research area, a variety of APR techniques have +been proposed and continuously achieved promising results +in terms of the number of fixed bugs in the literature [6], [7]. +However, it is fundamentally difficult to achieve high +precision for generated patches due to the weak program +specifications [8]. Existing APR techniques usually leverage +the developer-written test cases as the criteria to assess the +correctness of the generated patches. In fact, a generated +patch passing the available test cases may not generalize +to other potential test cases, leading to a long-standing +challenge of APR (i.e., the overfitting issue) [8]. For example, +when a bug is detected in functionality, a patch can be sim- +ply generated by deleting the functionality and the available +• +Quanjun Zhang, Chunrong Fang, Weisong Sun, Yan Liu, Tieke He, +Xiaodong Hao and Zhenyu Chen are with the State Key Laboratory for +Novel Software Technology, Nanjing University, China. +E-mail: +quanjun.zhang@smail.nju.edu.cn, +fangchunrong@nju.edu.cn, +weisongsun@smail.nju.edu.cn, +MF21320104@smail.nju.edu.cn, +hetieke@nju.edu.cn, MF21320054@smail.nju.edu.cn, zychen@nju.edu.cn +• +*Chunrong Fang and Tieke He are the corresponding authors. +Manuscript received xxx xxx, 2022; revised xxx xxx, 2022. +test cases usually fail to exercise the deleted functionality +[9]. In this case, developers need to consume tremendous +time and effort to filter the overfitting patches, resulting in a +negative debugging performance when APR techniques are +applied in practice [10], [11]. +Thus, various automated patch correctness assessment +(APCA) techniques have been proposed to determine +whether a generated patch is indeed correct or not [12]. +According to extracted features, the traditional APCA tech- +niques can be categorized into two groups: static and dy- +namic ones [13]. Static techniques tend to analyze the code +changed patterns or code similarity based on the syntactic +and semantic features. For example, Tan et al. [14] define +a set of generic forbidden transformations (e.g., the above- +mentioned functionality deleting) for the buggy program. In +contrast, dynamic techniques usually execute the plausible +patches against extra test cases generated by automated test +generation tools (e.g., Evosuite [15] and Randoop [16]). For +example, Xiong et al. [17] generate new test cases and de- +termine patch correctness based on the behavior similarity +of the test case executions. However, the static techniques +may suffer from prediction precision problems, while it is +pretty time-consuming for dynamic techniques to generate +additional test cases and execute all patched programs [13]. +Recently, inspired by large-scale patch benchmarks being +released [6], [7], some learning-based APCA techniques +have been proposed to assess patch correctness by em- +bedding buggy and patched code snippets [12], [18], [19]. +For example, Lin et al. [20] leverage the abstract syntax +tree (AST) path to represent the patch and build a deep +learning classifier to predict the correctness of the patch. +arXiv:2301.12453v1 [cs.SE] 29 Jan 2023 + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +2 +Similarly, He et al. [18] extract code features at the AST +level statically and train a probabilistic model to perform +patch prediction. However, despite outstanding prediction +results, existing learning-based APCA techniques mainly +employ complex code-aware features (e.g., AST path in [20]) +or manually-designed code features (e.g., 202 code features +in [18]), which are costly to conduct and extract in practice. +In this work, we propose, APPT, the first Automated Pre- +trained model-based Patch correcTness assessment tech- +nique, which employs the pre-training and fine-tuning to +address the above limitation of prior work. We first adopt +the large pre-trained model as the encoder stack to extract +code representations. We then employ bidirectional LSTM +layers to capture rich dependency information between the +buggy and patched code snippets. Finally, we build a deep +learning classifier to predict whether the patch is overfitting +or not. APPT treats only the source code tokens as the +input and automatically extracts code features using a well- +trained encoder stack, getting rid of the need for code-aware +features and manually-designed features. Although APPT is +conceptually general and can be built on various pre-trained +models, we have implemented APPT as a practical APCA +tool based on the BERT model. Our experimental results +on 1,183 Defects4J patches indicate that APPT improves the +state-of-the-art technique CACHE by 3.6% accuracy, 1.2% +precision, 4.8% recall, 2.9% F1-score and 3.1% AUC. We +conduct an additional investigation on 49,694 real-world +patches from five different patch benchmarks and the results +show that APPT exceeds 99% in accuracy, precision, re- +call, F1-score and AUC metrics, outperforming the existing +representation learning techniques. We also adopt different +pre-trained models to further investigate the generalization +ability of APPT. The results demonstrate that APPT with +advanced pre-trained models can enhance the prediction +performance. For example, precision and recall of APPT +can be improved by 3.0% and 2.6% when equipped with +GraphCodeBERT, which are 4.2% and 7.2% higher than the +state-of-the-art technique CACHE. +To sum up, we make the following major contributions: +• New Direction. This paper opens a new direction for +patch correctness prediction to directly utilize large pre- +trained models by pre-training and fine-tuning. Com- +pared with existing learning-based APCA techniques, +our approach does not need any additional efforts to +design and extract complex code features. +• Novel Technique. We propose APPT, a BERT-based +APCA technique that leverages the pre-training and +classifier to predict patch correctness. To the best of our +knowledge, we are the first to exploit fine-tuning the +pre-trained model for assessing patch correctness. +• Extensive Study. We conduct various empirical studies +to investigate and evaluate APPT on diverse patch +benchmarks. The results show that APPT achieves +significantly better overall performance than existing +learning-based and traditional APCA techniques. +• Available Artifacts. We release the relevant materials +(including source code, patches and results) used in the +experiments for replication and future research1. +1. All +artifacts +relevant +to +this +work +can +be +found +at +anonymouswebsite, accessed August 2022. +repair strategy +test suite +correct patch +plausible patch +developer +generated patch +overfitting patch +suspicious code +fault localization +deployment +Localization Phase +Repair Phase +buggy program +Verification Phase +Fig. 1: Overview of APR +2 +BACKGROUND +2.1 +Automated Program Repair +APR techniques’ primary objective is to identify and fix +program bugs automatically. Fig. 1 illustrates the workflow +of the typical APR technique, which is usually composed +of three steps: (1) the localization phrase utilizes off-the- +shelf fault localization techniques to recognize the suspi- +cious code elements (e.g., statements or methods) [21], [22]; +(2) the repair phrase then modifies these elements based +on a set of transformation rules to generate various new +program variants, also called candidate patches; (3) the ver- +ification phrase adopts the original test cases as the oracle +to check whether candidate patches execute as expected or +not. Specifically, a candidate patch passing the original test +cases is called a plausible patch. A plausible patch that is +semantically equivalent to the developer patch denotes a +correct patch; otherwise, it is an overfitting patch. +It is fundamentally challenging to ensure the correctness +of the plausible patches due to the weak specification of the +program behavior in practice. Existing studies have demon- +strated that manually identifying the overfitting patches is +time-consuming and may harm the debugging performance +of developers [10], [23]. Thus, various techniques have been +proposed to validate patch correctness automatically. Ac- +cording to whether the dynamic execution or machine learn- +ing is required [13], we categorize them into three main cat- +egories: static-based techniques, dynamic-based techniques +and learning-based techniques. +• Static-based APCA techniques. These techniques aim to +prioritize correct patches over overfitting ones by static code +features, such as code-deleting program transformations. +• Dynamic-based APCA techniques. These techniques aim +to filter out overfitting patches by executing extra test cases, +which are generated based on fixed or patched programs. +According to whether the correct patches are required, these +techniques can be further categorized into dynamic with +oracle-based ones and dynamic without oracle-based ones. +• Learning-based APCA techniques. These techniques aim +to predict the correctness of plausible patches enhanced +by machine learning techniques. They usually extract the +manually-designed code features and then adopt a classi- +fier to perform patch prediction [18]. Some techniques are + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +3 +proposed to adopt code embedding techniques to extract +code features automatically [20], which are also denoted as +representation learning-based APCA techniques. +Recently, an increasing number of research efforts have +attempted to use machine learning techniques to learn from +existing patch benchmarks for predicting potential patch +correctness, achieving promising results. In this work, we +adopt the large pre-trained model (i.e., BERT) to encode +plausible patches and train a deep learning classifier to +predict patch correctness. Compared to existing techniques, +our paper is the first work to predict patch correctness by +pre-training and fine-tuning the pre-trained model. +2.2 +Pre-trained Model +Recently, Pre-trained language models (e.g., BERT) have +significantly improved performance across a wide range of +natural language processing (NLP) tasks, such as machine +translation and text classification [24]–[26]. Typically, the +models are pre-trained to derive generic language represen- +tations by self-supervised training on large-scale unlabeled +data and then are transferred to benefit multiple down- +stream tasks by fine-tuning on limited data annotation. +Existing pre-trained models usually adopt the encoder- +decoder architectures, where an encoder encodes an input +sequence as a fixed-length vector representation, and a +decoder generates an output sequence based on the in- +put representation. Encoder-only models (e.g., BERT [24]) +usually pre-train a bidirectional transformer in which each +token can attend to each other. Encoder-only models are +good at understanding tasks (e.g., code search), but their +bidirectionality nature requires an additional decoder for +generation tasks, where this decoder initializes from scratch +and cannot benefit from the pre-training tasks. Decoder-only +models (e.g., GPT [25]) are pre-trained using unidirectional +language modeling that only allows tokens to attend to +the previous tokens and itself to predict the next token. +Decoder-only models are good at auto-regressive tasks like +code completion, but the unidirectional framework is sub- +optimal for understanding tasks. Encoder-decoder models +(e.g., T5 [26]) often make use of denoising pre-training objec- +tives that corrupt the source input and require the decoder +to recover them. Compared to encoder-only and decoder- +only models that favor understanding and auto-regressive +tasks, encoder-decoder models can support generation tasks +like code summarization. In this work, we treat the patch +correctness assessment as a binary classification task and we +consider encoder-only models to get embeddings of code +snippets according to existing work [27]. +Inspired by the success of pre-trained models in NLP, +many recent attempts have been adopted to boost numer- +ous code-related tasks (e.g., code summarization and code +search) with pre-trained models (e.g., GraphCodeBERT) +[28], [29]. Despite the promising results, little work aims +to explore the capabilities of pre-trained models in sup- +porting patch correctness assessment. In this work, BERT is +selected to exploit pre-trained models for automated patch +correctness assessment, as it has been widely adopted in +various code-related tasks and is quite effective for classi- +fication tasks [28], [29]. Two advanced BERT-style models +(i.e., CodeBERT and GraphCodeBERT) are also selected to +investigate the generalization ability of APPT. +3 +APPROACH +Fig. 2 presents the overall framework of our approach. +Generally, APPT accepts a buggy program and a plausible +patch that passes the available test cases as inputs. APPT ex- +tracts the buggy code snippet and its corresponding patched +code snippet, and adopts four strategies to truncate the +code tokens. APPT then uses the pre-trained BERT model +for embedding the truncated tokens. After obtaining the +representations for the buggy and patched code snippets, +APPT uses four pre-defined functions for integrating the +representations. Finally, APPT adopts a deep learning clas- +sifier to return the final result (i.e., correct or overfitting). +3.1 +Code Extraction +Given a buggy program, existing APR tools may return +a plausible patch p (if it exists) that passes all available +test cases. Code extraction phrase aims to take the returned +patch and the buggy program as the inputs, and output the +corresponding buggy and patched code tokens (shown in +Fig. 2(a)). +Specifically, we get the buggy and patched code snippets +(i.e., Cb and Cp) by parsing the patch file. Firstly, we select +removed and added lines as the buggy and patched lines, +marked with “+” and ‘-’, respectively. Secondly, to keep +the context information about the plausible patch, we keep +unchanged lines (i.e., without +” and ‘- in the beginning) as +part of each code snippet. Finally, the buggy (or patched) +code snippet are made up by the buggy (patched) lines and +common context part. +We treat the buggy (or patched) code snippet as se- +quences of tokens and utilize a subword tokenization +method to address out-of-vocabulary (OOV) problem by +further breakdowning identifiers into their subtokens [30] +when tokenizing the code snippet. In this work, we keep the +original tokenization vocabulary instead of building a new +vocabulary using byte pair encoding (BPE) algorithm as we +want APPT to inherit the natural language understanding +ability and start learning prediction from a good initial +point. +After the buggy (or patched) code tokens are extracted, +we attempt to take them as the inputs into the token em- +bedding phrase. However, pre-trained models are usually +limited to a particular input length. For example, BERT +can only take input sequences up to 512 tokens in length. +We further truncate the inputs whose length is longer than +512 after tokenization. Following existing work [31], we use +different methods to truncate the method pair. +• head-only: keep the first 512 tokens in Cb and Cp. +• tail-only: keep the last 512 tokens in Cb and Cp. +• mid-only: select 512 tokens in the middle of in Cb and +Cp. +• hybrid: select the first 256 and the last 256 tokens in Cb +and Cp. +In our experiment, we use the head-only method to +truncate the code tokens by default. We also discuss the +impact of different truncation methods in Section 5.3.2. +Finally, the buggy and patched code tokens (i.e., Tb and Tp) +are extracted based on Cb and Tp to fit the maximum length +limit of BERT. + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +4 +Plausible Patch +Buggy Code Vector +Patched Code Vector +con +- +× +mix ++ +Code Change +Vector +Correct +or +Overfitting +Token +Truncation +Head-Tail +Mid +Head +Tail +(b) Token Embedding +(a) Code Extraction +(c) Patch Classification +Classifier +Encoder Stack +Self-Attention +Feed-Forward +Layer-Norm +Encoder 1 +Encoder 2 +Encoder 12 +… +Word Embedding +Truncated Tokens +Linear +Softmax +Linear +LSTM +LSTM +LSTM Stack +Vector +Integration +Patched Lines +Buggy Lines +Context Part +Patched Code Snippet +Buggy Code Snippet +Buggy Tokens +Patched Tokens +Tokenizer +Code +Tokenization +Code Snippets +Parsed Patch +Fig. 2: Overview of APPT +3.2 +Token Embedding +Token Embedding phrase takes the buggy (or patched) code +tokens (i.e., Tb or Tp) as input and embeds it into the +buggy (or patched) vector (i.e., Eb or Ep) as output (shown +in Fig. 2(b)). APPT implements a stack of twelve layers +of encoder blocks to extract the hidden state of the code +snippet. Each encoder block consists of three components. +The first part is a multi-head self-attention layer to learn +long-range dependencies in the input code tokens. The +second part is a simple, position-wise fully connected feed- +forward neural network, which can linearly transform the +token embedding for better feature extraction. The third part +is a residual connection around each component, followed +by a layer normalization to ensure the stability of code token +embeddings distribution. +In particular, the self-attention mechanism computes +the representation of each code token by considering the +position relationship between the code tokens. The self- +attention mechanism mainly relies on three main vectors, +query Q, key K, and value V , by mapping a query and +a set of key-value pairs to an output vector. We employ a +scaled dot-product self-attention to calculate the attention +scores of each token by taking the dot product between all of +the query vectors and key vectors. The attention scores are +then normalized to probabilities using the softmax function +to get the attention weights. Finally, the value vectors can +be updated by taking a dot product between the value +vectors and the attention weight vectors. The self-attention +operation is computed using three matrices Q, K and V as +follows: +Attention(Q, K, V ) = softmax +�QKT +√dk +� +V +(1) +To capture richer semantic meanings of the input code +tokens, we further use a multi-head mechanism to real- +ize the self-attention, which allows the model to jointly +attend the information from different code representation +subspaces at different positions. For d-dimension Q, K, and +V , we split those vectors into h heads where each head +has d/h-dimension. After all of the self-attention operations, +each head will then be concatenated back again to feed into a +fully-connected feed-forward neural network including two +linear transformations with a ReLU activation in between. +The multi-head mechanism can be summarized by the fol- +lowing equation: +MultiHead(Q, K, V ) = Concat (head1, . . . , headh) W O +(2) +where headi = Attention(QW Q +i , KW Q +i , V W Q +i ) and W O +is used to linearly project to the expected dimension after +concatenation. Therefore, the encoder stack can take an +input code snippet and output a real-valued vector for each +code token within the code snippet based on the context. +3.3 +Patch Classification +After the embedding vectors of the buggy and patched code +snippets (i.e., Eb and Ep) are extracted by the encoder stack, +patch classification phrase first integrates the two vectors into a +single input vector (i.e., Econ) and then adopts a deep learn- +ing classifier to predict the patch correctness automatically +(shown in Fig. 2(c)). +3.3.1 +Representations Integration +Given two vectors Eb and Ep with n dimensions repre- +senting the buggy and patched code snippets, respectively, +we integrate the two vectors into one code changed vector +for patch classification. In detail, we leverage different ap- +proaches to integrate them to characterize the differences +between Eb and Ep from diverse aspects, such as an vector- +wise concatenation operation Econ, element-wise addition +operation Eadd , element-wise subtraction operation Esub, +Hadamard product Epro. We also attempt to capture crossed +features between the two vectors by concatenating the above +integrated vectors Emix. The integration approaches are +selected due to their promising results in previous studies +[12], [32], which are listed as follows: +(1) Econ is a concatenation operation between Eb and Ep +on vector-wise level with 2n dimension (i.e., Econ = +Eb +� Ep). +(2) Eadd is an addition operation between Eb and Ep on +element-wise level with n dimensions (i.e., Eadd = Eb+ +Ep). +(3) Esub is a subtraction operation between Eb and Ep on +element-wise level with n dimensions (i.e., Esub = Eb− +Ep). +(4) Epro is a Hadamard product operation between Eb +and Ep on element-wise level with n dimensions (i.e., +Esub = Eb ⊙ Ep). +(5) Emix is a concatenation over Econ, Eadd, Esub and +Epro on vector-wise level with 5n dimension (i.e., +Emix = Econ +� Eadd +� Esub +� Esub). + +1902IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +5 +3.3.2 +LSTM Stack +After the embedding vector (e.g., Econ) of the changed +code tokens is extracted, APPT aims to determine the given +patch’s correctness based on a deep learning classifier. To +extract more hidden code change features, we further feed +the code changed vector into a Long Short-Term Memory +(LSTM) stack. The LSTM stack has two bidirectional LSTM +layers, the output of which is a new state generated by +concatenating the hidden states from both directions at +a time. LSTM is a specialized recurrent neural network +(RNN) for modeling long-term dependencies of sequences. +A common LSTM gate unit is composed of a cell, an input +gate, an output gate and a forget gate. Thanks to the gated +mechanism, LSTM is well-suited to extract the contextual +semantic features containing token sequential dependencies +and has been widely used in various kinds of tasks, such +as vulnerability detection [33], fault localization [34], and +automated program repair [35]. +In APPT, the LSTM stack computes a mapping from an +input code changed vector x = (x1, ..., xT ) (e.g., Econ) to +an output vector z = (z1, ..., zT ) by calculating the network +gate unit activations. We implement the gated mechanism +by leveraging the input gates and forget gates to control +the propagation of cell states. Specifically, when updating +the cell state, the input gates decide what new information +from the current input to be included in the cell states (i.e., +Equation 3), and forget gates decide what information to +be excluded from the cell states (i.e., Equation 4). Based on +new and forgetting information, cell states as the memory +of the LSTM unit can be updated (i.e., Equation 5). The +output gate then determines the value for the next hidden +state by point-wise multiplication of the output gate (i.e., +Equation 6). Finally, the value of the current cell state passed +through tanh function (i.e., Equation 7), by which the output +of LSTM stack is calculated (i.e., Equation 8). +it = sigmoid (Wixxt + Wihht−1 + bi) +(3) +ft = sigmoid (Wfxxt + Wfhht−1 + bf) +(4) +ct = ft ⊙ ct−1 + it ⊙ tanh (Wgxxt + Wghht−1 + bg) +(5) +ot = sigmoid (Woxxt + Wohht−1 + bo) +(6) +ht = ot ⊙ tanh (ct) +(7) +zt = Wzhht + bz +(8) +where the W terms denote weight matrices (e.g., Wix is the +matrix of weights from the input gate to the input), the b +terms denote bias vectors (e.g., bi is the input gate bias +vector) and ⊙ denotes element-wise multiplication of the +vectors. +3.3.3 +Classifier +After the computation of all LSTM iterations, the embedding +vectors of changed code tokens are further fed to a designed +deep learning classifier to predict the patch correctness. The +classifier is composed of two fully connected layers followed +by a binary predictor. In APPT, we apply a standard softmax +function to obtain the probability distribution over correct- +ness. A patch is labeled as correct if its probability of being +correct is larger than that of being incorrect; otherwise, it is +considered overfitting. +In particular, for patch p, z denotes its output of the last +iteration in the LSTM stack, which is further linearly trans- +formed into a real number as Equation 9, where W ∈ Rd×1, +b ∈ R, and n denotes the number of class (i.e., correct and +overfitting). We then leverage softmax function to normalize +the output of patch p as Equation 10, where s denotes the +correct or overfitting probability of patch p predicted by the +model . +yi = Wzi + bi +∀i ∈ 1 . . . n +(9) +s (yi) = +exp {yi} +�n +i=1 exp {yj} +(10) +3.4 +Training +To train the network, we calculate the loss to update the +neural weights based on its predicted result and ground +truth. We use the cross-entropy loss, which has been widely +used in some classification tasks and patch prediction stud- +ies [20], [36]. In particular, gi ∈ {0, 1} denotes whether the +i-th patch is correct or overfitting. The cross-entropy loss +compares a target gi with a prediction s in a logarithmic +and hence exponential fashion. The objective function is +computed in Equation 11, which is minimized constantly +in the training to update the parameters in our model. +L = +� +i +−[gi · log(s) + (1 − gi) · log(1 − s)] +(11) +We employ the dropout technique to improve the robust- +ness of APPT and the Adam approach [37] to optimize the +objective function. +4 +EXPERIMENT +4.1 +Research Questions +The empirical study is conducted to answer the following +research questions. +RQ1: How does APPT perform compared with existing +state-of-the-art representation learning-based APCA +techniques? +RQ2: How does APPT perform compared with existing +state-of-the-art traditional and learning-based APCA +techniques? +RQ3: To what extent do the different choices affect the +overall effectiveness of APPT? +RQ3.1: To what extent do the token truncation choices +affect the overall effectiveness of APPT? +RQ3.2: To what extent do the vector concatenation +choices affect the overall effectiveness of APPT? + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +6 +TABLE 1: APR tools in small benchmark +Category +APR Tools +Heuristic-based +jGenProg [41], jKali [41], jMutRepair [41], SimFix [42], ARJA [38], GenProg-A [38], Kali-A [38], RSRepair-A [38], CapGen [43]. +Constraint-based +DynaMoth [44], Nopol [45], ACS [46], Cardumen [47], JAID [48], SketchFix [49]. +Template-based +kPAR [50], FixMiner [51], AVATAR [52], TBar [5], SOFix [53], HDRepair [54]. +Learning-based +SequenceR [55]. +RQ3.3: To what extent do the pre-trained model +choices affect the overall effectiveness of APPT? +RQ1 aims to compare APPT with 16 representation learning +techniques to explore to what extent APPT outperforms +these techniques, including three classifiers multiplied (de- +cision tree, logistic regression, and naive Bayes) by five rep- +resentation methods (BERT, code2vec, code2seq, Doc2Vec, +and CC2Vec) from Tian et al. [12], and the most recent +technique CACHE from Lin et al. [20]. RQ2 is designed to +investigate the effectiveness of APPT by comparing it with +both dynamic and static techniques. The latest learning- +based APCA technique, ODS, is also evaluated in our study. +RQ3 focuses on impact analysis of APPT, which is further +refined into three sub-RQs. In detail, RQ3.1 explores how +the four token truncation choices affect the effectiveness of +APPT. RQ3.2 explores how the five vector concatenation +methods affect the effectiveness of APPT. RQ3.3 replaces +BERT with advanced CodeBERT and GraphCodeBERT to +investigate the impact of the pre-trained models on the +effectiveness of APPT. +4.2 +Dataset +With the rapid development of APR research in the last +decades, a broad range of repair techniques has been pro- +posed [38]–[40], resulting in a growing number of patches +across many benchmarks being released [7], [13]. The large- +scale patch benchmarks enable deep learning-based pre- +diction techniques to learn the distribution of correct and +overfitting patches for patch correctness assessment. In this +study, we adopt two patch datasets based on the recent +studies [12], [13], [20], a small one containing 1,183 Defects4J +labeled patches and a large one containing 50,794 real-world +labeled patches. +On the small dataset, we mainly focus on the released +patches from Defects4J [56], which is the most widely- +adopted benchmark in APR research [7]. We select the +benchmarks released by two recent large-scale studies, i.e., +Wang et al. [13] and Tian et al. [12]. Specifically, the first +benchmark [13] includes the labeled patches provided by +Liu et al. [7], Xiong et al. [17] and Defects4J developers [56]. +The second benchmark [12] includes the labeled patches +from Liu et al. [7] and also considers the patches generated +by some well-known APR tools that are not included in Liu +et al. [7] to better explore the overfitting problem, i.e., JAID +[48], SketchFix [49], CapGen [43], SOFix [53] and SequenceR +[55]. To avoid the data leakage issue in the two benchmarks, +a filtering process is also conducted to discard duplicate +patches. In particular, given a patch whose all the blank +spaces are removed, the left text information is compared +with that from the other patches. If two patches are iden- +tical concerning their text information, they are considered +TABLE 2: Datasets used in our experiment +Datasets +Subjects +# Correct +# Overfitting +Total +Small +Tian et al. [12] +468 +532 +1,000 +Wang et al. [13] +248 +654 +902 +Our Study +532 +648 +1,183 +Large +ManySStuBs4J [57] +51,433 +0 +51,433 +RepairThemAll [6] +900 +63,393 +64,293 +Our Study +25,589 +24,105 +49,694 +duplicates, resulting in 1,183 patches in our small dataset. +The patches are generated by 22 distinct APR tools, which +can be divided into four categories, i.e., heuristic-based, +constraint-based, template-based, and learning-based tech- +niques. The detailed information on these covered APR tools +is presented in Table 1, where the first column lists the four +repair technique categories and the second column list the +corresponding repair techniques. +On the large dataset, we further consider a variety of +patches generated from other benchmarks, to evaluate the +generality of APPT. Recently, existing studies demonstrate +that APR techniques may overfit Defects4J in terms of +repairability [6], [11]. Thus, some other benchmarks have +been conducted to evaluate the performance of APR tech- +niques, such as Bugs.jar [58], IntroclassJava [59], BEARS +[60] and QuixBugs [61], providing substantial patches on +the large dataset. In this work, we consider a large patch +dataset released by a recent study [20] to investigate the +generality of APPT. The large patch dataset includes the +labeled patches provided from RepairThemAll framework +[6] and ManySStuBs4J [57]. In particular, RepairThemAll +framework [6] contains 64,293 patches using 11 Java test- +suite-based repair tools and 2,141 bugs from five diverse +benchmarks. However, there exists an imbalanced dataset +issue as over 98.6%2 (63,393/64,293) of generated patches +are actually labeled as incorrect. Recent studies have re- +vealed that a well-balanced dataset is essential when investi- +gating deep learning-based prediction techniques [12], [18]. +To compensate the lack of correct patches, the large patch +dataset then includes ManySStuBs4J [57], which provides +simple bug-fix changes mined from 1,000 popular open- +source Java projects. The bug-fix changes are correct fix +attempts of real-world bugs and thus are considered cor- +rect patches in our experiment. Finally, a large balanced +patch dataset is built from the RepairThemAll framework +and ManySStuBs4J by discarding duplicate patches and +filtering the ones from small student-written programming +assignments (e.g., IntroClassJava). The dataset involves all +2. The +RepairThemAll +Framework. +https://github.com/ +program-repair/RepairThemAll, accessed August 2022 + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +7 +TABLE 3: Compared APCA techniques in our experiment. +with Oracle Required +without Oracle Required +Dynamic-based +Evosuite [15], Randoop [16], DiffTGen [62], Daikon [63] +PATCH-SIM [17], E-PATCH-SIM [17], R-Opad [63], E-Opad [63] +Static-based +� +ssFix [64], CapGen [43], Anti-patterns [14], S3 [65] +Learning-based +� +ODS [18], Random Forest [12], +Embedding learning [12], CACHE [20], Our proposed APPT +denotes the representation learning techniques. +available patches generated on RepairThemAll framework +and ManySStuBs4J, resulting in 49,694 patches after dedu- +plication. +Statistics on the two datasets are shown in Table 2. Table +2 has two main rows representing the two datasets, each of +which has three sub-rows. The first and second sub-rows +list the two sources in the corresponding dataset. The third +column lists the filtered patches used in our experiment +from the two sources. We also present the number of correct, +overfitting and total patches in the last three columns. +4.3 +Baselines +Various APCA techniques have been proposed in the litera- +ture to validate patch correctness. Following existing studies +[17], [20], we attempt to select state-of-the-art techniques +designed for Java language as Java is the most targeted +language in APR community [7] and the existing patches +of real-world bugs are usually available in Java language +[12]. We first consider the recent empirical study by Wang et +al. [13] to identify existing APCA techniques. We then select +recent advanced studies [12], [20] that are not included in +Wang et al. [13]. +In general, following existing work [13], [20], the existing +APCA techniques can be categorized into static, dynamic +and learning-based APCA techniques according to whether +test execution is needed or deep learning techniques are +adopted (mentioned in Section 2). Meanwhile, according +to whether the ground-truth patch is required, they can be +further categorized into two categories (i.e., with or without +oracle). Particularly, similar to our proposed method APPT, +CHCHE and embedding learning techniques adopt repre- +sentation models to embed changed code and a deep learn- +ing classier to predict patch correctness. Such techniques +can be further considered as representation learning APCA +techniques. +The details of the selected APCA techniques are illus- +trated in Table 3. The first column lists three APCA cate- +gories. The second and third columns list whether the oracle +information is equipped. We also list the representation +learning techniques (e.g., APPT) in the light gray box. We +summarize the selected techniques as follows. +4.3.1 +Dynamic-based APCA Techniques +Dynamic-based techniques are designed to distinguish cor- +rect patches from overfitting patches based on the outcome +or the execution traces of the original or generated test cases. +Simple Test Generation. The overfitting issue is preva- +lent in the repair process due to the weak adequacy of +existing test cases. Thus, researchers use test case generation +tools to generate extra test cases based on the fixed program +and check whether or not the generated patches that pass +the original test cases can pass the extra test cases [23], +[66]. In this work, we adopt Evosuite [15] and Randoop [16] +as the test case generation tools, as they have been widely +investigated in previous studies. +DiffTGen. Xin et al. [62] identify overfitting patches by +executing test cases generated by an external test generator +(i.e., Evosuite). Different from simple test generation gener- +ating test cases randomly, DiffTGen generates test cases to +uncover the syntactic differences between the patched and +buggy program. A plausible patch is regarded as overfitting +if the output of the patched program is not the same as +that of the correct program. DiffTGen needs a human- +written patch as a reference and requires providing human- +amenable testing information for the developers to provide +oracles the generated test cases. +Daikon. Daikon is a dynamic-based technique based on +the program invariant with oracle information. Yang et al. +[63] adopt the program invariant to explore the differences +between an overfitting and a correct patch. A patch is +considered correct if its inferred invariant is identical to that +of the ground-truth. If there exists a different comparison, +the patch is considered overfitting. +PATCH-SIM. Xiong et al. [17] consider the execution +traces of the passing tests on the buggy and patched pro- +grams are likely to be similar, while the execution traces +of failing tests on the buggy and patched programs are +likely different. Based on the concept, they approximate the +correctness of a patch based on the execution trace without +the oracle information. PATCH-SIM adopts Randoop to +generate additional test cases to collect dynamic execution +information. In this work, we also replace Randoop with +Evosuite to comprehensively explore the impact of test +generation techniques (denoted as E-PATCH-SIM). +Opad. Yang et al. [67] adopt fuzzing testing to generate +new test cases and employ two test oracles (crash and +memory-safety) to enhance the validity checking of patches. +The original implementation of Opad is not designed for +Java language and uses American Fuzz Lop (AFL) as the +fuzzing technique. In this work, following recent studies +[13], [20], we replace AFL with Randoop and Evosuite to +generate new test cases on the Java programs and denote +them as R-Opad and E-Opad, respectively. +4.3.2 +Static-based APCA Techniques +Static-based techniques usually adopt static analysis tools to +extract some designed static features and then check patch +correctness based on such features. + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +8 +ssFix. ssFix [64] is a static-based technique that utilizes +token-based syntax representation to generate patches with +a higher probability of correctness. ssFix first performs a +syntactic code search to find code snippets from a codebase +that is syntax-related to the context of a bug to generate +correct patches, and then prioritizes the patches based on +the modification types and the modification sizes. +CapGen. Wen et al. [43] propose three aspects of context +information (i.e., genealogy contexts, variable contexts and +dependency contexts) embedded in an AST node and its +surrounding codes to prioritize correct patches over overfit- +ting ones. In this work, following recent studies [13], [20], +we extract the three context information as static features to +investigate patch correctness assessment. +Anti-patterns. Tan et al. [14] define a set of rules that +essentially capture disallowed modifications to the buggy +program, and a patch is overfitting if it falls into the rules. +A recent study [13] has shown that the manually-defined +anti-patterns may have false positives for correct patches, +resulting in destructive effects in patch correctness predic- +tion. +S3. Le et al. [65] assume that a correct patch is often +syntactically and semantically close to a buggy code snippet. +Thus, they adopt six syntactic features (i.e., AST differenc- +ing, cosine similarity and locality of variables and constants) +and semantic features (i.e., model counting, output coverage +and anti-patterns) to measure the distance between a candi- +date patch and the buggy code snippet. +4.3.3 +Learning-based APCA Techniques +Learning-based techniques can predict whether a plausible +patch is correct or not based on machine learning tech- +niques. +ODS. Ye et al. [18] first extract 202 code features at the +abstract syntax tree level and then use supervised learning +to learn a probabilistic model automatically. The results +show that ODS can achieve better prediction performance +than the dynamic-based technique PATCH-SIM with a faster +speed. +CACHE. Lin et al. [20] propose a context-aware APCA +technique CACHE by taking both the changed code snippet +and the correlated unchanged code snippet into considera- +tion. CACHE first parses the patched code snippet into AST +representation and then utilizes the AST path technique to +capture the structure information. +Random Forest. Wang et al. [13] investigate the effective- +ness of adopting deep learning models to predict patch cor- +rectness based on eight static features (two from ssFix, three +from S3, and three from CapGen). To integrate the static +features, six widely-used classification models (including +Random Forest, Decision Table, J48, Naive Bayes, Logistic +Regression, and SMO) are adopted. The results demonstrate +that Random Forest can achieve both superior precision and +recall performance. In this work, following existing work +[20], we also adopt Random Forest to predict the patch +correctness based on the integrated static features. +Embedding Learning. Tian et al. [12] propose to leverage +representation learning techniques to produce embedding +for buggy and patched code snippets and then adopt su- +pervised learning classifies to predict patch correctness. In +particular, nine representation learning APCA techniques +are evaluated, involving three embedding techniques (i.e., +CC2vec, BERT and Doc2Vec) and three classifiers (logistic +regression, decision tree and naive bayes). +4.4 +Model Selection +To the best of our knowledge, APPT is the first automated +patch correctness prediction technique by fine-tuning the +existing pre-trained model. In this paper, we adopt BERT +as the encoder stack due to its powerful performance in +previous work [24]. +Specifically, BERT is pre-trained on large amounts of text +data with two self-supervised goals, i.e., masked language +modeling (MLM) and next sentence prediction (NSP). MLM +aims to let the model predict the masked words by masking +15% of words in each sentence randomly. NSP aims to +further improve the model’s ability to understand the rela- +tionship between two sentences by letting the model predict +whether the given sentence pair is continuous. The model +then can be fine-tuned to adapt to some specific downstream +tasks and has achieved remarkable state-of-the-art results +on a variety of natural language processing tasks, such as +question answering and language inference. +There exist two model architectures at different sizes, i.e., +BERTbase and BERTlarge [24]. The former has 12 layers and +12 attention heads, and the embedding size is 768, while +the latter has a double layer number and 16 attention heads, +and the embedding size is changed to 1024. In this paper, we +do not modify the vocabulary size and use the pre-trained +BERTbase as the fine-tuning starting point instead of starting +from scratch. +In this paper, APPT is conceptually and practically gen- +eralizable to various pre-trained models. We also select +CodeBERT and GraphCodeBERT as the encoder stack to +evaluate the scalability of APPT. CodeBERT and Graph- +CodeBERT share the same model architecture as BERT, +while utilizing paired natural language and programming +language to pre-train the model to support code-related +tasks (mentioned in Section 8.2.2). +4.5 +Evaluation Metrics +We evaluate the prediction performance of various APCA +approaches by accuracy, precision, recall, F1-score and AUC +metrics, which have been widely adopted in patch correct- +ness assessment research and other classification tasks [12], +[20]. Given the number of true positives (TPs, a TP refers +to an overfitting patch that is identified as overfitting), false +positives (FPs, a FP refers to a correct patch that is identified +as overfitting), false negatives (FNs, a FN refers to an over- +fitting patch is identified as correct) and true negatives (TNs, +a TN refers to a correct patch that is identified as correct), +the metrics are defined as follows: +• Accuracy: the proportion of correctly reported (whether +the patch is correct or not) patches. Accuracy measures +the probability that the prediction of APCA techniques is +correct. +Accuracy = +TP + TN +TP + FP + FN + TN +(12) +• Precision: the proportion of real overfitting patches over +the reported overfitting patches. Precision measures how + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +9 +much we can trust the APCA techniques when it predicts +a patch as overfitting. +Precision = +TP +TP + FP +(13) +• Recall: the proportion of reported overfitting patches +over all the real overfitting patches. Recall measures the +ability of the APCA techniques to find all the overfitting +patches in the dataset. +Recall = +TP +TP + FN +(14) +• F1-score: twice the multiplication of precision and recall +divided by the sum of them. F1-score measures the trade-off +between precision and recall by taking their harmonic mean. +F1-score = 2 ∗ Precision ∗ Recall +Precision + Recall +(15) +• AUC: the entire two-dimensional area underneath the +entire receiver operating characteristic curve. AUC mea- +sures the probability that the classifier will rank a randomly +chosen overfitting patch higher than that of a randomly +chosen correct patch. The higher the AUC, the better the +APCA techniques is at predicting real overfitting patches as +overfitting and real correct patches as correct. +AUC = +� I +�Poverfitting , Pcorrect +� +M × N +I +�Poverfitting, Pcorrect +� = +� +� +� +1, Poverfitting > Pcorrect +0.5, Poverfitting = Pcorrect +0, Poverfitting < Pcorrect +(16) +where M and N denote the number of overfitting and +correct patches, while Poverfitting and Pcorrect denote the pre- +diction probability for the overfitting and correct patches. +4.6 +Implementation Details +All of our approaches are built based on PyTorch frame- +work3. We use the Hugging Face4 implementation version +of BERT in our work. Considering previous work recom- +mendation [26], [40], we utilize “bert-base-uncased” (refer +to BERTbase) as the initial point, as the base version is quite +lightweight to employ in practice with comparable effec- +tiveness compared against the large version. There exist 12 +layers of transformer blocks and 12 self-attention heads in +the “bert-base-uncased” model. The optimizer is Adam [37] +with 5e − 5 learning rate. The batch size is 16 and dropout +rate is 0.5. We train for most 50 epochs and the max length +of the input is set to 512 due to model limitation. +All the training and evaluation of our methods are +conducted on one Ubuntu 18.04.3 server with two Tesla +V100-SXM2 GPUs. +5 +RESULTS AND ANALYSIS +5.1 +RQ1: Comparing with Representation Learning- +based APCA Techniques +5.1.1 +Experimental Design +As discussed in Section 4.3, APPT, CACHE and embedding +learning techniques (i.e., techniques within the light gray +3. PyTorch. https://pytorch.org/, accessed August 2022 +4. Hugging Face. https://huggingface.co/, accessed August 2022 +box in Table 3) can be categorized as representation learning +APCA techniques. In this section, we aim to explore the per- +formance of APPT when compared with these representa- +tion learning techniques. In particular, embedding learning +techniques [12] mainly adopt embedding models (i.e., BERT, +Doc2Vec, and CC2Vec) to embed buggy and patched code +fragments, and then train classification models (i.e., Deci- +sion Tree, Logistic Regression, and Naive Bayes) to predict +patch correctness. Following previous study [20], we also +consider two additional embedding models (i.e., code2vec +and code2seq) in the experiment. Meanwhile, CACHE can +also be considered as a representation learning technique, +which incorporates the context information in embedding +code changes, and trains a deep learning classifier to predict +the patch correctness. +In total, 16 representation learning techniques are con- +sidered in our experiment, involving five embedding tech- +niques multiplied by three classification models, and one +context-aware representation learning technique CACHE. +Following the previous study [12], we perform a 5-fold +cross-validation on both the small and large datasets for +comparison. +5.1.2 +Results +Comparison results against the existing representation +learning techniques are presented in Table 4 to Table 5 for +the both small and large dataset. The first column lists the +three classifiers and the second column lists the five embed- +ding approaches. The remaining columns list the detailed +values of accuracy, precision, recall, F1-score and AUC met- +rics, respectively. We present the most recent representation +learning work CACHE and our APPT in the bottom part of +Table 4 and Table 5. It can be observed that APPT achieves +the best performance under each experimental setting. +On the small dataset, APPT is around 3.6%, 1.2%, 4.8%, +2.9% and 3.1% higher than the state-of-the-art technique +CACHE in terms of all metrics (i.e., 79.0% vs. 75.4% for +accuracy, 80.7% vs. 79.5% for precision, 81.3% vs. 76.5% for +recall, 80.9% vs. 78.0% for F1-score, and 83.4% vs. 80.3% +for AUC). Compared with all representation learning tech- +niques, APPT achieves the best performance in terms of +accuracy, precision, F1-score and AUC metrics. In particular, +the values of APPT on the accuracy and precision metrics +are 79.0% and 80.7%, respectively, while the optimal values +of all other techniques are 75.4% and 79.5%. This suggests +that APPT can generally achieve the most accurate predic- +tions, and the patches identified as overfitting by APPT are +of high confidence to be overfitting. Regarding recall, the +values of CC2vec and code2vec can sometimes exceed those +of APPT since they tend to classify most patches as overfit- +ting (e.g., CC2vec with Naive Bayes classifies 1,051 out of +1,183 patches as overfitting and thus achieves a high recall +of 94.6%). However, these techniques achieve relatively low +precision (e.g., CC2vec with Naive Bayes classifier has only +72.2% for recall). On the contrary, APPT can achieve a high +recall exceeding 81% while maintaining a high precision of +79.5%. +On the large dataset, we can find APPT achieves over +99% for the five metrics, outperforming all existing ap- +proaches. For example, APPT reaches 99.9% in terms of +AUC, which is 1.0% higher than the second highest value + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +10 +TABLE 4: Effectiveness of APPT compared with representation learning-based APCA techniques on the small dataset +Classifier +Embedding +Accuracy +Precision +Recall +F1-score +AUC +Decision Tree +BERT +63.5% +65.3% +70.9% +67.9% +63.7% +CC2vec +66.1% +69.4% +68.0% +68.7% +66.5% +code2vec +65.1% +68.1% +68.3% +68.1% +64.4% +code2seq +60.1% +63.5% +64.0% +63.7% +60.0% +Doc2Vec +61.2% +64.5% +65.3% +64.8% +60.8% +Logistic Regression +BERT +64.8% +66.5% +72.4% +69.2% +68.7% +CC2vec +64.9% +62.4% +90.1% +73.7% +68.6% +code2vec +66.8% +68.6% +72.9% +70.6% +70.2% +code2seq +60.7% +63.3% +67.6% +65.3% +63.1% +Doc2Vec +63.7% +65.7% +70.8% +68.0% +68.9% +Na¨ıve Bayes +BERT +61.6% +64.8% +65.7% +65.0% +64.7% +CC2vec +60.0% +58.3% +94.6% +72.2% +58.1% +code2vec +57.7% +58.1% +81.5% +67.8% +55.6% +code2seq +57.0% +59.0% +70.5% +64.2% +60.6% +Doc2Vec +64.1% +65.8% +72.4% +68.7% +67.0% +CACHE +75.4% +79.5% +76.5% +78.0% +80.3% +APPT +79.0% +80.7% +81.3% +80.9% +83.4% +TABLE 5: Effectiveness of APPT compared with representation learning techniques on the large dataset +Classifier +Embedding +Accuracy +Precision +Recall +F1-score +AUC +Decision Tree +BERT +95.7% +93.9% +97.4% +95.6% +95.9% +CC2vec +95.6% +95.4% +95.7% +95.5% +95.7% +code2vec +95.0% +93.2% +96.6% +94.9% +95.4% +code2seq +92.2% +91.0% +93.2% +92.3% +92.4% +Doc2Vec +85.1% +84.2% +85.3% +84.7% +85.3% +Logistic Regression +BERT +82.4% +83.6% +79.4% +81.4% +91.0% +CC2vec +91.2% +96.1% +85.4% +90.4% +95.0% +code2vec +89.6% +88.6% +90.2% +89.4% +95.0% +code2seq +91.5% +90.5% +92.2% +91.4% +96.0% +Doc2Vec +90.4% +91.9% +88.0% +89.9% +96.1% +Na¨ıve Bayes +BERT +68.2% +80.3% +45.7% +58.2% +74.6% +CC2vec +78.4% +94.8% +58.6% +72.5% +92.4% +code2vec +61.4% +68.7% +37.4% +48.4% +69.3% +code2seq +70.3% +76.8% +55.5% +64.5% +78.9% +Doc2Vec +81.2% +86.4% +75.5% +78.9% +88.9% +CACHE +98.6% +98.9% +98.2% +98.6% +98.9% +APPT +99.1% +99.1% +99.1% +99.1% +99.9% +obtained from the most recent technique CACHE (i.e., +98.9%). This suggests that APPT is more capable of dis- +tinguishing correct and overfitting patches than CACHE. +Besides, the improvement against CACHE for accuracy, +precision, recall and F1-score metrics achieves 0.5%, 0.3%, +0.9% and 0.5%, respectively. We also find that the perfor- +mance achieved on the large dataset is commonly higher +than that achieved on the small dataset. For example, the +average value among the five metrics increases from 81.06% +to 99.26%, resulting in a 22.5% improvement rate. Based on +our analysis on the two datasets, the possible reason for this +improvement is that bugs on the large dataset are usually +simple. We observe that all ManySStuBs4J patches on the +large dataset are single-line operations, while patches on +the small dataset usually cross multiple lines (e.g., more +than 40% of Defects4J developer patches are multiple line +patches [20]). It is easy for the neural networks to learn +the correctness distribution of such simple code changes. +Meanwhile, the difference in patch scale between the two +datasets may be the second reason. We find there exist 49,694 +patches on the large dataset, which is 42 times larger than +that of the small dataset. The amount of training data is +often the single most dominant factor that determines the +performance of the neural networks [68]. More available +patches benefit the neural networks to learn diverse code +changes better. +Answer to RQ1: Overall, our analysis on representation +learning techniques reveals that (1) APPT can outper- +form a state-of-the-art representation learning technique +CACHE under all metrics and datasets. (2) on the small +dataset, APPT achieves 79.0% for accuracy and 83.4% for +AUC, which surpass CACHE by 3.6% and 3.1%. (3) on the +large dataset, APPT exceeds 99% on all metrics, yet none of +existing representation learning techniques achieves that. +5.2 +RQ2: Comparing with Traditional and Learning- +based APCA Techniques +5.2.1 +Experimental Design +In this section, we aim to further compare the proposed +method APPT with the existing APCA techniques. We select +the remaining techniques mentioned in Section 4.3 (except + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +11 +TABLE 6: Effectiveness of APPT compared with the traditional and learning-based APCA technique +Category +APCA +Accuracy +Precision +Recall. +F1-score +Dynamic-based +w-oracle +Evosuite +65.9% +99.1% +53.5% +69.5% +Randoop +51.3% +97.4% +33.8% +50.2% +DiffTGen +49.6% +97.4% +30.6% +46.6% +Daikon +76.1% +89.9% +73.7% +81.0% +wo-oracle +R-Opad +34.9% +100.0% +10.2% +18.5% +E-Opad +37.7% +100.0% +14.7% +25.6% +PATCH-SIM +49.5% +83.0% +38.9% +53.0% +E-PATCH-SIM +41.7% +82.1% +25.8% +39.3% +Static-based +Anti-patterns +47.6% +85.5% +33.5% +48.1% +S3 +69.7% +79.3% +78.9% +79.0% +ssFix +69.2% +78.9% +78.8% +78.8% +CapGen +68.0% +78.3% +77.4% +77.8% +Learning-based +Random Forest +72.5% +87.0% +89.1% +88.0% +ODS +88.9% +90.4% +94.8% +92.5% +APPT +90.4% +91.5% +96.0% +93.6% +representation learning techniques discussed in RQ1). In +total, 14 APCA techniques are considered in the experiment, +involving four static techniques (Anti-patterns, ssFix, Cap- +Gen and S3), eight dynamic techniques (Evosuite, Randoop, +DiffTGen, Daikon, R-Opad, E-Opad, PATCH-SIM and E- +PATCH-SIM) and two learning techniques (Random Forest +and ODS). +As it is time-consuming to run all the techniques (espe- +cially for dynamic and learning ones), following the existing +work [20], we reuse the released results from the recent work +[13], [18], [20]. We collect the detailed results of all selected +APCA techniques from Lin et al. [20], which are concluded +based on 902 patches (i.e., Wang et al. [13] in Table 2) and +a 10-fold cross-validation. To fairly compare with all the +state-of-the-art techniques, we perform our experiment in +the same experimental setting. +5.2.2 +Results +The experiment results are listed in Table 6. The first two +columns list the selected techniques and their corresponding +categories. The remaining columns list the detailed values of +accuracy, precision, recall and F1-score metrics. +Compared with traditional dynamic-based and static- +based APCA techniques, we can find that APPT reaches +90.4%, 96.0% and 93.6% in terms of accuracy, recall and +F1-score, respectively. Specifically, APPT achieves the best +overall performance with the three metrics, and none of +the previous techniques exceeds 90%. As for precision, more +than 91% of patches reported by APPT are indeed overfit- +ting patches, which is better than all static-based techniques +and three dynamic-based techniques (i.e., Daikon, PATCH- +SIM, and E-PATCH-SIM). Although some dynamic ones +have higher precision values, it is time-consuming to gen- +erate additional test cases and collect run-time information. +More importantly, the recall of these techniques is usually +low (e.g., 10.3% for R-Opad), or the ground-truth oracle is +needed (e.g., Evosuite and Randoop techniques), limiting +the application of such techniques in practice. +Compared with learning-based techniques, we find that +APPT still performs better than a state-of-the-art technique +ODS with respect to all four metrics (90.4% vs. 88.9% for +accuracy, 91.5% vs. 90.4% for precision, 96.0% vs. 94.8% for +recall, 93.6% vs. 92.5% for F1-score, respectively). Overall, +the improvement against Random Forest and ODS reaches +4.5%∼17.9% and 1.1%∼1.5%. Considering that it is expen- +sive for ODS to extract hundreds of manually-designed +code features at AST level, our approach simply adopting +the pre-trained model to encode a sequence of tokens is +even more promising. We also highlight this direction of +integrating code-aware features (e.g., code edits and AST +representation) with pre-trained models for patch correct- +ness assessment. +Answer to RQ2: Overall, our comparison results reveal +that, (1) APPT can achieve remarkable performance com- +pared to exiting static-based techniques with a high re- +call reaching 96.0%. (2) APPT can achieve higher pre- +cision than a state-of-the-art dynamic-based technique +PATCH-SIM by 8.5%. (3) compared with existing learning- +based techniques, APPT can achieve the best performance +among all metrics. +5.3 +RQ3: The Impact Analysis +5.3.1 +Experimental Design +To further explore how different fine-tuning choices affect +the prediction performance of pre-trained models, we first +consider and replace the head-only token truncation with +other truncation methods, such as hybrid, mid-only and tail- +only token truncation. We then adopt different methods to +merge the buggy method vector and patched method vec- +tor, such as concatenate, additional, subtraction, and prod- +uct operation. We also mix the above-mentioned merged +vectors as an additional concatenation method. Recently, +following the BERT model architecture, researchers use +some code-related pre-trained tasks to capture the semantic +connection between natural language and programming +language, so as to further adapt these pre-training models +for programming language. Thus, we replace the BERT with +two advanced models pre-trained with the programming +language, i.e., CodeBERT [28] and GraphCodeBERT [29]. +5.3.2 +RQ3.1 Results: The Impact of Token Truncation +Choice +Table 7 presents the prediction results under different trun- +cation choices. The first column lists the two datasets. The + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +12 +TABLE 7: Effectiveness of APPT with different truncation choices. +Dataset +Truncation +Accuracy +Precision +Recall +F1-score +AUC +small +APPThybrid +79.04% +80.67% +81.34% +80.92% +83.43% +APPThead +79.72% +80.84% +83.17% +81.76% +82.55% +APPTmid +75.48% +78.27% +78.41% +77.85% +81.34% +APPTtail +73.20% +76.00% +76.40% +75.38% +78.45% +large +APPThybrid +99.13% +99.09% +99.13% +99.11% +99.86% +APPThead +99.04% +99.17% +98.86% +99.01% +99.54% +APPTmid +97.36% +96.62% +98.17% +97.35% +98.18% +APPTtail +97.85% +98.28% +97.30% +97.77% +99.49% +TABLE 8: Effectiveness of APPT with different concatenation choices. +Dataset +Truncation +Accuracy +Precision +Recall +F1-score +AUC +small +APPTconcat +79.04% +80.67% +81.34% +80.92% +83.43% +APPTaddition +69.83% +70.24% +80.12% +73.83% +75.44% +APPTsubtraction +71.38% +72.42% +77.27% +74.72% +75.59% +APPTproduct +63.27% +62.37% +96.32% +74.81% +66.46% +APPTmix +80.90% +82.21% +83.18% +82.64% +83.46 +large +APPTconcat +99.13% +99.09% +99.13% +99.11% +99.86% +APPTaddition +98.96% +98.80% +99.07% +98.93% +99.81% +APPTsubtraction +97.31% +99.14% +95.29% +97.17% +99.46% +APPTproduct +98.82% +98.88% +98.69% +98.78% +99.78% +APPTmix +99.10% +98.99% +99.17% +99.08% +99.79% +second column lists the four truncation choices, i.e., head- +only, mid-only, tail-only and hybrid. The remaining columns +list the detailed values of accuracy, precision, recall and F1- +score and AUC metrics. +On the small dataset, we can find that the head-only +approach achieves the optimum performance for accuracy +(79.72%), precision (80.84%), recall (80.84%) and F1-score +(81.76%), while the hybrid approach achieves the optimum +AUC score (83.43%). The mid-only approach, considering +the middle tokens in the buggy and patched methods, +achieves the third-best performance for all metrics, followed +by the tail-only approach. Similar performance can be ob- +served on the large dataset. For example, the head-only and +hybrid approaches have the best performance in all metrics, +while the mid-only and tail-only ones are the following. The +results demonstrate that the head-only approach extracting +the beginning code tokens is effective in distinguishing the +buggy and patched code snippets for the pre-trained model. +5.3.3 +RQ3.2 Results: The Impact of The Vector Concate- +nation Choice +Table 8 presents the prediction results under different con- +catenation choices. The first column lists the two datasets. +The second column lists the five concatenation choices, i.e., +concat, addition, subtraction, product and mix. The remain- +ing columns list the detailed values of accuracy, precision, +recall and F1-score and AUC metrics. +On the small dataset, although conceptually simple, +APPTconcat can obtain 79.04%, 80.67%, 81.34%, 80.92% and +83.43% for accuracy, precision, recall, F1-score and AUC +metrics, four of which are highest among all investigated +concatenation methods. APPTproduct has the highest recall +score (96.32%), while it performs worse than APPTconcat +by 15.77%, 18.30%, 6.11% and 16.97% for the other four +metrics. APPTaddition and APPTsubtraction perform the ad- +dition and subtraction operation for buggy and patched +vectors, and have similar performance for all metrics. Mean- +while, a mixed method APPTmix that applies these different +comparison functions to represent the changed embedding +vector can achieve better results than APPTconcat, which +is also consistent with the existing study results [12], [32]. +Such results indicate that the pre-trained model can better +capture the code change information by integrating differ- +ent concatenation ways. On the large dataset, APPTconcat +achieves the best performance in accuracy, F1-score and +AUC metrics, while APPTsubtraction and APPTmix perform +best in precision and recall respectively. The difference in +performance is similar as the methods have relatively high +metric values. For example, all metric values are higher than +99% for APPTconcat and APPTmix. +5.3.4 +RQ3.3 Results: The Impact of Pre-trained Model +Choice +Table 9 demonstrates the predicted performance of three +pre-trained models. The first column lists the two datasets. +The second column lists the three models , i.e., BERT, +CodeBERT, and GraphCodeBERT. The remaining columns +list the detailed values of accuracy, precision, recall and F1- +score and AUC metrics. +Generally speaking, all of the adopted models achieve a +higher performance than state-of-the-art technique CACHE +on all metrics. For example, on the small dataset, BERT, +CodeBERT and GraphCodeBERT reach 80.9%, 83.3%, and +83.5% with respect to the F1-score, which is 2.9%, 5.3%, +and 5.5% higher than CACHE, respectively. A similar im- +provement can also be observed on the large dataset. This +demonstrates the model choice may not impact the per- +formance dramatically, and pre-trained models can consis- +tently achieve state-of-the-art performance. +Specifically, to compare the performance of different pre- +trained models, we can observe that both CodeBERT and +GraphCodeBert achieve a better value for all metrics on the + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +13 +TABLE 9: Effectiveness of APPT with different pre-trained models. +Dataset +Model +Accuracy +Precision +Recall +F1-score +AUC +small +APPTbert +79.04% (↑ 3.6) +80.67% (↑ 1.2) +81.34% (↑ 4.8) +80.92% (↑ 2.9) +83.34% (↑ 3.1) +APPTcodebert +81.49% (↑ 6.1) +82.10% (↑ 2.6) +84.73% (↑ 8.2) +83.35% (↑ 5.3) +85.32% (↑ 5.0) +APPTgraphcodebert +81.83% (↑ 6.4) +83.68% (↑ 4.2) +83.63% (↑ 7.2) +83.47% (↑ 5.5) +85.79% (↑ 5.5) +large +APPTbert +99.13% (↑ 0.5) +99.09% (↑ 0.2) +99.13% (↑ 0.9) +99.11% (↑ 0.5) +99.86% (↑ 1.0) +APPTcodebert +99.57% (↑ 1.0) +99.71% (↑ 0.8) +99.40% (↑ 1.2) +99.55% (↑ 1.0) +99.89% (↑ 1.0) +APPTgraphcodebert +99.61% (↑ 1.0) +99.61% (↑ 0.7) +99.59% (↑ 1.4) +99.60% (↑ 1.0) +99.90% (↑ 1.0) +↑ denotes performance improvement against state-of-the-art technique CACHE. +small dataset. This superior performance also generalizes +to large datasets, where CodeBERT and GraphCodeBert +have better or competitive (e.g., AUC) performance on the +metrics. One possible explanation for this is that BERT +is designed for natural language processing tasks, while +CodeBERT and GraphCodeBERT regard a source code as +a sequence of tokens or graph representation and then pre- +train models on source code to support code-related tasks. +This indicates that although pre-trained models in NLP +can achieve state-of-the-art performance for assessing patch +correctness, the adoption of pre-trained models targeting +source code can further boost the improvement. +Answer to RQ3: The performance under different choices +demonstrates that: (1) the beginning code tokens can rep- +resent the buggy and patched code snippets well for the +pre-trained model; (2) the concat of buggy and patched +vectors is better than other methods to distinguish the +changed code snippets, while the integration of differ- +ent concatenation ways can achieve optimum results. (3) +advanced pre-trained models can provide a stable even +better performance. +6 +DISCUSSION +6.1 +Threats to Validity +To facilitate the replication and verification of our exper- +iments, we have made the relevant materials (including +source code, trained models, and patch data) available. +Despite that, our study still faces some threats to validity, +listed as follows. +The first threat to validity lies in the patch benchmark. +We focus on the Defects4J database with reproducible real +faults and collect 1,183 patches generated by existing APR +tools. However, the patch benchmark may not consider all +available APR tools. To address this, following the latest +work [20], we include the 22 APR tools covering four cate- +gories. It should be worth noting that although the learning- +based category contains only SequenceR, it contains 73 +patches, which is the largest number for a single APR tool +[20]. We also mitigate the potential bias by using multiple +evaluation metrics to exhaustively assess the APCA tech- +niques. Further, we adopt another large benchmark contain- +ing 49,694 real-world patches to evaluate the generalization +ability of the studied techniques. Overall, to the best of our +knowledge, the used patch benchmarks are the largest set +explored in the literature on patch correctness assessment. +The second threat to validity is that the performance of +APPT may not generalize to other pre-trained models. We +select BERT in our experiment due to its powerful perfor- +mance in recent code-related works. However, it is unclear +whether the conclusions in our experiment (discussed in +Section 5) can be maintained when using other pre-trained +models. We have mitigated the potential threat by using +CodeBERT and GraphCodeBERT to demonstrate the per- +formance of APPT under different pre-trained models. The +investigated pre-trained models include both code-related +ones (e.g., CodeBERT) and natural language-specific ones +(e.g., BERT). We also rely on two diverse patch benchmarks +to ensure the generality of the experimental conclusions. +The last threat to validity is the implementation of the +baselines. In our work, we compare APPT against a wide +range of APCA techniques with different categories. Imple- +menting these baselines may introduce a potential threat +to the internal validity. To mitigate this threat, following the +recent work [20], we conduct the experiment under the same +setting and reuse the released results from the original work +[12], [13], [20]. Further, we carefully check the reused results +and publicly release all our materials for further verification. +6.2 +Comparison with BATS +In our work, following some recent APCA work [12], [13], +30 related APCA techniques with different categories (i.e., 16 +representation learning-based ones, 9 dynamic-based ones, +4 static-based ones and 2 learning-based ones) are compared +in our experiment (discussed in Section 5). To the best of our +knowledge, the selected baselines are the largest set on patch +correctness prediction in the literature. However, there may +exist other possible techniques that could have been used. +For example, the recent BATS [19] predicts patch correctness +based on the similarity of failing test cases, which can be +complementary to the state-of-the-art APCA techniques. We +do not include BATS in our experiment (discussed in Section +5) because it requires historical test cases as the search space +for searching similar cases, which are not available in our +dataset. +We then perform an additional evaluation by assessing +APPT on the dataset provided in BATS. However, BATS +fails to assess some plausible patches as it considers only +historical test cases with the similarity which are higher than +a threshold. For example, BATS with 0.8 threshold value is +able to predict only 8.9% (114/1278) of the plausible patches. +Thus, we compare APPT against BATS with 0.0 threshold +value, which can perform prediction for all patches. We also +compare APPT against BATS with 0.8 threshold value, as +it achieves the best recall, F1-score and AUC performance +among all threshold values. The results are presented in +Table 10. The first column lists APPT and BATS (with 0.0 and + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +14 +TABLE 10: Comparison with a state-of-the-art learning- +based APCA technique BATS. +APCA +#Patch +Accurancy +Preciosn +Recall +F1-score +BATS (0.0) +1278 (1278) +52.50% +48.81% +62.82% +54.94% +BATS (0.8) +114 (1278) +67.54% +63.16% +84.21% +72.18% +APPA +1278 (1278) +85.05% +83.39% +84.38% +83.88% +0.8 threshold values, respectively). The second column lists +the number of predicted patches. Each cell is represented as +x(y), where x is the number of patches predicted by APPT +and BATS and y is the total number of patches in the dataset. +The remaining columns list the detailed performance under +the metrics. We can find APPT achieves 83.39%∼85.05%, +improving the metrics by 21.56%∼34.58% when compared +with BATS (threshold is set to 0.0). When the threshold of +BATS is set to 0.8, APPT can still improve the metrics by +12.40% on average while predicting 91.1% more plausible +patches. Overall, the results demonstrate that APPT per- +forms better than BATS in terms of the number of predicted +patches and the prediction metrics. +7 +IMPLICATION AND GUIDELINE +Based on the observations in our experiment, we can sum- +marize the following essential practical guidelines for future +patch correctness assessment studies. +Simple features can work. Our study demonstrates that +APPT, representing source code as a sequence of tokens, +performs even better than the existing learning techniques +(e.g., CACHE) considering complex code-aware characteris- +tics (e.g., abstract syntax tree). Also, the token sequences can +already outperform manually-designed static features (e.g., +the line number) and time-consuming dynamic features +(e.g., code coverage) in this work. Such observations indi- +cate that simple features, such as code sequences, should not +be just ignored and a systematic study to explore the impact +of different code representations is needed in the future. In +fact, they should be considered and even integrated with +different features (e.g., data flow graph) to design more +advanced patch correctness assessment techniques. +The quality of the training dataset is important. We +can find that APPT achieves 91.5% precision in Table 4 +while the precision is decreased by 10.8% in Table 6. Similar +performance can also be observed in Lin et al. [20]. The +results show that more training data cannot always lead to +better performance for patch correctness assessment. It is +crucial to automatically select the most informative training +set that represents the whole patch benchmarks to optimize +the prediction accuracy. For example, it is interesting to +explore how the number of patches is distributed across fix +patterns and how to select balanced patches for each fix +pattern. Future work can also be conducted to investigate +training data selection approaches targeting specific bug +benchmarks under prediction or even specific bug types +under prediction. +Pre-trained model-based APCA techniques require +more attention. Our results show that the BERT-based APPT +performs even better than the state-of-the-art APCA tech- +niques. Also, the CodeBERT-based and GraphCodeBERT- +based APPT can further enhance the prediction effective- +ness. Such observation motivates future researchers to in- +vestigate more advanced APCA techniques by employing +different pre-trained models. For example, it is interesting +to propose domain-specific pre-trained models by designing +repair-related pre-training tasks. Meanwhile, thorough eval- +uations are recommended to explore how different features, +such as bug types and fix patterns, influence the perfor- +mance of pre-trained models in patch correctness prediction. +8 +RELATED WORK +In this paper, we adopt pre-trained language models to +predict patch correctness generated by off-the-shelf auto- +mated program repair tools. Our work is related to auto- +mated program repair, patch correctness assessment and +pre-trained models. We have introduced the existing work +about patch correctness assessment in Section 4.3. Thus, in +this section, we focus on and discuss the existing work on +automated program repair techniques (Section 8.1) and pre- +trained models (Section 8.2). +8.1 +Automated Program Repair +Over the past decade, researchers have proposed a variety of +techniques to generate patches based on different hypothe- +ses [1], [69]. Following recent work [2], [7], [11], we cate- +gorize them into four main categories: heuristic-based [38], +[41], [70], constraint-based [44], [45], [71], template-based +[5], [51], [52] and learning-based repair techniques [35], [39], +[40], [72]. +• Heuristic-based repair techniques. These techniques usu- +ally use a heuristic algorithm to find a valid patch by +iteratively exploring a search space of syntactic program +modifications [38], [41], [70]. Among them, GenProg [70] +proposed in the early days has been considered a seminal +work in this field, which uses genetic programming to +search for correct repairs. GenProg represents candidate re- +pairs as sequences of edits to source code and evaluate them +by the execution results of test cases. Those candidates that +pass more test cases are considered to have a higher fitness +and are iteratively applied to produce new candidates based +on mutation and crossover operations. The recent SimFix +technique [42] utilizes code change operations from existing +patches across different projects and similar code snippets +within the buggy project to build two search spaces. Then, +the intersection of the above two search spaces is further +used to search the final patch using basic heuristics. +• Constraint-based repair techniques. These techniques +mainly focus on repairing conditional statements, which +can repair more than half of the bugs repaired by existing +APR approaches [44], [45], [47]. In detail, these techniques +transform the patch generation into a constraint-solving +problem, and use a solver to obtain a feasible solution. +For example, Nopol [45] relies on an SMT solver to solve +the condition synthesis problem after identifying potential +locations of patches by angelic fix localization and collecting +test execution traces of the program. Among them, ACS [46] +refining the ranking of ingredients for condition synthesis +is considered one of the most advanced constraint-based +repair techniques [7]. +• Template-based repair techniques. These techniques gener- +ate patches by designing pre-defined fix patterns to mutate + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +15 +buggy code snippets with the retrieved donor code [5], [51], +[52]. For example, Liu et al. [5] revisit the repair performance +of repair patterns using a systematic study that evaluates the +effectiveness of a variety of fix patterns summarized from +the literature. Among them, the recent PraPR technique [73] +is able to generate plausible and correct patches for 148 and +43 real bugs, respectively, which is the largest number of +bugs reported as fixed for Defects4J when published. +• Learning-based repair techniques. These techniques at- +tempt to fix bugs enhanced by machine learning techniques +[30], [35], [39], [74]–[76] and are getting increasing attention +recently. For example, Tufano et al. [75] extensively evalu- +ate the ability of neural machine translation techniques to +generate patches from bug-fixes commits in the wild. Li et +al. [35] adopt a tree-based RNN encoder-decoder model (i.e., +DLFix) to learn code contexts and transformations from pre- +vious bug fixes. Lutellier et al. [39] propose a new context- +aware NMT architecture (i.e., CoCoNut) that represents the +buggy source code and its surrounding context separately, +to automatically fix bugs in multiple programming lan- +guages. +In our experiment, we select 22 representative APR tools +(e.g., SimFix, ACS, and SEQUENCER) from the four cate- +gories, representing state-of-the-art techniques in the corre- +sponding category. Then we evaluate APPT on the plausible +patches (i.e., passing the original test cases) generated by +these APR techniques. +8.2 +Pre-trained Model +Our approach is inspired by the application of pre-trained +models in NLP and code-related tasks. In this section, we +first introduce the existing studies about pre-trained models +in NLP (Section 8.2.1) and SE (Section 8.2.2). We then discuss +the application of pre-trained models to some code-related +tasks in SE (Section 8.2.3). +8.2.1 +Pre-trained Model in NLP +Recent work has demonstrated substantial gains on many +NLP tasks and benchmarks by pre-training on a large corpus +of text followed by fine-tuning on a specific task. For exam- +ple, Devlin et al. [24] propose a new language representation +model BERT to pre-train deep bidirectional representations +from the unlabeled text by jointly conditioning on both left +and right contexts in all layers. To explore the landscape +of transfer learning techniques for NLP, Raffel et al. [26] +propose a text-to-text transfer transformer T5 by introducing +a unified framework that converts all text-based language +problems into a text-to-text format. Brown et al. [25] pro- +pose an autoregressive language model GPT-3 without any +gradient updates or fine-tuning, with tasks and few-shot +demonstrations specified purely via text interaction with the +model. +In this work, we choose BERT to encode a given plau- +sible patch into a fixed-length representation vector as the +input of the deep learning classifier, due to the powerful +performance of BERT in previous work [77]. +8.2.2 +Pre-trained Model in SE +Inspired by the application of pre-trained models in NLP, +many researchers apply the pre-trained model to code- +related tasks. Instead of designing new network architec- +tures, SE researchers usually adopt existing architectures +in NLP and design some code-aware pre-training tasks +(e.g., code-AST prediction and bimodal dual generation) +to learn representations of the source code. Then the pre- +trained models are further fine-tuned to some diversified +code-related tasks such as code-code (clone detection, de- +fect detection, cloze test, code completion, code refinement, +and code-to-code translation), text-code (natural language +code search, text-to-code generation), and code-text (code +summarization) scenarios. +For example, Feng et al. [28] present a bimodal pre- +trained model (CodeBERT) for natural language and pro- +gramming languages by masked language modeling and +replaced token detection to support code search and code +documentation generation tasks. Guo et al. [29] present +the first pre-trained model (GraphCodeBERT) that leverages +code structure to learn code representation to improve code +understanding tasks (i.e., code search, clone detection, code +translation, and code refinement). Guo et al. [27] present +UniXcoder, a unified cross-modal pre-trained model for +programming language. UniXcoder utilizes mask attention +matrices with prefix adapters to control the behavior of the +model and leverages cross-modal contents such as AST and +code comment to enhance code representation. In contrast +to most studies pre-training a large-scale model from scratch +costly, we attempt to boost patch correctness assessment on +top of the existing pre-trained language model fine-tuning +paradigm. +In this work, to further explore the generalization ability +of APPT, we select other BERT-like models (i.e., CodeBERT +and GraphCodeBERT) as the encoder stack due to their +powerful performance in the code-related tasks. +8.2.3 +Applications of Pre-trained Model in SE +In addition to the above-mentioned typical code-related +tasks (e.g., automatic bug-fixing, injection of code mutants, +generation of asserts in tests and code summarization in +[78]), researchers have also applied pre-trained models to +some other domains (e.g., code completion, and program +repair) in SE. +For example, Cinisell et al. [77] evaluate the performance +of the BERT model in the task of code completion at different +granularity levels, including single tokens, one or multiple +entire statements. The results show that the model achieves +promising results superior to state-of-the-art n-gram mod- +els, and the model learns better on some specific datasets +(e.g., Android) when code abstraction is used. Ciborowska +et al. [79] apply BERT to the bug localization problem with +the goal of improved retrieval quality, especially on bug +reports where straightforward textual similarity would not +suffice. Recently, Salza et al. [80] investigate how transfer +learning can be applied to code search by pre-training and +fine-tuning a BERT-based model on combinations of natural +language and source code. Mashhadi et al. [81] propose +a novel pre-trained model-based APR technique by fine- +tuning CodeBERT on the ManySStuBs4J benchmark and +find the approach generates fix codes for different types of +bugs with comparable effectiveness and efficacy compared +with state-of-the-art APR techniques. + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. XXX, NO. XXX, XXX 2022 +16 +Although there exist some SE tasks (e.g., code review +and bug localization) benefitting from pre-trained models, +in this work, we perform the first application of pre-trained +models to predict the generated patch correctness in auto- +mated program repair. +9 +CONCLUSION +In this work, we present APPT, a novel automated patch +correctness prediction technique based on the pre-training +model and classifier. We first adopt the off-the-shelf pre- +trained model as the encoder stack and LSTM stack to +enhance the dependency relationships among the buggy +and patched code snippets. Then we build a deep learning +classifier by two fully connected layers and a standard +softmax function to predict whether the patch is overfitting +or not. We conduct experiments on both patch datasets and +show that APPT significantly outperforms state-of-the-art +learning-based and traditional APCA techniques. We fur- +ther demonstrate that APPT is generalizable to various pre- +trained models. Based on these observations, some impli- +cations and guidelines on improving the existing learning- +based techniques (e.g., the usage of simple features and pre- +trained models) are provided. We highlight the direction +of applying pre-trained models to predict patch correctness +automatically. +ACKNOWLEDGMENTS +This +work +is +supported +partially +by +the +National +Key +Research +and +Development +Program +of +China +(2021YFB1715600), the National Natural Science Founda- +tion of China (61932012, 62141215), and the Program B +for Outstanding PhD Candidate of Nanjing University +(202201B054). +REFERENCES +[1] +L. Gazzola, D. Micucci, and L. Mariani, “Automatic software re- +pair: A survey,” IEEE Transactions on Software Engineering (TSE’17), +vol. 45, no. 1, pp. 34–67, 2017. +[2] +S. Benton, X. Li, Y. Lou, and L. 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King2† +1Theoretical Division, Los Alamos National Laboratory, +Los Alamos, New Mexico 87545, USA +2D-Wave Systems, Burnaby, British Columbia, Canada, V5G 4M9, Canada +3Vector Institute, University of Toronto, +Toronto, Ontario, M5G 1M1, Canada and +4Department of Physics and Astronomy, +University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada +(Dated: January 6, 2023) +Abstract +Topological phases of spin liquids with constrained disorder can host a kinetics of fractionalized +excitations. +However, spin-liquid phases with distinct kinetic regimes have proven difficult to +observe experimentally. Here we present a realization of kagome spin ice in the superconducting +qubits of a quantum annealer, and use it to demonstrate a field-induced kinetic crossover between +spin-liquid phases. Employing fine control over local magnetic fields, we show evidence of both +the Ice-I phase and an unconventional field-induced Ice-II phase. In the latter, a charge-ordered +yet spin-disordered topological phase, the kinetics proceeds via pair creation and annihilation of +strongly correlated, charge conserving, fractionalized excitations. As these kinetic regimes have +resisted characterization in other artificial spin ice realizations, our results demonstrate the utility +of quantum-driven kinetics in advancing the study of topological phases of spin liquids. +1 +arXiv:2301.01853v1 [cond-mat.stat-mech] 4 Jan 2023 + +INTRODUCTION +Dynamics in crystals typically proceeds via motion of topological defects such as dislo- +cation gliding [1]. One might expect the kinetics of disordered systems to be naturally free. +But in spin liquids, where disorder is present but constrained, kinetics often also proceeds +through defects or excitations are endowed with a conserved topological charge [2]. For +instance, frustrated spin systems, such as pyrochlore [3, 4] or square [5–7] spin ices, remain +disordered at low temperature, leading to a Pauling residual entropy, and their disorder is +constrained by the so-called ice rule [8]. There, kinetics consists of creation/annihilation and +walks of localized violations of the ice rule, in the form of emergent magnetic monopoles [9] +that conserve a topological charge. +Among spin ices, the kagome ice model [10–18] has been widely studied because it mim- +ics a remarkable variety of natural and artificial systems, from rare-earth pyrochlores [3], to +nanomagnetic fabrications [19], gravitationally trapped colloids [15], and many other sys- +tems [20–28]. Kagome spin ice can in principle manifest various unusual phases [13–15], but +the large energy scales of artificial implementations pose an experimental challenge; thor- +ough measurements of these phases and the physical conditions driving the phase-to-phase +transition are scarce. +Here we present a kagome qubit ice realized in a superconducting quantum annealer. +Using this experimental platform, we study its field-induced spin-liquid phases and quantum- +activated kinetics. We experimentally establish that topological constraints affecting the +dynamics proceeds via charge-conserving fractionalized excitations. +Using thousands of +programmable external magnetic fields, we detune the system from its more common ice- +rule-obeying “Ice-I” phase into a field-induced “Ice-II” phase, which exhibits charge order +while remaining spin-disordered. +RESULTS +Kagome spin ice +Kagome spin ice consists of magnetic dipoles as classical binary Ising spins arranged +along the edges of a hexagonal lattice and therefore on the sites of a kagome lattice. They +point from one triangular “ice vertex” (kagome plaquette) to another (Fig. 1a). We can +2 + +thus introduce the notion of a magnetic charge for a vertex, defined as the number of +spins pointing toward the vertex minus those pointing away from it. Because of the odd +coordination, a vertex can host only nonzero, odd charges q = −3, −1, 1, 3 (Fig. 1b). +The simplest magnetic kagome model includes interactions only among spins impinging +on the same vertex. Since not all pairs of spins at a vertex can simultaneously assume an +energy-minimizing head-to-tail configuration, the system is frustrated. The ground state is +therefore an extensively degenerate ensemble of disordered spins obeying the (pseudo-) ice +rule: frustration is minimized when each vertex has two spins pointing in and one pointing +out, or vice-versa. This ice manifold is often called the Ice-I phase, and can be thought as +a spin liquid forming an overall neutral plasma of disordered ±1 magnetic charges. In the +Ice-II phase [13, 14, 29], disordered spins still still obey the ice rule but charges are ordered +in an ionic lattice [30–33]. +Kagome qubit ice +In this work, we realize kagome spin ice in a quantum annealer. Its superconducting flux +qubits are described by the transverse field Ising Hamiltonian +HQ = −Γ +� +i +ˆσx +i + J +� � +i +hiˆσz +i + +� +ij +Jijˆσz +i ˆσz +j +� +, +(1) +where ˆσx and ˆσz are Pauli matrices on the qubits, J is an energy prefactor on the classi- +cal Ising Hamiltonian, hi are per-qubit programmable longitudinal fields [? ], and Jij are +programmable two-qubit couplers. Γ is a transverse field entangling the Pauli matrices and +thus controls quantum fluctuations. +Kagome spin ice can be mapped to a classical Ising model [34], and therefore to the +Hamiltonian of Eq. (1). Consider alternating A and B vertices pointing up (△) and down +(▽) respectively in Fig. 1a. We assign an Ising spin value si = +1 if it points into the +A vertex, and si = −1 if it points into the B vertex (Fig. 1b). (Compare with the Then, +standard kagome ice corresponds to the Hamiltonian +HI = J +� +⟨i,j⟩ +sisj + +� +i +hisi +(2) +where each nearest-neighbor spin is coupled antiferromagnetically. +3 + +We then embed the kagome lattice in the graph of available two-body couplers, as shown +in Fig. 1c, by modifying an embedding of a Z2 lattice gauge theory into the transverse-field +Ising model [35]. Each kagome site is represented by a ferromagnetic three-qubit chain, and +nearest-neighbor chains are coupled antiferromagnetically with two physical couplers. (Three +qubits are needed for each kagome lattice site because it is not possible to directly couple +two arbitrarily-chosen qubits.) We use h and J (with no index) to denote the total field +on a three-qubit chain and the total coupling between two neighboring chains, respectively, +obtaining the kagome qubit ice (KQI) Hamiltonian +HKQI = −˜Γ +� +i +˜σx +i + J +� +h +� +i +˜σz +i + J +� +⟨i,j⟩ +˜σz +i ˜σz +j +� +, +(3) +where ˜Γ = Γ3/J2 +FM is an effective transverse field on the three-qubit chains for a ferromag- +netic chain coupling JFM [36], ˜σi denotes a logical moment represented by a three-qubit +chain, and indices i and j are also over three-qubit chains, rather than individual qubits. +Phases +When Γ = ˜Γ = 0 and h = 0, the extensively degenerate ground state manifold of HKQI +corresponds to that of HI, which is the commonly seen Ice-I phase [19]. But we can go beyond +this regime. In nanoscopic realizations, another phase of lower entropy is possible [30, 31, 33, +37]. In such systems, it is driven by the long range nature of the dipolar interactions [13, 14]. +It still has disordered ice-rule obeying spins, but with charges ordered in an ionic lattice +where A and B vertices have opposite charge. While the spins remain disordered, though at +lower entropy [13, 14, 38], their disorder is topologically constrained: it can be mapped to a +dimer cover model [12, 38] and considered a case of classical topological order [2, 29, 39, 40]. +This is often called the Ice-II phase, and its topological nature should show a topologically +protected kinetics. (Note also that Ice-II can also be considered a broken symmetry phase +with unsaturated order parameter in the context of magnetic fragmentation [18, 41, 42].) +Indeed, the kinetics in the Ice-I phase is not gapped: It is possible to flip a single spin— +or indeed an extensive number of single spins—without violating the ice rule and thus +without creating an excitation (see also Supplementary Informations). Thus, the system +can kinetically explore the phase from within the local low energy manifold. +Instead, in the Ice-II phase any individual spin flip disrupts the charge balance, thus +4 + +creating an excitation. Therefore [2] the kinetics of the Ice-II phase must proceed either +via pair creation, motion, and annihilation of gapped excitations, or else via cooperative, +ungapped flips of entire loops of head-to-tail spins which do not alter the charge distribution. +Such kinetics was never probed in previous realizations of kagome ice because the Ice-II phase +has proved very hard to reach [30, 31, 33, 37] (see Supplementary Informations). Fortunately, +the quantum annealer offers another route: we can induce it by the field h, acting on σz, +and then we can study field-induced Ice-II kinetics. +If we define a staggered charge qs on a vertex such that qs = −q for A vertices and qs = q +for B vertices, then the field h determines the vertex energies ε−3, ε−1, ε+1, ε+3 for vertices +with qs = −3, −1, 1, and 3 respectively, as shown in Fig. 2 (see also SI). +For 0 < h/J < 4, ε+1 has the lowest energy, leading to the charge-ordered, spin-disordered +Ice-II phase as the ground state. Within this window, Fig. 2 shows a regime crossover at +h/J = 2. The lowest excitations are charge-order violations upsetting the ionic crystals of +charges when 0 < h/J < 2, and ice rule violations when 2 < h/J < 4. The two types of +excitations are degenerate at h/J = 2 where the excitation gap is highest. +Then, for h/J > 4, the ground state degeneracy vanishes, replaced by an ordered state +in which all A and B vertices have charge −3 and 3, respectively. +To estimate pseudo-equilibrium properties of the kagome qubit ice in these different +phases, we begin with a random spin state and repeatedly expose the system to quantum +fluctuations as described by Eq. (1), by cycling the transverse field Γ on and off. +An +appropriate magnitude of transverse field drives the kinetics of this kagome qubit ice without +erasing the state memory, as previously demonstrated in square ice [7]. After each exposure, +we read out a classical spin state. This leads to a sequence of states amenable to statistics +(see SI). +Fig. 3 summarizes experimental results for varying h/J. Fig. 3a shows real-space samples, +represented as vertex charges, for increasing values of h. At h = 0 we see the expected +disordered charge plasma of the Ice-I phase. Increasing h first leads to ionic ordering of the +charge ( the Ice-II phase) eventually giving way to a polarized state in which the longitudinal +field overcomes the ice rule, forming ionic crystals of ±3 charge, and all spins have value +si = 1. +Fig. 3b shows the corresponding result in reciprocal space via the Fourier transform of the +spins defined as S(q) ∝ � +ij eiq(ri−rj) (⟨sisj⟩ − ⟨si⟩⟨sj⟩). Our sign convention for the spins +5 + +leads to the appearance of peaks only in the Ice-II phase and its proximity, and the formation +of pinch points in the topologically protected region with h/J = 2.5. In Fig. 3c, cuts of the +Fourier transforms through the high-symmetry points in the extended Brillouin zone clearly +show growing peaks at K in the proximity of the Ice-II phase. These peaks correspond +to the expected logarithmic divergence of the dipolar correlations [12] (see also Fig. 5 in +ref [12], obtained from a dimer model). They, and the pinch points, follow therefore from the +topological properties induced on the phase by the charge ordering. From an implementation +point of view, S(q) reveals a highly symmetric system in which the multi-qubit embedding +of kagome spins preserves isotropy. This is an important advance over previous work [7]. +Fig. 3d plots the charge order parameter, defined as one third the average staggered +charge of a vertex. The two two broad plateaus at ±1/3 correspond to the Ice-II phases. +Fig. 3e confirms the high ice-rule obedience throughout the Ice-I and Ice-II phases, which +breaks down at h/J > |4| where, from Fig. 2, the lowest energy vertex no longer obeys the +ice rule. +Topologically protected quasi-classical kinetics +These measurements validate the annealer’s effectiveness as an experimental platform +for probing phases of the Ising kagome spin ice system near a low-temperature thermal +equilibrium. Because consecutive output states are separated dynamically by a relatively +short exposure to a relatively weak transverse field Γ (compared to J), we can also probe +the quasi-classical kinetics. +As mentioned above, in the Ice-II ground state a single spin flip always corresponds +to fractionalized excitations, as either violations of the Ice-II charge-order constraint, or +violations of the kagome ice rule (Fig. 2). We can define a topological charge (or t-charge) +as qt = q + 1, qt = q − 1 for A and B vertices respectively. In the Ice-II charge-ordered +ground state, the topological charge is zero on all vertices. Instead, excitations of the Ice-II +phase are topologically charged. Their t-charge is conserved: flipping a spin creates a pair of +fractional excitations of t-charges ±2 and zero net t-charge. Further flips can separate the +t-charges, which can then annihilated when meeting other, opposite ones. This situation of +paired fractional excitations is very reminiscent of square and pyrochlore ice [7, 43], although +here the topological charge is not the magnetic charge. +6 + +To probe the thermal and quantum-activated kinetics of the Ice-I and Ice-II phases, we +compare QA output samples. Between consecutive samples, the qubits are exposed to the +a transverse field for 1 µs, and at the same time J is dropped. This protocol is depicted +in Fig. 4a. Since the system is in a thermal bath at 12 mK, this allows both quantum and +thermal fluctuations to drive dynamics [7]. +In agreement with the description above, our results show a kinetics of fractionalized +excitations, that can be created and annihilated in pairs of opposite topological charge, +and more rarely a kinetics consisting of flips of entire loops of spins—which can always +be construed mathematically as creations followed by annihilation of topologically charged +pairs. +Fig. 4b shows two representative samples from each of h/J = 0.5, 2.5, 4, correspond- +ing roughly to the boundaries and the middle of the field-induced Ice-II phase. Ice-rule +and charge-order violations are shown as triangles. Between the two samples, we highlight +the spins that flip during the exposure to fluctuations, as well as the motion of fractional +excitations. +At h/J = 0.5 the charge order is fragile and we are close to the Ice-I phase. We see many +excitations popping up erratically, and they are charge order violations, due to their small +energy cost (Fig. 2). +At h/J = 2.5 we see far fewer excitations, and the kinetics consists of their wandering. We +also see flipping of closed loops of spins. One fractional excitation escapes off the boundary, +one appears from the boundary, and one moves to another location through a chain of flipped +spins. This picture is consistent with the large energy gap shown in Fig. 2, which suppresses +pair creation of excitations. +At h/J = 4, we again see a regime in which excitations can appear at low cost; these +cheap excitations are now ice-rule violations, in contrast to the charge-order violations seen +near the Ice-I phase, consistent with the energetics (see Fig. 2). +To quantify the creation/annihilation and motion of fractional excitations, we consider +the subgraph of the honeycomb lattice whose edges correspond to flipped spins (Fig. 4c– +d) between consecutive states. +We measure the degrees (valencies) of honeycomb sites +in this graph. A closed loop of flipped spins results in only degree-two honeycomb sites. +Conversely, an open chain of flipped spins will have degree two in the interior, and degree +one on the ends. This can involve the motion of a fractional excitation, with or without +7 + +creation/annihilation. +In general, degree-two spins correspond to motion of excitations, +while degree-one spins correspond to creation/annihilation. +Fig. 4e shows that the system is overall most active around h/J = 0 and h/J = 4, which +corresponds to points of degeneracy (see Fig. 2) where excitations are cheapest. The plot of +the relative frequency of excitation motion over pair creation/annihilation shows a maximum +around h/J = 2, the point of maximum gap: where excitations are most expensive, kinetics +consists mostly of their random walk, much like monopoles in square or pyrochlore spin ice. +The non-monotonicity of the curves in Fig. 3e shows that in kagome qubit ice, by tuning +the gap of the phase, the topological protection of the kinetics can be controlled, from a hard +to distinguish soup of excitations at h/J = 0, 4, to a clear picture of creation/annihilation +and motion of fractionalized excitations around the value h/J = 2. +DISCUSSION +We have realized kagome qubit spin ice in 2742 superconducting flux qubits of a quantum +annealing processor and explored its field-induced spin-liquid ice phases. We have studied +the quantum-activated, topologically protected kinetics of the Ice-II phase and shown that +it proceeds via creation/annihilation and propagation of charge-conserving fractionalized +excitations. We emphasize that quantum fluctuations are used here only to drive kinetics, +but can be employed in the future to study entangled states. Furthermore, the kagome +antiferromagnet in a transverse field Γ has a rich ground-state phase diagram [44] arising +from high-order perturbations in Γ, which may be probed in future work. +Our results +demonstrate that quantum annealers are capable of implementing exotic programmable +phases of frustrated spin sliquids, whose gap and topologically-protected kinetic regimes +can be finely tuned. +8 + +a +A +A +A +A +B +B +B +B +A +A +A +A +B +B +B +B +c +A +B +B +B +Ising spin +(flux qubit) +FM +AFM +b ++ +­ +A +B +Ising +1 ++ +­ +B +A +Ising –1 ++3 ++1 +­1 +­3 +Ice­rule +obeying +Ice­rule +breaking +8.5 mm +d +FIG. 1: Kagome qubit ice. A, Kagome spin ice consists of magnetic dipoles on the edges of +a hexagonal lattice, which point in or out of triangular plaquettes (vertices) of the dual kagome +lattice (gray lines). B, Each vertex in a given configuration has a nonzero charge: ±1-charged +vertices satisfy the kagome ice rule; ±3-charged vertices do not. Denoting triangles pointing up +and down by A and B respectively, one can map dipoles to Ising spins according to whether not +the dipole points into an A triangle. C, In the kagome qubit ice, each kagome site is realized +using a ferromagnetically-coupled three-qubit chain. Sites impinging on the same triangular ice +vertex are coupled antiferromagnetically, leading to geometric frustration. D, Optical image of the +superconducting quantum annealing processor in a sample holder. 2742 qubits are used to realize +a 913-spin kagome ice. +9 + +0 +1 +2 +3 +4 +5 +6 +−4 +−2 +0 +2 +4 +Ice­II, +first excitation is +charge violation +Ice­II, +first excitation is +ice rule violation +Ordered +phase +gap maximized +h/J +Energies +ε−3 +ε+3 +ε−1 +ε+1 +Ice­I +FIG. 2: Ice vertex energies (normalized to J). In the Ice-I phase (h = 0), the ice-rule vertex +states ε+1 and ε−1 are degenerate. Detuning h leads energetic preference towards the (staggered) ++1-charged configurations. Within the ice region 0 ≤ h/J ≤ 4, the energy gap is maximized at +h/J = 2, where charge-imbalance excitations are degenerate with ice-rule excitations. +10 + +2π +π +0 +−π +−2π +2π +π +0 +−π +−2π +qx (r.l.u.) +qy +2π +π +0 +−π +−2π +2π +π +0 +−π +−2π +qx (r.l.u.) +qy +2π +π +0 +−π +−2π +2π +π +0 +−π +−2π +qx (r.l.u.) +qy +2π +π +0 +−π +−2π +2π +π +0 +−π +−2π +qx (r.l.u.) +qy +0 +2 +4 +intensity (arb. units) +Γ +K +Γ′ +h/J = 0 +h/J = 0.5 +h/J = 2.5 +h/J = 4.5 +−4 +0 +4 +−1 +0 +1 +Charge order parameter +h/J +−4 +0 +4 +0 +1 +Ice rule obedience +h/J +Γ +K +Γ′ +Γ +0 +2 +4 +6 +S(q) +0 +0.5 +2.5 +4.5 +h/J +a) +b) +c) +d) +e) +FIG. 3: Field-induced charge phases and qubit ice structure. A, Charge states for varying +external field h. +At h = 0, vertices have no energetic preference between −1 and +1 charge +(light blue and red respectively), leading to charge disorder. As h increases, A and B vertices +energetically favor −1 and +1 charge respectively, leading to long-range order in the staggered +charge. Eventually h polarizes the sites, leading to a preponderance of −3 and +3 charged vertices +(dark blue and red respectively). B, Fourier transforms S(q) calculated from QA experimental +output, with Brillouin zone in gray. C, Cuts of S(q) for varying h/J through high-symmetry +points Γ, K, and Γ′ (shown in b) show the effect of the longitudinal field on peak height at K and +pinch-point width at Γ′. D, Charge order parameter. E, Proportion of vertices obeying the ice +rule. +11 + +FIG. 4: Kinetics and field-induced topological protection. A, Within the quantum annealer, +the kinetics is driven by a reverse anneal protocol wherein the qubits (Eq. (1)) are exposed to +quantum fluctuations (Γ) and thermal fluctuations (T/J ) for a duration of 1 µs between projected +classical output states. B, Quantum annealer output samples. 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Roy, Next-Generation Topology of D-Wave Quan- +tum Processors (2020), 2003.00133, URL http://arxiv.org/abs/2003.00133. +Acknowledgments +The authors aknowledge the contributions of technical staff at D-Wave, without whom +this work would not be possible. The work of ALB and CN was carried out under the +auspices of the U.S. DoE through the Los Alamos National Laboratory, operated by Triad +National Security, LLC (Contract No. 892333218NCA000001). JC acknowledges support +from the Natural Sciences and Engineering Research Council (NSERC), the Shared Hier- +archical Academic Research Computing Network (SHARCNET), Compute Canada, Google +Quantum Research Award, and the Canadian Institute for Advanced Research (CIFAR) AI +chair program. Resources used in preparing this research were provided, in part, by the +Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring +the Vector Institute. +15 + +Competing interests +The authors declare no competing interests. +Author Contributions +J.C. first proposed the idea of a Kagome embedding in a D-Wave QA to A.D.K.. A.D.K., +A.L.-B., C.N., and J.C. conceived the project. C.N., A.D.K. contributed to the design of +the experiments. A.D.K. and K.B. realized the embedding. J.R. performed supporting mea- +surements. A.D.K., A.L.-B. performed QA experiments. A.D.K. performed data analysis. +C.N. provided the theoretical framework for experiment design and result interpretation. +C.N. drafted the manuscript with A.D.K. All authors contributed to the final version of the +manuscript. +Methods +The QA processor used in this work was a D-Wave Advantage QPU (Fig. 1d) housed in +Burnaby, BC, Canada, operating at T = 12 mK and accessed remotely. The QPU contains +5627 operable superconducting flux qubits of which we used 2739 to implement our kagome +qubit spin ice. The architecture is discussed in Ref. [45]. +In quantum annealing, the Hamiltonian (1) in the Main Text is controlled by an annealing +parameter s ranging from 0 to 1: +HQ(s) = −Γ(s) +� +i +σx +i + J (s) +� � +i +hiσz +i + +� +ij +Jijσz +i σz +j +� +, +(4) +where Γ(0) ≫ J (0) and Γ(1) ≈ 0 ≪ J (1). +Thus a typical “forward anneal”, in which s is ramped linearly for the duration of anneal +time ta (s = t/ta) begins in an easily-prepared superposition ground state and ends in a +low-energy state of a classical Ising Hamiltonian. For simulating spin systems, it has proven +useful [7, 46] to employ a “quantum evolution Monte Carlo” method, in which a chain of +classical samples S0, . . . Sk is generated. To generate Si, the system is initialized in state +S0 at the end of the anneal (s = 1), then “reverse annealed” back to some intermediate +s∗, paused at s∗ to allow equilibration for some time tp, then quickly quenched back to +16 + +s = 1. Although this method can be used to estimate observables from a transverse field +Ising model at s∗ [46], here we just use quantum fluctuations as a driver of mixing dynamics +between low energy state in the kagome ice system. +In this work we generate chains of k = 128 samples, starting with a random initial state +S0. To estimate equilibrium properties (Fig. 3) we use tp = 256 µs and discard the first 64 +samples of each chain (and the random initial state) as Monte Carlo burn-in. For dynamics +inquiries (Fig. 4) we use tp = 1 µs. In both cases we interrogate Hamiltonian (4) using +s∗ = 0.32, which was chosen to give an appropriate amount of mixing in one microsecond +(smaller s leads to faster mixing since both Γ/J and T/J are larger [7]). When statistical +quantities are estimated, we take the average of 200 repeated experimental iterations; each +iteration includes a call to the QPU for each value of h probed. +Graph embedding +The qubits in the QA processor are intercoupled in a “Pegasus” layout [47], in which a +qubit is coupled to up to 15 other qubits. From these available couplers we select a geometry +that represents a kagome graph using three qubits per kagome spin as depicted in Fig. 1c. +We show the full embedded lattice in Fig. 6. The kagome embedding does not require the +use of all qubits, and it is possible to embed a defect-free lattice with no site vacancies, +despite the existence of some inoperable qubits (empty circle in Fig. 6c). +Since ferromagnetic chains are sometimes broken, they are majority-voted to provide an +unambiguous mapping from the qubit system to the kagome system. We run all experiments +presented herein with Jij = 0.9 for AFM couplers and Jij = −1.5 for FM couplers. This +choice of ferromagnetic coupling is sufficient to guarantee that chains are almost never broken +in QPU output, despite the frustration in the system. +Disorder suppression +In this application we perform many experiments on a single programmed lattice, whose +classical ground state is highly degenerate. Under such conditions it is appropriate to refine +the general-purpose QA calibration by exploiting symmetries in the system. +For example, when h = 0 each qubit should have average magnetization ⟨si⟩ = 0. Thus we +17 + +tune per-qubit flux offsets to balance qubits at zero for h = 0, then use the same flux offsets +when h ̸= 0. In this experiment, we are not interested in probing boundary conditions. +Rather, we want to simulate the thermodynamic limit of an infinite system. In an infinite +system, for any fixed h, the correlation of two neighboring kagome sites ⟨sisj⟩ is the same. +Thus we fine-tune the AFM couplers to promote this property. Since the three-qubit FM +chains are almost never broken, we do not fine-tune the FM couplers. Similarly, for any +fixed h ̸= 0, the magnetization of each qubit should be equal; we fine-tune the per-qubit +fields hi to promote this property (maintaining the property that the average +1 +N +� +i hi does +not change from the nominal value h). These calibration refinements are performed before +collecting the analyzed data. Fig. 7 shows an example of this refinement for J = 0.9, h = 0.6, +with the magnetizations and correlations achieved, and the programmed values that achieve +them. +18 + +0 +0.5 +1 +0 +2 +4 +6 +8 +10 +s +Energy scale (GHz) +Γ(s) +J (s) +a) +b) +0 +s∗ +1 +relaxation: tp +reverse anneal: +tq(1 − s∗) +readout quench: +tq(1 − s∗) +readout/wait: tw +Time +Annealing parameter s +FIG. 5: Quantum annealing schedule and protocol. a, Transverse field Γ(s) and Ising energy +scale J (s) as a function of annealing parameter s. Note that the total coupling between two three- +qubit chains is 1.8J . b, Quantum evolution Monte Carlo method. A sequence of classical readout +states is generated by repeated exposure to quantum fluctuations and thermal fluctuations. +19 + +a) +b) +c) +d) +FIG. 6: Embedding of the kagome lattice into the qubit graph. a–b, Each kagome site is +represented by three qubits, coupled together ferromagnetically in a chain (Jij = −1.5). The entire +qubit graph and embedding are shown in a with green and orange lines representing FM and AFM +couplers respectively; b shows a detailed zoom, with operable and inoperable qubits represented +by filled and empty circles respectively. c–d, The embedding shown in a realizes a 729-site kagome +lattice, which can be viewed as Ising spins (c), or magnetic dipoles (d). +20 + +−0.34 −0.34 −0.33 −0.33 −0.32 +0 +200 +400 +Avg. qubit magnetization +Occurrences +−0.35 −0.34 −0.34 −0.33 −0.33 +0 +200 +400 +600 +Avg. coupler spin-spin correlation +Occurrences +0.89 +0.89 +0.9 +0.9 +0.91 +0 +100 +200 +300 +Adjusted coupling +Occurrences +0.6 +0.8 +1 +1.2 +0 +200 +400 +600 +800 +Adjusted longitudinal eld +Occurrences +-2e-5 +0 +2e-5 +0 +100 +200 +300 +Qubit ux-bias oset (h = 0) +Occurrences +a) +b) +c) +d) +e) +boundary +bulk +FIG. 7: Suppressed disorder with fine-tuned Hamiltonian terms. Example data are shown +for nominal AFM coupling values of J = 0.9 and local fields of h = 0.6. a–b, Over 100 iterations, +tightly-concentrated average qubit magnetizations and spin-spin correlations of coupled pairs in- +dicate a balanced degenerate ice system. c–e, This is achieved by small adjustements of couplers +(c), adjustment of fields (d), and qubits are balanced using flux-bias offsets at h = 0 that are also +used at nonzero h. Note the two modes in d, where boundary spins are assigned roughly half the +field of bulk spins, in accordance with their degree in the graph, to achieve similar magnetizations. +21 + diff --git a/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf b/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b686b2b7e6cf6ec166cebd0c3a15d8f896c832e9 --- /dev/null +++ b/k9E4T4oBgHgl3EQftQ0b/content/2301.05222v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8895aa7191331e6286306505abc8fa0f1da65c2411ef8c8491534510665dd096 +size 4787740 diff --git a/k9E4T4oBgHgl3EQftQ0b/vector_store/index.faiss b/k9E4T4oBgHgl3EQftQ0b/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a0cb7de15d530517740ca1c862d26974a6bcd249 --- /dev/null +++ b/k9E4T4oBgHgl3EQftQ0b/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:831eb5a494da9ef6c600e83ec88c4f956696d33c85328f80fe03ac66640ae04f +size 5373997 diff --git a/n9E5T4oBgHgl3EQfjw-t/content/tmp_files/2301.05658v1.pdf.txt b/n9E5T4oBgHgl3EQfjw-t/content/tmp_files/2301.05658v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b526cb0b23837fd4593fcfd528fbab6e56180ba --- /dev/null +++ b/n9E5T4oBgHgl3EQfjw-t/content/tmp_files/2301.05658v1.pdf.txt @@ -0,0 +1,1268 @@ +arXiv:2301.05658v1 [cs.CC] 13 Jan 2023 +Streaming Lower Bounds and Asymmetric +Set-Disjointness +Shachar Lovett ∗ +Computer Science Department +University of California San Diego +slovett@ucsd.edu +Jiapeng Zhang † +Department of Computer Science +University of Southern California +jiapengz@usc.edu +January 16, 2023 +Abstract +Frequency estimation in data streams is one of the classical problems in streaming +algorithms. +Following much research, there are now almost matching upper and +lower bounds for the trade-off needed between the number of samples and the space +complexity of the algorithm, when the data streams are adversarial. However, in the +case where the data stream is given in a random order, or is stochastic, only weaker +lower bounds exist. In this work we close this gap, up to logarithmic factors. +In order to do so we consider the needle problem, which is a natural hard problem +for frequency estimation studied in (Andoni et al. 2008, Crouch et al. 2016). Here, +the goal is to distinguish between two distributions over data streams with t samples. +The first is uniform over a large enough domain. The second is a planted model; a +secret ”needle” is uniformly chosen, and then each element in the stream equals the +needle with probability p, and otherwise is uniformly chosen from the domain. It is +simple to design streaming algorithms that distinguish the distributions using space +s ≈ 1/(p2t). It was unclear if this is tight, as the existing lower bounds are weaker. We +close this gap and show that the trade-off is near optimal, up to a logarithmic factor. +Our proof builds and extends classical connections between streaming algorithms +and communication complexity, concretely multi-party unique set-disjointness. We +introduce two new ingredients that allow us to prove sharp bounds. The first is a lower +bound for an asymmetric version of multi-party unique set-disjointness, where players +receive input sets of different sizes, and where the communication of each player is +normalized relative to their input length. The second is a combinatorial technique that +allows to sample needles in the planted model by first sampling intervals, and then +sampling a uniform needle in each interval. +∗Research supported by NSF awards 1953928 and 2006443. +†Research supported by NSF CAREER award 2141536. +1 + +1 +Introduction +The needle problem is a basic question studied in the context of streaming algorithms +for stochastic streams [AMS99, AMOP08, GH09, CMVW16, BVWY18]. +The goal is to +distinguish, using a space-efficient single-pass streaming algorithm, between streams +sampled from two possible underlying distributions. +Setting notations, we let t denote the number of samples, s the space of the streaming +algorithm, n the domain size, and p ∈ (0, 1) the needle probability. The two underlying +distributions are: +• Uniform: sample t uniform elements from [n]. +• Planted: Let x ∈ [n] be uniformly chosen (the “needle”). Sample t elements, where +each one independently with probability p equals x, and otherwise is sampled +uniformly from [n]. +We will assume that n = Ω(t2) so that with high probability, all elements in the stream +(except for the needle in the planted model) are unique. The question is what space is +needed to distinguish between the two models with high probability. +Sample-space tradeoffs for the needle problem. +We start with describing some basic +streaming algorithms for the needle problem, in order to build intuition. First, note that +we need p = Ω(1/t) as otherwise the two distributions are statistically close, because with +high probability the needle never appears in the planted model. +One possible algorithm is to check if there are two adjacent equal elements in the +stream. This requires t = Θ(1/p2) samples and space s = Θ(log n). Another possible +algorithm is to store the entire stream in memory, and check for a repeated element. This +algorithm requires less samples, t = Θ(1/p), but more space, s = t log n. Note that in both +cases, we get a sample-space tradeoff of st = Θ((log n)/p2). One can interpolate between +these two basic algorithms, but the value of the product st remains the same in all of them. +This motivated the following conjecture, given explicitly in [CMVW16] and implicitly in +[AMOP08]. +Conjecture 1.1 (Sample-space tradeoff for the needle problem). Any single-pass streaming +algorithm which can distinguish with high probability between the uniform and planted models, +where p is the needle probability, t the number of samples and s the space, satisfies p2st = Ω(1). +The best result to date towards Conjecture 1.1 is by Andoni et al. [AMOP08] who +showed that p2.5st1.5 = Ω(1) (this bound is indeed weaker since p = Ω(1/t)). Guha et +al. [GH09] claimed to prove Conjecture 1.1 but later a bug was discovered in the proof, +as discussed in [CMVW16]. In this paper we establish Conjecture 1.1 up to logarithmic +factors. We can also handle streaming algorithms which pass over the data stream multiple +times, scaling linearly in the number of passes. +Theorem 1.2 (Main theorem). Any ℓ-pass streaming algorithm which can distinguish with high +probability between the uniform and planted models, where p is the needle probability, t the number +of samples and s the space, satisfies ℓp2st log(t) = Ω(1). +2 + +1.1 +Application: lower bound for frequency estimation in stochastic +streams +For many streaming problems, the current state-of-the-art streaming algorithms space +requirements are known to be tight (up to poly-logarithmic terms) in the adversarial +model, where the streams arrive in an adversarial order. Following a sequence of works +on the random-order model [MP80, DLOM02, GM07, CCM08, CJP08, AMOP08, GM09], +Crouch et al. [CMVW16] initiated the study of stochastic streams, where the streams are +sampled from some underlying distribution. The question is if in this model one can attain +better streaming algorithms compared to the adversarial model, utilizing the stochastic +nature of the streams; or whether the existing lower bounds can be strengthened to this +model as well. +The needle problem we described is an example of a problem in the +stochastic model. +A basic problem in the streaming literature, starting with the pioneering work of +[AMS99], is that of estimating the frequency moments of a stream. Given a stream x1, . . . , xt +of elements from [n], let fx denote the number of times an element x appears in the stream. +The k-th frequency moment of the stream is +Fk = +� +x∈[n] +f k +x. +In the adversarial model, there are matching upper and lower bounds of ˜Θ(n1−2/k) 1 on +the space needed for a streaming algorithm to approximate Fk [CKS03, IW05]. It was +conjectured by [CMVW16] that the same lower bound also holds in the stochastic model. +They showed that the result of [AMOP08] gives a somewhat weaker lower bound of +˜Ω(n1−2.5/k) space, and that Conjecture 1.1, if true, implies the tight bound of ˜Ω(n1−2/k). +Theorem 1.2 thus verifies their conjecture, up to logarithmic terms, which still implies a +lower bound of ˜Ω(n1−2/k). We refer to [CMVW16] for further details. +We note another related application, communicated to us by David Woodruff. Mc- +Gregor et al. [MPTW12] studied streaming algorithms based on sub-sampling a data +stream. In particular, one of the problems they studied is that of frequency estimation. +They designed space-efficient streaming algorithms based on sub-sampling, and also gave +matching lower bounds, based on the results of Guha et al. [GH09]. However, as later +a bug was found in this latter work, the journal version of McGregor et al. [MPTW16] +removed the lower bounds. Using Theorem 1.2 the claimed lower bounds hold, up to a +logarithmic factor. +1.2 +Proof approach +We prove Theorem 1.2 by a reduction to the unique set-disjointness problem in communi- +cation complexity. This is a common technique used to prove lower bounds for streaming +algorithms [CKS03, BYJKS04, AMOP08, GH09, BVWY18, KPW21]. +1We use ˜Θ, ˜Ω to ignore poly-logarithmic terms. +3 + +The basic idea is to partition the stream samples into intervals I1, . . . , Ik and consider +the stream distribution where we place a single needle uniformly in each interval, and +sample the other elements in the stream uniformly. It is straightforward to show that any +streaming algorithm which can distinguish this distribution from the uniform distribution +using space s, can be used to construct a communication protocol that solves the k-party +unique set-disjointness problem, where player i gets a set of size |Ii|, and where each player +sends s bits. If for example we take the intervals to be of equal size |I1| = . . . = |Ik| = t/k, +then using existing tight lower bounds for multi-party unique set-disjointness, one can +prove tight sample-space lower bounds in the adversarial model2. This was the approach +taken by many of the previous works in this area [CKS03, BYJKS04, AMOP08, GH09, +BVWY18, KPW21]. Our plan is to extend this approach to the stochastic model. However, +this presents two new challenges. +First, a simple calculation shows that the number of needles is k ≈ pt with high +probability, but the gaps between needles are not uniform; for example, the two closest +needles have a gap of ≈ p2t. This necessitates taking intervals of very different lengths, if +we still plan toplace one needle perinterval. In turn, thisrequiresprovinglowerboundson +multi-party unique set-disjointness when the players receive inputs of different lengths. In +this model, it no longer makes sense to measure the total communication of the protocols. +Instead, we develop a new measure, which normalizes the communication of each player +relative to their input length. We expand on this in Section 1.3. +The second challenge is that using a single partition of the stream by intervals, and then +planting a uniform needle in each interval, cannot induce the planted needle distribution. +Instead, we need to carefully construct a distribution over sets of intervals, such that if +then one places a uniform needle in each interval, the resulting stream distribution mimics +exactly the planted distribution. We expand on this in Section 1.4. +1.3 +Multi-party unique set-disjointness with different set sizes +We start by defining the standard multi-party unique set-disjointness problem. Let k ≥ 2 +denote the number of players. The players inputs are sets S1, . . . , Sk ⊂ [n]. They are +promised that one of two cases hold: +• Disjoint: the sets S1, . . . , Sk are pairwise disjoint. +• Unique intersection: there is a common element x ∈ S1 ∩ . . . ∩ Sk, and the sets +S1 \ {x}, . . . , Sk \ {x} are pairwise disjoint. +Their goal is to distinguish which case is it, while minimizing the communication3. +Observe that under any of the two promise cases, one of the players’ inputs has size +|Si| ≤ n/k + 1. A simple protocol is that such a player sends their input, which allows the +2Concretely, the total communication of the protocol is ks, whereas the lower bound for k-party unique +set-disjointness is Ω(t/k). Thus ks = Ω(t/k). Taking k = pt gives p2st = Ω(1). +3Formally, we consider randomized multi-party protocols in the blackboard model, where at each turn one +of the players writes a message on a common blackboard seen by all the players. +4 + +other players to solve the problem on their own. This simple protocol sends O(n/k · log n) +bits. This can be further improved to O(n/k) bits using the techniques of [HW07]. A line +of research [AMS99, BYJKS04, CKS03, Gro09, Jay09, YZ22] studied lower bounds. A tight +lower bound was first achieved by [Gro09]. +Theorem 1.3 ([Gro09, Jay09]). Any randomized communication protocol which solves the k-party +unique set-disjointness problem must send Ω(n/k) bits. +As discussed in Section 1.2, we need a fine-grained variant of the unique set- +disjointness problem, where the set sizes are fixed and can be different between the +players. +Definition 1.4 (Fixed-size multi-party unique set-disjointness). Let s1, . . . , sk ≥ 1. The +[s1, . . . , sk]-size k-party unique set-disjointness problem is a restriction of the k-party unique +set-disjointness problem to input sets of size |Si| = si. +Consider protocols for the [s1, . . . , sk]-size k-party unique set-disjointness problem. For +any i ∈ [k], one option is that the i-th player sends their input to the rest of the players, +which requires sending ci = Ω(si) bits. If the input sizes s1, . . . , sk are very different, it no +longer makes sense to consider the total amount of bits sent by the players. Instead, we +should normalize the number of bits sent by the i-th player ci by its input length si. We +prove that with this normalization, the simple protocols are indeed optimal. +Towards this, we make the following definition: +a k-party protocol Π is called +[c1, . . . , ck]-bounded if in any transcript of Π, the i-th player sends at most ci bits. +Theorem 1.5 (Lower bound for fixed-size multi-party unique set-disjointness). Let Π +be a randomized k-party [c1, . . . , ck]-bounded protocol, which solves with high probability the +[s1, . . . , sk]-size k-party unique set-disjointness problem, where � si ≤ n/2. Then +� +i∈[k] +ci +si += Ω(1). +We conclude this subsection with three comments. First, the condition � si ≤ n/2 is +a technical condition emerging from the proof technique; it suffices for our application, +and we believe that it can be removed in future work. +Next, it is known that the hard case for the standard multi-party unique set-disjointness +problem is when all the sets have about the same size, namely when s1 = . . . = sk = +Θ(n/k). In this case Theorem 1.5 implies � ci = Ω(n/k) which recovers Theorem 1.3. +Last, we prove Theorem 1.5 by constructing a hard distribution over inputs, and +then proving a lower bound for deterministic protocols under this distribution. +The +hard distribution is a natural one, the uniform distribution over inputs of sizes s1, . . . , sk. +For details see Theorem 2.13. Moreover, we show (Claim 2.15) that Theorem 1.5 and +Theorem 2.13 are in fact equivalent. +5 + +1.4 +Efficient reduction of the needle problem to multi-party unique +set-disjointness +We establish Theorem 1.2 by reducing lower bounds for the needle problem to lower +bounds for the unique set-disjointness, and then applying Theorem 1.5 (Theorem 2.13 +more precisely). To do so, we need a way of mapping inputs to the unique set-disjointness +problem to inputs for a streaming algorithm. A natural way to do so, taken for example by +[AMOP08], is to partition the stream into intervals and assign one to each player. We follow +the same approach but generalize it, so we can use it to simulate the planted distribution +of the needle problem by random inputs to the unique set-disjointness problem. +Recall that n denotes the domain size, t the number of samples and p the needle +probability. Our goal will be to simulate the planted distribution using inputs to multi- +party unique set-disjointness. In order to do so, we define interval systems. +Definition 1.6 (Interval systems). An interval system F is a family of pairwise disjoint non- +empty intervals F = {I1, . . . , Ik} with I1, . . . , Ik ⊂ [t]. +Given an interval system F, we define a planted distribution Planted[F] over streams +X ∈ [n]t as follows: +1. Sample uniform needle x ∈ [n]; +2. In each interval I ∈ F sample uniform index i ∈ I and set Xi = x; +3. Sample all other stream elements uniformly from [n]. +Using Theorem 1.5, we prove a space lower bound for streaming algorithms that can +distinguish between the uniform distribution and the planted distribution for F. Here is +where we exploit the fact that we can prove lower bounds for unique set-disjointness also +when the set sizes vary between the players. We use the following notation: given an +interval system F, its value is val(F) = � +I∈F +1 +|I|. +Lemma 1.7. Let F be an interval system. Any streaming algorithm which with high probability +distinguishes between Planted[F] and the uniform distribution must use space +s = Ω +� +1 +val(F) +� +. +In order to complete the reduction, we need to simulate the planted distribution using +planted distributions for interval systems F. Clearly, this cannot be done using a single +interval system, and hence we need to consider randomized interval systems. +A randomized interval system F is a distribution over interval systems F. The planted +distribution Planted[F] for F is defined by first sampling F ∼ F and then X ∼ Planted[F]. +The value of F is val(F) = EF ∼F[val(F)]. We can extend Lemma 1.7to randomized interval +systems. +6 + +Lemma 1.8. Let F be a randomized interval system. Any streaming algorithm which with high +probability distinguishes between Planted[F] and the uniform distribution must use space +s = Ω +� +1 +val(F) +� +. +To prove the lower bound for the needle problem, we need Planted[F] to simulate +exactly the planted distribution; we call such randomized interval systems perfect. +Definition 1.9 (Perfect randomized interval systems). A randomized interval system F is +called perfect if Planted[F] is distributed exactly as the planted distribution. +In light of Lemma 1.8, we need a perfect randomized interval system F with as low a +value as possible. It is relatively simple to show that if F is perfect then val(F) = Ω(p2t). +The following theorem gives a construction nearly matching the lower bound. +Theorem 1.10. There exists a perfect randomized interval system F with val(F) = O (p2t log(t)). +Theorem 1.2 now follows directly by combining Lemma 1.8 and Theorem 1.10. +1.5 +Related works +In a seminal work, Miltersen et al. [MNSW95] first observed connections between asym- +metric communication complexity and its applications to data structures in the cell probe +model. Since then, several works [BR00, JKKR03, PT06, BIPW10, CKLM18] proved data +structure lower bounds and streaming lower bounds via connections to asymmetric com- +munication complexity lower bounds. To the best of our knowledge, all these works built +on two-party communication problems. In contrast, we consider multi-party communi- +cation complexity in this work. It is interesting to ask if multi-party communication can +provide more applications to data structure and streaming lower bounds. +Other than connections to data structure lower bounds and streaming lower bounds, +Dinur et al. [DDKS16] studied the needle problem in cryptography. It would be interesting +to explore more connections between our work and cryptography. +Acknowledgements. +We thank David Woodruff for helpful discussions about streaming +algorithms, and for insightful comments on an earlier version of this paper. +Paper organization. +We prove lower bounds for multi-party unique set-disjointness with +fixed set sizes (Theorem 1.5) in Section 2. We design an efficient reduction using interval +systems (Lemmas 1.7 and 1.8) in Section 3. We combine both to prove our lower bound +for the needle problem (Theorem 1.2) in Section 4. We discuss open problems in Section 5. +2 +Lower bounds for asymmetric unique set-disjointness +We prove Theorem 1.5 in this section. +First, we recall some definitions and fix some +notations. +7 + +Notations. +it will be convenient to identify sets with their indicator vectors; thus, we +identify X ∈ {0, 1}n with the set {i : Xi = 1} ⊂ [n]. Let k ≥ 2 denote the number of +players. The players inputs are X = (X1, . . . , Xk), where Xi = (Xi(1), . . . , Xi(n)) ∈ {0, 1}n. +It will be convenient to also define Xj = (X1(j), . . . , Xk(j)) ∈ {0, 1}k, the j-th coordinate +for all the players for j ∈ [n]. In this section use boldface to denote random variables (such +as X, W ) to help distinguish them from non-random variables. +Protocols. +Let Π be a protocol. Given an input X, we denote by Π(X) the transcript of +running Π on X. We assume that every transcript also has an output value which is a bit +determined by the transcript (for example, the last bit sent). A protocol solves a decision +problem under input distribution ν with error δ, if it outputs the correct answer with +probability at least 1−δ when the inputs are sampled from ν. We will prove lower bounds +on protocols that solve unique set-disjointness under a number of input distributions. As +such, we may assume unless otherwise specified that the protocols are deterministic. +Finally, recall that we call k-party protocol Π is called [c1, . . . , ck]-bounded if in any +transcript of Π, the i-th player sends at most ci bits. +multi-party unique set-disjointness. +The k-party unique set-disjointness problem is +defined on inputs coming from two promise sets: +• Disjoint: F 0 = {X ∈ ({0, 1}n)k : ∀j ∈ [n], |Xj| ≤ 1}, +• Unique intersection: F 1 = {X ∈ ({0, 1}n)k : ∃j ∈ [n], |Xj| = k, ∀j′ ̸= j, |Xj′| ≤ 1}. +Towards proving Theorem 1.5, our first step is to consider unique set-disjointness +under product distribution which assign weight asymmetrically between the players. +2.1 +Lower bounds for product asymmetric distributions +Let ν be a distribution over [k]. We denote by νn the distribution over W ∈ [k]n, where +we sample Wj ∼ ν independently for all j ∈ [n]. We define two distributions, µ0 +prob[ν] +supported on F 0 and µ1 +prob[ν] supported on F 1. +Definition 2.1 (Disjoint asymmetric distribution). Let X ∈ ({0, 1}n)k be sampled as follows: +1. Sample W ∼ νn. +2. For each j ∈ [n], if Wj = i then we sample Xi(j) ∈ {0, 1} uniformly, and set Xi′(j) = 0 +for all i′ ̸= i. +We denote by µ0 +prob[ν] the marginal distribution of X, and note that it is supported on F 0. +Definition 2.2 (Unique intersection asymmetric distribution). Let Y ∈ ({0, 1}n)k be sampled +as follows: +8 + +1. Sample X ∼ µ0 +prob[ν]. +2. Sample j ∈ [n] uniformly. +3. If j = j then we set Y j = 1k and Y j′ = Xj′ for all j′ ̸= j. +We denote by µ1 +prob[ν] the marginal distribution of Y , and note that it is supported on F 1. +We denote by µprob[ν] the mixture distribution, where we sample b ∈ {0, 1} uniformly, +and then sample X ∼ µb +prob[ν]. Our main technical result is a communication lower bound +on protocols which solve unique set-disjointness under input distribution µprob[ν]. We will +later reduce the fixed set size case to this model. +Theorem 2.3. Fix n, k ≥ 1. Let ν be a distribution on [k]. Let Π be a [c1, . . . , ck]-bounded k-party +deterministic protocol which solves the unique set-disjointness problem under input distribution +µprob[ν] with error 2%. Then +� +i∈[k] +ci +ν(i) = Ω(n). +We note that Theorem 2.3 is a generalization of the lower bound for symmetric case +[Gro09, Jay09], where ν(i) = 1/k for all i ∈ [k]. +In this case Theorem 2.3 gives that +� +i ci = Ω(n/k). +2.1.1 +Information theory framework +We will use information theory to prove Theorem 2.3. Although we assume that Π has +small error with respect to both µ0 +prob[ν] and µ1 +prob[ν], we will only study its information +complexity with respect to µ0 +prob[ν]. +Below we let W ∈ [k]n, X ∈ ({0, 1})n be jointly +samples as in Definition 2.1. The following observation will play an important role. +Observation 2.4. Conditioned on W = W, the random variables (Xi(j) : i ∈ [k], j ∈ [n]) are +independent. +We start by giving a general bound for individual communication based on informa- +tion theory, which assumes only the existence of such W under which X1, . . . , Xk are +independent. +Lemma 2.5. Let Π be a k-party protocol which is [c1, . . . , ck]-bounded. Assume joint random +variables (W , X), where X = (X1, . . . , Xk) are the players inputs, and such that for every value +W for W , the random variables X1|W = W, . . . , Xk|W = W are independent. Then for each +i ∈ [k] we have +ci ≥ I(Xi : Π(X)|W ). +9 + +Proof. We first set up some notations. We denote by π a possible transcript for Π, and let +π si] ≤ exp(−si/6). +Let E denote the event that |Xi| > si for some i ∈ [k]. Then +Pr[E] ≤ +� +i∈[k] +exp(−si/6). +We first analyze the case that Pr[E] ≥ 1%. In this case, since ci ≥ 1 by assumption, and +since 1 +x ≥ C exp(−x/6) for some absolute constant C > 0 for all x ≥ 1, we get +� +i∈[k] +ci +si +≥ C +� +i∈[k] +exp(−si/6) ≥ C Pr[E] = Ω(1). +From now on we assume Pr[E] < 1%. +We now design the protocol Π′. First, each player checks if their input Xi satisfies +|Xi| > si. If so, the protocol aborts. This requires each player to send one bit, and by +assumption it aborts with probability at most 1%. Otherwise, each player extends their +input Xi to a new input Yi ∈ {0, 1}n of size |Yi| = si as follows. +Before the protocol starts, the players agree ahead of time on pairwise disjoint subsets +T1, . . . , Tk with |Ti| = si, supported in the last n/2 coordinates (so they do not overlap the +inputs X1, . . . , Xk). Now, the i-th player adds arbitrary si − |Xi| elements from Ti to their +set Xi; we denote the new input Yi ∈ {0, 1}n. Note that Y = (Y1, . . . , Yk) satisfies the same +promise as X = (X1, . . . , Xk); namely, either they are pairwise disjoint, or they have a +common element and except for it they are pairwise disjoint. +15 + +We would like to apply Π to Y . However we cannot quite yet; while it is true that +Y ∈ F 0 +size[s] or Y ∈ F 1 +size[s], its distribution is not uniform in the sets. However, here we +can apply Claim 2.14 to make the distribution of Y uniform in the respective family. The +players use public randomness to sample a permutation Σ on [n] and apply it to Y . Now +we can apply Π(Σ(Y )) which would give the correct with error 2% by assumption. The +proof now follows from Theorem 2.12. +3 +Interval systems +Recall that our plan is to use the lower bounds for multi-party unique set-disjointness in +order to prove lower bounds for streaming algorithms for the needle problem. In order +to effectively embed the inputs for unique set-disjointness inside streams, we introduce a +combinatorial construct that we call interval systems. +Definition 3.1 (Interval). An interval is a non-empty set of the form I = {a, a + 1, . . . , b} for +some a ≤ b. +Definition 3.2 (Interval systems). A [t]-interval system is a set F = {I1, . . . , Ik} of k pairwise +disjoint intervals supported in [t]. If we want to specify the number of intervals, we say F is a +[t, k]-interval system. +Definition 3.3 (Randomized interval systems). A randomized [t]-interval system F is a distri- +bution over [t]-interval systems F. Similarly, a randomized [t, k]-interval system F is a distribution +over [t, k]-interval systems F. +Next, we define for an interval system a corresponding distribution over sets T ⊂ [t]. +Definition 3.4 (Set distribution for interval systems). Let F be a [t]-interval system. We denote +by Sets(F) the distribution over sets T ⊂ [t] obtained by choosing uniformly one element from each +interval I ∈ F. +If F is a randomized [t]-interval system, then we define Sets(F) as follows: first sample F ∼ F +and then sample T ∼ Sets(F). +Observe that if F is a randomized [t, k]-interval system, then Sets(F) is a distribution +over k-sets in [t] (a k-set is a set of size k). Our goal will be to simulate the uniform +distribution over k-sets in [t]. We call such randomized interval systems perfect. +Definition 3.5 (Perfectinterval systems). A randomized [t, k]-interval system F is called perfect +if Sets(F) is the uniform distribution over all k-sets in [t]. +There are many ways to construct perfect randomized [t, k]-interval systems. +For +example, a naive way is to sample k uniform coordinates i1, . . . , ik ∈ [t], and then take +the distribution over F = {{i1}, . . . , {ik}}. However, for an efficient reduction, we would +need interval systems with as long intervals as possible. Technically, the efficiency of the +reduction will be controlled by the following notion of value of interval systems. +16 + +Definition 3.6 (Value of interval systems). Let F be a [t]-interval system. Its value is +val(F) = +� +I∈F +1 +|I|. +If F is a randomized [t]-interval system then its value is +val(F) = EF ∼F [val(F)] . +In order to prove strong lower bounds on streaming algorithms, we would need a +perfect distribution over [t, k]-intervals with as low a value as possible. The following +claim gives a lower bound for this. +Claim 3.7. Let F be a [t, k]-interval system. Then +val(F) ≥ k2 +t . +Proof. Let F = {I1, . . . , Ik} where |Ii| = si. We have � si ≤ t, and val(F) = � 1 +si. This +expression is minimized when all the si are the equal, and hence +val(F) ≥ k · +k +� si +≥ k2 +t . +Our main technical result in this section is a construction of a perfect randomized +[t, k]-interval system with value close to optimal. We do so by designing a randomized +algorithm that samples [t, k]-interval systems. We will show that its output distribution is +perfect, and of value close to the minimum given by Claim 3.7. +It will be convenient to make the following definition of “shifting” an interval or an +interval system. For an interval I = [a, b] and an integer c, define I + c = [a + c, b + c]. For +an interval system F = {I1, . . . , Ik} define F + c = {I1 + c, . . . , Ik + c}. +Algorithm 1: SampleIntervalSystem +Input: t ≥ 1, k ≥ 0 with k ≤ t +Output: [t, k]-interval system F +1 if k = 0 then +2 +return F = {} +3 else if k = 1 then +4 +return F = {[t]} +5 else +6 +Let s = ⌈t/2⌉ +7 +Sample j ∈ {0, . . . , k} with probability Pr[j = j] = (s +j)(t−s +k−j) +(t +k) +8 +Compute F1 = SampleIntervalSystem(s, j) +9 +Compute F2 = SampleIntervalSystem(t − s, k − j) +10 +return F = F1 ∪ (F2 + s) +11 end +17 + +We denote by F[t, k] the randomized [t, k]-interval system obtained by running +SampleIntervalSystem(t, k). +Claim 3.8. F[t, k] is perfect. +Proof. The proof is by induction on k, t. If k = 0 or k = 1 this is clear from the base cases of +the algorithm. If k ≥ 2, then we sample the number of elements j in the interval [s] with +the same probability as a uniform k-set in [t] would. By induction, the distribution F[s, j] +of F1 is a perfect randomized [s, j] interval system; and the distribution F[t − s, k − j] of +F2 is a perfect randomized [t − s, k − j] interval system. The claim follows. +We next analyze the value of F[t, k]; to simplify the analysis, we restrict to the case t +is a power of two. This suffices for our application, and we expect the bound to extend +to general t with minimal modifications. We assume below that all logarithms are in base +two. +Lemma 3.9. Assume t is a power of two. Then val(F[t, k]) ≤ k2 log(2t) +t +. +In order to prove Lemma 3.9, we will need the following technical claim, computing +first and second moments for the distribution over j in the algorithm. +Claim 3.10. Let t, k ≥ 1, t even, and 0 ≤ j ≤ k. Define p(t, k, j) = (t/2 +j )( t/2 +k−j) +(t +k) +. Then +k +� +j=0 +p(t, k, j) · j = k +2 +and +k +� +j=0 +p(t, k, j) · j2 ≤ k(k + 1) +4 +. +Proof. Let s = t/2. Let T be a uniform subset of [t] of size k. Then p(t, k, j) = Pr[|T ∩[s]| = j]. +Hence +k +� +j=0 +p(t, k, j) · j = ET + +� +i∈[s] +1[i ∈ T] + + = +� +i∈[s] +Pr[i ∈ T] = s · k +2s = k +2 +and +k +� +j=0 +p(t, k, j) · j2 = ET + + � +i,j∈[s] +1[i ∈ T] · 1[j ∈ T] + + = +� +i,j∈[s] +Pr[i, j ∈ T] += s · k +2s + s(s − 1) k(k − 1) +2s(2s − 1) ≤ k +2 + k(k − 1) +4 += k(k + 1) +4 +. +18 + +Proof of Lemma 3.9. Let f(t, k) = t · val(F[t, k]). +We have f(t, 0) = 0, f(t, 1) = 1 and +f(t, k) = 0 if k > t. The definition of f(t, k) for k ≥ 2 is recursive. Let p(t, k, j) = (t/2 +j )( t/2 +k−j) +(t +k) +. +Then +val(F[t, k]) = +k +� +j=0 +p(t, k, j) (val(F[t/2, j]) + val(F[t/2, k − j])) . +which implies +f(t, k) = 4 +k +� +j=0 +p(t, k, j)f(t/2, j). +It will be instructive to compute f(t, 2): +f(t, 2) = +t +t − 1 + t − 2 +t − 1f(t/2, 2) ≤ 2 + f(t/2, 2) ≤ 2 log(t). +We will prove by induction that +f(t, k) ≤ k2 + k(k − 1) log(t). +We already verified this for k = 0, 1, 2. For k ≥ 3 we have by induction: +f(t, k) ≤ 4 +k +� +j=0 +p(t, k, j) +� +j2 + j(j − 1) log(t/2) +� +. +Applying Claim 3.10 gives +f(t, k) ≤ k(k + 1) + k(k − 1) log(t/2) += 2k + k(k − 1) log(t) +≤ k2 + k(k − 1) log(t). +Finally we get +val(F[t, k]) = f(t, k) +t +≤ k2 + k(k − 1) log(t) +t +≤ k2 log(2t) +t +. +Our application for streaming algorithms for the needle problem has an additional +restriction, that the total length of the intervals in the interval system be bounded away +from t. We refer to such interval systems as valid. +Definition 3.11 (Valid interval systems). A [t]-interval system F is called valid if � +I∈F |I| ≤ +t/2. A randomized [t]-interval system F is called valid if all [t]-interval systems F in its support +are valid. +19 + +We next show how to refine a an interval system to obtain a valid randomized interval +system, while preserving the sets distribution, and without increasing the value too much. +Lemma 3.12. Assume k ≤ t/6. Let F be a [t, k]-interval system. Then there exists a randomized +[t, k]-interval system F such that: +1. Sets(F) = Sets(F) +2. val(F) ≤ 5 · val(F) +3. F is valid +Proof. Let F = {I1, . . . , Ik}. Given an interval Ii define ℓi = min(3, |Ii|). Partition Ii into ℓi +intervals {Ii,a : a ∈ [ℓi]} of as equal length as possible, and observe that +|Ii| +5 ≤ |Ii,a| ≤ |Ii| +3 + 1 +∀a ∈ [ℓi]. +Let pi,a = |Ii,a| +|Ii| . We define a randomized [t, k]-interval system F, where for each i ∈ [k] +independently, we replace Ii with one of its sub-intervals. Concretely, we choose a ∈ [ℓi] +with probability pi,a and replace Ii with Ii,a. We now prove the claims. +1. Observe that sampling a uniform element x ∈ Ii can equivalently be sampled by first +sampling a ∈ [ℓi] with probability pi,a, and then sampling a uniform element x ∈ Ii,a. +This implies that Sets(F) = Sets(F). +2. Since |Ii,a| ≥ |Ii|/5 for all i, a, the claim holds for any F ′ in the support of F, and +hence also for F. +3. Since |Ii,a| ≤ (|Ii| + 1)/2 for all i, a, we have for any F ′ = {I1,a1, . . . , Ik,ak} in the +support of F ′ that +� +i∈[k] +|Ii,ai| ≤ k + 1 +3 +� +i∈[k] +|Ii| ≤ k + t +3 ≤ t +2 +where the last inequality follows since we assume k ≤ t/6. +Lemma 3.12 applies also to randomized [t, k]-interval systems, by applying it to any +interval system in their support. The following lemma summarizes all the facts we would +need by applying it to F[t, k]. +Lemma 3.13. Let k, t ≥ 1. Assume t is a power of two and k ≤ t/6. Then there exists a valid +perfect randomized [t, k]-interval system F with +val(F) ≤ 10k2 log(t) +t +. +20 + +4 +Lower bound for the needle problem +We prove Theorem 1.2 in this section, by combining our lower bound for unique set- +disjointness with fixed set sizes (Theorem 2.13) with the efficient reduction given by +interval systems (Lemma 3.13). +First, we recall the parameters: n denotes the size of the domain, t the number of +samples and p the needle probability. We assume throughout that n = Ω(t2) is large +enough. We would denote by k the number of needles in a stream in the planted model, +where k ∼ Bin(t, p). We denote by Uniform the uniform distribution over [n]t. +First, we show how to prove lower bounds when k is fixed. Given a [t, k]-interval +system F = {I1, . . . , Ik}, we will assume in this section that the intervals are sorted in +order, namely that I1 comes before I2, which comes before I3, and so on. We define its +corresponding sizes as +Sizes(F) = (|I1|, . . . , |Ik|). +We recall the definition of a planted stream distribution from the introduction, where we +now present it more formally. +Definition 4.1 (Planted distribution for interval systems). Let F be a [t]-interval system. we +define a planted distribution Planted[F] over streams X ∈ [n]t as follows: +1. Sample uniform needle x ∈ [n]; +2. In each interval I ∈ F sample uniform index aI ∈ I and set XaI = x; +3. For all j ∈ [n] \ {aI : I ∈ F}, sample Xj ∈ [n] uniformly. +For F a randomized [t]-interval system, we define its planted distribution Planted[F] by first +sampling F ∼ F and then X ∼ Planted[F]. +We start by formalizing and proving Lemma 1.7. Given a streaming algorithm ALG +and two distributions D0, D1 over streams, we say that ALG distinguishes between D0, D1 +with error δ if, at the end of running the algorithm, the last player can guess if the input +was sampled from D0 or D1 and be correct with probability at least 1 − δ. A streaming +algorithm is an ℓ-pass streaming algorithm if it makes ℓ passes over the data stream. +Lemma 4.2. Let F be a [t, k]-interval system and set s = Sizes(F). +Let ALG be an ℓ-pass +streaming algorithm which distinguishes between Planted[F] and Uniform with error 0.5% and +uses space s. Then there is a communication protocol Π which solves the unique set-disjointness +problem under input distribution µsize[s], in which each player sends ℓs bits, and has error 1%. +Proof. Let X = (X1, . . . , Xk) ∈ ({0, 1}n)k be the input to the players, where we assume +X ∼ µb +size[s] for some b ∈ {0, 1}. The goal of the players is to figure out b. +Let F = {I1, . . . , Ik}. Let J1, . . . , Jk be a partition of [t], where Ii ⊂ Ji. As a first step, +each player individually constructs a stream Yi ∈ [n]Ji based on their input Xi. The i-th +player generates their stream as follows: +21 + +1. For each j ∈ Ji \ Ii, sample Yi(j) ∈ [n] uniformly. +2. Let Si = {j ∈ [n] : Xi(j) = 1}, where |Si| = si be assumption. Let Li ∈ [n]si be a +random permutation of Si. Set (Yi(j) : j ∈ Ii) = Li. +Let Y = Y1 ◦ · · · ◦ Yk ∈ [n]t be the concatenation of the streams. The players simulate +running ALG on the stream, where each player simulates it on their part of the stream, +and send the internal memory of the streaming algorithm to the next player. At the end +of each pass, the last player sends the internal memory back to the first player. Thus each +player sends at most ℓs bits. To conclude, we need to show that this allows to distinguish +between b = 0 and b = 1. +To conclude, we compute the distribution of Y based on the value of b, and show that +when b = 0 the distribution of Y is close to uniform, and when b = 1 it is close to the +planted distribution Planted[F]. Thus by assumption the algorithm distinguishes between +these two cases, which is our goal. +First, if b = 0 then X1, . . . , Xk are uniform sets of sizes s1, . . . , sk in [n], conditioned on +being pairwise disjoint. Thus the elements of Y are uniform among all choices of t distinct +elements in n. Since we assume n = Ω(t2), the statistical distance between Y and Uniform +is at most t2/n, which can be made as small as we want, say 0.1%. +Similarly, if b = 1 then X1, . . . , Xk are uniform conditioned on having a unique intersec- +tion. Similarly, the assumption n = Ω(t2) implies that the the statistical distance between +Y and Planted[F] can be made as small as we want, say 0.1%. +Overall, as we assume that ALG can distinguish between Uniform and Planted[F] with +error 0.5%, then it also distinguishes between the distributions of Y for b = 0 and b = 1 +with slightly larger error 1%. +Combining Lemma 4.2 with Theorem 2.13, we obtain the following corollary which +formalizes Lemma 1.7. +Lemma 4.3. Let F be a valid [t, k]-interval system. Let ALG be an ℓ-pass streaming algorithm +which distinguishes between Planted[F] and Uniform with error 0.5% and uses space s. Then +ℓs = Ω +� +1 +val(F) +� +. +Proof. Let Π be the protocol obtained by Lemma 4.2, which solves unique set-disjointness +under inputs distribution µsize[s] for s = Sizes(F) = [s1, . . . , sk], and where each player +sends at most ℓs bits. Since F is valid we have � si ≤ t/2. Theorem 2.13 then gives +� +i∈[k] +ℓs +si += Ω(1). +Recalling the definition of val(F) = � +i∈[k] +1 +si, we can rephrase this as ℓs·val(F) = Ω(1). +The following lemma, which formalizes Lemma 1.8, generalizes Lemma 4.3 to ran- +domized interval systems. +22 + +Lemma 4.4. Let F be a valid randomized [t]-interval system. Let ALG be an ℓ-pass streaming +algorithm which distinguishes between Planted[F] and Uniform with error 0.1% and uses space s. +Then +ℓs = Ω +� +1 +val(F) +� +. +Proof. Sample F ∼ F. Since val(F) = E[val(F)], by Markov’s inequality we have +Pr +F [val(F) > 2val(F)] ≤ 50%. +Next, let err(F) denote the error of ALG in distinguishing Planted[F] from Uniform. Since +Planted[F] is a mixture of Planted[F], then the average of err(F) is the error of ALG in +distinguishing Planted[F] from Uniform, which we assume is 0.1%. Thus +Pr +F [err(F) > 0.5%] ≤ 20%. +Overall, there is some choice of F in the support of F such that val(F) ≤ 2val(F) and +err(F) ≤ 0.5%. The lemma follows by applying Lemma 4.3 to F. +We now in place to finally prove Theorem 1.2, giving sample-space lower bounds for +any streaming algorithm that solves the needle problem. +Proof of Theorem 1.2. Let ALG be an ℓ-pass streaming algorithm which can distinguish with +high probability between the uniform and planted needle distribution using t samples. +As the inputs are stochastic, we may repeat it a few times to decrease its error. Thus, by +increasing t by a constant multiplicative factor, we may assume that the error is at most +0.1% and that t is a power of two. +For k ≤ t let Fk be the valid perfect randomized [t, k]-interval system given by +Lemma 3.13. We construct a randomized [t]-interval system F by sampling k ∼ Bin(t, p) +and taking Fk. Observe that Planted[F] is identical to the planted needle distribution. If +ALG uses s bits of space then Lemma 4.4 gives that +ℓs = Ω +� +1 +val(F) +� +. +To conclude the proof we just need to compute val(F). +For any fixed k we have by +Lemma 3.13 that +val(Fk) ≤ 10k2 log(t) +t +. +Since k ∼ Bin(t, p) we have E[k2] = p(1 − p)t + p2t2. Since we assume p = Ω(1/t), the +dominant term is the quadratic term, and hence E[k2] = Θ(p2t2). Thus we get +val(F) = O(p2t log(t)). +Rearranging the terms concludes the proof, since it gives ℓp2st log(t) = Ω(1). +23 + +5 +Open problems +We proved in Theorem 1.2 near-tight bound for the sample vs space complexity needed for +the needle problem, which proves similar near-tight bounds for the frequency estimation +in stochastic streams problem. It still remains open to prove sharp bounds, removing the +remaining logarithmic factor. We propose the following natural conjecture. +Conjecture 5.1. Any ℓ-pass streaming algorithm which can distinguish with high probability +between the uniform and planted models, where p is the needle probability, t the number of samples, +s the space and n the domain size, satisfies ℓp2st = Ω(1). +Another natural conjecture is to remove the artificial restriction of � si ≤ n/2 from +Theorem 1.5. We need it because we do not prove the theorem directly, but rather via a +reduction to the asymmetric product distribution case. 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Manuscript, 2022. +5 +27 + diff --git a/oNFST4oBgHgl3EQfMTgQ/content/tmp_files/2301.13743v1.pdf.txt b/oNFST4oBgHgl3EQfMTgQ/content/tmp_files/2301.13743v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..31992f7886e05e0b029e2768da6ac4460462740e --- /dev/null +++ b/oNFST4oBgHgl3EQfMTgQ/content/tmp_files/2301.13743v1.pdf.txt @@ -0,0 +1,1128 @@ +Zero-shot-Learning Cross-Modality Data +Translation Through Mutual Information +Guided Stochastic Diffusion +Zihao Wang1∗, Yingyu Yang2, +Maxime Sermesant2, Hervé Delingette2, Ona Wu1 +1 A. A. Martinos Center for Biomedical Imaging@MGH, +Harvard University. +149 13th St, Charlestown, MA 02129 +2 Centre Inria d’Université Côte d’Azur. +2004 Rte des Lucioles, 06902 Valbonne, France +zwang63@mgh.harvard.edu +February 1, 2023 +Abstract +Cross-modality data translation has attracted great interest in image +computing. Deep generative models (e.g., GANs) show performance im- +provement in tackling those problems. Nevertheless, as a fundamental +challenge in image translation, the problem of Zero-shot-Learning Cross- +Modality Data Translation with fidelity remains unanswered. This paper +proposes a new unsupervised zero-shot-learning method named Mutual +Information guided Diffusion cross-modality data translation Model (MID- +iffusion), which learns to translate the unseen source data to the target +domain. The MIDiffusion leverages a score-matching-based generative +model, which learns the prior knowledge in the target domain. We propose +a differentiable local-wise-MI-Layer (LMI) for conditioning the iterative +denoising sampling. The LMI captures the identical cross-modality fea- +tures in the statistical domain for the diffusion guidance; thus, our method +does not require retraining when the source domain is changed, as it does +not rely on any direct mapping between the source and target domains. +This advantage is critical for applying cross-modality data translation +methods in practice, as a reasonable amount of source domain dataset +is not always available for supervised training. We empirically show the +advanced performance of MIDiffusion in comparison with an influential +group of generative models, including adversarial-based and other score- +matching-based models. +∗corresponding: zihao.wang@ieee.org +1 +arXiv:2301.13743v1 [cs.CV] 31 Jan 2023 + +Figure 1: The proposed local-wise mutual information (LMI) guidance for +zero-shot stochastic diffusion-based modality translation. Unlike the classical +score-matching model, the proposed method conditions on a statistical measure: +local-wise mutual information. It measures the LMI between target image F +and its perturbed image Ft at time t during training time (left sub-figure) for +the forward SDE. At test time (right sub-figure), it calculates the LMI between +source image G and the sampled image ˆFt at time t for the reverse SDE. +1 +Introduction +One often wants to use existing resources to solve new problems without having +to develop similar resources for new data modalities to maximize economic +utility. For example, software systems to perform a specific task, e.g. brain +parcellation, may be designed to work with a particular data modality, e.g. MRI. +Ideally, one would be able to translate the new data modality, e.g. CT, into the +specific data modality and be able to analyze the new data with the existing +software. This method is often termed cross-modality data translation, which +has been of particular interest to the generative learning research community +in the past few years. Two major types of methods have been proposed: (1) +generative adversarial-based translation and (2) mapping-based translation. +Those two approaches correspond to the two major machine learning model +types: generative and discriminative learning. +The mapping-based method +directly learns the mapping between different modalities through regression +between the source and target modal data points. However, paired datasets +of different modalities are not always available in many scenarios. Also, slight +pixel-wise misalignments between different modalities may lead to inaccurate +translation because mapping-based translation models are designed to build a +direct mapping between the source and target modalities. These issues have led +to recent research focused on generative adversarial-based translation, which +learns the cross-modality mapping through cycle-consistency training, which +does not require paired datasets. +Two conceptual metrics have been proposed to evaluate the translation quality: +2 + +Target data Fo +Perturbed data Ft +Source data G +SampleddataF +LMI Layers LMI(Fo; Fo, Ft) +LMI Layers LMI(G ; G, Ft) +Perturbed Image Ft +Sampled Image Ft +Convolutional Layers +ConvolutionalLayerssynthesis realism (see Def. 1) and faithfulness (see Def. 2)[41]. Translation +faithfulness refers to how well the translation results maintain the context of the +source data intact. Translation realism means how consistent the translation +results are with the range of the target modality. Ensuring those two metrics in +a Zero-shot-Learning way is challenging as a Zero-shot-trained model does not +have any information from the source domain. +Mapping-based methods[60, 16, 26, 18, 43] usually favor translation faithful- +ness over realism. Many early mapping-based modality translations methods +rely on pixel-level modeling between the source and target modality. Generative +adversarial network (GAN) [33, 5, 68, 62, 59, 36, 4] based methods were proposed +due to various shortcomings of the previous mapping-based approaches. The +generative model is usually used for modeling the target modality directly, thus +achieving translation realism[1, 69, 8]. This method usually involves complicated +adversarial architecture design and modality task-specific loss function design +for different translation tasks [41]. Although the generative adversarial-based +translation does not need access to paired datasets for training, it still requires +source domain data, which may be challenging to collect, leading to insufficient +samples to balance the cycle-consistency training. Recent works [15, 14, 31, 6] +show that score-based generative models achieve better generation performance +than that GAN-based models. Meng et. al. [41] proposed SDEdit, which em- +ploys a Stochastic Diffusion Model (SDM) [57] to perform image translation that +balanced faithfulness and realism in the way of Zero-shot-Learning. Different +from conventional GAN or mapping-based models, which are nearly end-to-end +generation, an SDM is a score-based generative model [23] that relies on iter- +atively denoising a diffusion sequence driven by an SDE. Given a datapoint +from the source domain, SDEdit perturbs the datapoint with Gaussian noise +(perturbation-based guidance). Then a reverse SDE is used by SDEdit to gradu- +ally project the disturbed datapoint to the manifold of the target domain. SDEdit +has been shown to be superior to GAN-based translation models in terms of +parsimony of model structure and complexity of loss function. Although SDEdit +overcomes the drawbacks of previous GAN-based translation methods, it remains +limited in cross-modality data translation applications. The perturbation-based +guidance assumes the source and target domains can be perturbed by noise, +which may not be true in specific translation tasks (e.g., microscopy brightfield → +darkfield translation). In addition, SDEdit requires optimizing initial time t0 to +find the best interval for perturbation.[41] Muzaffer et. al. [46] proposed SynDiff +to tackle this issue by introducing a cycle-consistent architecture devised with a +bilateral diffusion process. The advantage of SynDiff is that the transformation +is semantically consistent, yet the computational cost of computing a bilateral +diffusion process is doubled. In fact, although it is claimed to be an unsupervised +method, SynDiff needs to pre-train a generator to estimate a paired source image +for training purposes, which requires an additional generation module; thus, the +model’s overall performance is also influenced by the pre-training quality of the +generator. +We propose a new zero-shot unsupervised learning-based cross-modality data +translation method based on a stochastic diffusion process. Instead of seeking any +3 + +conditioning guidance from the data domain, our model leverages the statistical +feature-wise homogeneity for conditioning the diffusion process (see Fig. 1). +This allows us to bridge the source and target domain by using their local-wise +statistical attributions for cross-modality data translation. The proposed method +overcomes the shortcomings of current modality translation approaches: +(1) Unlike GAN-based strategies, the proposed method (MIDiffusion) neither +needs inversion optimization in the test step nor requires additional adversarial +architectures and loss functions design (cycle consistency, etc.). +(2) In contrast to the current SDEdit method, our framework does not require +the optimization of the hyper-parameter t0, which is a user-selected parameter, +to achieve balance between realism and faithfulness.[41] +(3) MIDiffusion is a fully unsupervised Zero-shot-Learning model that can +translate unseen modalities into target modalities. +(4) MIDiffusion also empirically shows that the training of an additional +generator [46] for diffusion process guidance is not necessary to achieve semantic +consistency. +2 +Related Works +Generative Learning-based Cross-modality Data Translation A plethora +of works performs cross-modality data translation through conditional generative +adversarial networks (cGANs) [72, 71, 58, 54, 77, 4]. Those cGANs-based cross- +modality data translation usually requires the generation of a supervised signal +for conditioning the generator network (e.g., Zhuge et. al. [4] used a register to +generate the conditional signal). Some other GANs-based cross-modality data +translation methods use cycle consistency training to swap the features between +different domains [52, 29, 21, 50]. +Kernel-based Generative Learning The proposed method falls in the +scope of transition-kernel-based generative learning [56, 51, 65, 55, 10, 35]; +specifically, it belongs to score-based generative learning [24, 23, 61, 27]. The +score-based generative models show comparable data modeling performance to +those generative adversarial methods [33, 5, 68, 62, 59, 36, 8, 4, 28]. +Zero-shot Learning The cross-modality data translation problem without +access to the source modal data leads to a Zero-shot-Learning-based cross- +modality data translation problem. One challenge of learning-based methods is +that their modeling ability is restricted when dealing with unseen data classes +[66, 9, 34]. Zero-shot-Learning is a robust learning scheme to deal with such +cases when training and test classes are disjoint. One traditional approach of +Zero-shot-Learning tries to find a direct projection from image feature space to +a semantic space through discriminative methods [48, 2] or generative models +[39, 63]. Another popular way is to combine different modalities (such as images, +texts, attributes, etc.) and learn a non-linear multi-modal embedding [11, 38], +to help unseen class recognition. For example, Lin et. al. [37] used a GAN- +based model to learn a multi-modal consistent semantic representation, and the +disentangled domain-invariant features are extracted for unsupervised zero-shot +4 + +image-to-image translation. +Zero-shot Learning-based Cross-modality Data Translation The GAN- +based zero-shot-learning cross-modality data translation models usually modify +the latent representation of a pre-trained GAN model (a.k.a. GAN inversion +[75, 53, 8, 1]). It needs additional optimization in the testing step, which is +computationally expensive and time-consuming. +Perturbation-diffusion-based cross-modality data translation allows Zero- +shot learning for cross-modality data translation tasks [13, 23, 32, 57]. The +perturbation-diffusion-based methods perform excellently when the numerical +features of source and target domains are consistent [41]. Nevertheless, they may +fail when the cross-domain appearance features are significantly distinct. +3 +Theroy +3.1 +Cross-modality Data Translation +Mathematically, the cross-modality data translation task between two modalities +G ∈ V and F ∈ U can be formalized as: +ˆF ∈ U : ˆF = ΦF (G) +(1) +, where ΦF is an operator that maps the data G in the source domain V to the +corresponding data ˆF, which is ideally the same as F in the target domain U. +Specifically, this generalized form has been broadly applied in image synthesis, +stroke painting, image registration, segmentation, etc. [49] In this paper, we +will mainly focus on 2D image space. However, the proposed method can be +extended to 1D or 3D signals without loss of generality. +In zero-shot cross-modality data translation, only samples in target domain +F ∈ U are available during the training phase. Given the target samples, the +aim of Zero-shot-Learning is to learn ΦF without seeing the G in training step. +Since no source samples are accessible in the training phase, it is important to +build and use auxiliary information for domain transfer [64]. +3.2 +Mutual Information +Mutual Information (MI) measures the dependence of two random variables +X, Y : +MI(X, Y ) = +�� +p(x, y) log p(x, y) +p(x)p(y)dxdy +(2) +MI is useful in cross-modality data processing tasks as the statistical features +are assumed to be identical. It has been applied to tackle many unsupervised +learning problems such as cross-modality data retrieval[25], data representations +[22, 74, 17], domain adaptation [42], and cross-modal clustering [40] etc.. +A particular case of MI is using MI for measuring a random variable itself: +MI(X, X), which is called Entropy. +5 + +3.3 +Score Matching and Its Denoising Equivalent +Different from variational inference-based or likelihood-based training, which +attempts to approximate the true probability distribution log p(x) of the data, +the score-based models learn to represent the distribution log p(x) through its +partial derivative ∂ log p(x) +∂x +information (a.k.a score function). The maximization +process between a learning model sθ(x) and ∂ log p(x) +∂x +needs to get an explicit +form q(x) of the accurate distribution p(x), which usually remains unknown. +Instead of seeking any form q(x) of the distribution of the target dataset, the +denoising score matching method [61] sidesteps the searching of q(x) by directly +estimating the score function through [61, 27]: +arg min +θ +Eqσ(x,ˆx)[1 +2||sθ(x) − ∂ log qσ(ˆx|x) +∂ˆx +||2] +(3) +, where ∂ log qσ(ˆx|x) +∂ˆx +is the gradient between the clean data x and a noise- +polluted observation ˆx; as long as the noise is driven by a Gaussian kernel: +∂ log qσ(ˆx|x) +∂ˆx += +x−ˆx +σ2 , the learning model s(x) aims to learn the process of +denoising.[61] +3.4 +Score-based Generative Modeling with SDEs +Song et. al. [57] generalized the above denoising score matching-based models +into an SDE framework and unified the generative process by sampling with +a reverse SDE [3], which is also a diffusion process. The neural network sθ is +trained in an implicit denoising form by treating the noise-adding steps as a +diffusion process. The diffusion process driven by a standard normal distribution +N(0, 1) in a potential Ut(x) can be modeled by an SDE: +dxt = Ut(x)dt + σtdwt, +t ∈ [0, T] +(4) +, where σt ∈ [0, ∞) controls the magnitude of the input noise, wt denotes +a Wiener process, and t ∈ [0, T] is the time start from 0 with infinitesimal +incremental to t. We use a static potential Ut(x) = 0 (a.k.a "Variance Exploding +SDE (VE-SDE)" [57]) for modality translation here, which models the magnitude +of the data x0 ( the target modality F0). Adding noise with Eq. 4 is a similar +step when using a Gaussian transition kernel to perform score matching in Eq. +3. The training target of a diffusion process defined by the SDE 4 shares a +similar training target as Eq. 3 yet with an expectation of time t sampled +uniformly between [0, T]: Et∼U(0,T ); also the whole process becomes a multi-step +conditioning q(xt|x0) instead of a single step q(ˆx|x), +arg min +θ +Et∼U(0,T ){Ex0∼p0(x) +Ext∼qσt(xt,x0)[1 +2||sθ(x) − ∂ log qσt(xt|x0) +∂xt +||2]} +(5) +6 + +Figure 2: Example data of different modalities for the three different datasets; +from left to right are: CuRIOUS dataset of FLAIR imaging, CuRIOUS dataset of +T1-weighted (T1-w) imaging, Gold Atlas dataset of MR imaging, Gold Atlas +dataset of CT imaging, IXI dataset of PD-weighted imaging, IXI dataset of T1-w +imaging. +Figure 3: Illustration of an example of the proposed operator functioning for +1D functions X and Y in the neighborhood δ of point xi. (I) δ-neighborhood +of the two functions (upper Y , lower X) at point yi and xi. (II) functions are +processed by the proposed operators defined in Def. 5 and Def. 4. (III) compute +the Mutual Information between the paired segments in step (II) using kernel +density estimation. (IV) select the maximum value of MI in (III) as LMIδ(xi, yi) +at point xi between X and Y . +As long as the matched score model sθ ∼ ∂ log p(x) +∂x +learns the distribution +score ∂ log p(x) +∂x +, we can employ a reverse SDE [57, 3, 41] to sample datapoints +by inferring a backward in time t : T → 0 dynamic process: +dxt = −dσ2 +t +dt [∂ log p(x) +∂x +]tdt + +� +dσ2 +t +dt dwt, +(6) += −dσ2 +t +dt sθ(xt, t)dt + +� +dσ2 +t +dt dwt, +tT →0 ∈ [0, T] +(7) +, where wt is a Wiener process with infinitesimal negative incremental of time +dt. +7 + +FLAIR +T1w +MR +CT +PD4 +Methods +4.1 +Diffusion for Cross-modality Data Translation +We can solve the cross-modality data translation problem by adapting the target +data F generation task into the framework of score-matching and then using +a perturbed source domain G to guide (conditioning) the iterative diffusion +process [41, 46, 7, 47, 73, 32, 6]. Ideally, we want the generated data ˆF to +follow the semantic meaning of the guided data G and share the features as +the ground truth data point F in the target domain, balancing realism and +faithfulness.[41, 13] +4.1.1 +Cross-modality Data Translation Fidelity +Definition 1 The generated image ˆF shows realism means it is well translated +into the target domain U: ˆF ∈ U. +Definition 2 The generated image ˆF remains faithfulness means it is faith- +fully translated from the guided data: +ˆF ∼ G ∈ V, where ∼ is a similarity +measure. +The relationship between translation realism and faithfulness is related to the +domain correlations U and V. +We say a translation achieves high fidelity +if and only if the translation achieves a balance point between the realism +and faithfulness. However, when the difference in the appearance feature set +between the source domain G and target domain F becomes too large, a balance +point between the two sides may neither be easy to capture nor exist for a +satisfying translation. This leads current distribution perturbing-based methods +(e.g., SDEdit, etc. [41, 61, 27]) unsuitable for unsupervised cross-modality data +translation when the numerical features between the two domains have huge +differences. +4.2 +Mutual Information Guidance in Diffusion Generation +In the Zero-shot-Learning-based translation task, we do not have access to the +data in the source domain during the training process. Nevertheless, the local +statistical features between the source and target modalities are assumed to be +identical. MI maximization has been proven an effective method to empower +neural networks to learn non-linear representations [22]. To capture those shared +representations and use the extracted information for generation guidance, we +propose using MI to measure the local statistical representations in the iterative +denoising process. +4.2.1 +Local-wise MI +To obtain the semantic information in the data for guidance, we need to convert +the original data to statistical representation, as MI is a statistical measure. +8 + +Given a datapoint X, for point xi ∈ X at position i, the local-wise statistical +information at i can be captured through the probability density function (PDF) +pδxi (·) of the neighborhood area δxi of xi. +Without loss of generality, for the other points xj inside the neighborhood +area δxi of xi, we can obtain the local-wise statistical information through the +PDF pδxj (·); j ∈ δi. +Definition 3 The local-wise MI (LMI) from data point X to data point Y at +point xi is defined through: +LMIδ(xi, yj) = +sup +�� +pδ(x, y) log +pδ(x, y) +pδxi (x)pδyj (y)dxdy, ∀yj ∈ δxi +(8) +In the forward steps (training), we can use Eq. 8 as a reference signal to condition +each diffusion step; this is achieved by computing the LMI(X0, Xt), t ∈ [0, T] +during the training process of the scoring neural network. +Theorem 1 The upper bound of the LMI from data point X to datapoint Y at +location i is: LMIδ(xi, yi) ≤ LMIδ(xi, xi) , which is the optimum informative +match between X and Y at point xi. +The Thm. 1 indicates that the LMI achieves maximum (statistical similarity) +when: pδ(X) = pδ(Y ), ∀δ ∈ X. Thus the LMI always achieves maximum at the +same position between X0 and Xt in the training step; Yet, when pδ(X) ̸= pδ(Y ), +the LMI achieves the local maximum at position j, which is located in the +neighborhood δyi of yi. +We have defined the LMI for conditioning the iterative diffusion process. +However, the computational cost of LMI is very high as a statistical measure. +The iterative training of the score model requires many steps of discrete time +conditioning. It is unrealistic to use a sliding window or patch-wise looping to +compute the LMI between each time steps. To overcome this bottleneck of +applying LMI in score matching models, we developed an efficient method to +compute the LMI in both of the perturbing and denoising steps. +4.2.2 +Differentiable local-wise MI Layer +Definition 4 Let X(i) be a function defined in Rn, then Cδ(X) ∈ Rn is a +periodic extension of X in neighborhood K · δ, K ∈ Z: +Cδ(X) +:= +� +� +� +� +� +X +for i ∈ δ +is periodic of δ +for i ∈ K · δ +0 +otherwise +(9) +9 + +Definition 5 Let Y (i) be a function defined in Rn, then Bδ(Y ) ∈ Rn is a +k−steps (k ∈ Zn) sliding extension of Y in neighborhood k · δ: +Bδ(Y ) := JτY (i − τ) +s.t. +Jτ = +� +1 +τ − δ +2 ≤ i < τ + δ +2 +0 +else +(10) +for τ : 0 → δ with incremental ∆τ = δ/k. +Definition (5) and (4) can be understood as a ’segment-wise’ linear inter- +polation and nearest-neighbor interpolation [67] of functions Y and X in the +neighborhood δ. A demonstration of the operator applied for 1D functions de- +fined in ( Def. 4) and (Def. 5) is shown in Fig. 3 (II), for which the two functions +X and Y (Fig. 3 (I)) are processed by the operators C and B respectively. +Proposition 1 With two given functions F ∈ U and G ∈ U in R2, the LMI +(Def. 8) between G and F can be computed through the following operator S: +LMI(G, F) = max +δ (K(Bδ(F), Cδ(G)) log K(Bδ(F), Cδ(G)) +KBδ(F)KCδ(G))) +δ ∈ G, F +(11) +where K is a kernel density estimator, which approximates the PDF. +The Prop. 1 computes the maximum LMI in both of the training and testing +steps, which are LMIδ(F, F) and LMIδ(G, F) respectively. The operator given +in Prop. 1 can be accelerated by memory copying and parallel reduction in +GPGPU [12]. +Algorithm 1: Mutual Information Guided Diffusion +Inputs: G, sθ, time tT →0 ∈ [0, T], step size dt +Output: Generated high fidelity data ˆF +Initialize all parameters, variables; +while t < T do +z ∼ N(0, 1); +t ← t + dt; +ϵ = σ2 +t − σ2 +t−dt; +ˆFt+1 ← ˆFt + ϵsθ( ˆFt, LMI(G; G, ˆFt), t) + √ϵz ; +// Euler +discretization step, guided by G +end +4.3 +Embedding the Conditioner into SDE +In the training step (see left sub-figure of Fig. 1), we can condition the noise +perturbing process by embedding the defined LMI between a datapoint F and +10 + +Table 1: Quantitative evaluation of the MIDiffusion in comparison with three +baselines on three different cross-modality data translation tasks. The best- +performing values on the same task being compared are bolded in black italics +for highlighting. sup=supervised; unsup=unsupervised +Datasets +Methods +Modalities +SSIM +Tar↑ +SSIM +Src↑ +MSE↓ +MI↑ +PSNR↑ +FID↓ +GoldAtlas +CycleGAN[76] +(sup, few-shot 2%) +CT→MR +0.04 +0.03 +614.02 +1.16 +20.53 +202.43 +MR→CT +0.03 +0.02 +819.59 +1.13 +19.08 +281.35 +StyleGAN [30] +(unsup, inversion) +CT→MR +0.13 +0.04 +788.76 +1.09 +20.09 +213.47 +MR→CT +0.08 +0.07 +570.91 +1.12 +21.17 +170.83 +SDEdit [41] +(unsup) +CT→MR +0.003 +0.01 +766.40 +1.11 +19.50 +237.27 +MR→CT +0.01 +0.04 +996.71 +1.10 +18.58 +223.44 +MIDiffusion +(unsup) +CT→MR +0.06 +0.11 +523.18 +1.08 +21.66 +245.82 +MR→CT +0.12 +0.08 +392.35 +1.17 +23.03 +194.35 +CuRIOUS +CycleGAN +(sup, few-shot ~6%) +T1→FLAIR +-0.006 +0.81 +1747.13 +1.08 +16.04 +186.59 +FLAIR→T1 +0.005 +0.02 +3145.05 +1.05 +13.82 +331.89 +StyleGAN +(unsup, inversion) +T1→FLAIR +0.003 +0.12 +1880.62 +1.04 +15.83 +261.47 +FLAIR→T1 +-0.003 +0.19 +1570.83 +1.05 +16.62 +229.73 +SDEdit +(unsup) +T1→FLAIR +0.011 +0.01 +1558.22 +1.04 +16.42 +131.70 +FLAIR→T1 +0.005 +0.01 +2165.42 +1.03 +15.14 +141.89 +MIDiffusion +(unsup) +T1→FLAIR +0.07 +-0.08 +1226.40 +1.08 +17.65 +146.77 +FLAIR→T1 +0.15 +0.23 +1175.11 +1.08 +18.02 +157.98 +IXI +CycleGAN +(sup, few-shot 11%) +PD→T1 +0.12 +0.14 +1154.19 +1.17 +17.65 +141.95 +T1→PD +0.16 +0.16 +876.99 +1.19 +18.86 +113.67 +StyleGAN +(unsup, inversion) +PD-T1 +0.02 +0.06 +6609.13 +1.08 +10.17 +266.52 +T1→PD +0.21 +0.37 +2319.78 +1.14 +14.65 +199.12 +SDEdit +(unsup) +PD-T1 +0.09 +0.06 +1619.14 +1.15 +16.19 +68.6 +T1→PD +0.10 +0.06 +1753.82 +1.16 +15.95 +80.81 +MIDiffusion +(unsup) +PD-T1 +0.11 +0.19 +1652.81 +1.17 +16.35 +129.12 +T1→PD +0.18 +0.26 +1301.91 +1.13 +17.13 +132.46 +its disturbed data Ft in to the time-dependent-scoring model sθ, which is guided +by G: +arg min +θ +Et∼U(0,T ){Ex0∼p0(x)Ext∼qσt(xt,x0) +[1 +2|sθ( ˆFt, LMI(F; F, Ft), t) − ∂ log qσt(xt|x0) +∂xt +|2]} +(12) +In the sampling step (see right sub-figure of Fig. 1), recall that the generative +process is an iterative solution of the reverse SDE 6. We can inject the proposed +conditioning procedure into the SDE 6: +d ˆFt = −dσ2 +t +dt sθ( ˆFt, LMI(G; G, ˆFt), t)dt+ +� +dσ2 +t +dt dwt, +tT →0 ∈ [0, T] +(13) +and use an naive Euler-Maruyama numerical solver (see Algorithm 1) to solve +the SDE. +11 + +5 +Experiments and Evaluation +We show the following features of the proposed method in this section: (1) +The proposed zero-shot unsupervised learning method outperforms GAN-based +cross-modality data translation models in both the zero-shot and few-shot +supervised tracks. (2) Our method achieves high cross-modality data translation +fidelity compared with the state-of-the-art diffusion method. (3) The proposed +LMI-guided diffusion model enables cross-modality data translation semantic +consistency in the practical application. +5.1 +Evaluation Datasets +We introduce three public datasets that cover different cross-modality data +translation tasks. Fig. 2 shows six samples collected from three different datasets: +(1) The Gold Atlas dataset includes CT and T1-weighted (T1w) and T2-weighed +(T2w) magnetic resonance imaging (MRI) data from 19 male patients. The pelvic +area was the subject of the imaging. All the CT images had been deformably +registered to the corresponding MRI [45]. The datasets were collected from three +different sites [44]. We resampled all the datasets using the SimpleITK package +to a uniform voxel size (0.875mm × 0.875mm × 3mm), and selected slices every +15mm interval to build the training set (993 CT-MR slice pairs from 15 patients) +and the testing set (227 CT-MR slice pairs from 4 patients). +(2) The publicly available CuRIOUS dataset consists of 22 subjects with low- +grade glioma [70]. The original dataset includes T1w and FLAIR MRI scan pairs, +as well as unregistered ultrasound scans. All the images were collected during +routine clinical exams. The original scans had been resampled to 256 × 256 × 288 +voxels at an isotropic voxel size of 0.5mm3 [19]. We chose the T1w and FLAIR +MRI scan pairs to build our cross-modality image translation task. All the +volume pairs were resampled to 128 × 128 × 288 voxels with a spacing size +of 1.0mm × 1.0mm × 0.5mm by the SimpleITK package. As a result, 1168 +FLAIR-T1w slice pairs were picked in every 5mm from the original voxels to +generate the training (952 pairs from 17 subjects) and testing (216 pairs from 5 +subjects) datasets. +(3) IXI dataset1 includes pre-aligned 600 images from normal subjects. The +full dataset includes five different modalities: T1w, T2w, PD-weighted, magnetic +resonance angiograms (MRA), and Diffusion-weighted MRI. In our experiment, +we perform PD-T1w modal translation tasks. A training set of 300 slices (sampled +from 100 subjects) and a testing set of 75 slices (sampled from 25 subjects) were +generated using a subset of the IXI dataset . +5.2 +Baselines +We select two representative GAN-based image translation and synthesis ap- +proaches and one state-of-the-art diffusion-based method as baselines for com- +parison. We first compare our zero-shot-Learning-based method with a few-shot +1https://brain-development.org/ixi-dataset/ +12 + +Figure 4: Qualitative performance of different methods on CuRIOUS, GoldAtlas, +and IXI datasets. The top row shows the original images. The second to the +last rows show: CycleGAN (few-shot sup), StyleGAN(unsup zero-shot with +GAN inversion), SDEdit(zero-shot unsup learning with SDE-based perturbing +guidance), and the proposed MIDiffusion (unsup zero-shot with LMI guidance). +learning-based CycleGAN translation model [76]. The CycleGAN in this exper- +iment will be allowed to see both the full target domain dataset and a small +group (about 2% for Gold Atlas dataset, 6% for CuRIOUS datasetand 11% for +IXI dataset) of the source domain dataset. However, our proposed method will +see only the data in the target domain. +The second baseline method is a GAN inversion-based approach. A StyleGAN2- +ADA [30] is allowed to see the target domain training data. The out-domain +guided generation is performed through 5000 steps of optimization of inversion in +the latent space of the trained StyleGAN2-ADA [75]. An extra encoder network +is introduced in the generation step for inversion. +The third baseline model is SDEdit [41], which works for an ablation study. +The SDEdit is also a diffusion model-based translation method but uses a dis- +tribution perturbation guidance. Whenever possible, we use a default training +setup for all the baselines provided in the papers. The experiments were imple- +mented based on publicly available open-source code in the same experiment +environment. +5.3 +Quantitative Performance +We evaluate the different methods based on five metrics, including three image +quality measures: SSIM (structural similarity index measure), PSNR (peak +signal-to-noise ratio), MSE (mean square error), and two statistical similarity +measures: FID (Fréchet Inception Distance) [20] and MI (mutual information). +To study translation faithfulness (Def. 2) of each method, we report the SSIM +between the translation results ˆF and the guidance data G (SSIM-Src) and the +target data F (SSIM-Tar). The FID and MI are used to evaluate the translation +13 + +4similarity (Def. 1) statistically between ˆF and F for different methods. The FID +evaluates the generation similarity in the feature level between the ˆF and F. +The MI assesses the statistical similarity between the paired ˆF and F images. +We show the performance based on those metrics of the four methods in Table. +1. +5.3.1 +Comparison with few-shot training +We compare our method with the supervised few-shot trained CycleGAN (source +domain F ∈ U is visible) on each dataset. We notice that the size of the source +datasets for the few-shot training differs between different datasets. Regarding +the translation errors (measured by SSIM, MSE, and PSNR), the proposed zero- +shot trained MIDiffusion model outperforms the few-shot trained CycleGAN +model on both the GoldAtlas (2% of training data from source, 100% from +target) and CuRIOUS (6% of training data from source„ 100% from target) +datasets. However, this advantage of the MIDiffusion vanishes on the IXI dataset +for the MSE and PSNR metrics. This can be attributed to the 11% training +data from the source domain being sufficient for the CycleGAN to surpass the +performance of the zero-shot-learning models. +5.3.2 +Comparison with zero-shot training +As shown in Table 1, the proposed method performs better than the other +one-shot training-based methods in terms of the SSIM-Src, SSIM-Tar, and +MSE, which means the generated data of MIDiffusion achieves higher semantic +faithfulness with regard to both the source and target domain. Our model also +outperforms the other methods in terms of the PSNR and MI. This implies +our method shows translation realism 1 with respect to the target data in +pairwise comparison. In terms of similarity between the generated dataset and +the target dataset, we see that the SDEdit has an overall lower FID score. Yet, +in considering our generation target of fidelity translation, the SDEdit fails in +faithfulness translation. Thus the translation results of SDEdit performs worse. +In contrast, the proposed method achieves good realism (lowest FID score apart +from SDEdit) while keeping the semantic meaning from the guidance to the +target (higher SSIM, lower MSE). The cross-modality data translation method +with the highest fidelity is therefore the MIDiffusion. +5.4 +Qualitative Performance +Fig. 4 shows translated results from different methods on the test dataset as +well as the groundtruth (top rows). The first two columns in each dataset group +represent different modalities, and the last two columns correspond to their +zoomed-in details. Overall, the proposed method achieves the best translation +fidelity among the compared methods. The anatomical structures are more +faithfully represented in the MIDiffusion translation results compared to the +results from the baseline models. The SDEdit and StyleGAN fail to translate +14 + +the source images with identical features. In addition, the few-shot trained +CycleGAN clearly underperforms the zero-shot trained MIDiffusion model when +trained with insufficient source data on the CuRIOUS (2%) and GoldAtlas (6%) +dataset. This observation is consistent with the quantitative results shown in +Table 1. Overall, the MIDiffusion method outperforms the baseline methods in +both anatomical consistency and appearance similarity. +6 +Conclusion +This paper presents a novel local-wise mutual information-guided diffusion model +named MIDiffusion for cross-modality data translation. Different from cur- +rent cycle-consistency training, the MIDiffusion does not require seeing the +source dataset for training. Unlike GAN inversion methods that require itera- +tive optimization during the generation, the MIDiffusion does not need online +optimization in the test steps.Our method introduces a new conditioner that +achieves high-fidelity generation but does not need any extra training on the +main diffusion flow. 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Express, 12(12):7526–7543, Dec 2021. +22 + diff --git a/oNFST4oBgHgl3EQfMTgQ/content/tmp_files/load_file.txt b/oNFST4oBgHgl3EQfMTgQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d73501240b078cd2c5fa7a4bf33f0c3bcfa26c2 --- /dev/null +++ b/oNFST4oBgHgl3EQfMTgQ/content/tmp_files/load_file.txt @@ -0,0 +1,1209 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf,len=1208 +page_content='Zero-shot-Learning Cross-Modality Data Translation Through Mutual Information Guided Stochastic Diffusion Zihao Wang1∗, Yingyu Yang2, Maxime Sermesant2, Hervé Delingette2, Ona Wu1 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Martinos Center for Biomedical Imaging@MGH, Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 149 13th St, Charlestown, MA 02129 2 Centre Inria d’Université Côte d’Azur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 2004 Rte des Lucioles, 06902 Valbonne, France zwang63@mgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='edu February 1, 2023 Abstract Cross-modality data translation has attracted great interest in image computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Deep generative models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=', GANs) show performance im- provement in tackling those problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Nevertheless, as a fundamental challenge in image translation, the problem of Zero-shot-Learning Cross- Modality Data Translation with fidelity remains unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This paper proposes a new unsupervised zero-shot-learning method named Mutual Information guided Diffusion cross-modality data translation Model (MID- iffusion), which learns to translate the unseen source data to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The MIDiffusion leverages a score-matching-based generative model, which learns the prior knowledge in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We propose a differentiable local-wise-MI-Layer (LMI) for conditioning the iterative denoising sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The LMI captures the identical cross-modality fea- tures in the statistical domain for the diffusion guidance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' thus, our method does not require retraining when the source domain is changed, as it does not rely on any direct mapping between the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This advantage is critical for applying cross-modality data translation methods in practice, as a reasonable amount of source domain dataset is not always available for supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We empirically show the advanced performance of MIDiffusion in comparison with an influential group of generative models, including adversarial-based and other score- matching-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' ∗corresponding: zihao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='wang@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='org 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='13743v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='CV] 31 Jan 2023 Figure 1: The proposed local-wise mutual information (LMI) guidance for zero-shot stochastic diffusion-based modality translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Unlike the classical score-matching model, the proposed method conditions on a statistical measure: local-wise mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' It measures the LMI between target image F and its perturbed image Ft at time t during training time (left sub-figure) for the forward SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' At test time (right sub-figure), it calculates the LMI between source image G and the sampled image ˆFt at time t for the reverse SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1 Introduction One often wants to use existing resources to solve new problems without having to develop similar resources for new data modalities to maximize economic utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' For example, software systems to perform a specific task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' brain parcellation, may be designed to work with a particular data modality, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Ideally, one would be able to translate the new data modality, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' CT, into the specific data modality and be able to analyze the new data with the existing software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This method is often termed cross-modality data translation, which has been of particular interest to the generative learning research community in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Two major types of methods have been proposed: (1) generative adversarial-based translation and (2) mapping-based translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Those two approaches correspond to the two major machine learning model types: generative and discriminative learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The mapping-based method directly learns the mapping between different modalities through regression between the source and target modal data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' However, paired datasets of different modalities are not always available in many scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Also, slight pixel-wise misalignments between different modalities may lead to inaccurate translation because mapping-based translation models are designed to build a direct mapping between the source and target modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' These issues have led to recent research focused on generative adversarial-based translation, which learns the cross-modality mapping through cycle-consistency training, which does not require paired datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Two conceptual metrics have been proposed to evaluate the translation quality: 2 Target data Fo Perturbed data Ft Source data G SampleddataF LMI Layers LMI(Fo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Fo, Ft) LMI Layers LMI(G ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' G, Ft) Perturbed Image Ft Sampled Image Ft Convolutional Layers ConvolutionalLayerssynthesis realism (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1) and faithfulness (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 2)[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Translation faithfulness refers to how well the translation results maintain the context of the source data intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Translation realism means how consistent the translation results are with the range of the target modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Ensuring those two metrics in a Zero-shot-Learning way is challenging as a Zero-shot-trained model does not have any information from the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Mapping-based methods[60, 16, 26, 18, 43] usually favor translation faithful- ness over realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Many early mapping-based modality translations methods rely on pixel-level modeling between the source and target modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Generative adversarial network (GAN) [33, 5, 68, 62, 59, 36, 4] based methods were proposed due to various shortcomings of the previous mapping-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The generative model is usually used for modeling the target modality directly, thus achieving translation realism[1, 69, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This method usually involves complicated adversarial architecture design and modality task-specific loss function design for different translation tasks [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Although the generative adversarial-based translation does not need access to paired datasets for training, it still requires source domain data, which may be challenging to collect, leading to insufficient samples to balance the cycle-consistency training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Recent works [15, 14, 31, 6] show that score-based generative models achieve better generation performance than that GAN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Meng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [41] proposed SDEdit, which em- ploys a Stochastic Diffusion Model (SDM) [57] to perform image translation that balanced faithfulness and realism in the way of Zero-shot-Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Different from conventional GAN or mapping-based models, which are nearly end-to-end generation, an SDM is a score-based generative model [23] that relies on iter- atively denoising a diffusion sequence driven by an SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Given a datapoint from the source domain, SDEdit perturbs the datapoint with Gaussian noise (perturbation-based guidance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Then a reverse SDE is used by SDEdit to gradu- ally project the disturbed datapoint to the manifold of the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' SDEdit has been shown to be superior to GAN-based translation models in terms of parsimony of model structure and complexity of loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Although SDEdit overcomes the drawbacks of previous GAN-based translation methods, it remains limited in cross-modality data translation applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The perturbation-based guidance assumes the source and target domains can be perturbed by noise, which may not be true in specific translation tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=', microscopy brightfield → darkfield translation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' In addition, SDEdit requires optimizing initial time t0 to find the best interval for perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [41] Muzaffer et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [46] proposed SynDiff to tackle this issue by introducing a cycle-consistent architecture devised with a bilateral diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The advantage of SynDiff is that the transformation is semantically consistent, yet the computational cost of computing a bilateral diffusion process is doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' In fact, although it is claimed to be an unsupervised method, SynDiff needs to pre-train a generator to estimate a paired source image for training purposes, which requires an additional generation module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' thus, the model’s overall performance is also influenced by the pre-training quality of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We propose a new zero-shot unsupervised learning-based cross-modality data translation method based on a stochastic diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Instead of seeking any 3 conditioning guidance from the data domain, our model leverages the statistical feature-wise homogeneity for conditioning the diffusion process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This allows us to bridge the source and target domain by using their local-wise statistical attributions for cross-modality data translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The proposed method overcomes the shortcomings of current modality translation approaches: (1) Unlike GAN-based strategies, the proposed method (MIDiffusion) neither needs inversion optimization in the test step nor requires additional adversarial architectures and loss functions design (cycle consistency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (2) In contrast to the current SDEdit method, our framework does not require the optimization of the hyper-parameter t0, which is a user-selected parameter, to achieve balance between realism and faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [41] (3) MIDiffusion is a fully unsupervised Zero-shot-Learning model that can translate unseen modalities into target modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (4) MIDiffusion also empirically shows that the training of an additional generator [46] for diffusion process guidance is not necessary to achieve semantic consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 2 Related Works Generative Learning-based Cross-modality Data Translation A plethora of works performs cross-modality data translation through conditional generative adversarial networks (cGANs) [72, 71, 58, 54, 77, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Those cGANs-based cross- modality data translation usually requires the generation of a supervised signal for conditioning the generator network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=', Zhuge et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [4] used a register to generate the conditional signal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Some other GANs-based cross-modality data translation methods use cycle consistency training to swap the features between different domains [52, 29, 21, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Kernel-based Generative Learning The proposed method falls in the scope of transition-kernel-based generative learning [56, 51, 65, 55, 10, 35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' specifically, it belongs to score-based generative learning [24, 23, 61, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The score-based generative models show comparable data modeling performance to those generative adversarial methods [33, 5, 68, 62, 59, 36, 8, 4, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Zero-shot Learning The cross-modality data translation problem without access to the source modal data leads to a Zero-shot-Learning-based cross- modality data translation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' One challenge of learning-based methods is that their modeling ability is restricted when dealing with unseen data classes [66, 9, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Zero-shot-Learning is a robust learning scheme to deal with such cases when training and test classes are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' One traditional approach of Zero-shot-Learning tries to find a direct projection from image feature space to a semantic space through discriminative methods [48, 2] or generative models [39, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Another popular way is to combine different modalities (such as images, texts, attributes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=') and learn a non-linear multi-modal embedding [11, 38], to help unseen class recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' For example, Lin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [37] used a GAN- based model to learn a multi-modal consistent semantic representation, and the disentangled domain-invariant features are extracted for unsupervised zero-shot 4 image-to-image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Zero-shot Learning-based Cross-modality Data Translation The GAN- based zero-shot-learning cross-modality data translation models usually modify the latent representation of a pre-trained GAN model (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' GAN inversion [75, 53, 8, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' It needs additional optimization in the testing step, which is computationally expensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Perturbation-diffusion-based cross-modality data translation allows Zero- shot learning for cross-modality data translation tasks [13, 23, 32, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The perturbation-diffusion-based methods perform excellently when the numerical features of source and target domains are consistent [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Nevertheless, they may fail when the cross-domain appearance features are significantly distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 3 Theroy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='1 Cross-modality Data Translation Mathematically, the cross-modality data translation task between two modalities G ∈ V and F ∈ U can be formalized as: ˆF ∈ U : ˆF = ΦF (G) (1) , where ΦF is an operator that maps the data G in the source domain V to the corresponding data ˆF, which is ideally the same as F in the target domain U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Specifically, this generalized form has been broadly applied in image synthesis, stroke painting, image registration, segmentation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [49] In this paper, we will mainly focus on 2D image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' However, the proposed method can be extended to 1D or 3D signals without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' In zero-shot cross-modality data translation, only samples in target domain F ∈ U are available during the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Given the target samples, the aim of Zero-shot-Learning is to learn ΦF without seeing the G in training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Since no source samples are accessible in the training phase, it is important to build and use auxiliary information for domain transfer [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='2 Mutual Information Mutual Information (MI) measures the dependence of two random variables X, Y : MI(X, Y ) = �� p(x, y) log p(x, y) p(x)p(y)dxdy (2) MI is useful in cross-modality data processing tasks as the statistical features are assumed to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' It has been applied to tackle many unsupervised learning problems such as cross-modality data retrieval[25], data representations [22, 74, 17], domain adaptation [42], and cross-modal clustering [40] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='. A particular case of MI is using MI for measuring a random variable itself: MI(X, X), which is called Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='3 Score Matching and Its Denoising Equivalent Different from variational inference-based or likelihood-based training, which attempts to approximate the true probability distribution log p(x) of the data, the score-based models learn to represent the distribution log p(x) through its partial derivative ∂ log p(x) ∂x information (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='a score function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The maximization process between a learning model sθ(x) and ∂ log p(x) ∂x needs to get an explicit form q(x) of the accurate distribution p(x), which usually remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Instead of seeking any form q(x) of the distribution of the target dataset, the denoising score matching method [61] sidesteps the searching of q(x) by directly estimating the score function through [61, 27]: arg min θ Eqσ(x,ˆx)[1 2||sθ(x) − ∂ log qσ(ˆx|x) ∂ˆx ||2] (3) , where ∂ log qσ(ˆx|x) ∂ˆx is the gradient between the clean data x and a noise- polluted observation ˆx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' as long as the noise is driven by a Gaussian kernel: ∂ log qσ(ˆx|x) ∂ˆx = x−ˆx σ2 , the learning model s(x) aims to learn the process of denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [61] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='4 Score-based Generative Modeling with SDEs Song et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [57] generalized the above denoising score matching-based models into an SDE framework and unified the generative process by sampling with a reverse SDE [3], which is also a diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The neural network sθ is trained in an implicit denoising form by treating the noise-adding steps as a diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The diffusion process driven by a standard normal distribution N(0, 1) in a potential Ut(x) can be modeled by an SDE: dxt = Ut(x)dt + σtdwt, t ∈ [0, T] (4) , where σt ∈ [0, ∞) controls the magnitude of the input noise, wt denotes a Wiener process, and t ∈ [0, T] is the time start from 0 with infinitesimal incremental to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We use a static potential Ut(x) = 0 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='a "Variance Exploding SDE (VE-SDE)" [57]) for modality translation here, which models the magnitude of the data x0 ( the target modality F0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Adding noise with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 4 is a similar step when using a Gaussian transition kernel to perform score matching in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The training target of a diffusion process defined by the SDE 4 shares a similar training target as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 3 yet with an expectation of time t sampled uniformly between [0, T]: Et∼U(0,T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' also the whole process becomes a multi-step conditioning q(xt|x0) instead of a single step q(ˆx|x), arg min θ Et∼U(0,T ){Ex0∼p0(x) Ext∼qσt(xt,x0)[1 2||sθ(x) − ∂ log qσt(xt|x0) ∂xt ||2]} (5) 6 Figure 2: Example data of different modalities for the three different datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' from left to right are: CuRIOUS dataset of FLAIR imaging, CuRIOUS dataset of T1-weighted (T1-w) imaging, Gold Atlas dataset of MR imaging, Gold Atlas dataset of CT imaging, IXI dataset of PD-weighted imaging, IXI dataset of T1-w imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Figure 3: Illustration of an example of the proposed operator functioning for 1D functions X and Y in the neighborhood δ of point xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (I) δ-neighborhood of the two functions (upper Y , lower X) at point yi and xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (II) functions are processed by the proposed operators defined in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5 and Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (III) compute the Mutual Information between the paired segments in step (II) using kernel density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (IV) select the maximum value of MI in (III) as LMIδ(xi, yi) at point xi between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' As long as the matched score model sθ ∼ ∂ log p(x) ∂x learns the distribution score ∂ log p(x) ∂x , we can employ a reverse SDE [57, 3, 41] to sample datapoints by inferring a backward in time t : T → 0 dynamic process: dxt = −dσ2 t dt [∂ log p(x) ∂x ]tdt + � dσ2 t dt dwt, (6) = −dσ2 t dt sθ(xt, t)dt + � dσ2 t dt dwt, tT →0 ∈ [0, T] (7) , where wt is a Wiener process with infinitesimal negative incremental of time dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 7 FLAIR T1w MR CT PD4 Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='1 Diffusion for Cross-modality Data Translation We can solve the cross-modality data translation problem by adapting the target data F generation task into the framework of score-matching and then using a perturbed source domain G to guide (conditioning) the iterative diffusion process [41, 46, 7, 47, 73, 32, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Ideally, we want the generated data ˆF to follow the semantic meaning of the guided data G and share the features as the ground truth data point F in the target domain, balancing realism and faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [41, 13] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='1 Cross-modality Data Translation Fidelity Definition 1 The generated image ˆF shows realism means it is well translated into the target domain U: ˆF ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Definition 2 The generated image ˆF remains faithfulness means it is faith- fully translated from the guided data: ˆF ∼ G ∈ V, where ∼ is a similarity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The relationship between translation realism and faithfulness is related to the domain correlations U and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We say a translation achieves high fidelity if and only if the translation achieves a balance point between the realism and faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' However, when the difference in the appearance feature set between the source domain G and target domain F becomes too large, a balance point between the two sides may neither be easy to capture nor exist for a satisfying translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This leads current distribution perturbing-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=', SDEdit, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' [41, 61, 27]) unsuitable for unsupervised cross-modality data translation when the numerical features between the two domains have huge differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='2 Mutual Information Guidance in Diffusion Generation In the Zero-shot-Learning-based translation task, we do not have access to the data in the source domain during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Nevertheless, the local statistical features between the source and target modalities are assumed to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' MI maximization has been proven an effective method to empower neural networks to learn non-linear representations [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' To capture those shared representations and use the extracted information for generation guidance, we propose using MI to measure the local statistical representations in the iterative denoising process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='1 Local-wise MI To obtain the semantic information in the data for guidance, we need to convert the original data to statistical representation, as MI is a statistical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 8 Given a datapoint X, for point xi ∈ X at position i, the local-wise statistical information at i can be captured through the probability density function (PDF) pδxi (·) of the neighborhood area δxi of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Without loss of generality, for the other points xj inside the neighborhood area δxi of xi, we can obtain the local-wise statistical information through the PDF pδxj (·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' j ∈ δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Definition 3 The local-wise MI (LMI) from data point X to data point Y at point xi is defined through: LMIδ(xi, yj) = sup �� pδ(x, y) log pδ(x, y) pδxi (x)pδyj (y)dxdy, ∀yj ∈ δxi (8) In the forward steps (training), we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 8 as a reference signal to condition each diffusion step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' this is achieved by computing the LMI(X0, Xt), t ∈ [0, T] during the training process of the scoring neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Theorem 1 The upper bound of the LMI from data point X to datapoint Y at location i is: LMIδ(xi, yi) ≤ LMIδ(xi, xi) , which is the optimum informative match between X and Y at point xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1 indicates that the LMI achieves maximum (statistical similarity) when: pδ(X) = pδ(Y ), ∀δ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Thus the LMI always achieves maximum at the same position between X0 and Xt in the training step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Yet, when pδ(X) ̸= pδ(Y ), the LMI achieves the local maximum at position j, which is located in the neighborhood δyi of yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We have defined the LMI for conditioning the iterative diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' However, the computational cost of LMI is very high as a statistical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The iterative training of the score model requires many steps of discrete time conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' It is unrealistic to use a sliding window or patch-wise looping to compute the LMI between each time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' To overcome this bottleneck of applying LMI in score matching models, we developed an efficient method to compute the LMI in both of the perturbing and denoising steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='2 Differentiable local-wise MI Layer Definition 4 Let X(i) be a function defined in Rn, then Cδ(X) ∈ Rn is a periodic extension of X in neighborhood K · δ, K ∈ Z: Cδ(X) := � � � � � X for i ∈ δ is periodic of δ for i ∈ K · δ 0 otherwise (9) 9 Definition 5 Let Y (i) be a function defined in Rn, then Bδ(Y ) ∈ Rn is a k−steps (k ∈ Zn) sliding extension of Y in neighborhood k · δ: Bδ(Y ) := JτY (i − τ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Jτ = � 1 τ − δ 2 ≤ i < τ + δ 2 0 else (10) for τ : 0 → δ with incremental ∆τ = δ/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Definition (5) and (4) can be understood as a ’segment-wise’ linear inter- polation and nearest-neighbor interpolation [67] of functions Y and X in the neighborhood δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' A demonstration of the operator applied for 1D functions de- fined in ( Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 4) and (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 3 (II), for which the two functions X and Y (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 3 (I)) are processed by the operators C and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Proposition 1 With two given functions F ∈ U and G ∈ U in R2, the LMI (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 8) between G and F can be computed through the following operator S: LMI(G, F) = max δ (K(Bδ(F), Cδ(G)) log K(Bδ(F), Cδ(G)) KBδ(F)KCδ(G))) δ ∈ G, F (11) where K is a kernel density estimator, which approximates the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1 computes the maximum LMI in both of the training and testing steps, which are LMIδ(F, F) and LMIδ(G, F) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The operator given in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1 can be accelerated by memory copying and parallel reduction in GPGPU [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Algorithm 1: Mutual Information Guided Diffusion Inputs: G, sθ, time tT →0 ∈ [0, T], step size dt Output: Generated high fidelity data ˆF Initialize all parameters, variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' while t < T do z ∼ N(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' t ← t + dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' ϵ = σ2 t − σ2 t−dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' ˆFt+1 ← ˆFt + ϵsθ( ˆFt, LMI(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' G, ˆFt), t) + √ϵz ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' // Euler discretization step, guided by G end 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='3 Embedding the Conditioner into SDE In the training step (see left sub-figure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1), we can condition the noise perturbing process by embedding the defined LMI between a datapoint F and 10 Table 1: Quantitative evaluation of the MIDiffusion in comparison with three baselines on three different cross-modality data translation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The best- performing values on the same task being compared are bolded in black italics for highlighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' sup=supervised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' unsup=unsupervised Datasets Methods Modalities SSIM Tar↑ SSIM Src↑ MSE↓ MI↑ PSNR↑ FID↓ GoldAtlas CycleGAN[76] (sup, few-shot 2%) CT→MR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='03 614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='16 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='53 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='43 MR→CT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='02 819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='13 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='08 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='35 StyleGAN [30] (unsup, inversion) CT→MR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='04 788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='09 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='09 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='47 MR→CT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='07 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='12 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='17 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='83 SDEdit [41] (unsup) CT→MR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='01 766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='11 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='50 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='27 MR→CT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='04 996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='10 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='58 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='44 MIDiffusion (unsup) CT→MR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='11 523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='08 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='66 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='82 MR→CT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='08 392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='17 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='03 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='35 CuRIOUS CycleGAN (sup, few-shot ~6%) T1→FLAIR 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='04 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='83 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='47 FLAIR→T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='19 1570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='83 1.' 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16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='42 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='70 FLAIR→T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='01 2165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='03 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='14 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='89 MIDiffusion (unsup) T1→FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='08 1226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='08 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='65 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='17 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='52 T1→PD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='37 2319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='14 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='65 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='12 SDEdit (unsup) PD-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='06 1619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='15 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='19 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='6 T1→PD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='06 1753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='16 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='95 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='81 MIDiffusion (unsup) PD-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='19 1652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='35 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='12 T1→PD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='26 1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='13 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='13 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='46 its disturbed data Ft in to the time-dependent-scoring model sθ, which is guided by G: arg min θ Et∼U(0,T ){Ex0∼p0(x)Ext∼qσt(xt,x0) [1 2|sθ( ˆFt, LMI(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' F, Ft), t) − ∂ log qσt(xt|x0) ∂xt |2]} (12) In the sampling step (see right sub-figure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1), recall that the generative process is an iterative solution of the reverse SDE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We can inject the proposed conditioning procedure into the SDE 6: d ˆFt = −dσ2 t dt sθ( ˆFt, LMI(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' G, ˆFt), t)dt+ � dσ2 t dt dwt, tT →0 ∈ [0, T] (13) and use an naive Euler-Maruyama numerical solver (see Algorithm 1) to solve the SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 11 5 Experiments and Evaluation We show the following features of the proposed method in this section: (1) The proposed zero-shot unsupervised learning method outperforms GAN-based cross-modality data translation models in both the zero-shot and few-shot supervised tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (2) Our method achieves high cross-modality data translation fidelity compared with the state-of-the-art diffusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (3) The proposed LMI-guided diffusion model enables cross-modality data translation semantic consistency in the practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='1 Evaluation Datasets We introduce three public datasets that cover different cross-modality data translation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 2 shows six samples collected from three different datasets: (1) The Gold Atlas dataset includes CT and T1-weighted (T1w) and T2-weighed (T2w) magnetic resonance imaging (MRI) data from 19 male patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The pelvic area was the subject of the imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' All the CT images had been deformably registered to the corresponding MRI [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The datasets were collected from three different sites [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We resampled all the datasets using the SimpleITK package to a uniform voxel size (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='875mm × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='875mm × 3mm), and selected slices every 15mm interval to build the training set (993 CT-MR slice pairs from 15 patients) and the testing set (227 CT-MR slice pairs from 4 patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (2) The publicly available CuRIOUS dataset consists of 22 subjects with low- grade glioma [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The original dataset includes T1w and FLAIR MRI scan pairs, as well as unregistered ultrasound scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' All the images were collected during routine clinical exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The original scans had been resampled to 256 × 256 × 288 voxels at an isotropic voxel size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='5mm3 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We chose the T1w and FLAIR MRI scan pairs to build our cross-modality image translation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' All the volume pairs were resampled to 128 × 128 × 288 voxels with a spacing size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='0mm × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='0mm × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='5mm by the SimpleITK package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' As a result, 1168 FLAIR-T1w slice pairs were picked in every 5mm from the original voxels to generate the training (952 pairs from 17 subjects) and testing (216 pairs from 5 subjects) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' (3) IXI dataset1 includes pre-aligned 600 images from normal subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The full dataset includes five different modalities: T1w, T2w, PD-weighted, magnetic resonance angiograms (MRA), and Diffusion-weighted MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' In our experiment, we perform PD-T1w modal translation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' A training set of 300 slices (sampled from 100 subjects) and a testing set of 75 slices (sampled from 25 subjects) were generated using a subset of the IXI dataset .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='2 Baselines We select two representative GAN-based image translation and synthesis ap- proaches and one state-of-the-art diffusion-based method as baselines for com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We first compare our zero-shot-Learning-based method with a few-shot 1https://brain-development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='org/ixi-dataset/ 12 Figure 4: Qualitative performance of different methods on CuRIOUS, GoldAtlas, and IXI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The top row shows the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The second to the last rows show: CycleGAN (few-shot sup), StyleGAN(unsup zero-shot with GAN inversion), SDEdit(zero-shot unsup learning with SDE-based perturbing guidance), and the proposed MIDiffusion (unsup zero-shot with LMI guidance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' learning-based CycleGAN translation model [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The CycleGAN in this exper- iment will be allowed to see both the full target domain dataset and a small group (about 2% for Gold Atlas dataset, 6% for CuRIOUS datasetand 11% for IXI dataset) of the source domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' However, our proposed method will see only the data in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The second baseline method is a GAN inversion-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' A StyleGAN2- ADA [30] is allowed to see the target domain training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The out-domain guided generation is performed through 5000 steps of optimization of inversion in the latent space of the trained StyleGAN2-ADA [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' An extra encoder network is introduced in the generation step for inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The third baseline model is SDEdit [41], which works for an ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The SDEdit is also a diffusion model-based translation method but uses a dis- tribution perturbation guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Whenever possible, we use a default training setup for all the baselines provided in the papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The experiments were imple- mented based on publicly available open-source code in the same experiment environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='3 Quantitative Performance We evaluate the different methods based on five metrics, including three image quality measures: SSIM (structural similarity index measure), PSNR (peak signal-to-noise ratio), MSE (mean square error), and two statistical similarity measures: FID (Fréchet Inception Distance) [20] and MI (mutual information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' To study translation faithfulness (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 2) of each method, we report the SSIM between the translation results ˆF and the guidance data G (SSIM-Src) and the target data F (SSIM-Tar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The FID and MI are used to evaluate the translation 13 4similarity (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1) statistically between ˆF and F for different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The FID evaluates the generation similarity in the feature level between the ˆF and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The MI assesses the statistical similarity between the paired ˆF and F images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We show the performance based on those metrics of the four methods in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='1 Comparison with few-shot training We compare our method with the supervised few-shot trained CycleGAN (source domain F ∈ U is visible) on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' We notice that the size of the source datasets for the few-shot training differs between different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Regarding the translation errors (measured by SSIM, MSE, and PSNR), the proposed zero- shot trained MIDiffusion model outperforms the few-shot trained CycleGAN model on both the GoldAtlas (2% of training data from source, 100% from target) and CuRIOUS (6% of training data from source„ 100% from target) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' However, this advantage of the MIDiffusion vanishes on the IXI dataset for the MSE and PSNR metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This can be attributed to the 11% training data from the source domain being sufficient for the CycleGAN to surpass the performance of the zero-shot-learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='2 Comparison with zero-shot training As shown in Table 1, the proposed method performs better than the other one-shot training-based methods in terms of the SSIM-Src, SSIM-Tar, and MSE, which means the generated data of MIDiffusion achieves higher semantic faithfulness with regard to both the source and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Our model also outperforms the other methods in terms of the PSNR and MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This implies our method shows translation realism 1 with respect to the target data in pairwise comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' In terms of similarity between the generated dataset and the target dataset, we see that the SDEdit has an overall lower FID score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Yet, in considering our generation target of fidelity translation, the SDEdit fails in faithfulness translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Thus the translation results of SDEdit performs worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' In contrast, the proposed method achieves good realism (lowest FID score apart from SDEdit) while keeping the semantic meaning from the guidance to the target (higher SSIM, lower MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The cross-modality data translation method with the highest fidelity is therefore the MIDiffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='4 Qualitative Performance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 4 shows translated results from different methods on the test dataset as well as the groundtruth (top rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The first two columns in each dataset group represent different modalities, and the last two columns correspond to their zoomed-in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Overall, the proposed method achieves the best translation fidelity among the compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The anatomical structures are more faithfully represented in the MIDiffusion translation results compared to the results from the baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The SDEdit and StyleGAN fail to translate 14 the source images with identical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' In addition, the few-shot trained CycleGAN clearly underperforms the zero-shot trained MIDiffusion model when trained with insufficient source data on the CuRIOUS (2%) and GoldAtlas (6%) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This observation is consistent with the quantitative results shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Overall, the MIDiffusion method outperforms the baseline methods in both anatomical consistency and appearance similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 6 Conclusion This paper presents a novel local-wise mutual information-guided diffusion model named MIDiffusion for cross-modality data translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Different from cur- rent cycle-consistency training, the MIDiffusion does not require seeing the source dataset for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Unlike GAN inversion methods that require itera- tive optimization during the generation, the MIDiffusion does not need online optimization in the test steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content='Our method introduces a new conditioner that achieves high-fidelity generation but does not need any extra training on the main diffusion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' This feature is different from supervised guidance [15], which needs to train an extra model to pose conditional guidance on the diffusion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' The proposed diffusion model is more robust than another diffusion model (SDEdit) in terms of translation faithfulness, thanks to the LMI guidance signal of MIDiffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Limitations However, the MIDiffusion requires hundreds of times iterative solutions of SDE, which takes dozens of seconds for translating a single image on a GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Effectively reducing the number of sampling steps without compromising the translation fidelity would be meaningful for future work.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' Express, 12(12):7526–7543, Dec 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFST4oBgHgl3EQfMTgQ/content/2301.13743v1.pdf'} diff --git a/ptAyT4oBgHgl3EQfl_gG/content/tmp_files/2301.00461v1.pdf.txt b/ptAyT4oBgHgl3EQfl_gG/content/tmp_files/2301.00461v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c96ff10c9ace21ff4c5a0b4abfd4c1c08e72e1c --- /dev/null +++ b/ptAyT4oBgHgl3EQfl_gG/content/tmp_files/2301.00461v1.pdf.txt @@ -0,0 +1,2915 @@ +arXiv:2301.00461v1 [math.PR] 1 Jan 2023 +The GHP scaling limit of uniform spanning trees of dense graphs +Eleanor Archer∗ +Matan Shalev† +January 3, 2023 +Abstract +We consider dense graph sequences that converge to a connected graphon and prove that the GHP +scaling limit of their uniform spanning trees is Aldous’ Brownian CRT. Furthermore, we are able to +extract the precise scaling constant from the limiting graphon. As an example, we can apply this to +the scaling limit of the uniform spanning trees of the Erd¨os-R´enyi sequence (G(n, p))n≥1 for any fixed +p ∈ (0, 1], and sequences of dense expanders. A consequence of GHP convergence is that several associated +quantities of the spanning trees also converge, such as the height, diameter and law of a simple random +walk. +1 +Introduction +Uniform spanning trees (USTs) are fundamental objects in probability theory and computer science, with +close connections to many other areas of mathematics including electrical network theory [20], loop erased +random walks [32] and random interlacements [18], to name but a few. +It was recently shown in [7], building on the work of [31], that the universal metric measure space scaling +limit of USTs of a large class of graphs is Aldous’ Brownian continuum random tree (CRT). The purpose +of the present paper is to extend this result to sequences of dense graphs encoded by graphons. Due to a +transitivity assumption in previous papers, these USTs are not covered by the results of [31] and [7], but +here we establish that the CRT is nevertheless still the scaling limit. In addition we are able to express the +precise scaling factor in terms of the encoding graphon, making the result more precise than that in [7] and +demonstrating that the notion of graphon convergence is enough to fully determine the UST scaling limit. +The CRT, introduced by Aldous [1, 2, 3], is a well-known object in probability theory, and is perhaps +best-known as the scaling limit of critical finite variance Galton–Watson trees. We do not attempt to give a +full introduction here; we will give a formal definition in Section 3 and we refer to the survey of Le Gall [23] +for further background. +A weighted graph (G, w) is a graph G = (V, E) in which we assign to each edge e ∈ E a non-negative +weight we. In this paper, we will work with sequences of weighted graphs with no loops or multiple edges +in which we ∈ [0, 1] for each e ∈ E. In the case where all edge-weights are equal to 1, we say that the graph +is simple. We extend the definition of vertex degree to weighted graphs by defining deg v to be the sum of +the weights of the edges emanating from v. +The uniform spanning tree of a weighted graph (G, w) is a random spanning tree chosen from the set +of all spanning trees of G where each spanning tree t is chosen with probability proportional to � +e∈t we. +We will say that such a sequence (Gn)n≥1 of weighted graphs is dense if there exists δ > 0 such +that ∆n := minv∈Gn deg(v) ≥ δ#V (Gn) for all n. The notion of convergence of dense graph sequences is +naturally captured by objects known as graphons, introduced by Lov´asz and Szegedy [25] and also Borgs, +∗ ´Equipe Modal’X, Universit´e Paris Nanterre, Batiment G, 200 Avenue de la R´epublique, 92000 Nanterre, France. Email: +eleanor.archer@parisnanterre.fr +†School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel. Email: matanshalev@mail.tau.ac.il +1 + +Chayes, Lov´asz, S´os and Vesztergombi [10] for this purpose. See also [14] for a very quick introduction. A +graphon W is a symmetric measurable function from [0, 1]2 to [0, 1] and can be thought of as (roughly) +the continuum analogue of an adjacency matrix. Using this viewpoint, there is a natural notion of distance +between discrete graphs and graphons, known as the cut-distance, which we will define in Section 2.1. This +allows us to consider the notion of convergence to a given graphon W. +Graphons are commonly used in combinatorics and computer science to analyze large dense graphs. +For example, they have been used in extremal graph theory [12], mean-field games [11], analysis of large +graphs [21], and to study the thermodynamic limit of statistical physics systems [27, 13], to give a very +non-exhaustive list. +Given a graphon W, define a constant +αW = +1 +�� +[0,1]2 W(x, y)dxdy +�2 · +� +[0,1] +�� +[0,1] +W(x, y)dy +�2 +dx. +(1) +Note it follows immediately from Jensen’s inequality that αW ≥ 1, with equality if and only if W is +constant almost everywhere. We also say that a graphon W is connected if for all A ⊂ [0, 1] of positive +Lebesgue measure, it holds that +� +A +� +AC W(x, y)dxdy > 0. +The main result of the present paper is the following. Below, the GHP distance refers to the Gromov +Hausdorff Prohorov distance between metric measure spaces; we define it in Section 2.8. +Theorem 1.1. Let (Gn)n≥1 be a dense sequence of deterministic weighted graphs converging to a connected +graphon W, where each Gn has n vertices. For each n ≥ 1, let Tn be a uniform spanning tree of Gn. Denote +by dTn the corresponding graph-distance on Tn and by µn the uniform probability measure on the vertices of +Tn. Then +� +Tn, +√αW +√n dTn, µn +� +(d) +−→ (T , dT , µ) +where αW is defined as in (1), (T , dT , µ) is the CRT equipped with its canonical mass measure µ and +(d) +−→ +denotes convergence in distribution with respect to the GHP distance between metric measure spaces. +A single graphon can also encode sequences of random graphs G(k, W)k≥1 and H(k, W)k≥1 with k nodes, +obtained by sampling k uniform vertices x1, . . . , xk in [0, 1], and either adding an edge of weight 1 between +nodes i and j with probability W(xi, xj) (this is the sequence G(k, W)k≥1), or instead adding an edge +of weight W(xi, xj) (this is the sequence H(k, W)k≥1). We will deduce the following as a consequence of +Theorem 1.1. +Corollary 1.2. Let W be a connected graphon. Suppose that there exists δ > 0 such that the minimal degree +of G(n, W) is at least δn with probability tending to 1 as n → ∞. For each n ≥ 1, let Tn be a uniform +spanning tree of G(n, W). Denote by dTn the corresponding graph-distance on Tn and by µn the uniform +probability measure on the vertices of Tn. Then +� +Tn, +√αW +√n dTn, µn +� +(d) +−→ (T , dT , µ) +where (T , dT , µ) is the CRT equipped with its canonical mass measure µ and +(d) +−→ denotes convergence in +distribution with respect to the GHP distance between metric measure spaces. +Moreover, the same statement holds for H(n, W) in place of G(n, W). +For example, this applies to the Erd¨os-R´enyi sequence (G(n, p))n≥1 for any fixed p ∈ (0, 1], which is the +sequence (G(n, W))n≥1 when W is the graphon that is p (almost) everywhere, and in which case αW = 1. +2 + +Theorem 1.1 shows that graphons contain enough information to determine the scaling limit of USTs, +or in other words that the GHP scaling limit is continuous with respect to the topology induced by the +cut-distance. In [16], the authors show an analogous result for the Benjamini-Schramm local limit of the +USTs appearing in Theorem 1.1, and show that the local limit can be characterized as a multi-type critical +branching process conditioned to survive, where the offspring distributions are encoded by the limiting +graphon. Additionally, the authors show that continuity also holds for the total number of spanning trees +of Gn, after being properly renormalized. However, they also give an example to show that this is no longer +true under weaker assumptions. +Note that convergence of a graph sequence to a connected graphon does not automatically imply that the +graph sequence must be dense, and in fact the local limit result for USTs of dense graphs obtained in [16] does +not require this assumption. There, the authors assume only that the limiting graphon is non-degenerate, +meaning that +degW (x) := +� +[0,1] +W(x, y)dy > 0 ∀x ∈ [0, 1], +(2) +and that the graph sequence is connected. In fact this implies that “most” vertices have high degree; see [16, +Theorem 2.7 and Definition 2.6] for a precise statement. This is enough to prove a local limit statement since +with high probability, the local limit will not see the exceptional vertices of low degree. On the other hand, +the GHP scaling limit is a global statement and therefore we require more uniform control of the underlying +graphs. One can easily see this through a simple counterexample: let Gn denote the complete graph on +n − n2/3 vertices, and attach a stick of length n2/3 to one vertex of the complete graph. The graphs still +converge to the graphon that is 1 everywhere, and the local limit of UST(Gn) is once again the Poisson(1) +Galton–Watson tree conditioned to survive. On the other hand, the only non-trivial compact scaling limit +is a single stick, and not the CRT. One can also construct similar counterexamples with minimum degree at +least +n +γn for any sequence γn → ∞, meaning that the assumption of linear minimal degree is indeed necessary. +Since the local limit of the CRT is well-known [1, Section 6] to be Aldous’ self-similar CRT (SSCRT), one +can also ask whether the operations of taking scaling limits and local limits commute. In general, answering +this question seems quite non-trivial, as the multitype branching process appearing as the local limit is very +non-homogeneous and the offspring distributions of successive generations are not independent. However, a +special case arises when the sequence (Gn)n≥1 is regular. In this case the local limit is a Poisson(1) Galton– +Watson tree conditioned to survive, which is well-known to rescale to the SSCRT; moreover we will show in +Remark 7.3 that the constant αW must be equal to 1, which entails that +1 +αW is equal to the variance of the +Poisson(1) offspring distribution, and from which we can deduce that the operations do indeed commute in +this case. +For non-regular graph sequences, the question seems a bit more subtle. While the expected number of +non-backbone neighbours of the root vertex of the local limit is indeed 1, the variance is not necessarily +equal to +1 +αW . For example, for the complete bipartite graph K 2n +3 , n +3 , one can calculate using [16, Definition +1.2] that the variance of the offspring number of the root vertex is equal to 3 +2, but +1 +αW is equal to 8 +9. This +does not preclude the possibility that the operations commute, since the variance in subsequent generations +may converge to +1 +αW in the appropriate sense. For K 2n +3 , n +3 we can in fact apply results of Miermont [28] (the +local limit in this case is in fact a Galton–Watson tree with two alternating types: Poi(2) and Poi( 1 +2)) to +deduce that the operations do commute. However, in the general case the local limit is a Galton–Watson +tree with uncountably many types, for which, to the best of our knowledge, scaling limits are not covered +by the existing Galton–Watson tree literature. +Finally, we note that in [6], the authors consider similar dense graph sequences, but do not assume +that the sequence converges to a graphon. Under this weaker assumption, they prove that the diameter +of UST(Gn) is of order √n with high probability. We cannot hope to prove a scaling limit result under +the same hypotheses, since one can, for example, connect two copies of Kn/2 by a single edge, in which +case the diameter is still of order √n but the scaling limit is not the CRT. However, when the graphs are +well-connected, we can obtain the scaling limit. +In this paper we in fact prove the following theorem. In what follows, for a given γ > 0 we say that +a graph G is a γ-expander if for all U ⊂ V (G), the number of edges between U and V (G) \ U is at least +γ|U|(|V (G)| − |U|). +3 + +Theorem 1.3. Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of connected γ-expanders, where +each Gn has n vertices and minimal degree at least δn. For each n ≥ 1, let Tn be a uniform spanning tree +of Gn. Denote by dTn the corresponding graph-distance on Tn and by µn the uniform probability measure on +the vertices of Tn. Then there exists a sequence (αn)n≥1, satisfying 1 ≤ αn ≤ δ−1 for all n ≥ 1, such that +� +Tn, +√αn +√n dTn, µn +� +(d) +−→ (T , dT , µ) +as n → ∞ where (T , dT , µ) is the CRT equipped with its canonical mass measure µ and +(d) +−→ denotes conver- +gence in distribution with respect to the GHP distance between metric measure spaces. +In fact the theorem holds slightly more generally, see Remark 1.4, but the above assumptions make the +proof more straightforward. Clearly one cannot hope for convergence of the parameter αn without making +stronger assumptions, since one can alternate graphs from sequences with different limiting values of αn. +For example, for the sequence of complete graphs αn → 1, but if Gn is instead the complete bipartite graph +K n +3 , 2n +3 , then αn → 9 +8. +As well as the convergence of the rescaled diameter, it follows directly from the GHP convergence of +Theorem 1.3 that we also have convergence of the rescaled height and rescaled simple random walk on +UST(Gn). More formally, the following three convergences hold in distribution. +1. +√αn Diam(Tn) +√n +(d) +→ Diam(T ). +2. +√αn Height(Tn) +√n +(d) +→ Height(T ). +3. If Xn is a simple random walk on Tn, then the quenched law of +� √αn +√n Xn(2n3/2α−1/2 +n +t) +� +t≥0 converges +in distribution to the quenched law of Brownian motion on the CRT. It also follows that the associated +mixing times converge on the same time scale. +See [7, Section 1.3] for further details of why these three properties follow from GHP convergence. In the +settings of Theorem 1.1 and Corollary 1.2, we can replace αn with αW in the above three statements. +1.1 +Proof strategy +Clearly, in order to prove the main theorems, it suffices to first prove Theorem 1.3 and then show that the +graph sequence is an expander sequence and that αn → αW under the additional assumption of Theorem 1.1. +We will prove Theorem 1.3 in two steps using the lower mass bound criterion of [8]. In particular, by +[7, Theorem 6.5], in order to prove the GHP convergence of Theorem 1.3 it is enough to prove the following +two statements. +(A) The convergence holds in a finite-dimensional sense (this will be formally stated in Theorem 3.1). +(B) The lower mass bound condition holds; that is, if mn(η) = infx∈UST(Gn) +� +|B(x,η√n)| +n +� +, then for every +η > 0 the sequence mn(η)−1 is tight (this will be formally stated in Claim 6.4). +The second condition will follow quite straightforwardly from minor adaptations of the arguments in [7]. +The bulk of this paper is devoted to proving the first condition. In fact, this condition is equivalent to the +joint convergence, for all k ≥ 1, of the set of +�k +2 +� +distances between k points chosen uniformly at random in +UST(Gn). +This type of convergence was previously proved for USTs of sequences of high-dimensional graphs in [31]. +This is a different class of graphs and includes the assumption of transitivity. Their proof uses Wilson’s +algorithm, which is a method for sampling USTs one branch at a time by running loop erased random walks +4 + +(LERWs). In their proof, they couple Wilson’s algorithm on Gn with Wilson’s algorithm on the complete +graph and prove that the set of +�k +2 +� +distances on the two graphs must have the same scaling limit. +Our proof, by contrast, is more direct. +We also use Wilson’s algorithm, but we work directly with +UST(Gn) and use the Laplacian random walk representation of LERWs to sample each branch. By tightly +controlling the capacity of loop-erased random walks, we are able to directly compute the probability that +a given branch exceeds a given length, and show that this converges to the analogous quantity for the CRT +using Aldous’ stick-breaking construction. +Remark 1.4. As demonstrated by the examples and discussion above Theorem 1.3, the assumption of linear +minimal degree is necessary in order to obtain convergence in the GHP topology. In order to keep the exposi- +tion clean, we prove both conditions (A) and (B) above under these assumptions. However, the assumption +is not really necessary for condition (A). The proof would work unchanged if we allow o(n) vertices to have +degrees less than √n, for example (since the loop-erased random walk that we analyze in Section 5 will never +hit this set, whp). In fact, we believe that it may be possible to adapt our proof of condition (A) (Theorem 3.1) +to work under the original assumptions of [31], but this would require one to keep track of several additional +messy details, and would not add further insight. +1.2 +Organization of the paper +This paper is organized as follows. In Section 2 we give the necessary background, including an introduction +to graphons, USTs and the topologies of interest. In Section 3 we introduce a general framework for stick- +breaking constructions of trees, and state Aldous’ stick-breaking construction of the CRT. In Section 4 we +give some precise random walk estimates and we apply these with the Laplacian random walk representation +in Section 5 to obtain estimates for the first steps of Wilson’s algorithm. In Section 6 we use these estimates +to couple stick-breaking on the CRT with Wilson’s algorithm and prove that the two processes are very +similar when n is large enough. This proves condition (A) above. We also explain how (B) can be deduced +from the results of [7] which in fact establishes Theorem 1.3. Finally, in Section 7 we prove Theorem 1.1 and +Corollary 1.2. +1.3 +Acknowledgments +We would like to thank Asaf Nachmias and Jan Hladky for suggesting to look at graphons and for many +helpful comments. This research is supported by ERC consolidator grant 101001124 (UniversalMap), and +by ISF grant 1294/19. EA was partially supported by the ANR ProGraM grant. +2 +Background +2.1 +Graphons +A graphon is a symmetric measurable function [0, 1]2 → [0, 1]. As mentioned in the introduction, graphons +were introduced by Borgs, Chayes, Lov´asz, S´os, Szegedy and Vesztergombi [25, 10] in order to characterize +dense graph limits. To understand why this definition is natural, we define the graphon representation +of a discrete graph G as follows. Suppose that G is a simple graph with n vertices. Number the vertices +from v1 to vn and partition the interval [0, 1] into a sequence of intervals (Ii)n +i=1, where Ii = +� i−1 +n , i +n +� +for +each 1 ≤ i ≤ n. We define the graphon WG : [0, 1]2 → [0, 1] by (e.g. see [25, Section 7.1]) +WG((x, y)) = +1{v⌈nx⌉∨1 ∼ v⌈ny⌉∨1} +∀ (x, y) ∈ [0, 1]2. +If G is a weighted graph, we instead define +WG((x, y)) = w(v⌈nx⌉∨1, v⌈ny⌉∨1) +∀ (x, y) ∈ [0, 1]2, +where w(vi, vj) represents the weight of the edge joining vi and vj (and is zero if there is no such edge). +5 + +Note that, given only G, this definition of WG is not unique, since it depends on the ordering of the +vertices. Therefore, in order to define a metric on the space of graphons, we will instead consider equivalence +classes of graphons. In particular, given two graphons W1 and W2 the cut-distance between them is defined +as (e.g. see [25, Equation (8.16)]) +δ□(W1, W2) = inf +ϕ ||W ϕ +1 − W2||□, +where the infimum is taken over all measure-preserving automorphisms of [0, 1], where W ϕ is defined by +W ϕ(x, y) = W(ϕ(x), ϕ(y)), and where the cut-norm of a measurable function U : [0, 1]2 → [−1, 1] is given +by +||U||□ = +sup +S,T ∈B([0,1]) +���� +� +x∈S +� +y∈T +U(x, y)dxdy +����. +We therefore say that a sequence of deterministic graphs (Gn)n≥1 converges to a graphon W if +δ□(WGn, W) → 0 as n → ∞. +Remark 2.1. Graphons can in fact be defined as functions from Ω2 → [0, 1], where Ω is any probability +space, see [25, Chapter 13], but since all probability spaces are isomorphic, this does not provide much greater +generality. +We will make use of the following lemma. +Lemma 2.2. [9, Lemma 7]. Let W be a connected graphon. Then, for every α ≤ 1/2 there exists some +constant β = β(W, α) such that for every set A with α ≤ µ(A) ≤ 1/2 we have +� +A +� +AC W(x, y)dxdy > β. +2.1.1 +Random graphs and graphons +A graphon W can be used to define a random graph with n vertices in two ways. +1. Sample x1, . . . , xn i.i.d. uniformly on [0, 1]. We define a random simple graph on {1, . . . , n} by +joining nodes i and j with probability W(xi, xj), independently for each (unordered) pair (i, j). We +denote the resulting random graph G(n, W). +2. Sample x1, . . . , xn i.i.d. uniformly on [0, 1]. We define a random weighted graph on {1, . . . , n} by +adding an edge between i and j of weight W(xi, xj) for each (unordered) pair (i, j). We denote the +resulting random graph H(n, W). +In both constructions, note that we can use a single graphon to define a whole sequence of random graphs. +The following lemma tells that in either case, the cut-distance between a random sample of G(k, W) or +H(k, W) and W goes to zero w.h.p. as k → ∞. +Lemma 2.3. [25, Lemma 10.16]. Fix a graphon W and for k ≥ 1, let G(k, W) and H(k, W) be defined as +above. Then, δ□(WG(k,W), W) and δ□(WH(k,W), W) both tend to 0 in probability as k → ∞. +In particular this means that results we prove for USTs of deterministic sequences of graphs extend +automatically to sequences of the form G(k, W)k≥1 or H(k, W)k≥1 under the assumptions of Corollary 1.2. +For example, the classical Erd¨os-R´enyi graphs G(n, p) for n ≥ 1, p ∈ [0, 1] correspond to the graphs +G(n, Wp) where Wp is the graphon that is equal to p everywhere. +For further background and applications of graphons, we refer to [25, Part 3]. +6 + +2.2 +Mixing times +Let G be a connected weighted graph with n vertices, with weights (w(x, y))x,y∈V (G), and with no loops or +multiple edges. A random walk on G is the Markov Chain (Xm)m≥0 such that, for all vertices x, y ∈ V (G), +and all m ≥ 1, +P(Xm = y | Xm−1 = x) = +w(x, y) +� +z∼x w(x, z), +where z ∼ x means that z is a neighbour of x. Due to periodicity considerations, it is sometimes more +convenient to instead use the notion of a lazy random walk. This is defined by +P(Xm = y | Xm−1 = x) = +w(x, y) +2 � +z∼x w(x, z) ∀y ∼ x and P(Xm = x | Xm−1 = x) = 1 +2 +for all m ≥ 1. +For each t ≥ 0 let pt denote the t-step transition density of a lazy random walk, i.e. +pt(x, y) = +P(Xt = y | X0 = x) for all x, y ∈ V (G). We define the mixing time of G as +tmix(G) = min +� +t ≥ 0 : max +x,y∈G|pt(x, y) − π(x)| ≤ 1 +4 +� +, +(3) +(see [24, Equation (4.31)]), where π denotes the stationary measure on G. +We will also need the notion of total variation distance between two probability measures on µ and ν +on a finite subset X ⊂ V (G). This is defined by +dTV(µ, ν) = max +A⊂X |µ(A) − ν(A)| . +Furthermore, by [24, Section 4.5], we have for any k ≥ 1, any t ≥ ktmix and any vertex x that +dTV(pt(x, ·), π(·)) ≤ 2−k. +(4) +2.3 +Expanders +We will use the following definition of an expander graph. +Definition 2.4. ([16, Definition 2.1]). For any γ > 0, a loopless weighted graph G is a γ-expander if for +all U ⊂ V (G), we have that w(U, V (G) \ U) ≥ γ|U|(V (G) − |U|) where w(A, B) = � +v∈A,u∈B w(v, u). +Although we give the definition for loopless graphs, note that adding loops to a graph does not change the +law of its UST, since loops can never appear in a UST. Note that often in the literature a slightly different +definition of expander is used, involving the Cheeger constant. We are using the definition above as it fits +more naturally into the framework of dense graphs (as we will later show in Claim 7.1) and is the same +definition used to consider the local limit in [16]. +The main property of expanders that we will use is as follows. +Claim 2.5. Let γ > 0 and let G be a γ-expander with n ≥ 2 vertices. Then, provided that n is large enough +(depending on only γ), we have that +tmix(G) ≤ 64 +γ4 log n. +Proof. Note that it follows from Definition 2.4 that G has minimal degree at least γ +2 n. First note that by +[24, Theorem 12.4] that +tmix(G) ≤ trel log +�8n +γ +� +, +7 + +where trel is the relaxation time of G. By the Cheeger inequality (see [4, 5, 19, 22] for various proofs), +1 +trel +is lower bounded by Φ(G)2/2, where +Φ(G) = +min +S⊂V (G),π(S)≤1/2 +w(S, V (G) \ S) +� +v∈S deg v +. +Note that π(S) ≤ 1/2 implies that (|V (G)| − |S|) ≥ nγ +4 . Since G is a γ-expander, it follows that +Φ(G) ≥ γ|S|(|V (G)| − |S|) +� +v∈S deg v +≥ γ|S|(|V (G)| − |S|) +|S|n +≥ γ2 +4 . +Combining all the inequalities gives the result. +2.4 +Loop-erased random walk and Wilson’s algorithm +We now describe Wilson’s algorithm [32] which is a widely-used algorithm for sampling USTs. A walk +X = (X0, . . . XL) of length L ∈ N is a sequence of vertices where (Xi, Xi+1) ∈ E(G) for every 0 ≤ i ≤ L − 1. +For an interval J = [a, b] ⊂ [0, L] where a, b are integers, we write X[J] for {Xi}b +i=a. Given a walk X, we +define its loop erasure Y = LE(X) = LE(X[0, L]) inductively as follows. We set Y0 = X0 and let λ0 = 0. +Then, for every i ≥ 1, we set λi = 1 + max{t | Xt = Yλi−1} and if λi ≤ L we set Yi = Xλi. We halt this +process once we have λi > L. When X is a random walk on the weighted graph G starting at some vertex +v and terminated when hitting another vertex u (L is now random), we say that LE(X) is a loop erased +random walk (LERW) from v to u. +To sample a UST of a finite connected weighted graph G we begin by fixing an ordering of the vertices of +V = (v1, . . . , vn). First let T1 be the tree containing v1 and no edges. Then, for each i > 1, sample a LERW +from vi to Ti−1 and set Ti to be the union of Ti−1 and the LERW that has just been sampled. We terminate +this algorithm with Tn. Wilson [32] proved that Tn is distributed as UST(G). An immediate consequence +is that the path between any two vertices in UST(G) is distributed as a LERW between those two vertices. +This was first shown by Pemantle [30]. +2.5 +Laplacian random walk +Here we outline the Laplacian random walk representation of the LERW (see [26, Section 4.1] for full details) +and its application to Wilson’s algorithm. Take a finite, weighted, connected graph G and suppose we have +sampled Tj for some j ≥ 1 using Wilson’s algorithm as described above. We now sample a LERW from vj+1 +to Tj. Denote this LERW by (Ym)m≥0. Also let X denote a random walk on G. For a set A ⊂ G, let τA +denote the hitting time of A by X, and τ + +A denote the first return time to A by X. The Laplacian random +walk representation of Y says that, conditionally on Tj and on the event {(Ym)i +m=0 ∩ Tj = ∅}, we have for +any i ≥ 0 that +P +� +Yi+1 = v +�� (Ym)i +m=0 +� += PYi +� +X1 = v +��� τTj < τ+ +∪i +m=0{Ym} +� += +PYi(X1 = v)Pv +� +τTj < τ∪i +m=0{Ym} +� +PYi +� +τTj < τ + +∪i +m=0{Ym} +� +. +Clearly this is only non-zero when v /∈ �i +m=0{Ym}. We can now extrapolate this to ask about the law of +(Ym)i+H +m=i+1 for some H ≥ 1, given (Ym)i +m=0. In particular, if u0, u1, . . . , uH is a simple path in Gn, where +{u1, . . . , uH−1} is disjoint from �i +m=0{Ym} ∪ Tj and u0 = Yi, then +P +� +(Ym)i+H +m=i+1 = (um)H +m=1 +�� (Ym)i +m=0 +� += Pu0 +� +(Xm)H +m=1 = (um)H +m=1 +� +C((Ym)i +m=0, Tj, (um)H +m=1)), +where +C((Ym)i +m=0, Tj, (um)H +m=1) = +H +� +h=1 +Puh +� +τTj < τ∪i +m=0{Ym}∪ ∪h−1 +m=1{um} +� +Puh−1 +� +τTj < τ+ +∪i +m=0{Ym}∪ ∪h−1 +m=1{um} +�. +8 + +2.6 +Capacity and closeness +Recall that G is a connected weighted graph with n vertices with minimal degree at least δn. The capacity +of a set of vertices of G quantifies how difficult it is for a random walk to hit the set. Let (Xi)i≥0 be a +random walk on G and for U ⊂ V (G), let τU = inf{i ≥ 0 : Xi ∈ U}. Given k ≥ 0 we define the k-capacity +of U by Capk(U) = Pπ(τU ≤ k). +Here we collect some useful facts about the capacity. +Lemma 2.6. Let A ⊂ V (G) and k ≥ 1. Then +Capk(A) ≤ kπ(A) ≤ k|A| +δn . +(5) +Moreover, if k|A| ≤ δ3n +2 , then +Capk(A) ≥ kπ(A) +2 +≥ δk|A| +2n . +(6) +Proof. The upper bound follows from a union bound. The lower bound follows from the Bonferroni inequal- +ities and the lower bound on the degree, which imply that +Capk(A) ≥ kπ(A) − +�k|A| +δn +�2 +≥ δk|A| +2n . +We will also use the following claim. +Claim 2.7. Let tmix = tmix(G). Let A ⊂ V (G), let M ≥ (log n)2tmix and suppose that (log n)2 · tmix|A| ≤ n. +Then, provided n is large enough, +sup +u∈V (G)\A +|Pu(τA ≤ M) − CapM(A)| ≤ 3 log n · tmix|A| +δn +. +Proof. Let X be a random walk started at u ∈ G. Clearly, for any t ≥ 0, the first t steps of X can be +coupled with the first t non-repeat steps of a lazy random walk ˜X. Therefore, first run a lazy random walk +started from u until time T = 2 log n · tmix. Let N denote the total number of non-repeat jumps of this +lazy random walk. The distribution of ˜Xt is almost stationary by (4). Moreover, we have that 0 ≤ N ≤ T +deterministically. To sample (Xt)M +t=0, we first couple it with the first N steps of ( ˜Xt)T +t=0 as explained above, +and then run X for a further M − N steps. Under this coupling, we therefore have from a union bound that +Pu(τA ≤ M) ≤ 2 log n · tmix|A| +δn ++ Pπ(τA ≤ M) + 2−2 log n ≤ CapM(A) + 3 log n · tmix|A| +δn +. +Similarly, +Pu(τA ≤ M) ≥ Pπ(τA ≤ M − T ) − 2−2 log n ≥ CapM(A) − 3 log n · tmix|A| +δn +. +In order to obtain lower bounds on capacity, we define the k-closeness of two sets U and W by +Closek(U, W) = Pπ(τU < k, τW < k). +(7) +Corollary 2.8. For any disjoint sets U, W ⊂ G, we have that +sup +v∈G\(U∪W) +Pv(τU < k, τW < k) ≤ 2k2|U||W| +δ2n2 +. +In particular, Closek(U, W) ≤ 2k2|U||W| +δ2n2 +. +9 + +Proof. Note that +sup +v∈G\(U∪W) +Pv(τU < k, τW < k) ≤ +sup +v∈G\(U∪W) +{Pv(τU < τW < k) + Pv(τW < τU < k)} +≤ +sup +v∈G\(U∪W),u∈U,w∈W +{Pv(τU < k)Pu(τW < k) + Pv(τW < k)Pw(τU < k)} +≤ 2k2|U||W| +(δn)2 +. +2.7 +Random variables +Here we present two elementary results that will be useful in Section 6. +Claim 2.9. Let ε > 0 and let 0 < a < b with b − a ≤ ε. Let Xa ∼ U([0, a]) and Xb ∼ U([0, b]). Then, we +can couple Xa and Xb such that P(|Xa − Xb| > ε) < ε. +Proof. We take Xb = b +aXa. Then, |Xb − Xa| = | b−a +a +· Xa| ≤ |b − a| ≤ ε. +Lemma 2.10. For any L > 0, let XL be the random variable on (0, ∞) satisfying +P(XL > x) = exp +� +−(x + L)2 − L2 +2 +� +. +Then for any δ > 0, there exists η = η(δ, L) > 0 such that the following holds. Let Y be another random +variable on (0, ∞), and suppose that for all x > 0, +|P(XL > x) − P(Y > x)| < η. +(8) +Then this implies that we can couple XL and Y so that P(|XL − Y | > δ) < δ. +Furthermore, for any δ, L1 and L2 with L1 < L2, there exists η = η(δ, L1, L2) such that we can couple +XL and Y as described above for every L ∈ [L1, L2]. +Proof. Note that we can couple XL and Y by first sampling U ∼ Uniform([0, 1]) and setting +XL(ω) = sup +x≥0 +{P(XL ≥ x) ≥ U(ω)}, +Y (ω) = sup +x≥0 +{P(Y ≥ x) ≥ U(ω)}. +Now choose Kδ,L < ∞ so that P(XL ≥ Kδ,L) < δ. Wlog assume that δ < 1 and Kδ,L > 1, otherwise decrease +or increase them if necessary. Note that, for all 0 ≤ x < Kδ,L, we have that +exp +� +−(x + L)2 − L2 +2 +� +− exp +� +−(x + δ + L)2 − L2 +2 +� +≥ δ(x + L) exp +� +−(x + δ + L)2 − L2 +2 +� +≥ Mδ,L, +where Mδ,L = δL exp +� +− (2Kδ,L+L)2−L2 +2 +� +> 0. +Now suppose that (8) holds and η < Mδ,L. Then, for any 0 ≤ x < Kδ,L we have that +P(Y ≥ x + δ) ≤ P(XL ≥ x + δ) + η ≤ P(XL ≥ x) − Mδ,L + η ≤ P(XL ≥ x). +Therefore, under the coupling, we have for any x < Kδ,L that +{XL ≤ x} ⇔ {P(XL ≥ x) ≤ U} ⇒ {P(Y ≥ x + δ) ≤ U} ⇔ {Y ≤ x + δ}. +Similarly, {XL ≥ x} ⇒ {Y ≥ x − δ}. Therefore, under this coupling we have that +P(|XL − Y | > δ) ≤ P(XL ≥ Kδ,L) < δ, +as required. +For the second claim, note that for every L′ > L we also have that P(XL′ ≥ Kδ,L) < δ. Therefore for the +interval [L1, L2] we can simply use Kδ,L1 and Mδ,L1 on the whole interval. +10 + +2.8 +GHP topology +Here we define the GHP topology. We use the framework of [29, Sections 1.3 and 6] and work in the space Xc +of equivalence classes of metric measure spaces (mm-spaces) (X, d, µ) such that (X, d) is a compact metric +space and µ is a Borel probability measure on it, and we say that (X, d, µ) and (X′, d′, µ′) are equivalent +if there exists a bijective isometry φ : X → X′ such that φ∗µ = µ′ (here φ∗µ is the pushforward measure +of µ under φ). To ease notation, we will represent an equivalence class in Xc by a single element of that +equivalence class. +First recall that if (X, d) is a metric space, the Hausdorff distance dH between two sets A, A′ ⊂ X is +defined as +dH(A, A′) = max{sup +a∈A +d(a, A′), sup +a′∈A′ d(a′, A)}. +For ε > 0 and A ⊂ X we also let Aε = {x ∈ X : d(x, A) < ε} be the ε-fattening of A in X. If µ and ν are +two measures on X, the Prohorov distance between them is given by +dP (µ, ν) = inf{ε > 0 : µ(A) ≤ ν(Aε) + ε and ν(A) ≤ µ(Aε) + ε for any closed set A ⊂ X}. +Definition 2.11. Let (X, d, µ) and (X′, d′, µ′) be elements of Xc. The Gromov-Hausdorff-Prohorov +(GHP) distance between (X, d, µ) and (X′, d′, µ′) is defined as +dGHP((X, d, µ), (X′, d′, µ′)) = inf{dH(φ(X), φ′(X′)) ∨ dP (φ∗µ, φ′ +∗µ′)}, +where the infimum is taken over all isometric embeddings φ : X → F, φ′ : X′ → F into some common +metric space F. +Recall that our aim in this paper is to prove distributional convergence with respect to the GHP topology. +Given an mm-space (X, d, µ) and a fixed m ∈ N we define a measure νm((X, d, µ)) on R( +m +2) to be the law of +the +�m +2 +� +pairwise distances between m i.i.d. points drawn according to µ. Each law P on Xc therefore defines +random measures (νm)m≥2 and annealed measures (˜νm)m≥2 on R( +m +2), given by +˜νm(P) := +� +Xc +νm((X, d, µ))dP. +In [7] we rephrased a result of [8, Theorem 6.1] in the distributional setting to characterize GHP conver- +gence in terms of convergence of the measures (˜νm)m≥2 and a volume condition. To state the version that +we will use in this paper, given c > 0 and an mm-space (X, d, µ) we define +mc((X, d, µ)) = inf +x∈X{µ(B(x, c))} +(cf [8, Section 3]). +In the proof of the next proposition we will also make reference to the (coarser) Gromov-Prohorov +topology, which is defined as follows. +Definition 2.12. Let (X, d, µ) and (X′, d′, µ′) be elements of Xc. The Gromov-Prohorov (GP) distance +between (X, d, µ) and (X′, d′, µ′) is defined as +dGP((X, d, µ), (X′, d′, µ′)) = inf{dP (φ∗µ, φ′ +∗µ′)}, +where the infimum is taken over all isometric embeddings φ : X → F, φ′ : X′ → F into some common +metric space F. +The key result is as follows. +Proposition 2.13. Let (X, d, µ) be an element of Xc with law P such that µ has full support almost surely. +Let ((Xn, dn, µn))n≥1 be a sequence in Xc with respective laws (Pn)n≥1 and suppose that: +11 + +(a) For all m ≥ 0, ˜νm(Pn) → ˜νm(P) as n → ∞. +(b) For any c > 0, the sequence +� +mc((Xn, dn, µn))−1� +n≥1 is tight. +Then (Xn, dn, µn) +(d) +→ (X, d, µ) with respect to the GHP topology. +Proof. First we show that part (a) and (b) together imply that (Xn, dn, µn) +(d) +−→ (X, d, µ) with respect to the +GP topology, by verifying the two conditions of [15, Corollary 3.1]. The second condition of [15, Corollary +3.1] is precisely (a). To verify the first condition we further use [15, Theorem 3] (recall that by Prohorov’s +Theorem the relative compactness of the measures is equivalent to their tightness) and verify conditions (i) +and (ii) there (see also Proposition 8.1 in [15]). Condition (i) is just saying that ˜ν2 is a tight sequence of +measures on R, which follows from (a). Lastly, (b) directly implies condition (ii). +Therefore, by [7, Theorem 6.5], the spaces convergence with respect to the GHP topology. +3 +Stick-breaking construction of trees +Our first goal will be to prove condition (a) of Proposition 2.13 which is equivalent to the following statement. +Theorem 3.1. Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of γ-expanders, where each Gn +has n vertices and minimal degree at least δn. Denote by dTn the graph distance on Tn and by (T , d, µ) the +CRT. Then there exists a sequence (βn)n≥1, satisfying +√ +δ ≤ βn ≤ 1 for all n ≥ 1, such that for any fixed +k ≥ 1, if {x1, . . . , xk} are uniformly chosen independent vertices of Gn, then the distances +dTn(xi, xj) +βn +√n +converge jointly in distribution to the +�k +2 +� +distances in T between k i.i.d. points drawn according to µ. +To prove this theorem, we will use Aldous’ stick-breaking construction of the CRT which is particularly +well adapted to dealing with the pairwise distances between a set of k uniform points. Our strategy will be +to show that the first k steps of Wilson’s algorithm on Gn closely approximate those of this stick-breaking +process when n is large. In this section we briefly recall the stick-breaking construction of the CRT and some +of its key properties. +We start with a more general description of how one can construct a sequence of trees from sticks on the +real line. +Definition 3.2. (Stick-breaking construction of a tree sequence). +Set y0 = z0 = 0, and suppose that we +have a sequence of points y1, y2, . . . ∈ [0, ∞) and z1, z2, . . . ∈ [0, ∞) such that yi−1 < yi and zi ≤ yi for all +i ≥ 1. Construct trees as follows. Start by taking the line segment [y0, y1) at time 1. This is T (2) (as it +contains two marked points). We proceed inductively. At time i ≥ 2, take the interval [yi−1, yi) and attach +the base of the interval [yi−1, yi) to the point on T (i) corresponding to zi−1. This gives a new tree with i + 1 +marked points (in bijection with the set (yj)i +j=0), which we call T (i+1). +Given two such sequences and any k ≥ 2 we define SB(k)((y0, y1, y2, . . .), (z0, z1, z2, . . .)) or equivalently +SB(k)((y0, y1, y2, . . . , yk−1), (z0, z1, z2, . . . , zk−2)) to be equal to the tree T (k). +In general, the sequence of trees constructed in this way above may not converge, but Aldous showed +that by choosing the points in the right way, we can in fact construct the CRT via stick-breaking. +Proposition 3.3. [1, Process 3]. Set Y0 = Z0 = 0, let (Y1, Y2, . . .) denote the ordered set of points of +a non-homogeneous Poisson process on [0, ∞) with intensity t dt, and let Zi be chosen uniformly on the +interval [0, Yi) for each i ≥ 1. Construct the sequence (T (k))∞ +k=2 as in Definition 3.2. Then the closure of the +limit of T (k) is equal in distribution to the CRT. Moreover, if one stops the process after k − 1 steps, then +the resulting tree T (k) has the same distribution as the subtree spanned by k uniform points in the CRT, and +the points corresponding to the set (Yi)k−1 +i=0 can be identified with k uniform points in the CRT. +12 + +In particular, the set of +�k +2 +� +pairwise distances between points corresponding (Yi)k +i=1 is equal in distribution +to the set of +�k +2 +� +pairwise distances between k uniform points in the CRT. +The following proposition will be important for the comparison with Wilson’s algorithm later on. It can +be verified by a direct computation. +Proposition 3.4. Define the sequence (Y1, Y2, . . .) as in Proposition 3.3. Then for any k ≥ 1 and any x ≥ 0, +P +� +Yk+1 − Yk ≥ x +�� (Yi)k +i=0 +� += exp +� +−1 +2 +� +(Yk + x)2 − Y 2 +k +�� +. +The following lemma will also be useful. +Lemma 3.5. There exists a function f : [0, ∞) × N → [0, 1] such that for every k ∈ N we have that +limC→∞ f(C, k) → 0, and such that if Yk is as in Proposition 3.3, then +P +� +C−1 ≤ Yk ≤ C +� +≥ 1 − f(C, k). +Proposition 3.6. Let (y0, y1, y2, . . .), (z0, z1, z2, . . .) and (y′ +0, y′ +1, y′ +2, . . .), (z′ +0, z′ +1, z′ +2, . . .) be the inputs to two +separate stick-breaking processes as defined in Definition 3.2. Fix any k ≥ 1 and let T (k+1) and T (k+1)′ be +the trees formed after k steps of the processes. Let d and d′ denote distances on T (k+1) and T (k+1)′. +Fix some ε > 0 and suppose that the following holds. +(i) |yi − y′ +i| ≤ ε for all i ≤ k and |zi − z′ +i| ≤ ε for all i ≤ k − 1, +(ii) |zi − yj| ≥ 3ε for all i ≤ k − 1, j ≤ k. +Then, for all 0 ≤ i, j ≤ k, it holds that +|d(yi, yj) − d′(y′ +i, y′ +j)| ≤ 2kε. +Proof. When conditions (i) and (ii) hold, we have for all i ≤ k − 1, j ≤ k that yj ≤ zi ≤ yj+1 if and only if +y′ +j ≤ z′ +i ≤ y′ +j+1. We claim that this implies that |d(yi, yj) − d′(y′ +i, y′ +j)| ≤ 2kε for all i, j ≤ k + 1. Indeed, it +follows by construction that d(yi, yj) is the sum of lengths of at most k branch segments in T (k+1), and all +of their lengths can be written in the form |yj − yj−1|, |zj − yℓ| or |zj − zℓ|. Moreover, by construction, when +the conditions (i) and (ii) hold, d′(y′ +i, y′ +j) can be written as the same sum but replacing each zj with z′ +j and +replacing each yj with y′ +j. It therefore follows from the triangle inequality that |d(yi, yj)−d′(y′ +i, y′ +j)| ≤ 2kε. +4 +Random walk properties +In this section we prove some results on random walk hitting probabilities and capacity, which we will later +transfer to segments of LERW using the Laplacian random walk representation of Section 2.5. +Throughout the section we fix a small κ ∈ (0, 1 +32) and for n ≥ 1 we set Mn = nκ. In what follows we will +simply write M instead of Mn. +Notational remark. For the statements in this section, we will take a sequence of graphs satisfying the +assumptions of Theorem 1.3 which is therefore associated with two positive constants γ > 0 and δ > 0. In +this section we will treat these constants as fixed, and therefore o(·) and O(·) quantities may also depend on +γ and δ. +13 + +4.1 +Hitting probabilities +We start with some results on hitting probabilities. Let X be a (non-lazy) random walk on Gn for some +n ≥ 1. For a set A ⊂ V (Gn), we define +τA = inf{t ≥ 0 : Xt ∈ A}. +The main lemma is the following. +Lemma 4.1. Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of γ-expanders, where each Gn has +n vertices and minimal degree at least δn. Take κ and M as defined at the start of Section 4. Then there +exists a sequence (ηn)n≥1 with ηn → 0, depending only on δ and γ, such that for any disjoint A, B ⊂ Gn +satisfying |A| + |B| ≤ δ3 +2 n +1 +2 +2κ: +����Pπ(τA < τB) − +CapM(A) +CapM(A) + CapM(B) +���� ≤ +CapM(A)ηn +CapM(A) + CapM(B). +Proof. Let (Xi)M +i=1 be a random walk of length M. Then, by Bayes’ formula, Corollary 2.8 and the lower +bound in Lemma 2.6, +Pπ(τA < M | τA ∧ τB < M) = +CapM(A) +CapM(A) + CapM(B) − Pπ(τA ∨ τB < M) += +CapM(A) +CapM(A) + CapM(B) +� +1 + O +�δ−3M|B| +n +�� +. +Pπ(τA ∨ τB < M | τA ∧ τB < M) = +Pπ(τA ∨ τB < M) +CapM(A) + CapM(B) − Pπ(τA ∨ τB < M) +≤ Pπ(τA < M | τA ∧ τB < M)O +�δ−3M|B| +n +� +. +Therefore, combining these and applying Lemma 2.6: +Pπ(τA < τB | τA ∧ τB < M) = Pπ(τA < M | τA ∧ τB < M) + O(Pπ(τA ∨ τB < M | τA ∧ τB < M)) += +CapM(A) +CapM(A) + CapM(B) +� +1 + O +�δ−3M|B| +n +�� +. +It similarly follows from Claim 2.7 and the lower bound in Lemma 2.6 that uniformly over all u ∈ Gn\(A∪B), +Pu(τA < τB | τA ∧ τB < M) = +CapM(A) +CapM(A) + CapM(B) +� +1 + O +�δ−3M|B| +n ++ tmix · log n +δ2M +�� +. +Now we decompose time into intervals of length M. For each i ≥ 1, define the interval Ai by +Ai = [iM, (i + 1)M]. +We then have that, using Corollary 2.8: +Pπ(τA < τB) ≥ +∞ +� +i=0 +Pπ(τA < τB | τA∪B ∈ Ai)Pπ(τA∪B ∈ Ai) +≥ +∞ +� +i=0 +inf +u∈Gn\(A∪B) Pu(τA < τB | τA∪B ∈ A0)Pπ(τA∪B ∈ Ai) +≥ +CapM(A) +CapM(A) + CapM(B) +� +1 + O +�δ−3|B|M +n ++ tmix · log n +δ2M +�� +We deduce that, uniformly over all permitted A and B, +Pπ(τA < τB) ≥ +CapM(A) +CapM(A) + CapM(B)(1 − oδ,γ(1)), +(9) +14 + +where the oδ,γ(1) term is uniform over all A and B but may depend on δ and γ. Similarly, for an upper bound +on Pπ(τA < τB) we simply exchange the roles of A and B. We deduce that, uniformly over all permitted A +and B, +Pπ(τA < τB) = +CapM(A) +CapM(A) + CapM(B)(1 − oδ,γ(1)). +(10) +We will also need the following minor adaptation. +Lemma 4.2. Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of γ-expanders, where each Gn +has n vertices and minimal degree at least δn. Take κ and M as defined at the start of Section 4. Then, for +any disjoint A, B ⊂ Gn satisfying |A| + |B| ≤ δ3 +2 n +1 +2 +2κ, every u ∈ Gn \ (A ∪ B) and every v ∈ Gn \ A we +have that +Pu(τA < τB) = Pπ(τA < τB)(1 + o(n3κ−1/2t+ +mix)) +and +Pv +� +τA < τ + +B +� += Pπ(τA < τB)(1 + o(n3κ−1/2t+ +mix)). +Proof. We start by proving the first statement for a lazy random walk, since this is equivalent, and we denote +such a lazy random walk by X. Throughout this proof, we will also use the following notation. For a set +C ⊂ Gn and some time t ≥ 0 we write τ(C, t) for the first time s strictly larger than t such that Xs ∈ C. +Furthermore, write t+ +mix for log2 +2(n)tmix so that by (4) we have that for every u ∈ Gn, +dTV(pt+ +mix(u, ·), π(·)) ≤ n− log(2) log(n). +Now let u ∈ Gn \ A. We start with a lower bound on Pu(τA < τB). We have that +Pu(τA < τB) ≥ Pu +� +t+ +mix < τA < τB +� +≥ Pu +� +τ(A, t+ +mix) < τ(B, t+ +mix) +� +− Pu +� +τA∪B < t+ +mix < τ(A, t+ +mix) < τ(B, t+ +mix) +� +(11) +Note that by (4), the first term can be lower bounded by Pπ(τA < τB) − n− log(2) log(n). For the second term, +let us upper bound the probability of the event {τA∪B < t+ +mix < τ(A, tmix) < τ(B, t+ +mix)}. Using a union +bound we obtain +Pu +� +τA∪B < t+ +mix < τ(A, t+ +mix) < τ(B, t+ +mix) +� +≤ Pu +� +τA∪B < t+ +mix < τ(A, t+ +mix) < 2t+ +mix +� ++ Pu +� +τA∪B < t+ +mix and τ(A, 2t+ +mix) < τ(B, 2t+ +mix) +� +. +≤ |A|t+ +mix +δn ++ (|A| + |B|)t+ +mix +δn +· (Pπ(τA < τB) + n− log(2) log(n)). +Note that, by Lemma 2.6 and Lemma 4.1 we have that +|A|t+ +mix +Pπ(τA < τB)δn ≤ |A|t+ +mix +δn +· 2(|A| + |B|) +δ2|A| +≤ 2(|A| + |B|)t+ +mix +δ3n +, +so that, by Claim 2.5 +|A|t+ +mix +δn += Pπ(τA < τB) · Oγ(n2κ−1/2t+ +mix) = Pπ(τA < τB) · oγ(n3κ−1/2t+ +mix) +(12) +Substituting everything back into (11), we therefore deduce that +Pu(τA < τB) ≥ Pπ(τA < τB)(1 + o(n3κ−1/2t+ +mix)). +For an upper bound on Pu(τA < τB), we simply write +Pu(τA < τB) ≤ Pu +� +τA < t+ +mix +� ++ Pu +� +t+ +mix < τ(A, t+ +mix) < τ(B, t+ +mix) +� +≤ |A|t+ +mix +δn ++ Pπ(τA < τB) + n− log(2) log(n). +Using (12) again we obtain that +Pu(τA < τB) = Pπ(τA < τB)(1 + o(n3κ−1/2t+ +mix)). +For the second statement, it is again enough to prove it for the lazy random walk, replacing τ + +B with the +first hitting time of B after making at least one non-lazy step using the exact same proof. +15 + +4.2 +Capacity +Here we prove some similar properties for the capacity and closeness of a random walk. +In this section we can also introduce the sequence (αn)n≥1 appearing in Theorem 1.3. Given the graph +sequence (Gn)n≥1, take M = nκ as defined at the start of Section 4, let X be a random walk on Gn, and +for each n ≥ 1 set +αn = nEπ +� +CapM(X[0, nκ/2)) +� +Mnκ/2 +. +Proposition 4.3. Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of γ-expanders, where each +Gn has n vertices and minimal degree at least δn. Let u ∈ Gn and let (Xi)i≥0 denote a random walk on Gn +started at u. Take M = nκ as defined at the start of Section 4. Then for all sufficiently large n, +P +�����CapM(X[0, M)) − αnM 2 +n +���� ≥ αnM 2 +n +n−κ/16 +� +≤ 2M 2 +δn . +Proof. The proof is a simplified version of that of [31, Lemma 5.3]. +First recall from Lemma 4.2 that +t+ +mix = log2 +2(n)tmix. Let (Tj)nκ/2 +j=1 be a sequence of i.i.d random variables with distribution Bin(t+ +mix, 1/2). +Then, for all 1 ≤ j ≤ nκ/2 let +Bj = [(j − 1)nκ/2 + Tj, jnκ/2 − t+ +mix + Tj]. +Note that by (4) we have that for all j ≤ nκ/2, given X[0, jnκ/2], the starting point of Bj+1 is nearly +stationary. Also let (Xind,j)nκ/2 +j=1 denote a sequence of independent random walk segments each of length +nκ/2 − t+ +mix, and each started from stationarity. Note that, by (4), the segments (XBj)nκ/2 +j=1 can be coupled +with the segments (Xind,j)nκ/2 +j=1 so that the segments coincide for all j ≤ nκ/2 with probability at least +1 − nκ/2n− log2(n). +(13) +Note that the segments (Xind,j)j are i.i.d. and, by definition, +E +� +CapM(Xind,j) +� += E +� +CapM(Xind,j +[0,nκ/2)) +� ++ O +�Mt+ +mix +δn +� += αnMnκ/2 +n +� +1 + O +� t+ +mix +δnκ/2 +�� +. +(14) +Moreover, by a union bound, we also have the deterministic bound +CapM(Xind,j) ≤ Mπ(Xind,j) ≤ Mnκ/2 +δn +. +(15) +It therefore follows from a Hoeffding bound [17, Theorem 1] that there exist C < ∞, c > 0 such that for any +t > 0, +P + + +������ +nκ/2 +� +j=1 +CapM(Xind,j) − nκ/2E +� +CapM(Xind,1) +� +������ +≥ αnM 2t+ +mix +2n1+κ/8 + + ≤ 2 exp + + +−2nκ/2 + + +αnM2t+ +mix +2n1+5κ/8 +Mnκ/2 +δn + + +2 + + += 2 exp +� +−nκ/4(t+ +mix)2 α2 +nδ2 +2 +� +. +In particular, since it follows from (14) that +����nκ/2E +� +CapM(Xind,1) +� +− αnM 2 +n +���� ≤ O +�αnMnκ/2 +n +t+ +mix +δ +� +≪ αnM 2t+ +mix +n1+κ/8 +, +we deduce that +P + + +������ +nκ/2 +� +j=1 +CapM(Xind,j) − αnM 2 +n +������ +≥ αnM 2t+ +mix +n1+κ/8 + + ≤ 2 exp +� +−nκ/4(t+ +mix)2 α2 +nδ2 +2 +� +. +(16) +16 + +We would like to approximate the capacity of the whole segment X[iM, (i+1)M) by the sum of the capacities +of the smaller segments, but this is potentially a slight overestimate, since we are double-counting random +walk trajectories that hit more than one smaller segment. To account for this, we use the concept of closeness +defined in Section 2.6. For each J ≤ nκ/2, note that conditionally on (Xind,j)j≤J all being disjoint, which +happens with probability at least +1 − M 2 +δn , +(17) +we have by Corollary 2.8 that +CloseM(Xind,J, ∪j 0 and γ > 0. By Claim 2.5, this implies that tmix = O(log n). For each n, k ≥ 1, +T (k−1) +n +will denote the tree obtained after running Wilson’s algorithm on Gn on the vertex set (v1, . . . , vk−1). +Given such a sequence (Gn)n≥0, we set +αn = nEπ +� +CapM(X[0, nκ/2)) +� +Mnκ/2 +, +βn = +1 +√αn +, +(19) +where X is a random walk on Gn and κ is as defined at the start of Section 4. Lastly, if A ⊂ Gn, we will +use the notation τA to denote the hitting time of A for a random walk on Gn. +The goal of this section is to prove the forthcoming Proposition 5.2 for such a sequence of graphs, for +which we will need the following definition. +Definition 5.1. We say that a subgraph T ⊂ Gn is good if +1. T is a tree. +2. |T | ≤ n1/2+κ. +3. For every open connected subset A ⊂ T with |A| ≥ n3κ we have that |CapM(A) − αnM|A|/n| ≤ +αnM|A| +n +· n−κ/16. +17 + +Note that T (1) +n , the tree consisting of the first single vertex, is trivially good. +Proposition 5.2. Take any good subgraph T ⊂ Gn. Take any u ∈ V (Gn) \ T and let Y be a LERW started +at u and terminated when it hits T . +(1) Take any C ∈ (0, ∞) and any B ∈ (0, ∞). Suppose additionally that |T | ≤ B√n. Then +Pu +� +Y [0, Cβns√n] ∩ T = ∅ +� += exp +� +−(C + |T |/βn +√n)2 − (|T |/βn +√n)2 +2 +� +(1 + oB(1)). +(2) Let HT be the time that Y hits T . Then for any connected A ⊂ T with |A| ≥ n3κ and for all n2κ/M ≤ +i ≤ n1/2+κ/M, +Pu(YHT ∈ A | HT ∈ [iM, (i + 1)M)) = |A| +|T |(1 + o(1)), +where the o(1) is uniform over all n2κ/M ≤ i ≤ n1/2+κ/M. +(3) For any k ≥ 1 and B ∈ (0, ∞), if |T | ≤ B√n then +P +� +T (k) +n is good +��� T (k−1) +n += T +� += 1 − oB(1). +Proposition 5.2 will allow us to compare Wilson’s algorithm with the CRT stick-breaking process in +Section 6 and prove Theorem 1.3. +5.1 +Comparison of path probabilities +We will use the Laplacian random walk representation of LERW outlined in Section 2.5 to compare the +probability of different LERW trajectories. We first fix some n ≥ 1 and we run Wilson’s algorithm on Gn. +Let {v1, . . . , vn} denote the ordering of the vertices for this process. +Now fix some k ≥ 2, suppose we have run k − 2 steps of Wilson’s algorithm to form a tree spanned by +the vertices (v1, . . . , vk−1), which we denote by T (k−1) +n +. Let X denote a random walk on Gn, killed when +it hits T (k−1) +n +, and let (Ym)m≥0 denote its loop erasure. Recall from Section 2.5 that, if u0, u1, . . . , uH is a +simple path in Gn, where {u0, u1, . . . , uH−1} is disjoint from �L +m=0{Ym} ∪ T (k−1) +n +and u0 = YL, then +P +� +(Ym)L+H +m=L+1 = (um)H +m=1 +�� (Ym)L +m=0 +� += Pu0 +� +(Xm)H +m=1 = (um)H +m=1 +� +C((Ym)L +m=0, T (k−1) +n +, (um)H +m=1)), +(20) +where +C((Ym)L +m=0, T (k−1) +n +, (um)H +m=1) = +H +� +h=1 +Puh +� +τT (k−1) +n +< τ{Ym}L +m=0∪ {um}h−1 +m=0 +� +Puh−1 +� +τT (k−1) +n +< τ + +{Ym}L +m=0∪ {um}h−1 +m=0 +�. +(21) +At some points in this section, we will condition on an event of the form T (k−1) +n += T and X and Y will +respectively denote a random walk and a LERW, both killed when they hit T . For notational convenience, +if HT is the time that Y hits T , we set Ym = YHT for all m ≥ HT . +Remark 5.3. To prove Theorem 3.1, we should choose the vertices {v1, . . . , vk} uniformly on Gn. In fact +we will prove a result that holds for any choice of distinct {v1, . . . , vk}. +The strategy to prove Proposition 5.2 will be roughly as follows. Firstly, Lemma 5.4 enables us to control +the behavior of a first small segment of Y . This will enable us to give tight estimates for the constant C +defined by (21), which we do in Lemma 5.5. In Corollary 5.6 we combine this with Proposition 4.3 and +(20) to tightly control the capacity of LERW segments. In Lemma 5.7 we estimate the random walk hitting +measure on a good tree T , and in Corollary 5.8 we combine this with the estimates on the constant C to +plug into (20) and obtain analogous estimates for LERW. +18 + +Lemma 5.4. There exists N < ∞ such that for all n ≥ N the following holds. Let T ⊂ Gn be a subgraph +with |T | ≤ n +1 +2 +κ. Take any u ∈ V (Gn) \ T and let Y be a LERW started at u and terminated when it hits +T . Then +Pu +� +|Y | ≤ n2κ� +≤ 4n2κ(n2κ + |T |) +δ3n +. +Proof. For each K < n2κ, we will use the Laplacian random walk representation to bound the probability +that P(|Y | = K + 1 | |Y | > K). To this end, we have by (20) for every v ∈ T and every simple path ϕ of +length K with ϕ ∩ T = ∅ that +Pu +� +YK+1 = v +�� (Ym)K +m=0 = ϕ +� +≤ 1 +δn · +1 +PYK +� +τT < τ + +ϕ +�. +By Lemma 4.1 and Lemma 4.2, the second term can be bounded by estimating the capacities of T and of +ϕ. We claim that up to some constants depending on the minimal degree, they both can be estimated by +their sizes. Indeed, for any set of A size less than n1/2+κ, we have that δM|A| +2n +≤ CapM(A) ≤ |A|M +δn +when n +is large enough by Lemma 2.6. +Therefore, as ϕ is of size K < n2κ < n1/2+κ and T is of size at most n1/2+κ we have that +CapM(T ) +CapM(T ) + CapM(ϕ) ≥ +δ2|T | +2(n2κ + |T |). +Therefore, by Lemma 4.1, Lemma 4.2 and summing over all v ∈ T we have for all sufficiently large n that +Pu +� +YK+1 ∈ T +�� (Ym)K +m=0 = ϕ +� +≤ 4(n2κ + |T |) +δ3n +. +By a union bound, we can thus conclude that Pu +� +|Y | ≤ n2κ� +≤ 4n2κ(n2κ+|T |) +δ3n +as required. +We will also need the following result to control the constant C defined by (21). The reader should have +in mind that we will eventually apply the result with T = T (k−1) +n +. +Lemma 5.5. Take i ≤ n1/2+κ +M +. Then, for all simple paths {Ym}iM +m=0, all simple paths {um}H +m=0 such that +u0 = YiM and H ≤ M, and all connected subgraphs T ⊂ Gn such that {um}H−1 +m=1, {Ym}iM +m=0 and T are disjoint +and such that |T | ≤ n1/2+κ we have the following +(a) If uH /∈ T , then +��C((Ym)iM +m=0, T, (um)H +m=1) − 1 +�� = o(n5κ−1/2). +(b) If uH ∈ T and i ≥ n2κ/M, then +C((Ym)iM +m=0, T, (um)H +m=1) = [CapM(T ) + Cap((Ym)iM +m=0)](1 + o(1)) +CapM(T ) +. +Proof. Fix some 1 ≤ h ≤ H. In case (a), in order to bound a term appearing in the product in (21), we +would like to compare the probabilities +Puh +� +τT < τ{Ym}iM +m=0∪ {um}h−1 +m=0 +� +and Puh−1 +� +τT < τ+ +{Ym}iM +m=0∪ {um}h−1 +m=0 +� +. +(22) +By Lemma 4.2 and by the triangle inequality, we have that +���Puh +� +τT < τ{Ym}iM +m=0∪ {um}h−1 +m=0 +� +− Puh−1 +� +τT < τ+ +{Ym}iM +m=0∪ {um}h−1 +m=0 +���� +≤ Puh +� +τT < τ{Ym}iM +m=0∪ {um}h−1 +m=0 +� +(o(n3κ−1/2t+ +mix)). +19 + +In other words, +Puh +� +τT < τ{Ym}iM +m=0∪ {um}h−1 +m=0 +� +Puh−1 +� +τT < τ+ +{Ym}iM +m=0∪ {um}h−1 +m=0 +� = 1 + o(n3κ−1/2t+ +mix). +Hence we have that the product in (21) is bounded by +(1 + o(n3κ−1/2t+ +mix))nκ = 1 + o(n4κ−1/2t+ +mix). +To conclude, we use that tmix = o(nκ) as n → ∞ by Claim 2.5. +For (b), if uH ∈ T , then C((Ym)iM +m=0, T, (um)H +m=1) is instead equal to +1 +PuH−1 +� +τT < τ+ +{Ym}iM +m=0∪ {um}h−1 +m=0 +� +H−1 +� +h=1 +Puh +� +τT < τ{Ym}iM +m=0∪ {um}h−1 +m=0 +� +Puh−1 +� +τT < τ + +{Ym}iM +m=0∪ {um}h−1 +m=0 +�, +where the product is equal to C((Ym)iM +m=0, T, (um)H−1 +m=1) and is therefore 1 + o(1) by (a). Then note that by +Lemma 4.1 and Lemma 4.2, we have that +PuH−1 +� +τT < τ+ +{Ym}iM +m=0∪ {um}h−1 +m=0 +� += +CapM(T )(1 + o(1)) +CapM(T ) + CapM((Ym)iM +m=0 ∪ {um}h−1 +m=0) += +CapM(T )(1 + o(1)) +CapM(T ) + CapM((Ym)iM +m=0), +where the last equality holds since i ≥ n2κ/M and thus CapM({um}h−1 +m=0) ≤ M2 +δn = o( δiM2 +2n ), which is a lower +bound for CapM((Ym)iM +m=0)) by Lemma 2.6. This proves the claim. +Using the Laplacian random walk representation, we can now tightly control the probability that the +next LERW segment will have good capacity. +Corollary 5.6. Let T ⊂ Gn be a subgraph such that |T | ≤ n1/2+κ and let Y be a LERW trajectory started +at some u ∈ V (Gn) \ T and killed when it hits T . Then for each 0 ≤ i ≤ n1/2+κ +M +, and any LERW trajectory +{Ym}iM +m=0 = {ym}iM +m=0 disjoint from T we have +P +� +(Ym)(i+1)M +m=iM ∩ T = ∅ and +����CapM(Y [iM, (i + 1)M)) − αnM 2 +n +���� ≥ αnM 2 +n +n−κ/16 +���� {Ym}iM +m=0 = {ym}iM +m=0 +� +≤ 4M 2 +δn . +Proof. Let A be the set of simple paths (um)M +m=0 not intersecting T , with +���CapM(u[0, M)) − αnM2 +n +��� ≥ +αnM2 +n +n−κ/16 and with u0 = YiM. It follows from (20) that for any i ≥ 1, +P +� +(Ym)iM+M +m=iM+1 ∈ A +�� (Ym)iM +m=0 = (ym)iM +m=0 +� +≤ PYiM +� +(Xm)H +m=1 ∈ A +� +sup +(um)M +m=1∈A +� +C((ym)iM +m=0, T, (um)M +m=1)) +� +. +(23) +As (um)M +m=1 does not intersect T , by Lemma 5.5, the supremum is at most 2 for all sufficiently large n. +Therefore, by Proposition 4.3, we have that +P +�����CapM(Y [iM, (i + 1)M)) − αnM 2 +n +���� ≥ αnM 2 +n +n−κ/16 and Y [iM, (i + 1)M) ∩ T = ∅ +���� (Ym)iM +m=0 = (ym)iM +m=0 +� +≤ 4M 2 +δn . +In the next lemma we compute some hitting probabilities for a random walk. +20 + +Lemma 5.7. Let X be a random walk on Gn. Let T ⊂ Gn be a subgraph such that |T | ≤ n1/2+κ, and let A +be a connected subset of T . Then for any 0 ≤ i ≤ n1/2+κ +M +, for any simple path y[0, iM] on Gn disjoint from +T and any u /∈ T ∪ y[0, iM] +Pu(X[0, M] ∩ A ̸= ∅ and ̸ ∃0 < j < ℓ ≤ τA : Xj = Xℓ and ̸ ∃j ≤ τA : Xj ∈ y[0, iM] ∪ (T \ A)) += CapM(A)(1 + o(1)). +Proof. Upper bound. Recall that t+ +mix = log2 +2(n)tmix. First note that by Claim 2.7, we have that +Pu(X[0, M] ∩ A ̸= ∅ and ̸ ∃0 < j < ℓ ≤ τA : Xj = Xℓ and ̸ ∃j ≤ τA : Xj ∈ y[0, iM] ∪ (T \ A)) +≤ Pu(X[0, M] ∩ A ̸= ∅) +≤ CapM(A) + 3|A| log n · tmix +δn +≤ CapM(A) +� +1 + t+ +mix +δ2M +� +. +Here the final line follows because of Lemma 2.6, which implies that CapM(A) ≥ +δM|A| +2n +on the event +|T | ≤ n1/2+κ. +Lower bound. We first note that +Pu(X[0, M] ∩ A ̸= ∅) − Pu(X[0, M] ∩ A ̸= ∅ and ∃0 < j < ℓ < τA : Xj = Xℓ) +− Pu(X[0, M] ∩ A ̸= ∅ and ∃j < M : Xj ∈ y[0, iM] ∪ (T \ A)). +≤ Pu(X[0, M] ∩ A ̸= ∅ and ̸ ∃0 < j < ℓ ≤ τA : Xj = Xℓ and ̸ ∃j ≤ τA : Xj ∈ y[0, iM] ∪ (T \ A)) +We will now bound all three terms on the left hand side. +First, we lower bound Pu(X[0, M] ∩ A ̸= ∅) +by CapM(A) +� +1 − 3t+ +mix +δ2M +� +by Claim 2.7. +For the second term, we upper bound it by the product of the +probabilities for X to self intersect in M steps, and then to hit A in another M steps. This is upper bounded +by +M 2 +δn · M|A| +δn . +The third term can be bounded by the probability to hit y[0, iM] ∪ (T \ A) in at most M steps, and then to +hit A in at most another M steps, which is upper bounded by +M|A| +δn +· M(iM + |T |) +δn +. +We conclude that +CapM(A) +� +1 + O +�t+ +mix +M +�� +≤ CapM(A) +� +1 − t+ +mix +δ2M +� +− 2 · M|A| +δn +· M(iM + |T |) +δn +≤ Pu(X[0, M] ∩ A ̸= ∅ and ̸ ∃0 < j < ℓ ≤ τA : Xj = Xℓ and ̸ ∃j ≤ τA : Xj ∈ y[0, iM] ∪ (T \ A)). +We now have estimates for all the quantities appearing in (20). We combine these in the next corollary. +Corollary 5.8. Let T ⊂ Gn be a subgraph such that |T | ≤ nκ+1/2, and let A ⊆ T . Let Y be a LERW on +Gn killed when it hits T . For each n2κ +M ≤ i ≤ n1/2+κ +M +and for any simple path (ym)iM +m=0 not intersecting T , +satisfying +����CapM(y[jM, (j + 1)M)) − αnM 2 +n +���� ≤ αnM 2 +n +n−κ/16 for all n2κ +M ≤ j < i, +it holds that +P +� +Y hits T in time interval [iM, (i + 1)M) in set A +�� (Ym)iM +m=0 = (ym)iM +m=0 +� += CapM(A) +CapM(T ) +� +CapM(T ) + αniM 2 +n +� +(1 + o(1)). +21 + +Proof. First note that it follows from Corollary 2.8 that +CapM(y[0, iM)) = αniM 2(1 + o(1)) +n +. +Given 1 ≤ H < M, (ym)iM +m=0 and T , let ΓyiM→A,T,H denote the set of simple paths with u0 = yiM that first +hit T in the set A and at time H, and avoid (ym)iM +m=0 until that time. We can then write, using Lemma 5.5(b) +and Lemma 5.7: +P +� +Y hits T in time interval [iM, (i + 1)M) in set A +�� (Ym)iM +m=0 = (ym)iM +m=0 +� += +� +H 0. For every i ≤ Cβn +√n/M, let +EC,i be the event that +����CapM(Y [iM, (i + 1)M)) − αnM 2 +n +���� ≤ αnM 2 +n +n−κ/16 and Y [iM, (i + 1)M) ∩ T = ∅. +Write Eprefix for the event ∩i≤n2κ/MEC,i. Note that by Lemma 5.4 and Corollary 5.6 we have that +P(Eprefix) = 1 − o(1). +Note that, by Corollary 5.6 and Corollary 5.8, for any i ≥ n2κ/M, given Eprefix ∩n2κ/M≤j≤i EC,j, using +Corollary 5.8 we have +P +� +EC,i +�� Eprefix and ∩n2κ/M≤j≤i EC,j +� += 1 − +� +CapM(T ) + αniM 2 +n +� +(1 + o(1)) − O +�4M 2 +δn +� += 1 − +�αnM|T | +n ++ αniM 2 +n +� +(1 + o(1)). +Here the final line holds since T is good, i ≥ n2κ +M +and our conditioning on Eprefix ∩ ∩n2κ/M≤j≤iEC,j. +Then, write EC for the event that +Eprefix and {Y [0, Cβn +√n] ∩ T = ∅} and +� +∩Cβn +√n/M +i=n2κ/M EC,i +� +, +we have that, +P +� +EC +��� T (k−1) +n += T +� += +P +� +Eprefix +��� T (k−1) +n += T +� +· +22 + +Cβn +√n/M +� +i=n2κ/M +P +� +EC,i +�� Eprefix and ∩n2κ/M≤j n2κ/M, conditionally on +HT (k−1) +n +∈ (iM, (i + 1)M], we have that +P +� +YH +T (k−1) +n +∈ A +��� HT (k−1) +n +∈ (iM, (i + 1)M] +� += |A|(1 + o(1)) +|T (k−1) +n +| +, +as required. +(3) Given ε > 0, first choose C < ∞ so that the probability appearing in part (1) is at most ε. Then, on +the event Y [0, Cβn +√n] ∩ T ̸= ∅, we have that the probability that T (k) +n +is not good is upper bounded by +ε + o(1) by (24). Since ε > 0 is arbitrary this gives the result. +6 +Proof of Theorem 1.3 +In this section we prove Theorem 1.3. We start by using the estimates of the previous section to prove +Theorem 3.1. At the end of the section, we address the lower mass bound condition which completes the +proof of Theorem 1.3. +6.1 +Proof of Theorem 3.1 +In Definition 3.2 we defined how a sequence of trees can be constructed through a stick-breaking process. In +what follows next we outline how, for any k ≥ 1, Wilson’s algorithm on Gn can be used to give two sequences +(Yi)n +i=0 and (Zi)n−1 +i=0 such that SB(k)((Y0, Y1, . . . , Yk−1), (Z0, Z1, . . . , Zk−2)) is equal to the subtree obtained +after the first k − 1 steps of Wilson’s algorithm, and such that the +�k +2 +� +distances appearing in Theorem 3.1 +therefore match those between the points (Y0, Y1, . . . , Yk−1) in the stick-breaking construction. +Let Gn be a graph on n vertices and recall the definition of βn from (19). We will define a stick-breaking +process (Y n +i )n +i=0, (Zn +i )n−1 +i=0 which arises from Wilson’s Algorithm on Gn. To ease notation, we shall remove +the superscript and begin with Y0 = 0, Z0 = 0. We choose an ordering of the vertices of G, denoted by +{v1, . . . , vn}. Then, at the first step, we sample the UST path (using Wilson’s algorithm) from v2 to v1. We +23 + +denote this path by T (2) +n , and let Y1 be the length of this path divided by βn +√n. For every vertex z on this +path we say that z was added at the first step. Let k ≥ 2 and assume that we sampled T (k) +n +and Z0, . . . , Zk−2 +and Y0, . . . , Yk−1. For the kth step, take vk+1 and sample (again, using Wilson’s algorithm) its path to T (k) +n . +Denote this path Pk and set T (k+1) +n += Pk ∪ T (k) +n . For every vertex in Pk \ T (k) +n , we say that it was added +on the kth step. Let Yk = Yk−1 + +|Pk| +βn +√n. In order to define Zk−1, first let z be the vertex at which Pk hits +T (k) +n . If z is of the form vm for some m, set Zk−1 = Ym−1. Otherwise, let m < k be the step at which z was +added. Then, Ym−1 ≤ Zk−1 ≤ Ym and the exact value of Zk−1 is +Zk−1 = Ym − d(z, vm+1) +βn +√n +. +Furthermore, this way we can define a function I that identifies every v ∈ T (k) +n +with a point in [0, ∞). If +v was added at the mth step then we set I(v) = Ym − d(v,vm+1) +βn +√n +. +Throughout this section, we also let (Y ′ +i )i≥0 and (Z′ +i)i≥0 be the analogous quantities for stick-breaking +of the CRT, sampled as described in Proposition 3.3. +We will use the following claim. Recall the definition of “good” from Definition 5.1. +Claim 6.1. Assume that T (k−1) +n +is good and that |T (k−1) +n +| ∈ [C−1√n, C√n]. Let Imax = maxv∈T (k−1) +n +I(v). +Let Y be a LERW started from vk, and let HT (k−1) +n +be the time at which Y hits T (k−1) +n +. Let j ≤ n1/2+κ/M +and let Pd,j be the measure on [0, Imax] defined by +Pd,j(I(v)) = P +� +YH +T (k−1) +n += v +��� HT (k−1) +n +∈ [jεβn +√n/2, (j + 1)εβn +√n/2) +� +∀v ∈ T (k−1) +n +. +Then, for every ε > 0 there exists N ∈ N such that for all n > N and for all j ≤ n1/2+ε/M, the Prohorov +distance between the measure Pd,j and the uniform probability measure on [0, Imax] is less than ε. +Proof. We assume that C/βn is larger than ε, otherwise Imax is smaller than ε and there’s nothing to +prove. We also assume wlog that ε < 1. Decompose [0, Imax] into intervals of size ε by writing [0, Imax] = +∪i≤⌊Imax/ε⌋[iε, min{Imax, (i + 1)ε}]. Fix j, write Pd in place of Pd,j and denote by Pu the uniform measure +on [0, Imax]. Note that every interval I ⊂ [0, Imax] of length ε can be identified with the union of at most k +connected subsets of T (k−1) +n +such that the sum of their lengths is εβn +√n (which is much larger than n3ε). By +discarding those that are of length less than n3ε we can apply Proposition 5.2(2) to the remaining subsets +(by decomposing them and I into smaller intervals if necessary) to deduce that +Pd(I) = +ε +Imax +(1 + o(1)), +Pu(I) = +ε +Imax +. +Now take N large enough such that the o(1) error is bounded by ε. Then, take some set A in [0, Imax] and +let IA be the set of intervals of the form [iε, (i + 1)ε) intersecting A. Now, we have that +|IA| +ε +Imax +− ε ≤ |IA| +ε +Imax +(1 + o(1)) ≤ Pd(IA) ≤ |IA| +ε +Imax +(1 + o(1)) ≤ |IA| +ε +Imax ++ ε +and +Pd(A) ≤ Pd(IA) ≤ |IA| +ε +Imax ++ ε ≤ Pu(IA) + ε ≤ Pu(Aε) + ε, +Pu(A) ≤ Pu(IA) = |IA| +ε +Imax +≤ Pd(IA) + ε ≤ Pd(Aε) + ε. +Hence the Prohorov distance between these two measures is at most ε. +The main claim of this section is now as follows. +24 + +Claim 6.2. For every ε > 0 and k ≥ 1 there exists N such that for all n > N we can couple the stick-breaking +process for the CRT and for the UST such that |Yi − Y ′ +i | ≤ ε for all 0 ≤ i ≤ k − 1 and |Zi − Z′ +i| ≤ ε for every +0 ≤ i ≤ k − 2 with probability at least 1 − ε. +Proof. We prove the claim by induction. Clearly when k = 1 (i.e. for T (1) +n ) the statement holds trivially +since the tree is a single point and Y0 = Y ′ +0 = 0 by construction. Moreover since a tree consisting of a single +vertex is always good and since Z0 = Z′ +0 = 0, it also follows directly from Lemma 2.10, Proposition 3.4 and +Proposition 5.2 that the statement holds for k = 2 as well. +Now fix k ≥ 3 and suppose that the claim holds for all m < k. We will now show that the claim holds +also for k. That is, suppose that for every ε > 0 there exists N large enough such that for all n > N we can +successfully couple T (k−1) +n +with the CRT. It suffices to show that for every ε > 0, there exists 0 < ζ < ε/8 +such that if we condition on a successful coupling of the previous step with parameter ζ, then we can couple +(Yk−1, Zk−2) with (Y ′ +k−1, Z′ +k−2) such that |Yk−1 − Y ′ +k−1| < ε and |Zk−2 − Z′ +k−2| < ε with probability at least +1 − ε/2. +To this end, let ζ > 0 (its precise value will be chosen later) and suppose we have successfully coupled +(Yi)i≤k−2 and (Zi)i≤k−3 with (Y ′ +i )i≤k−2 and (Z′ +i)i≤k−3 as in the statement of the claim with parameter ζ. +Note that it follows directly by iterating Point 3 of Proposition 5.2 that T (k−1) +n +is good with probability at +least 1 − ε/3 for all sufficiently large n. Moreover, it therefore also follows from Lemma 3.5 that 0 < g(ε) ≤ +Yk−2 ≤ f(ε) with probability at least 1 − ε/3, for some functions f and g where g(ε) > 0 and f(ε) < ∞. +Hence we can assume that we coupled T (k−1) +n +with the CRT, that T (k−1) +n +is good and that g(ε) ≤ Yk−2 ≤ f(ε). +Under the coupling, we can write |T (k−1) +n +| +βn +√n += Yk−2 = Y ′ +k−2 + ε′ where ε′ ∈ [−ζ, ζ]. Therefore, since T (k−1) +n +is good, it follows from Proposition 5.2(1) with B = f(ε) that for any C ∈ (0, ∞), +P +� +Yk−1 − Yk−2 > C +��� T (k−1) +n +� += exp +� +−(C + Y ′ +k−2 + ε′)2 − (Y ′ +k−2 + ε′)2 +2 +� ++ o(1) += exp +� +−(C + Y ′ +k−2)2 − (Y ′ +k−2)2 +2 +�� +1 + e−Cε′ − 1 +� ++ o(1), +so +����P +� +Yk−1 − Yk−2 > C +��� T (k−1) +n +� +− exp +� +−(C + Y ′ +k−2)2 − (Y ′ +k−2)2 +2 +����� ≤ |1 − e−Cε′|e +−C2 +2 ++ o(1) ≤ Cζe +−C2 +2 ++ o(1). +(25) +The first term on the right hand side goes to 0 as ζ → 0 uniformly over C > 0. By Lemma 2.10, there exists +η depending on ε such that if f(ε) ≤ Yk−2 ≤ g(ε) and the right-hand side of (25) is smaller than η, then we +can couple Yk−1 − Yk−2 and Y ′ +k−1 − Y ′ +k−2 such that the probability that they are ε/4 close to one another +is at least 1 − ε/4. When this happens, by the triangle inequality, we have that |Yk−1 − Y ′ +k−1| < ε/2. We +therefore choose ζ small enough (and smaller than ε/8) and n large enough such that the right-hand side is +smaller than this η. +However, we note that Zk−2 is not independent of Yk−1 and we are required to couple the pair (Yk−1, Zk−2) +with (Y ′ +k−1, Z′ +k−2). To do so, we will decompose R+ into intervals of length ε/2, that is, we write R+ = �∞ +j=0 Ij +where Ij = [jε/2, (j+1)ε/2). Let Mk−1 (respectively M ′ +k−1) be the unique j such that Yk−1 ∈ Ij (respectively +Y ′ +k−1 ∈ Ij). By Lemma 2.10 and the discussion above, there exists a coupling of Mk−1 and M ′ +k−1 such that +the difference between them is at most 1 with probability 1 − ε/8. By Lemma 3.5, with probability at least +1 − ε/8 we have that M ′ +k−1 ≤ n1/2+ε − 1 for n large enough (and then so is Mk−1). +Then, given Mk−1, we sample Zk−2 according to its conditional law. By Claim 6.1, when n is large +enough, for every j ≤ n1/2+κ, conditionally on Mk−1 = j we have that the Prohorov distance between Zk−2 +and a uniform random variable on [0, Yk−2] is at most ζ. By Claim 2.9, the Prohorov distance between a +uniform random variable on [0, Yk−2] and Z′ +k−2 (recall that, given Y ′ +k−2, Z′ +k−2 is independent of Y ′ +k−1 and +hence of M ′ +k−1) is at most ζ. Therefore, the Prohorov distance between Z′ +k−2 and Zk−2 conditionally on +Mk−1 = j is at most 2ζ. Since ζ < ε/8, it follows that we can couple the pairs (Yk−1, Zk−2) and (Y ′ +k−1, Z′ +k−2) +such that |Yk−1 − Y ′ +k−1| < ε and |Z′ +k−2 − Zk−2| < ε with probability at least 1 − ε/2, as required. +25 + +Corollary 6.3. For every ε > 0 and k ≥ 1 there exists N such that for all n > N we can couple the +stick-breaking process for the CRT and for the UST such that, with probability at least 1 − ε, it holds for all +0 ≤ i, j ≤ k that +|d(yi, yj) − d′(y′ +i, y′ +j)| ≤ ε. +Proof. Take η > 0 and k ≥ 1. +We verify that there is a coupling such that each of the conditions of +Proposition 3.6 hold with high probability. +For the first condition note that, by Claim 6.2, we can couple the stick-breaking process for the CRT and +for the UST such that |Yi −Y ′ +i | ≤ η for all 0 ≤ i ≤ k and |Zi −Z′ +i| ≤ η for every 0 ≤ i ≤ k−1 with probability +at least 1 − η for all sufficiently large n. For the second condition, note that it follows from Proposition 3.3 +that we can choose δ = δ(η, k) > 0 such that |Z′ +i − Y ′ +j | ≥ 3η for all i ≤ k − 1, j ≤ k with probability at least +1 − δ, and such that δ ↓ 0 as η ↓ 0. +Therefore, it follows from Proposition 3.6 that under this coupling, it holds with probability at least +1 − η − δ that sup1≤i,j≤k |d(yi, yj) − d′(y′ +i, y′ +j)| ≤ 2kη. Given ε > 0, we can therefore choose η > 0 small +enough that 2kη + δ < ε in order to deduce the claim as stated. +Proof of Theorem 3.1. For k ≥ 1, let D(k) +n +denote the matrix of distances between k uniform points in +UST(Gn). Let D(k) denote the analogous matrix for the CRT. +We showed that for any k ≥ 1 and any ε > 0, we can couple T (k) +n +and the CRT so that ||D(k) +n −D(k)||∞ < ε +with probability at least 1 − ε for all sufficiently large n. Thus we have that ||D(k) +n +− D(k)||∞ converges to 0 +in probability and therefore D(k) +n +converges to D(k) in distribution, which is equivalent to the statement of +Theorem 3.1. +6.2 +Lower mass bound +To strengthen the convergence obtained in Theorem 3.1 to GHP convergence (and therefore prove Theo- +rem 1.3), it suffices to verify that Proposition 2.13(b) holds. Therefore, in our setting, it is enough to show +the following. +Claim 6.4 (Lower mass bound). Let (Gn)n≥1 be a dense sequence of deterministic graphs satisfying the +assumptions of Theorem 1.3. For each n ≥ 1, let Tn be a uniformly drawn spanning tree of Gn. Denote by +dTn the corresponding graph-distance on Tn and by µn the uniform probability measure on the vertices of Tn. +Then, for any c > 0 and any η > 0 there exists some ε > 0 such that for all n ∈ N +P +� +∃v ∈ Tn : |BTn(v, c√n)| ≤ εn +� +≤ η. +The results of [7] establish the lower mass bound for a sequence (Gn)n≥1 such that |Gn| = n for all n, +satisfying the following three conditions. +1. There exists θ < ∞ such that sup +n +sup +x∈Gn +√n +� +t=0 +(t + 1)pt(x, x) ≤ θ. +2. There exists α > 0 such that tmix(Gn) = o(n +1 +2 −α) as n → ∞. +3. Gn is transitive for all n. +For a graph sequence satisfying the assumptions of Theorem 1.3, note that the second condition is +immediately satisfied by Claim 2.5. The first condition is also satisfied since pt(x, x) ≤ +1 +δn for all x ∈ Gn +and all t ≥ 1. +However, we would like to relax the condition that Gn is transitive and instead require only that the +graphs are balanced; that is, that there exists a constant D < ∞ such that +maxv∈Gn deg v +minv∈Gn deg v ≤ D +26 + +for all n. As remarked at the end of [7], it is straightforward to extend the proof of the lower mass bound to +this setting by carrying the constant D through all of the computations in [7]; we do not provide the details +as they are not illuminating. Under the assumptions of Theorem 1.3, we can take D = δ−1 so this easily +verifies Claim 6.4 and therefore Proposition 2.13(b). Moreover, Theorem 3.1 ensures that Proposition 2.13(a) +is also fulfilled. Theorem 1.3 therefore follows directly. +7 +Proof of Theorem 1.1 and Corollary 1.2 +Recall from the introduction that that a graphon W is non-degenerate if the function +degW (x) := +� +[0,1] +W(x, y)dy +is defined and strictly positive for every x ∈ [0, 1], and that a non-degenerate graphon W is connected if +for every measurable A ⊂ [0, 1] we have that +� +A +� +AC W(x, y)dxdy > 0. +In order to verify Theorem 1.1 as consequence of Theorem 1.3, we need to verify that under the assump- +tions of Theorem 1.1, the graph sequence is an expander sequence and that αn → αW . +We start with the first of these. Recall the definition of a γ-expander sequence is given in Definition 2.4. +Claim 7.1. Let W : [0, 1]2 → [0, 1] be a connected graphon and let Gn be a sequence of weighted graphs with +minimal degree at least δn converging in cut-distance to W. Then there exists γ = γ(W, δ) > 0 such (Gn)n≥1 +is a γ-expander sequence. +Proof. Take U ⊂ Gn. We split the proof into two cases depending on whether |U| ≥ 1 +2δn or not. +Case 1: |U| ≤ 1 +2δn. Since Gn has minimal degree at least δn and the maximal weight of every edge is +1, it follows that there is a total weight of 1 +2δn emanating from every vertex leading to V (G) \ U, so that +w(U, V (G) \ U) ≥ 1 +2|U|δn ≥ 1 +2δ|U|(V (G) − |U|). +Case 2: |U| > 1 +2δn. By Lemma 2.2, there exists a constant β = β(W, δ) such that for every set U with +δ +2 ≤ µ(U) ≤ 1/2 we have +� +U +� +UC W(x, y) > β. +In particular, since Gn converges to W this implies that there exists N > ∞ such that for all such U and all +n ≥ N, +1 +n2 w(U, V (G) \ U) = +� +U +� +UC Wn(x, y) > β +2 . +Since |U|(V (G) − |U|) ≤ n2 trivially, this implies that e(U, V (G) \ U) ≥ β +2 |U|(V (G) − |U|). +We now turn to verifying the convergence αn → αW . Recall that in Section 4 we defined +αn = nEπ +� +CapM(X[0, nκ/2)) +� +Mnκ/2 +. +where X is a RW on Gn started from stationarity, and showed that under some assumptions, the sequence +� +UST(Gn), +√αndn +√n +, µn +� +(d) +→ (T , d, µ) +(26) +27 + +with respect to the GHP topology. We also let Uπn denote a random stationary vertex of Gn, and define +˜αn = nE[πn(Uπn)]. +In fact it is more convenient to deal with ˜αn rather than αn. This is sufficient as we show in the following +claim (we write the proof for completeness, but really it follows directly just from linearity of expectation +and Corollary 2.8). +Claim 7.2. Let (Gn)n≥0 be a sequence of weighted graphs on n vertices with minimal degree δn. Let αn and +˜αn be defined as above. Then αn = ˜αn(1 + o(1)) as n → ∞. +Proof. By the Bonferroni inequalities and linearity of expectation, and letting Z denote an independent RW +started from stationarity, we can write (recalling also from Lemma 2.6 that CapM(Uπ) ≥ Mδ +2n deterministi- +cally): +���E +� +CapM(X[0, nε/2)) +� +− nε/2E[CapM(Uπ)] +��� = +������ +E +� +CapM(X[0, nε/2)) +� +− +nε/2 +� +i=0 +E[CapM(Xi)] +������ +≤ Pπ +� +∃0 ≤ t1 < t2 ≤ M − 1 : Zt1 ∩ X[0, nε/2) ̸= ∅ and Zt2 ∩ X[0, nε/2) ̸= ∅ +� +≤ +�Mnε/2 +δn +�2 +≤ 2Mnε/2 +δ3n +· nε/2E[CapM(Uπ)]. +Similarly, then note that, since π(v) ≥ δ +n for all v ∈ Gn deterministically, +|E[CapM(Uπ)] − ME[π(Uπ)]| = +�����E[CapM(Uπ)] − +M−1 +� +i=0 +Pπ(Zi = Uπ) +����� +≤ Pπ(∃0 ≤ t1 < t2 ≤ M − 1 : Zt1 = Zt2 = Uπ) ≤ +�M +δn +�2 +≤ M +δ3n · ME[π(Uπ)]. +To conclude, we combine to get that +αn = +n +Mnε/2 E +� +CapM(X[0, nε/2)) +� += n +M E[CapM(Uπ)](1 + o(1)) = nE[π(Uπ)](1 + o(1)) = ˜αn(1 + o(1)), +as required. +It therefore follows that the convergence of (26) holds with the sequence (˜αn)n≥1 in place of (αn)n≥1. To +prove main convergence theorem, it is therefore sufficient to show that, under the assumptions of Theorem 1.1, +˜αn → αW , +(27) +where αW is as in (1). +Remark 7.3. Note that ˜αn is 1 when Gn is regular, so clearly (27) will entail that αW = 1 for a regular +graph sequence. +Our next goal is to show the following. +Claim 7.4. Let W : [0, 1]2 → [0, 1] be a connected graphon and let Gn be a sequence of graphs with stationary +measures πn converging in cut-distance to W. Then, +˜αn := nEπn[πn(v)] → +1 +�� +[0,1]2 W(x, y)dxdy +�2 · +� +[0,1] +�� +[0,1] +W(x, y)dy +�2 +dx. +28 + +Proof. We will begin with showing that +1 +n2 +� +v∈V +degGn(v) → +�� +[0,1]2 W(x, y)dxdy +� +. +(28) +Indeed, as Gn → W in the cut-distance, there exist φn measure-preserving automorphisms such that the +graphon representations Wn of Gn satisfy +sup +S,T ∈B([0,1]) +���� +� +S +� +T +W φn +n (x, y) − W(x, y)dxdy +���� → 0. +We henceforth write Wn in place of W φn +n . In particular, choosing S = T = [0, 1] we obtain that +� +[0,1] +� +[0,1] +Wn(x, y)dxdy → +� +[0,1] +� +[0,1] +W(x, y). +However, by the definition of a graphon representation of a graph given in Section 2.1, we also have that +� +[0,1] +� +[0,1] +Wn(x, y)dxdy = 1 +n2 +� +v∈V +degGn(v), +which establishes (28). Next, we will show that +1 +n3 +� +v∈Gn +degGn(v)2 → +� +[0,1] +�� +[0,1] +W(x, y)dy +�2 +dx +(29) +Note that in every graphon representation Wn of Gn we have that +1 +n3 +� +v∈Gn +degGn(v)2 = 1 +n +� +v∈Gn +�degGn(v) +n +�2 += 1 +n +� +v∈Gn +�� +[0,1] +Wn(xv, y)dy +�2 +, +where xv is some point in [0, 1] corresponding to v. Moreover, in the notation of Section 2.1, it follows from +the construction given there that +1 +n +� +v∈Gn +�� +[0,1] +Wn(xv, y)dy +�2 += +n +� +i=1 +� +Ii +�� +[0,1] +Wn(x, y)dy +�2 +dx = +� +[0,1] +�� +[0,1] +Wn(x, y)dy +�2 +dx. +To establish (29), it thus suffices to prove that +� +[0,1] +�� +[0,1] +Wn(x, y)dy +�2 +dx → +� +[0,1] +�� +[0,1] +W(x, y)dy +�2 +dx. +In other words, writing degW and degWn for the corresponding normalized degree functions of the graphons +W and Wn as defined in (2), we need to show that +� +[0,1] +� +degWn(x)2 − degW (x)2� +dx → 0. +(30) +As degWn and degW are measurable functions, we have that the set {x ∈ [0, 1] : degWn(x) > degW (x)} is +measurable. Denote this set by S. We have that +� +S +degWn(x)2 − degW (x)2dx = +� +S +(degWn(x) − degW (x))(degWn(x) + degW (x))dx +≤ 2 +� +S +(degWn(x) − degW (x))dx = 2 +� +S +� +[0,1] +Wn(x, y) − W(x, y)dydx → 0. +29 + +By symmetry and considering Sc we similarly have that +� +Sc degWn(x)2 − degW (x)2dx → 0, +from which we conclude that (30) and therefore (29) hold. Finally, given (28) and (29), note that +˜αn = n +� +v∈Gn +� +deg(v) +� +v∈Gn deg(v) +�2 += n · +1 +�� +v∈Gn deg(v) +�2 +� +v∈Gn +(deg(v)2) += +� +n2 · +1 +� +v∈Gn deg(v) +�2 +· +� � +v∈Gn +1 +n3 deg(v)2 +� +→ +1 +�� +[0,1]2 W(x, y)dxdy +�2 · +� +[0,1] +�� +[0,1] +W(x, y)dy +�2 +dx, +as required. +Proof of Theorem 1.1. We showed in Claim 7.1 that under the assumptions of Theorem 1.1, the graph +sequence in question is an expander sequence, so that Theorem 1.3 applies. In Claim 7.2 and Claim 7.4, +we showed that the sequence αn appearing in the conclusion of Theorem 1.3 converges to αW as n → ∞, +exactly as required. +Corollary 1.2 is a direct consequence of Theorem 1.1 and Lemma 2.3. +References +[1] D. Aldous. The continuum random tree. I. Ann. Probab., 19(1):1–28, 1991. +[2] D. Aldous. The continuum random tree. II. An overview. 167:23–70, 1991. +[3] D. Aldous. The continuum random tree iii. The Annals of Probability, pages 248–289, 1993. +[4] N. Alon. Eigenvalues and expanders. Combinatorica, 6(2):83–96, 1986. +[5] N. Alon and V. Milman. λ1, isoperimetric inequalities for graphs, and superconcentrators. Journal of +Combinatorial Theory, Series B, 38(1):73–88, 1985. +[6] N. Alon, A. Nachmias, and M. Shalev. The diameter of the uniform spanning tree of dense graphs. +Combinatorics, Probability and Computing, 31(6):1010–1030, 2022. +[7] E. Archer, A. Nachmias, and M. Shalev. 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ACM, New York, 1996. +31 + diff --git a/ptAyT4oBgHgl3EQfl_gG/content/tmp_files/load_file.txt b/ptAyT4oBgHgl3EQfl_gG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa31e2ff3ee3f94fa1020ec426446a25a0ae7f66 --- /dev/null +++ b/ptAyT4oBgHgl3EQfl_gG/content/tmp_files/load_file.txt @@ -0,0 +1,1295 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf,len=1294 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='00461v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='PR] 1 Jan 2023 The GHP scaling limit of uniform spanning trees of dense graphs Eleanor Archer∗ Matan Shalev† January 3, 2023 Abstract We consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees is Aldous’ Brownian CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Furthermore, we are able to extract the precise scaling constant from the limiting graphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' As an example, we can apply this to the scaling limit of the uniform spanning trees of the Erd¨os-R´enyi sequence (G(n, p))n≥1 for any fixed p ∈ (0, 1], and sequences of dense expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' A consequence of GHP convergence is that several associated quantities of the spanning trees also converge, such as the height, diameter and law of a simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 1 Introduction Uniform spanning trees (USTs) are fundamental objects in probability theory and computer science, with close connections to many other areas of mathematics including electrical network theory [20], loop erased random walks [32] and random interlacements [18], to name but a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' It was recently shown in [7], building on the work of [31], that the universal metric measure space scaling limit of USTs of a large class of graphs is Aldous’ Brownian continuum random tree (CRT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The purpose of the present paper is to extend this result to sequences of dense graphs encoded by graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Due to a transitivity assumption in previous papers, these USTs are not covered by the results of [31] and [7], but here we establish that the CRT is nevertheless still the scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In addition we are able to express the precise scaling factor in terms of the encoding graphon, making the result more precise than that in [7] and demonstrating that the notion of graphon convergence is enough to fully determine the UST scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The CRT, introduced by Aldous [1, 2, 3], is a well-known object in probability theory, and is perhaps best-known as the scaling limit of critical finite variance Galton–Watson trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We do not attempt to give a full introduction here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' we will give a formal definition in Section 3 and we refer to the survey of Le Gall [23] for further background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' A weighted graph (G, w) is a graph G = (V, E) in which we assign to each edge e ∈ E a non-negative weight we.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In this paper, we will work with sequences of weighted graphs with no loops or multiple edges in which we ∈ [0, 1] for each e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In the case where all edge-weights are equal to 1, we say that the graph is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We extend the definition of vertex degree to weighted graphs by defining deg v to be the sum of the weights of the edges emanating from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The uniform spanning tree of a weighted graph (G, w) is a random spanning tree chosen from the set of all spanning trees of G where each spanning tree t is chosen with probability proportional to � e∈t we.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We will say that such a sequence (Gn)n≥1 of weighted graphs is dense if there exists δ > 0 such that ∆n := minv∈Gn deg(v) ≥ δ#V (Gn) for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The notion of convergence of dense graph sequences is naturally captured by objects known as graphons, introduced by Lov´asz and Szegedy [25] and also Borgs, ∗ ´Equipe Modal’X, Universit´e Paris Nanterre, Batiment G, 200 Avenue de la R´epublique, 92000 Nanterre, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Email: eleanor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='archer@parisnanterre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='fr †School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Email: matanshalev@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='il 1 Chayes, Lov´asz, S´os and Vesztergombi [10] for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' See also [14] for a very quick introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' A graphon W is a symmetric measurable function from [0, 1]2 to [0, 1] and can be thought of as (roughly) the continuum analogue of an adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Using this viewpoint, there is a natural notion of distance between discrete graphs and graphons, known as the cut-distance, which we will define in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This allows us to consider the notion of convergence to a given graphon W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Graphons are commonly used in combinatorics and computer science to analyze large dense graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For example, they have been used in extremal graph theory [12], mean-field games [11], analysis of large graphs [21], and to study the thermodynamic limit of statistical physics systems [27, 13], to give a very non-exhaustive list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Given a graphon W, define a constant αW = 1 �� [0,1]2 W(x, y)dxdy �2 · � [0,1] �� [0,1] W(x, y)dy �2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (1) Note it follows immediately from Jensen’s inequality that αW ≥ 1, with equality if and only if W is constant almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We also say that a graphon W is connected if for all A ⊂ [0, 1] of positive Lebesgue measure, it holds that � A � AC W(x, y)dxdy > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The main result of the present paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Below, the GHP distance refers to the Gromov Hausdorff Prohorov distance between metric measure spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' we define it in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let (Gn)n≥1 be a dense sequence of deterministic weighted graphs converging to a connected graphon W, where each Gn has n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For each n ≥ 1, let Tn be a uniform spanning tree of Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Denote by dTn the corresponding graph-distance on Tn and by µn the uniform probability measure on the vertices of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then � Tn, √αW √n dTn, µn � (d) −→ (T , dT , µ) where αW is defined as in (1), (T , dT , µ) is the CRT equipped with its canonical mass measure µ and (d) −→ denotes convergence in distribution with respect to the GHP distance between metric measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' A single graphon can also encode sequences of random graphs G(k, W)k≥1 and H(k, W)k≥1 with k nodes, obtained by sampling k uniform vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , xk in [0, 1], and either adding an edge of weight 1 between nodes i and j with probability W(xi, xj) (this is the sequence G(k, W)k≥1), or instead adding an edge of weight W(xi, xj) (this is the sequence H(k, W)k≥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We will deduce the following as a consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let W be a connected graphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Suppose that there exists δ > 0 such that the minimal degree of G(n, W) is at least δn with probability tending to 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For each n ≥ 1, let Tn be a uniform spanning tree of G(n, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Denote by dTn the corresponding graph-distance on Tn and by µn the uniform probability measure on the vertices of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then � Tn, √αW √n dTn, µn � (d) −→ (T , dT , µ) where (T , dT , µ) is the CRT equipped with its canonical mass measure µ and (d) −→ denotes convergence in distribution with respect to the GHP distance between metric measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Moreover, the same statement holds for H(n, W) in place of G(n, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For example, this applies to the Erd¨os-R´enyi sequence (G(n, p))n≥1 for any fixed p ∈ (0, 1], which is the sequence (G(n, W))n≥1 when W is the graphon that is p (almost) everywhere, and in which case αW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 shows that graphons contain enough information to determine the scaling limit of USTs, or in other words that the GHP scaling limit is continuous with respect to the topology induced by the cut-distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In [16], the authors show an analogous result for the Benjamini-Schramm local limit of the USTs appearing in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1, and show that the local limit can be characterized as a multi-type critical branching process conditioned to survive, where the offspring distributions are encoded by the limiting graphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Additionally, the authors show that continuity also holds for the total number of spanning trees of Gn, after being properly renormalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' However, they also give an example to show that this is no longer true under weaker assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that convergence of a graph sequence to a connected graphon does not automatically imply that the graph sequence must be dense, and in fact the local limit result for USTs of dense graphs obtained in [16] does not require this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' There, the authors assume only that the limiting graphon is non-degenerate, meaning that degW (x) := � [0,1] W(x, y)dy > 0 ∀x ∈ [0, 1], (2) and that the graph sequence is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In fact this implies that “most” vertices have high degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' see [16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='7 and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6] for a precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This is enough to prove a local limit statement since with high probability, the local limit will not see the exceptional vertices of low degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' On the other hand, the GHP scaling limit is a global statement and therefore we require more uniform control of the underlying graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' One can easily see this through a simple counterexample: let Gn denote the complete graph on n − n2/3 vertices, and attach a stick of length n2/3 to one vertex of the complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The graphs still converge to the graphon that is 1 everywhere, and the local limit of UST(Gn) is once again the Poisson(1) Galton–Watson tree conditioned to survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' On the other hand, the only non-trivial compact scaling limit is a single stick, and not the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' One can also construct similar counterexamples with minimum degree at least n γn for any sequence γn → ∞, meaning that the assumption of linear minimal degree is indeed necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Since the local limit of the CRT is well-known [1, Section 6] to be Aldous’ self-similar CRT (SSCRT), one can also ask whether the operations of taking scaling limits and local limits commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In general, answering this question seems quite non-trivial, as the multitype branching process appearing as the local limit is very non-homogeneous and the offspring distributions of successive generations are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' However, a special case arises when the sequence (Gn)n≥1 is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In this case the local limit is a Poisson(1) Galton– Watson tree conditioned to survive, which is well-known to rescale to the SSCRT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' moreover we will show in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 that the constant αW must be equal to 1, which entails that 1 αW is equal to the variance of the Poisson(1) offspring distribution, and from which we can deduce that the operations do indeed commute in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For non-regular graph sequences, the question seems a bit more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' While the expected number of non-backbone neighbours of the root vertex of the local limit is indeed 1, the variance is not necessarily equal to 1 αW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For example, for the complete bipartite graph K 2n 3 , n 3 , one can calculate using [16, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2] that the variance of the offspring number of the root vertex is equal to 3 2, but 1 αW is equal to 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This does not preclude the possibility that the operations commute, since the variance in subsequent generations may converge to 1 αW in the appropriate sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For K 2n 3 , n 3 we can in fact apply results of Miermont [28] (the local limit in this case is in fact a Galton–Watson tree with two alternating types: Poi(2) and Poi( 1 2)) to deduce that the operations do commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' However, in the general case the local limit is a Galton–Watson tree with uncountably many types, for which, to the best of our knowledge, scaling limits are not covered by the existing Galton–Watson tree literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Finally, we note that in [6], the authors consider similar dense graph sequences, but do not assume that the sequence converges to a graphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Under this weaker assumption, they prove that the diameter of UST(Gn) is of order √n with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We cannot hope to prove a scaling limit result under the same hypotheses, since one can, for example, connect two copies of Kn/2 by a single edge, in which case the diameter is still of order √n but the scaling limit is not the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' However, when the graphs are well-connected, we can obtain the scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In this paper we in fact prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In what follows, for a given γ > 0 we say that a graph G is a γ-expander if for all U ⊂ V (G), the number of edges between U and V (G) \\ U is at least γ|U|(|V (G)| − |U|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of connected γ-expanders, where each Gn has n vertices and minimal degree at least δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For each n ≥ 1, let Tn be a uniform spanning tree of Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Denote by dTn the corresponding graph-distance on Tn and by µn the uniform probability measure on the vertices of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then there exists a sequence (αn)n≥1, satisfying 1 ≤ αn ≤ δ−1 for all n ≥ 1, such that � Tn, √αn √n dTn, µn � (d) −→ (T , dT , µ) as n → ∞ where (T , dT , µ) is the CRT equipped with its canonical mass measure µ and (d) −→ denotes conver- gence in distribution with respect to the GHP distance between metric measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In fact the theorem holds slightly more generally, see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='4, but the above assumptions make the proof more straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Clearly one cannot hope for convergence of the parameter αn without making stronger assumptions, since one can alternate graphs from sequences with different limiting values of αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For example, for the sequence of complete graphs αn → 1, but if Gn is instead the complete bipartite graph K n 3 , 2n 3 , then αn → 9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' As well as the convergence of the rescaled diameter, it follows directly from the GHP convergence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 that we also have convergence of the rescaled height and rescaled simple random walk on UST(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' More formally, the following three convergences hold in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' √αn Diam(Tn) √n (d) → Diam(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' √αn Height(Tn) √n (d) → Height(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' If Xn is a simple random walk on Tn, then the quenched law of � √αn √n Xn(2n3/2α−1/2 n t) � t≥0 converges in distribution to the quenched law of Brownian motion on the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' It also follows that the associated mixing times converge on the same time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' See [7, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3] for further details of why these three properties follow from GHP convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In the settings of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2, we can replace αn with αW in the above three statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 Proof strategy Clearly, in order to prove the main theorems, it suffices to first prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 and then show that the graph sequence is an expander sequence and that αn → αW under the additional assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 in two steps using the lower mass bound criterion of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In particular, by [7, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='5], in order to prove the GHP convergence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 it is enough to prove the following two statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (A) The convergence holds in a finite-dimensional sense (this will be formally stated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (B) The lower mass bound condition holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' that is, if mn(η) = infx∈UST(Gn) � |B(x,η√n)| n � , then for every η > 0 the sequence mn(η)−1 is tight (this will be formally stated in Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The second condition will follow quite straightforwardly from minor adaptations of the arguments in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The bulk of this paper is devoted to proving the first condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In fact, this condition is equivalent to the joint convergence, for all k ≥ 1, of the set of �k 2 � distances between k points chosen uniformly at random in UST(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This type of convergence was previously proved for USTs of sequences of high-dimensional graphs in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This is a different class of graphs and includes the assumption of transitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Their proof uses Wilson’s algorithm, which is a method for sampling USTs one branch at a time by running loop erased random walks 4 (LERWs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In their proof, they couple Wilson’s algorithm on Gn with Wilson’s algorithm on the complete graph and prove that the set of �k 2 � distances on the two graphs must have the same scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Our proof, by contrast, is more direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We also use Wilson’s algorithm, but we work directly with UST(Gn) and use the Laplacian random walk representation of LERWs to sample each branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' By tightly controlling the capacity of loop-erased random walks, we are able to directly compute the probability that a given branch exceeds a given length, and show that this converges to the analogous quantity for the CRT using Aldous’ stick-breaking construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' As demonstrated by the examples and discussion above Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3, the assumption of linear minimal degree is necessary in order to obtain convergence in the GHP topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In order to keep the exposi- tion clean, we prove both conditions (A) and (B) above under these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' However, the assumption is not really necessary for condition (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The proof would work unchanged if we allow o(n) vertices to have degrees less than √n, for example (since the loop-erased random walk that we analyze in Section 5 will never hit this set, whp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In fact, we believe that it may be possible to adapt our proof of condition (A) (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1) to work under the original assumptions of [31], but this would require one to keep track of several additional messy details, and would not add further insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2 Organization of the paper This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In Section 2 we give the necessary background, including an introduction to graphons, USTs and the topologies of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In Section 3 we introduce a general framework for stick- breaking constructions of trees, and state Aldous’ stick-breaking construction of the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In Section 4 we give some precise random walk estimates and we apply these with the Laplacian random walk representation in Section 5 to obtain estimates for the first steps of Wilson’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In Section 6 we use these estimates to couple stick-breaking on the CRT with Wilson’s algorithm and prove that the two processes are very similar when n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This proves condition (A) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We also explain how (B) can be deduced from the results of [7] which in fact establishes Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Finally, in Section 7 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 Acknowledgments We would like to thank Asaf Nachmias and Jan Hladky for suggesting to look at graphons and for many helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This research is supported by ERC consolidator grant 101001124 (UniversalMap), and by ISF grant 1294/19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' EA was partially supported by the ANR ProGraM grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 Graphons A graphon is a symmetric measurable function [0, 1]2 → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' As mentioned in the introduction, graphons were introduced by Borgs, Chayes, Lov´asz, S´os, Szegedy and Vesztergombi [25, 10] in order to characterize dense graph limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' To understand why this definition is natural, we define the graphon representation of a discrete graph G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Suppose that G is a simple graph with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Number the vertices from v1 to vn and partition the interval [0, 1] into a sequence of intervals (Ii)n i=1, where Ii = � i−1 n , i n � for each 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We define the graphon WG : [0, 1]2 → [0, 1] by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' see [25, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1]) WG((x, y)) = 1{v⌈nx⌉∨1 ∼ v⌈ny⌉∨1} ∀ (x, y) ∈ [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' If G is a weighted graph, we instead define WG((x, y)) = w(v⌈nx⌉∨1, v⌈ny⌉∨1) ∀ (x, y) ∈ [0, 1]2, where w(vi, vj) represents the weight of the edge joining vi and vj (and is zero if there is no such edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 5 Note that, given only G, this definition of WG is not unique, since it depends on the ordering of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Therefore, in order to define a metric on the space of graphons, we will instead consider equivalence classes of graphons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In particular, given two graphons W1 and W2 the cut-distance between them is defined as (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' see [25, Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='16)]) δ□(W1, W2) = inf ϕ ||W ϕ 1 − W2||□, where the infimum is taken over all measure-preserving automorphisms of [0, 1], where W ϕ is defined by W ϕ(x, y) = W(ϕ(x), ϕ(y)), and where the cut-norm of a measurable function U : [0, 1]2 → [−1, 1] is given by ||U||□ = sup S,T ∈B([0,1]) ���� � x∈S � y∈T U(x, y)dxdy ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We therefore say that a sequence of deterministic graphs (Gn)n≥1 converges to a graphon W if δ□(WGn, W) → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Graphons can in fact be defined as functions from Ω2 → [0, 1], where Ω is any probability space, see [25, Chapter 13], but since all probability spaces are isomorphic, this does not provide much greater generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We will make use of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' [9, Lemma 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let W be a connected graphon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, for every α ≤ 1/2 there exists some constant β = β(W, α) such that for every set A with α ≤ µ(A) ≤ 1/2 we have � A � AC W(x, y)dxdy > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 Random graphs and graphons A graphon W can be used to define a random graph with n vertices in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Sample x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , xn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' uniformly on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We define a random simple graph on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , n} by joining nodes i and j with probability W(xi, xj), independently for each (unordered) pair (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We denote the resulting random graph G(n, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Sample x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , xn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' uniformly on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We define a random weighted graph on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , n} by adding an edge between i and j of weight W(xi, xj) for each (unordered) pair (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We denote the resulting random graph H(n, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In both constructions, note that we can use a single graphon to define a whole sequence of random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The following lemma tells that in either case, the cut-distance between a random sample of G(k, W) or H(k, W) and W goes to zero w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' [25, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Fix a graphon W and for k ≥ 1, let G(k, W) and H(k, W) be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, δ□(WG(k,W), W) and δ□(WH(k,W), W) both tend to 0 in probability as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In particular this means that results we prove for USTs of deterministic sequences of graphs extend automatically to sequences of the form G(k, W)k≥1 or H(k, W)k≥1 under the assumptions of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For example, the classical Erd¨os-R´enyi graphs G(n, p) for n ≥ 1, p ∈ [0, 1] correspond to the graphs G(n, Wp) where Wp is the graphon that is equal to p everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For further background and applications of graphons, we refer to [25, Part 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2 Mixing times Let G be a connected weighted graph with n vertices, with weights (w(x, y))x,y∈V (G), and with no loops or multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' A random walk on G is the Markov Chain (Xm)m≥0 such that, for all vertices x, y ∈ V (G), and all m ≥ 1, P(Xm = y | Xm−1 = x) = w(x, y) � z∼x w(x, z), where z ∼ x means that z is a neighbour of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Due to periodicity considerations, it is sometimes more convenient to instead use the notion of a lazy random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This is defined by P(Xm = y | Xm−1 = x) = w(x, y) 2 � z∼x w(x, z) ∀y ∼ x and P(Xm = x | Xm−1 = x) = 1 2 for all m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For each t ≥ 0 let pt denote the t-step transition density of a lazy random walk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' pt(x, y) = P(Xt = y | X0 = x) for all x, y ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We define the mixing time of G as tmix(G) = min � t ≥ 0 : max x,y∈G|pt(x, y) − π(x)| ≤ 1 4 � , (3) (see [24, Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='31)]), where π denotes the stationary measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We will also need the notion of total variation distance between two probability measures on µ and ν on a finite subset X ⊂ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This is defined by dTV(µ, ν) = max A⊂X |µ(A) − ν(A)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Furthermore, by [24, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='5], we have for any k ≥ 1, any t ≥ ktmix and any vertex x that dTV(pt(x, ·), π(·)) ≤ 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 Expanders We will use the following definition of an expander graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' ([16, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For any γ > 0, a loopless weighted graph G is a γ-expander if for all U ⊂ V (G), we have that w(U, V (G) \\ U) ≥ γ|U|(V (G) − |U|) where w(A, B) = � v∈A,u∈B w(v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Although we give the definition for loopless graphs, note that adding loops to a graph does not change the law of its UST, since loops can never appear in a UST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that often in the literature a slightly different definition of expander is used, involving the Cheeger constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We are using the definition above as it fits more naturally into the framework of dense graphs (as we will later show in Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1) and is the same definition used to consider the local limit in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The main property of expanders that we will use is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let γ > 0 and let G be a γ-expander with n ≥ 2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, provided that n is large enough (depending on only γ), we have that tmix(G) ≤ 64 γ4 log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that it follows from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='4 that G has minimal degree at least γ 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' First note that by [24, Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='4] that tmix(G) ≤ trel log �8n γ � , 7 where trel is the relaxation time of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' By the Cheeger inequality (see [4, 5, 19, 22] for various proofs), 1 trel is lower bounded by Φ(G)2/2, where Φ(G) = min S⊂V (G),π(S)≤1/2 w(S, V (G) \\ S) � v∈S deg v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that π(S) ≤ 1/2 implies that (|V (G)| − |S|) ≥ nγ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Since G is a γ-expander, it follows that Φ(G) ≥ γ|S|(|V (G)| − |S|) � v∈S deg v ≥ γ|S|(|V (G)| − |S|) |S|n ≥ γ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Combining all the inequalities gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='4 Loop-erased random walk and Wilson’s algorithm We now describe Wilson’s algorithm [32] which is a widely-used algorithm for sampling USTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' A walk X = (X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' XL) of length L ∈ N is a sequence of vertices where (Xi, Xi+1) ∈ E(G) for every 0 ≤ i ≤ L − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For an interval J = [a, b] ⊂ [0, L] where a, b are integers, we write X[J] for {Xi}b i=a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Given a walk X, we define its loop erasure Y = LE(X) = LE(X[0, L]) inductively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We set Y0 = X0 and let λ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, for every i ≥ 1, we set λi = 1 + max{t | Xt = Yλi−1} and if λi ≤ L we set Yi = Xλi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We halt this process once we have λi > L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' When X is a random walk on the weighted graph G starting at some vertex v and terminated when hitting another vertex u (L is now random), we say that LE(X) is a loop erased random walk (LERW) from v to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' To sample a UST of a finite connected weighted graph G we begin by fixing an ordering of the vertices of V = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' First let T1 be the tree containing v1 and no edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, for each i > 1, sample a LERW from vi to Ti−1 and set Ti to be the union of Ti−1 and the LERW that has just been sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We terminate this algorithm with Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Wilson [32] proved that Tn is distributed as UST(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' An immediate consequence is that the path between any two vertices in UST(G) is distributed as a LERW between those two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This was first shown by Pemantle [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='5 Laplacian random walk Here we outline the Laplacian random walk representation of the LERW (see [26, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1] for full details) and its application to Wilson’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take a finite, weighted, connected graph G and suppose we have sampled Tj for some j ≥ 1 using Wilson’s algorithm as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We now sample a LERW from vj+1 to Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Denote this LERW by (Ym)m≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Also let X denote a random walk on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For a set A ⊂ G, let τA denote the hitting time of A by X, and τ + A denote the first return time to A by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The Laplacian random walk representation of Y says that, conditionally on Tj and on the event {(Ym)i m=0 ∩ Tj = ∅}, we have for any i ≥ 0 that P � Yi+1 = v �� (Ym)i m=0 � = PYi � X1 = v ��� τTj < τ+ ∪i m=0{Ym} � = PYi(X1 = v)Pv � τTj < τ∪i m=0{Ym} � PYi � τTj < τ + ∪i m=0{Ym} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Clearly this is only non-zero when v /∈ �i m=0{Ym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We can now extrapolate this to ask about the law of (Ym)i+H m=i+1 for some H ≥ 1, given (Ym)i m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In particular, if u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , uH is a simple path in Gn, where {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , uH−1} is disjoint from �i m=0{Ym} ∪ Tj and u0 = Yi, then P � (Ym)i+H m=i+1 = (um)H m=1 �� (Ym)i m=0 � = Pu0 � (Xm)H m=1 = (um)H m=1 � C((Ym)i m=0, Tj, (um)H m=1)), where C((Ym)i m=0, Tj, (um)H m=1) = H � h=1 Puh � τTj < τ∪i m=0{Ym}∪ ∪h−1 m=1{um} � Puh−1 � τTj < τ+ ∪i m=0{Ym}∪ ∪h−1 m=1{um} �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6 Capacity and closeness Recall that G is a connected weighted graph with n vertices with minimal degree at least δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The capacity of a set of vertices of G quantifies how difficult it is for a random walk to hit the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let (Xi)i≥0 be a random walk on G and for U ⊂ V (G), let τU = inf{i ≥ 0 : Xi ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Given k ≥ 0 we define the k-capacity of U by Capk(U) = Pπ(τU ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Here we collect some useful facts about the capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let A ⊂ V (G) and k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then Capk(A) ≤ kπ(A) ≤ k|A| δn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (5) Moreover, if k|A| ≤ δ3n 2 , then Capk(A) ≥ kπ(A) 2 ≥ δk|A| 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The upper bound follows from a union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The lower bound follows from the Bonferroni inequal- ities and the lower bound on the degree, which imply that Capk(A) ≥ kπ(A) − �k|A| δn �2 ≥ δk|A| 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We will also use the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let tmix = tmix(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let A ⊂ V (G), let M ≥ (log n)2tmix and suppose that (log n)2 · tmix|A| ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, provided n is large enough, sup u∈V (G)\\A |Pu(τA ≤ M) − CapM(A)| ≤ 3 log n · tmix|A| δn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let X be a random walk started at u ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Clearly, for any t ≥ 0, the first t steps of X can be coupled with the first t non-repeat steps of a lazy random walk ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Therefore, first run a lazy random walk started from u until time T = 2 log n · tmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let N denote the total number of non-repeat jumps of this lazy random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The distribution of ˜Xt is almost stationary by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Moreover, we have that 0 ≤ N ≤ T deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' To sample (Xt)M t=0, we first couple it with the first N steps of ( ˜Xt)T t=0 as explained above, and then run X for a further M − N steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Under this coupling, we therefore have from a union bound that Pu(τA ≤ M) ≤ 2 log n · tmix|A| δn + Pπ(τA ≤ M) + 2−2 log n ≤ CapM(A) + 3 log n · tmix|A| δn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Similarly, Pu(τA ≤ M) ≥ Pπ(τA ≤ M − T ) − 2−2 log n ≥ CapM(A) − 3 log n · tmix|A| δn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In order to obtain lower bounds on capacity, we define the k-closeness of two sets U and W by Closek(U, W) = Pπ(τU < k, τW < k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (7) Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For any disjoint sets U, W ⊂ G, we have that sup v∈G\\(U∪W) Pv(τU < k, τW < k) ≤ 2k2|U||W| δ2n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In particular, Closek(U, W) ≤ 2k2|U||W| δ2n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that sup v∈G\\(U∪W) Pv(τU < k, τW < k) ≤ sup v∈G\\(U∪W) {Pv(τU < τW < k) + Pv(τW < τU < k)} ≤ sup v∈G\\(U∪W),u∈U,w∈W {Pv(τU < k)Pu(τW < k) + Pv(τW < k)Pw(τU < k)} ≤ 2k2|U||W| (δn)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='7 Random variables Here we present two elementary results that will be useful in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let ε > 0 and let 0 < a < b with b − a ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let Xa ∼ U([0, a]) and Xb ∼ U([0, b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, we can couple Xa and Xb such that P(|Xa − Xb| > ε) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We take Xb = b aXa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, |Xb − Xa| = | b−a a Xa| ≤ |b − a| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For any L > 0, let XL be the random variable on (0, ∞) satisfying P(XL > x) = exp � −(x + L)2 − L2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then for any δ > 0, there exists η = η(δ, L) > 0 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let Y be another random variable on (0, ∞), and suppose that for all x > 0, |P(XL > x) − P(Y > x)| < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (8) Then this implies that we can couple XL and Y so that P(|XL − Y | > δ) < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Furthermore, for any δ, L1 and L2 with L1 < L2, there exists η = η(δ, L1, L2) such that we can couple XL and Y as described above for every L ∈ [L1, L2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that we can couple XL and Y by first sampling U ∼ Uniform([0, 1]) and setting XL(ω) = sup x≥0 {P(XL ≥ x) ≥ U(ω)}, Y (ω) = sup x≥0 {P(Y ≥ x) ≥ U(ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Now choose Kδ,L < ∞ so that P(XL ≥ Kδ,L) < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Wlog assume that δ < 1 and Kδ,L > 1, otherwise decrease or increase them if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that, for all 0 ≤ x < Kδ,L, we have that exp � −(x + L)2 − L2 2 � − exp � −(x + δ + L)2 − L2 2 � ≥ δ(x + L) exp � −(x + δ + L)2 − L2 2 � ≥ Mδ,L, where Mδ,L = δL exp � − (2Kδ,L+L)2−L2 2 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Now suppose that (8) holds and η < Mδ,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, for any 0 ≤ x < Kδ,L we have that P(Y ≥ x + δ) ≤ P(XL ≥ x + δ) + η ≤ P(XL ≥ x) − Mδ,L + η ≤ P(XL ≥ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Therefore, under the coupling, we have for any x < Kδ,L that {XL ≤ x} ⇔ {P(XL ≥ x) ≤ U} ⇒ {P(Y ≥ x + δ) ≤ U} ⇔ {Y ≤ x + δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Similarly, {XL ≥ x} ⇒ {Y ≥ x − δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Therefore, under this coupling we have that P(|XL − Y | > δ) ≤ P(XL ≥ Kδ,L) < δ, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For the second claim, note that for every L′ > L we also have that P(XL′ ≥ Kδ,L) < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Therefore for the interval [L1, L2] we can simply use Kδ,L1 and Mδ,L1 on the whole interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='8 GHP topology Here we define the GHP topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We use the framework of [29, Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 and 6] and work in the space Xc of equivalence classes of metric measure spaces (mm-spaces) (X, d, µ) such that (X, d) is a compact metric space and µ is a Borel probability measure on it, and we say that (X, d, µ) and (X′, d′, µ′) are equivalent if there exists a bijective isometry φ : X → X′ such that φ∗µ = µ′ (here φ∗µ is the pushforward measure of µ under φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' To ease notation, we will represent an equivalence class in Xc by a single element of that equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' First recall that if (X, d) is a metric space, the Hausdorff distance dH between two sets A, A′ ⊂ X is defined as dH(A, A′) = max{sup a∈A d(a, A′), sup a′∈A′ d(a′, A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For ε > 0 and A ⊂ X we also let Aε = {x ∈ X : d(x, A) < ε} be the ε-fattening of A in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' If µ and ν are two measures on X, the Prohorov distance between them is given by dP (µ, ν) = inf{ε > 0 : µ(A) ≤ ν(Aε) + ε and ν(A) ≤ µ(Aε) + ε for any closed set A ⊂ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let (X, d, µ) and (X′, d′, µ′) be elements of Xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The Gromov-Hausdorff-Prohorov (GHP) distance between (X, d, µ) and (X′, d′, µ′) is defined as dGHP((X, d, µ), (X′, d′, µ′)) = inf{dH(φ(X), φ′(X′)) ∨ dP (φ∗µ, φ′ ∗µ′)}, where the infimum is taken over all isometric embeddings φ : X → F, φ′ : X′ → F into some common metric space F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Recall that our aim in this paper is to prove distributional convergence with respect to the GHP topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Given an mm-space (X, d, µ) and a fixed m ∈ N we define a measure νm((X, d, µ)) on R( m 2) to be the law of the �m 2 � pairwise distances between m i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' points drawn according to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Each law P on Xc therefore defines random measures (νm)m≥2 and annealed measures (˜νm)m≥2 on R( m 2), given by ˜νm(P) := � Xc νm((X, d, µ))dP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In [7] we rephrased a result of [8, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1] in the distributional setting to characterize GHP conver- gence in terms of convergence of the measures (˜νm)m≥2 and a volume condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' To state the version that we will use in this paper, given c > 0 and an mm-space (X, d, µ) we define mc((X, d, µ)) = inf x∈X{µ(B(x, c))} (cf [8, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In the proof of the next proposition we will also make reference to the (coarser) Gromov-Prohorov topology, which is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let (X, d, µ) and (X′, d′, µ′) be elements of Xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The Gromov-Prohorov (GP) distance between (X, d, µ) and (X′, d′, µ′) is defined as dGP((X, d, µ), (X′, d′, µ′)) = inf{dP (φ∗µ, φ′ ∗µ′)}, where the infimum is taken over all isometric embeddings φ : X → F, φ′ : X′ → F into some common metric space F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The key result is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let (X, d, µ) be an element of Xc with law P such that µ has full support almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let ((Xn, dn, µn))n≥1 be a sequence in Xc with respective laws (Pn)n≥1 and suppose that: 11 (a) For all m ≥ 0, ˜νm(Pn) → ˜νm(P) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (b) For any c > 0, the sequence � mc((Xn, dn, µn))−1� n≥1 is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then (Xn, dn, µn) (d) → (X, d, µ) with respect to the GHP topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' First we show that part (a) and (b) together imply that (Xn, dn, µn) (d) −→ (X, d, µ) with respect to the GP topology, by verifying the two conditions of [15, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The second condition of [15, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1] is precisely (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' To verify the first condition we further use [15, Theorem 3] (recall that by Prohorov’s Theorem the relative compactness of the measures is equivalent to their tightness) and verify conditions (i) and (ii) there (see also Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 in [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Condition (i) is just saying that ˜ν2 is a tight sequence of measures on R, which follows from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Lastly, (b) directly implies condition (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Therefore, by [7, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='5], the spaces convergence with respect to the GHP topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 3 Stick-breaking construction of trees Our first goal will be to prove condition (a) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='13 which is equivalent to the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of γ-expanders, where each Gn has n vertices and minimal degree at least δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Denote by dTn the graph distance on Tn and by (T , d, µ) the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then there exists a sequence (βn)n≥1, satisfying √ δ ≤ βn ≤ 1 for all n ≥ 1, such that for any fixed k ≥ 1, if {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , xk} are uniformly chosen independent vertices of Gn, then the distances dTn(xi, xj) βn √n converge jointly in distribution to the �k 2 � distances in T between k i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' points drawn according to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' To prove this theorem, we will use Aldous’ stick-breaking construction of the CRT which is particularly well adapted to dealing with the pairwise distances between a set of k uniform points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Our strategy will be to show that the first k steps of Wilson’s algorithm on Gn closely approximate those of this stick-breaking process when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In this section we briefly recall the stick-breaking construction of the CRT and some of its key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We start with a more general description of how one can construct a sequence of trees from sticks on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (Stick-breaking construction of a tree sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Set y0 = z0 = 0, and suppose that we have a sequence of points y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' ∈ [0, ∞) and z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' ∈ [0, ∞) such that yi−1 < yi and zi ≤ yi for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Construct trees as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Start by taking the line segment [y0, y1) at time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This is T (2) (as it contains two marked points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We proceed inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' At time i ≥ 2, take the interval [yi−1, yi) and attach the base of the interval [yi−1, yi) to the point on T (i) corresponding to zi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' This gives a new tree with i + 1 marked points (in bijection with the set (yj)i j=0), which we call T (i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Given two such sequences and any k ≥ 2 we define SB(k)((y0, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' ), (z0, z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=')) or equivalently SB(k)((y0, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , yk−1), (z0, z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' , zk−2)) to be equal to the tree T (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In general, the sequence of trees constructed in this way above may not converge, but Aldous showed that by choosing the points in the right way, we can in fact construct the CRT via stick-breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' [1, Process 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Set Y0 = Z0 = 0, let (Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=') denote the ordered set of points of a non-homogeneous Poisson process on [0, ∞) with intensity t dt, and let Zi be chosen uniformly on the interval [0, Yi) for each i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Construct the sequence (T (k))∞ k=2 as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then the closure of the limit of T (k) is equal in distribution to the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Moreover, if one stops the process after k − 1 steps, then the resulting tree T (k) has the same distribution as the subtree spanned by k uniform points in the CRT, and the points corresponding to the set (Yi)k−1 i=0 can be identified with k uniform points in the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 12 In particular, the set of �k 2 � pairwise distances between points corresponding (Yi)k i=1 is equal in distribution to the set of �k 2 � pairwise distances between k uniform points in the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The following proposition will be important for the comparison with Wilson’s algorithm later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' It can be verified by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Define the sequence (Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=') as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then for any k ≥ 1 and any x ≥ 0, P � Yk+1 − Yk ≥ x �� (Yi)k i=0 � = exp � −1 2 � (Yk + x)2 − Y 2 k �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The following lemma will also be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' There exists a function f : [0, ∞) × N → [0, 1] such that for every k ∈ N we have that limC→∞ f(C, k) → 0, and such that if Yk is as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3, then P � C−1 ≤ Yk ≤ C � ≥ 1 − f(C, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let (y0, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' ), (z0, z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=') and (y′ 0, y′ 1, y′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' ), (z′ 0, z′ 1, z′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=') be the inputs to two separate stick-breaking processes as defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Fix any k ≥ 1 and let T (k+1) and T (k+1)′ be the trees formed after k steps of the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let d and d′ denote distances on T (k+1) and T (k+1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Fix some ε > 0 and suppose that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (i) |yi − y′ i| ≤ ε for all i ≤ k and |zi − z′ i| ≤ ε for all i ≤ k − 1, (ii) |zi − yj| ≥ 3ε for all i ≤ k − 1, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, for all 0 ≤ i, j ≤ k, it holds that |d(yi, yj) − d′(y′ i, y′ j)| ≤ 2kε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' When conditions (i) and (ii) hold, we have for all i ≤ k − 1, j ≤ k that yj ≤ zi ≤ yj+1 if and only if y′ j ≤ z′ i ≤ y′ j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We claim that this implies that |d(yi, yj) − d′(y′ i, y′ j)| ≤ 2kε for all i, j ≤ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Indeed, it follows by construction that d(yi, yj) is the sum of lengths of at most k branch segments in T (k+1), and all of their lengths can be written in the form |yj − yj−1|, |zj − yℓ| or |zj − zℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Moreover, by construction, when the conditions (i) and (ii) hold, d′(y′ i, y′ j) can be written as the same sum but replacing each zj with z′ j and replacing each yj with y′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' It therefore follows from the triangle inequality that |d(yi, yj)−d′(y′ i, y′ j)| ≤ 2kε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 4 Random walk properties In this section we prove some results on random walk hitting probabilities and capacity, which we will later transfer to segments of LERW using the Laplacian random walk representation of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Throughout the section we fix a small κ ∈ (0, 1 32) and for n ≥ 1 we set Mn = nκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In what follows we will simply write M instead of Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Notational remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For the statements in this section, we will take a sequence of graphs satisfying the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3 which is therefore associated with two positive constants γ > 0 and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In this section we will treat these constants as fixed, and therefore o(·) and O(·) quantities may also depend on γ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 Hitting probabilities We start with some results on hitting probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let X be a (non-lazy) random walk on Gn for some n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For a set A ⊂ V (Gn), we define τA = inf{t ≥ 0 : Xt ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The main lemma is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of γ-expanders, where each Gn has n vertices and minimal degree at least δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take κ and M as defined at the start of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then there exists a sequence (ηn)n≥1 with ηn → 0, depending only on δ and γ, such that for any disjoint A, B ⊂ Gn satisfying |A| + |B| ≤ δ3 2 n 1 2 +2κ: ����Pπ(τA < τB) − CapM(A) CapM(A) + CapM(B) ���� ≤ CapM(A)ηn CapM(A) + CapM(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let (Xi)M i=1 be a random walk of length M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, by Bayes’ formula, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='8 and the lower bound in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6, Pπ(τA < M | τA ∧ τB < M) = CapM(A) CapM(A) + CapM(B) − Pπ(τA ∨ τB < M) = CapM(A) CapM(A) + CapM(B) � 1 + O �δ−3M|B| n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Pπ(τA ∨ τB < M | τA ∧ τB < M) = Pπ(τA ∨ τB < M) CapM(A) + CapM(B) − Pπ(τA ∨ τB < M) ≤ Pπ(τA < M | τA ∧ τB < M)O �δ−3M|B| n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Therefore, combining these and applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6: Pπ(τA < τB | τA ∧ τB < M) = Pπ(τA < M | τA ∧ τB < M) + O(Pπ(τA ∨ τB < M | τA ∧ τB < M)) = CapM(A) CapM(A) + CapM(B) � 1 + O �δ−3M|B| n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' It similarly follows from Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='7 and the lower bound in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6 that uniformly over all u ∈ Gn\\(A∪B), Pu(τA < τB | τA ∧ τB < M) = CapM(A) CapM(A) + CapM(B) � 1 + O �δ−3M|B| n + tmix · log n δ2M �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Now we decompose time into intervals of length M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For each i ≥ 1, define the interval Ai by Ai = [iM, (i + 1)M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We then have that, using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='8: Pπ(τA < τB) ≥ ∞ � i=0 Pπ(τA < τB | τA∪B ∈ Ai)Pπ(τA∪B ∈ Ai) ≥ ∞ � i=0 inf u∈Gn\\(A∪B) Pu(τA < τB | τA∪B ∈ A0)Pπ(τA∪B ∈ Ai) ≥ CapM(A) CapM(A) + CapM(B) � 1 + O �δ−3|B|M n + tmix · log n δ2M �� We deduce that, uniformly over all permitted A and B, Pπ(τA < τB) ≥ CapM(A) CapM(A) + CapM(B)(1 − oδ,γ(1)), (9) 14 where the oδ,γ(1) term is uniform over all A and B but may depend on δ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Similarly, for an upper bound on Pπ(τA < τB) we simply exchange the roles of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We deduce that, uniformly over all permitted A and B, Pπ(τA < τB) = CapM(A) CapM(A) + CapM(B)(1 − oδ,γ(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (10) We will also need the following minor adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of γ-expanders, where each Gn has n vertices and minimal degree at least δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take κ and M as defined at the start of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, for any disjoint A, B ⊂ Gn satisfying |A| + |B| ≤ δ3 2 n 1 2 +2κ, every u ∈ Gn \\ (A ∪ B) and every v ∈ Gn \\ A we have that Pu(τA < τB) = Pπ(τA < τB)(1 + o(n3κ−1/2t+ mix)) and Pv � τA < τ + B � = Pπ(τA < τB)(1 + o(n3κ−1/2t+ mix)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We start by proving the first statement for a lazy random walk, since this is equivalent, and we denote such a lazy random walk by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Throughout this proof, we will also use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For a set C ⊂ Gn and some time t ≥ 0 we write τ(C, t) for the first time s strictly larger than t such that Xs ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Furthermore, write t+ mix for log2 2(n)tmix so that by (4) we have that for every u ∈ Gn, dTV(pt+ mix(u, ·), π(·)) ≤ n− log(2) log(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Now let u ∈ Gn \\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We start with a lower bound on Pu(τA < τB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' We have that Pu(τA < τB) ≥ Pu � t+ mix < τA < τB � ≥ Pu � τ(A, t+ mix) < τ(B, t+ mix) � − Pu � τA∪B < t+ mix < τ(A, t+ mix) < τ(B, t+ mix) � (11) Note that by (4), the first term can be lower bounded by Pπ(τA < τB) − n− log(2) log(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For the second term, let us upper bound the probability of the event {τA∪B < t+ mix < τ(A, tmix) < τ(B, t+ mix)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Using a union bound we obtain Pu � τA∪B < t+ mix < τ(A, t+ mix) < τ(B, t+ mix) � ≤ Pu � τA∪B < t+ mix < τ(A, t+ mix) < 2t+ mix � + Pu � τA∪B < t+ mix and τ(A, 2t+ mix) < τ(B, 2t+ mix) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' ≤ |A|t+ mix δn + (|A| + |B|)t+ mix δn (Pπ(τA < τB) + n− log(2) log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='1 we have that |A|t+ mix Pπ(τA < τB)δn ≤ |A|t+ mix δn 2(|A| + |B|) δ2|A| ≤ 2(|A| + |B|)t+ mix δ3n , so that, by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='5 |A|t+ mix δn = Pπ(τA < τB) · Oγ(n2κ−1/2t+ mix) = Pπ(τA < τB) · oγ(n3κ−1/2t+ mix) (12) Substituting everything back into (11), we therefore deduce that Pu(τA < τB) ≥ Pπ(τA < τB)(1 + o(n3κ−1/2t+ mix)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For an upper bound on Pu(τA < τB), we simply write Pu(τA < τB) ≤ Pu � τA < t+ mix � + Pu � t+ mix < τ(A, t+ mix) < τ(B, t+ mix) � ≤ |A|t+ mix δn + Pπ(τA < τB) + n− log(2) log(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Using (12) again we obtain that Pu(τA < τB) = Pπ(τA < τB)(1 + o(n3κ−1/2t+ mix)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For the second statement, it is again enough to prove it for the lazy random walk, replacing τ + B with the first hitting time of B after making at least one non-lazy step using the exact same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2 Capacity Here we prove some similar properties for the capacity and closeness of a random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In this section we can also introduce the sequence (αn)n≥1 appearing in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Given the graph sequence (Gn)n≥1, take M = nκ as defined at the start of Section 4, let X be a random walk on Gn, and for each n ≥ 1 set αn = nEπ � CapM(X[0, nκ/2)) � Mnκ/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take γ > 0 and δ > 0 and let (Gn)n≥1 be a dense sequence of γ-expanders, where each Gn has n vertices and minimal degree at least δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let u ∈ Gn and let (Xi)i≥0 denote a random walk on Gn started at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Take M = nκ as defined at the start of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then for all sufficiently large n, P �����CapM(X[0, M)) − αnM 2 n ���� ≥ αnM 2 n n−κ/16 � ≤ 2M 2 δn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' The proof is a simplified version of that of [31, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' First recall from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='2 that t+ mix = log2 2(n)tmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Let (Tj)nκ/2 j=1 be a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='d random variables with distribution Bin(t+ mix, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Then, for all 1 ≤ j ≤ nκ/2 let Bj = [(j − 1)nκ/2 + Tj, jnκ/2 − t+ mix + Tj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that by (4) we have that for all j ≤ nκ/2, given X[0, jnκ/2], the starting point of Bj+1 is nearly stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Also let (Xind,j)nκ/2 j=1 denote a sequence of independent random walk segments each of length nκ/2 − t+ mix, and each started from stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' Note that, by (4), the segments (XBj)nκ/2 j=1 can be coupled with the segments (Xind,j)nκ/2 j=1 so that the segments coincide for all j ≤ nκ/2 with probability at least 1 − nκ/2n− log2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (13) Note that the segments (Xind,j)j are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' and, by definition, E � CapM(Xind,j) � = E � CapM(Xind,j [0,nκ/2)) � + O �Mt+ mix δn � = αnMnκ/2 n � 1 + O � t+ mix δnκ/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (14) Moreover, by a union bound, we also have the deterministic bound CapM(Xind,j) ≤ Mπ(Xind,j) ≤ Mnκ/2 δn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (15) It therefore follows from a Hoeffding bound [17, Theorem 1] that there exist C < ∞, c > 0 such that for any t > 0, P \uf8eb \uf8ed ������ nκ/2 � j=1 CapM(Xind,j) − nκ/2E � CapM(Xind,1) � ������ ≥ αnM 2t+ mix 2n1+κ/8 \uf8f6 \uf8f8 ≤ 2 exp \uf8eb \uf8ec \uf8ed−2nκ/2 \uf8eb \uf8ed αnM2t+ mix 2n1+5κ/8 Mnκ/2 δn \uf8f6 \uf8f8 2\uf8f6 \uf8f7 \uf8f8 = 2 exp � −nκ/4(t+ mix)2 α2 nδ2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' In particular, since it follows from (14) that ����nκ/2E � CapM(Xind,1) � − αnM 2 n ���� ≤ O �αnMnκ/2 n t+ mix δ � ≪ αnM 2t+ mix n1+κ/8 , we deduce that P \uf8eb \uf8ed ������ nκ/2 � j=1 CapM(Xind,j) − αnM 2 n ������ ≥ αnM 2t+ mix n1+κ/8 \uf8f6 \uf8f8 ≤ 2 exp � −nκ/4(t+ mix)2 α2 nδ2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' (16) 16 We would like to approximate the capacity of the whole segment X[iM, (i+1)M) by the sum of the capacities of the smaller segments, but this is potentially a slight overestimate, since we are double-counting random walk trajectories that hit more than one smaller segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' To account for this, we use the concept of closeness defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content=' For each J ≤ nκ/2, note that conditionally on (Xind,j)j≤J all being disjoint, which happens with probability at least 1 − M 2 δn , (17) we have by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfl_gG/content/2301.00461v1.pdf'} +page_content='8 that CloseM(Xind,J, ∪j n, the vn−sG;E-periodic ring spectrum denotes the contractible spectrum ∗. We call sG;E +blue-shift number. +As far as we know, classical blue-shift phenomenon was discovered by Davis and Mahowald +[11] in 1984. They found that if G is a cyclic group of order 2, denoted by Z/2, then the construc- +tion tZ/2(infZ/2 +e +(−))Z/2 maps the v1-periodic 2-local ring spectrum bu, which denotes the connected +complex K-theory, to a v0-periodic spectrum K(Z2) which denotes the Eilenberg-Maclane spec- +trum for 2-adic integer. And they conjectured an extended result in which bu is replaced by the +spectrum BP⟨n⟩ of [22] and K(Z2) is replaced by BP⟨n − 1⟩. Let K(n) denote the n-th Morava K- +theory, then in 1994 Greenlees and Sadofsky [16, Theorem 1.1] found that tG(infG +e (K(n)))G ≃ ∗ for +any p-group G. In 1996, Hovey and Sadofsky [20] discovered that when G = Z/p, E is vn-periodic +and Landweber exact2, blue-shift number sZ/p;E is 1 for any prime p. In 1998, Ando-Morava- +Sadofsky [1] confirmed that Davis and Mahowald’s conjecture is true. Let T(n) be the telescope of +any vn-self map of a complex of type n3, then in 2004 Kuhn [24] proved that tG(infG +e (T(n)))G ≃ ∗ +for any p-group G. +For a finite group G, let SH(G) denote the G-equivariant stable homotopy category and +SH(G)c denote its full subcategory that consists of all compact objects of SH(G). +In 2017, +Balmer and Sanders [6] showed that classical blue-shift phenomenon, namely Conjecture 1.1, +for G = Z/p is closely related to the Zariski topology of Balmer spectrum Spc(SH(Z/p)c) of +SH(Z/p)c, which is a Z/p-equivariant analog of Devinatz-Hopkins-Smith’s work [10, 18]. To +compute the Zariski topology of Balmer spectrum Spc(SH(G)c), they proposed a new construc- +tion that replaces the functor (−)G in the construction tG(infG +e (−))G of classical blue-shift phe- +nomenon by the geometric fixed point functor ΦG(−), hence a new blue-shift phenomenon. +In 2019, Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton [7] obtained the Zariski topol- +ogy of Spc(SH(A)c) for an abelian group A by studying this new blue-shift phenomenon. To +unify classical blue-shift phenomenon and new blue-shift phenomenon to one framework, we +propose a general blue-shift phenomenon. To be precise, let N be a normal subgroup of G and +˜ΦN be the relative geometric N-fixed point functor from SH(G) to SH(G/N), then we consider a +more general functor ( ˜ΦN(tG(infG +e (−))))G/N which is obtained by replacing (−)G in the construc- +tion tG(infG +e (−))G by the functor ( ˜ΦN(−))G/N. For convenience, let TG,N(−) denote the functor +( ˜ΦN(tG(infG +e (−))))G/N : SH(e) → SH(e) from non-equivariant spectra to itself. In this paper, +we call TG,N(−) the generalized Tate construction for non-equivariant spectra. And for a non- +equivariant spectrum E, we call TG,N(E) the generalized Tate spectrum of E. Then the general +blue-shift phenomenon can be stated as follows. +1Usually vn-periodic means that vn is a unit in the homotopy ring π∗(E), but in this paper, we choose a less restrictive +definition 5.5 due to Hovey [21]. +2Details see [27] or Proposition 5.7. +3Details see Subsection 5.1. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +3 +Conjecture 1.2. (General blue-shift phenomenon) The functor TG,N(−) maps a vn-periodic +spectrum E to a vn−sG,N;E-periodic spectrum TG,N(E). In other words, this generalized Tate con- +struction reduces chromatic periodicity. +Remark 1.3. +(i) When N = G, TG,N(−) is the construction ΦG(tG(infG +e (−))) in Balmer and +Sanders’s new blue-shift phenomenon, details see Proposition 3.1. +(ii) When the family subgroups of G which do not contain N are {e}, one special case is that +G = Z/pj and N = Z/p, TG,N(−) is the construction tG(infG +e (−))G in classical blue-shift +phenomenon, details see Proposition 3.2. +The goal of this paper is to study this general blue-shift phenomenon, namely Conjecture 1.2. +And our main idea to explain this phenomenon is that since the homotopy group π∗(TG,N(E)) of +generalized Tate spectrum TG,N(E) is a graded ring, it must be isomorphic to a quotient of a free +graded ring by some relations. And we may reduce these relations like solving equations to obtain +vn−sG,N;E, then we need to prove the solution of vn−sG,N;E is invertible in π∗(TG,N(E)). +Inspired by Hopkins-Kuhn-Ravenel’s work [19], we use the roots of pj-series of formal group +law of E in π∗(TG,N(E)) to carry out our main idea. And we follow their assumption that E is a +p-complete and complex oriented spectrum with an associated formal group of height n. Recall +that a ring spectrum E is complex oriented if there exists an element x ∈ E2(CP∞) such that +the image i∗(x) of the map i∗ : E2(CP∞) → E2(CP1) induced by i : S 2 � CP1 ֒→ CP∞ is the +canonical generator of E2(S 2) � π0E. Such a class x is called a complex orientation of E. The +complex orientated E with the multiplication map µCP∞ : CP∞ ×CP∞ → CP∞ gives an associated +formal group law F over E∗: +x1 +F x2 = F(x1, x2) = µ∗ +CP∞(x) ∈ E∗(CP∞ × CP∞) = E∗[[x1, x2]], +where “[[]]” denotes the formal power series ring. For any integer m, the m-series of F is the +formal power series [m]E(x) = x +F x +F · · · +F x +�������������������������������������������� +m +∈ E∗[[x]]. Let vn denote the coefficient of xpn in +[p]E(x). Say that F +(i) has height ≥ n if vi = 0 for i < n; +(ii) has height exactly n if it has height ≥ n and vn ∈ E∗ is invertible. +When localized at p, such formal group laws are classified by height. +By using the Gysin sequence of S 1 → BZ/pj → CP∞ and the fact that [pj]E(x) is not a +zero divisor in E∗[[x]], one obtains that E∗(BZ/pj) � E∗[[x]]/([pj]E(x)). Besides, E∗(BZ/pj) +is a Hopf algebra over E∗ where the coalgebra structure is induced by the multiplication map +µBZ/pj : BZ/pj × BZ/pj → BZ/pj. To compute the roots of [pj]E(x) in a graded E∗-algebra4, +we recall a definition due to Hopkins-Kuhn-Ravenel. +Definition 1.4. ([19, Definition 5.5.]) Let R be a graded E∗-algebra and j be a natural number. +Then the set of E∗-algebra homomorphisms HomE∗−alg(E∗[[x]]/([pj]E(x)), R), denoted by pjF(R), +forms a group. +4In [19], a graded E∗-algebra means that a graded Hopf algebra over E∗, and we follow their notations. For a graded +E∗-algebra R, a root of [pj]E(x) in R is an element r ∈ R such that [pj]E(r) = 0 in R. + +4 +Yangyang Ruan +Remark 1.5. In other words, f ∗ ∈ HomE∗−alg(E∗[[x]]/([pj]E(x)), R) is an E∗-ring homomorphism +so that there is a one-one correspondence between f ∗ and its image f ∗(x). If we identify f ∗ with +its image f ∗(x), since f ∗([pj]E(x)) = [pj]E( f ∗(x)) = 0, then f ∗ is viewed as a root of [pj]E(x) in +R. And pjF(R) is viewed as a set of roots of [pj]E(x) in R. +If π∗(TG,N(E)) has an E∗-algebra structure, then by Remark 1.5 pjF(π∗(TG,N(E))) is viewed +as a set of roots of [pj]E(x) in π∗(TG,N(E)). +By simplifying the construction TG,N(−), +we identify the homotopy group π∗(TG,N(E)) with the G/N-equivariant homotopy group +πG/N +∗ +( ˜ΦN(F(EG, infG +e (E)))) of a G/N-spectrum +˜ΦN(F(EG, infG +e (E)), details see Proposition +3.2. Combining with Costenoble’s Theorem 3.3, we identify πG/N +∗ +( ˜ΦN(F(EG, infG +e (E)))) with +L−1 +N E∗(BG), where the multiplicatively closed set LN is generated by the set +MN = {χV ∈ E∗(BG) | V is any complex G-representation such that VN = 0} +of Euler classes. The work of [19] is one of the most important and profound results in the study +of the generalized cohomology of BG, and they showed that if G is an abelian group, E∗(BG) can +be computed and represented by a beautiful E∗-algebra. However, it is regrettable that by far, there +exsits no method to compute E∗(BG) for a general non-abelian group G. One of the difficulties +might lie in the fact that BG may not be an H-space for a non-abelian group G, in which case +E∗(BG) may not possess a coalgebra structure. The E∗-algebra structure is critical, so we take +G to be an abelian group A. Since BG is homotopy equivalent to the classifying space of the p- +Sylow group of G after localizing at p for a prime p, so without loss of generality we always work +p-locally and assume that A is an abelian p-group. Here we take N to be a subgroup C of A, and +obtain the homotopy group π∗(TA,C(E)). +Theorem 1.6. (The homotopy group of generalized Tate spectrum TA,C(E)) Let m be a positive +integer and E be a p-complete, complex oriented spectrum with an associated formal group of +height n. Let A be an abelian p-group of form Z/pi1 ⊕ · · · ⊕ Z/pim and C be its subgroup Z/pj1 ⊕ +· · · ⊕ Z/pjm with jk ≤ ik for 1 ≤ k ≤ m. There is a group homomorphism φ5 from A/C to A as +follows: +φ : Z/pi1−j1 ⊕ Z/pi2−j2 ⊕ · · · ⊕ Z/pim−jm → Z/pi1 ⊕ Z/pi2 ⊕ · · · ⊕ Z/pim +(w1, w2, · · · , wm) �→ (pi1−j1w1, pi2−j2w2, · · · , pim−jmwm). +Then +π∗(TA,C(E)) � L−1 +C E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)), +where the multiplicatively closed set LC is generated by the set +MC = {α(w1,···,wm) = [w1]E(x1) +F · · · +F [wm]E(xm) ∈ E∗(BA) | (w1, · · · , wm) ∈ A − imφ(A/C)}. +As pjF(π∗(TA,C(E))) is well-defined, then by Weierstrass Preparation Theorem 3.4, we have an +E∗-algebra isomorphism +η : E∗[[x]]/([pj]E(x)) → E∗[x]/(gj(x)) +5To describe the multiplicatively closed set LC, the group homomorphism φ : A/C → A arises, details see Lemma +3.18. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +5 +where gj(x) is the Weierstrass polynomial of [pj]E(x), which identifies the power series [pj]E(x) +with the polynomial gj(x) and their corresponding roots in π∗(TA,C(E)). To determinate the peri- +odicity of TA,C(E), we study the relation of roots and coefficients of gj(x) in π∗(TA,C(E)). +Let R be a commutative ring with 1 and f(x) be a polynomial over R. A polynomial f(x) in R[x] +can viewed as a polynomial map from R to R, which maps r ∈ R to f(r) ∈ R. We denote the set +of such polynomial maps by Pmap(R, R). To be more precise, Pmap(R, R) is the quotient R[x]/ ∼, +let [ f(x)] denote the equivalent class of f(x): f(x) ∼ g(x) if for any r ∈ R, f(r) = g(r). There is a +map λ : R[x] → Pmap(R, R) with λ( f(x)) = [ f(x)] for f(x) ∈ R[x], what conditions does R satisfy +with such that λ is injective? To serve our purpose here, we restrict ourself to a narrow version of +this question. Let R[x]n denote the set of polynomials of degree at most n and λR[x]n denote the +map that restricts λ to R[x]n, then the question now is what condition does R satisfy with so that +λR[x]n is injective ? A sufficient condition is that R has a set S in which the difference of any two +elements is not a zero divisor, and we call such S an |S |-tuple of R, see Lemma 4.4. Then if S +also is a subset of roots of a polynomial f(x) over R, we call such S an |S |-tuple of f(x) in R, see +Definition 4.5. And by using these two notions, we generalize the relation of roots and coefficients +of a polynomial over a commutative ring and obtain +Theorem 1.7. (Generalized relations of roots and coefficients of a polynomial) Let R be a +commutative ring with 1 and f(x) = a0 + a1x + · · · + amxm be a polynomial over R. Suppose that +R has an n-tuple {r1, r2, · · · , rn} of f(x). +(i) If n > m, then ai = 0 in R for 0 ≤ i ≤ m; +(ii) if n = m, then +ai = (−1)nan +� +1≤k1�k2�···�kn−i≤n +rk1rk2 · · · rkn−i in R for 0 ≤ i ≤ n − 1 and f(x) = an +n +� +i=1 +(x − ri); +(iii) if n ≤ m, then ai = +det(α0,··· ,αi−1,β,αi+1,··· ,αn−1) +det(α0,α1,··· ,αn−1) +in R for 0 ≤ i ≤ n − 1, where αi denotes +the column R-vector (ri +1, ri +2, · · · , ri +n)T for 0 ≤ i ≤ n − 1 and β denote the column R-vector +(− �m +i=n airi +1, − �m +i=n airi +2, · · · , − �m +i=n airi +n)T. +Remark 1.8. +(i) It is impossible for a nonzero polynomial over a field to have the number of +roots more than its degree, whereas it is possible for a nonzero polynomial over a commu- +tative ring, such as the nonzero polynomial x2 over Z[x1, x2]/(x2 +1, x2 +2). +(ii) To some extent, this theorem is a generalization of polynomial factorization. It is easy to see +that the first two cases of this theorem imply that f(x) has a polynomial factorization. The +third case just showed that if n ≤ m, one can obtain a factorization f(x) = an +�n +i=1(x − ri) +in R[x]/(am−n+1, am−n+2, · · · , am). +If R has a set S in which the difference of any two elements is invertible in R, we call such S +an invertible |S |-tuple of R. The first corollary of Theorem 1.7 shows that generalized relations +of roots and coefficients of a polynomial can be viewed in some sense as polynomial interpolation +over a commutative ring. + +6 +Yangyang Ruan +Corollary 1.9. Let R be a commutative ring with 1 and f(x) = a0+a1x+· · ·+amxm be a polynomial +over R. If R has an invertible n-tuple {r1, r2, · · · , rn} of f(x), then +f(x) = +n +� +j=1 +� +1≤i≤n,i�j +x − ri +rj − ri +(− +m +� +i1=n +ai1ri1 +j ), +when m < n, − �m +i1=n ai1ri1 +j denotes 0. +The other corollary of Theorem 1.7 gives a sufficient yet useful condition to guarantee the +vanishment of a commutative ring. +Corollary 1.10. (Vanishing ring condition) Let f(x) = a0 + a1x + · · · + amxm be a polynomial +over a commutative ring R with 1. R has an n-tuple {r1, r2, · · · , rn} of f(x) under the assumption +that R � 0. +(i) If n > m and 1 belongs to the ideal (a0, a1, · · · , an) of R, then R = 0; +(ii) if n ≤ m and 1 belongs to the ideal (a0 − det(β,α1,α2,··· ,αn−1) +det(α0,α1,··· ,αn−1) , a1 − det(α0,β,α2··· ,αn−1) +det(α0,α1,···,αn−1) , · · · , an − +det(α0,···,αi−1,β,αi+1,···,αn−1) +det(α0,α1,···,αn−1) +) of R, then R = 0. +The usefulness of Corollary 1.10 can be seen in Corollary 4.13 which includes a proof of Tate +vanishing result [16, Theorem 1.1] of Morava K-theory and a proof of ΦHKUG ≃ ∗ [9, Proposition +3.10] for a p-group G and a non-cyclic subgroup H. And our method greatly simplifies those +original proofs. +Studying the relation of roots and coefficients of a polynomial in R has a broad application +prospect in reducing the relations of R. The most common situation is that one obtains some +relations of R, like an n-tuple {r1, r2, · · · , rn} of f(x), then dedecates to reduce these relations to +get a desired relation, like the solution of ai. A useful application of Theorem 1.7 in dealing +with practical mathematical problems is the explanation of general blue-shift phenomenon which +is motivated by computing the Zariski topology of Balmer spectrum Spc(SH(G)c), details see +Section 2. +For a finite abelian p-group A, let rankp(A) denote the number of Z/p factors in the maximal +elementary abelian subgroup of A. Let ⟨E⟩ denote Bousfield class of E and E(k) denote k-th +Johnson-Wilson theory. By using pjF(π∗(TA,C(E))) and generalized relations of roots and coeffi- +cients of gj(x) in π∗(TA,C(E)), we have a partial answer of general blue-shift phenomenon 1.2 for +abelian cases. +Theorem 1.11. Let E be a p-complete, complex oriented spectrum with an associated formal +group of height n. Let A be a finite abelian p-group and C be its direct summand. If E is Landweber +exact, then +(i) TA,C(E) is Landweber exact; +(ii) TA,C(E) is vn−rankp(C)-periodic; +(iii) ⟨TA,C(E)⟩ = ⟨E(n − rankp(C))⟩. When k > n, E(n − k) = ∗. +Remark 1.12. +(i) By [21, Corollary 1.12], the assumption on E implies that ⟨E⟩ = ⟨E(n)⟩. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +7 +(ii) When A = C = Z/p and E = E(n), this theorem implies the corresponding case of [20, +Theorem 1.2], and gives an upper bound of BSm(Z/p; Z/p, e), that is BSm(Z/p; Z/p, e) ≤ +16, which implies [6, Proposition 7.1]. +(iii) When A = C = (Z/p)k and E is the n-th Morava E-theory En, this theorem implies [36, +Proposition 3.0.1]. +(iv) Note that ⟨TA,A(E(n))⟩ = ⟨E(n − rankp(A))⟩. If A = C = H/K is an abelian p-group, then +this theorem gives an upper bound of BSm(G; H, K), that is BSm(G; H, K) ≤ rankp(H/K)7, +which implies [7, Theorem 1.5]. +Let G be a finite p-group and N be its normal subgroup. To answer general blue-shift phe- +nomenon 1.2 for non-abelian cases, one of the most important problems that we have to deal with +is how to compute the roots of [pj]E(x) in π∗(TG,N(E)), and this problem is equivalent to how to +compute the roots of [pj]E(x) in E∗(BG). If G is an abelian group, we could define a homomor- +phism +ψpj +G : G → G +g �→ gpj, +and by the functorial property of the classifying space functor B, we have Bψpj +G = ψpj +BG and ψpj,∗ +BG : +E∗(BG) → E∗(BG) is an E∗-algebra homomorphism. Then the generators of the kernel of ψpj,2 +BG : +E2(BG) → E2(BG) are roots of [pj]E(x) in E∗(BG). But if G is a non-abelian group, then ψpj +G +need not be a homomorphism, so we can not use the functorial property of B to obtain a self-map +of BG. Inspired by Jackowski-Mcclure-Oliver’s work [23], we regard Bψpj +G as an unstable Adams +operation, which motivates us to give the following definition. +Definition 1.13. Let G be a finite p-group and G′ be the commutator group {aba−1b−1 | a, b ∈ G} +of G with a quotient homomorphism ǫ : G → G/G′. A self-map f : BG → BG is called an +unstable Adams operation of degree p if the following diagram +BG +Bǫ +−−−−−→ B(G/G′) +f +� +ψp +B(G/G′) +� +BG +Bǫ +−−−−−→ B(G/G′) +commutes up to homotopy. +Conjecture 1.14. Let G be a finite p-group and E be a p-complete complex-oriented spectrum +with an associated formal group of height n. Then there is an unstable Adams operation f : +BG → BG of degree p and E2( f(−)) = [p]E(−) : E2(BG) → E2(BG). +6Details see Section 2. +7Details see Section 2. + +8 +Yangyang Ruan +For a real number r, let ⌈r⌉ denote the least integer of no less than r. For a finite abelian group +A, let V(pj|A) denote the subgroup {a ∈ A | pja = 0}. Since ǫ(N) is a subgroup of G/G′, then the +quotient group G/G′/ǫ(N) can be canonically embedded in G/G′ by φ. +Theorem 1.15. (Theorem 7.3) Let E be a p-complete, complex oriented spectrum with an as- +sociated formal group of height n. Let G be a finite p-group and N be its normal subgroup. If +Conjecture 1.14 is true, then +sG,N;E ≥ max +j∈N+ ⌈ +logp |V(pj|G/G′)| − logp |V +Ä +pj|imφ(G/G′/ǫ(N)) +ä +| +j +⌉. +Our paper is organized as follows. The motivation from the computation of the Zariski topol- +ogy of Balmer spectrum of the G-equivariant stable homotopy category can be found in Section 2; +In Section 3, we compute the homotopy group of generalized Tate spectrum TA,C(E); In Section 4, +we generalize the relation of roots and coefficients of a polynomial in a commutative ring; In Sec- +tion 5, we recall the definition of algebraic periodicity and Landweber exactness for a spectrum; +Note that Theorem 1.11 is a corollary of Theorem 6.1, we give a detailed proof of Theorem 6.1 in +Section 6; In Section 7, we provide a possible way to deal with general blue-shift phenomenon for +non-abelian cases. +Acknowledgement: Firstly, I thank professor Kuhn Nicholas John for introducing me the prob- +lem of computing Balmer spectrum in an International Workshop on Algebraic Topology at Fudan +University in 2019. Secondly, I thank professor Stefan Schwede for teaching me lots of knowledge +about the G-equivariant stable homotopy category. Thirdly, as most our work is based on my PhD +thesis [39], I thank professor Xu An Zhao for his carefully reading my PhD thesis and making me +correct some vague arguments. Finally, I thank Long Huang and Ran Wang for carefully reading +my draft and suggesting lots of improvements. +2 +Motivation from the computation of the Zariski topology of +Balmer spectrum Spc(SH(G)c) +Our work is motivated by computing the Zariski topology of Balmer spectrum, this leads us to +Conjecture 1.2 and Theorem 1.11. so let us illustrate why Theorem 1.11 can be applied to compute +the Balmer spectrum. +SH(G)c has a symmetric monoidal structure whose tensor product and unit object are the smash +product of G-spectra and G-sphere spectrum S G respectively, which make it resembles a commu- +tative ring with a unit, so one could introduce the method of algebraic geometry and define “prime +ideal” and “spectrum” for it. In 2005, Blamer [4] defined the spectrum Spc(SH(G)c), which is +similar to the spectrum of a commutative ring with a unit, of SH(G)c as a set of all proper “prime +ideals” with Zarisiki topology, and now this spectrum is called Balmer spectrum. When G is the +trivial group e, SH(G) is the classical stable homotopy category SH(e). Hopkins and Smith [18] +classified all thick subcategories of SH(e)c by using the work of Ravenel [34] and Mitchell [30]. +In other words, they got the Balmer spectrum Spc(SH(e)c). Let K(0) and K(∞) denote the rational +and mod p Eilenberg-Maclane spectra K(Q), K(Z/p) respectively. Then all proper “prime ideals” + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +9 +of SH(e)c are of the following form +Cp,m = {X ∈ SH(e)c | K(m − 1)∗(X) = 0}. +For each p, there is a descending chain +(2.1) +Cp,1 ⊋ Cp,2 ⊋ · · · ⊋ Cp,∞ +due to [34, 30]. The topology space Spc(SH(e)c) can be described by the following diagram, +C2,∞ +C3,∞ +· · · +Cp,∞ +· · · +... +... +... +C2,n+1 +C3,n+1 +· · · +Cp,n+1 +· · · +C2,n +C3,n +· · · +Cp,n +· · · +... +... +... +C2,2 +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +C3,2 +▼ +▼ +▼ +▼ +· · · +Cp,2 +❧❧❧❧❧❧❧ +· · · +C0,1 +where the line between any two points denotes that there is an inclusion relation between the two +“prime ideals”. +The computation of Spc(SH(e)c) is one of the main tools used in applications of the nilpotence +theorem of Devinatz, Hopkins and Smith [10, 18] to global questions in stable homotopy theory. +Strickland [37] tried to generalize the non-equivariant case to the G-equivariant case. For any +subgroup H of a finite group G, Strickland used the geometric H-fixed point functor ΦH(−) : +SH(G) → SH(e) which resembles a “ring homomorphsim” to pull back Cp,m to obtain “prime +ideals” in SH(G)c, then got the G-equivariant “prime ideals” +PG(H, p, m) = (ΦH)−1(Cp,m) = {X ∈ SH(G)c | K(m − 1)∗ΦH(X) = 0}. +In 2017, Balmer and Sanders [6, Theorem 4.9 and 4.14] confirmed that all G-equivariant proper +“prime ideals” of SH(G)c are obtained by this way, which means that they determined the set +structure of Balmer spectrum Spc(SH(G)c). To compute the Zarisiki topology of Spc(SH(G)c), +it suffices to give an equivalent condition for any two “prime ideals” PG(K, q, l), PG(H, p, m) +of SH(G)c to have an inclusion relation PG(K, q, l) ⊆ PG(H, p, m). Balmer and Sanders [6, +Corollary 4.12 and 6.4] obtained two necessary conditions for the inclusion: one is p = q; the +other is that K is a subgroup of H up to G-conjucate, which is denoted by K ≤G H. Therefore, +the determination of Zariski topology of Spc(SH(G)c) can be reduced to the computation of the +following number +BSm(G; H, K) := min{l − m = i ∈ Z| PG(K, p, l) ⊆ PG(H, p, m)}. + +10 +Yangyang Ruan +There is an observation that l ≥ m which is due to Kuhn and Lloyd [25]. It suffices to prove that +for each l < m, there is a finite G-spectrum X such that X ∈ PG(K, p, l) and X � PG(H, p, m). +By Mitchell’s work [30], there is a non-equivariant finite spectrum Y such that Y ∈ Cp,m but +Y � Cp,m+1. Then we take X to be a G-spectrum Y with the trivial G-action, which finishes the +proof. +To determine BSm(G; H, K), some intuition for the relation PG(K, p, l) ⊆ PG(H, p, m) would +be helpful. From the descending chain 2.1 and the fact that ΦK(X) ∈ SH(e)c, it follows that +K(m − 1) ⊗ ΦK(X) = 0 ⇔ +m−1 +� +i=0 +K(i) ⊗ ΦK(X) = 0. +To transform the above equation into a more convenient form, we recall Bousfield’s [3] definition +of a non-equivariant spectrum E, ⟨E⟩ denotes the equivalence class of E: E ∼ F if for any spectrum +X ∈ SH(e), E∗X = 0 ⇔ F∗X = 0. And ⟨E⟩ is called Bousfield class of E. Due to Ravenel [34, +Theorem 2.1.], the Bousfield class ⟨�n +i=0 K(i)⟩ equals to the Bousfield class ⟨E(n)⟩. Then we have +for X ∈ SH(G)c, +m−1 +� +i=0 +K(i) ⊗ ΦK(X) = 0 ⇔ E(m − 1) ⊗ ΦK(X) = 0. +Thus for X ∈ SH(G)c, +K(m − 1) ⊗ ΦK(X) = 0 ⇔ E(m − 1) ⊗ ΦK(X) = 0. +PG(K, p, l) ⊆ PG(H, p, m) is equivalent to the fact that for X ∈ SH(G)c, E(l − 1)∗ΦK(X) = 0 +implies E(m − 1)∗ΦH(X) = 0. +The inclusion H ֒→ G provides a restriction functor resG +H : SH(G) → SH(H). Assume +that K ⊴ G, the surjective homomorphism G → G/K induces an inflation functor infG +G/K : +SH(G/K) → SH(G). Let ˜ΦK be the relative geometric K-fixed point functor from SH(G) to +SH(G/K). By [26, Chapter II. §9], we have resG/K +e +◦ ˜ΦK � ΦK and +0 = E(l − 1) ⊗ ΦK(X) = E(l − 1) ⊗ resG/K +e +◦ ˜ΦK(X) = resG/K +e +(infG/K +e +(E(l − 1)) ⊗ ˜ΦK(X)). +Let G/K+ denote the disjoint union of G/K and a point. By [5, 1.1 Theorem], we get resG/K +e +(−) � +G/K+ ⊗ (−) and +0 = resG/K +e +(infG/K +e +(E(l − 1)) ⊗ ˜ΦK(X)) = G/K+ ⊗ infG/K +e +(E(l − 1)) ⊗ ˜ΦK(X). +Let E(G/K) denote the Milnor construction, which is an infinite join G/K ∗ G/K ∗ · · · ∗ G/K, for +the group G/K. Then +0 = E(G/K)+ ⊗ infG/K +e +(E(l − 1)) ⊗ ˜ΦK(X). +Let �E(G/K) be the unreduced suspension of E(G/K) with one of the cone points as basepoint, +then we have +0 =F( �E(G/K), ΣE(G/K)+ ⊗ infG/K +e +(E(l − 1)) ⊗ ˜ΦK(X)). +(2.2) + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +11 +By [15, Corollary B.5], we have +F( �EG, ΣEG+ ⊗ −) � F(EG+, −) ⊗ �EG. +tG(kG) := F(EG+, kG)⊗ �EG is so-called classical Tate construction in the sense of Greenlees and +May [14] for a G-spectrum kG. Assume that K ⊴ H, we apply geometric H/K-fixed point functor +ΦH/K(−) to Formula 2.2. Since ΦH/K(−) preserves weak equivalences, we obtain +0 = ΦH/K(tG/K(infG/K +e +(E(l − 1)) ⊗ ˜ΦK(X))). +Note that for X ∈ SH(G), Y ∈ SH(G)c, tG(X) ⊗ Y � tG(X ⊗ Y) (details see [6, Remark 5.8]), we +have +0 = ΦH/K(tG/K(infG/K +e +(E(l − 1))) ⊗ ˜ΦK(X)). +From the facts that for any G/K-spectra X and Y, ΦH/K(X ⊗ Y) = ΦH/K(X) ⊗ ΦH/K(Y), and +ΦH/K ◦ ˜ΦK � ΦH, it follows that +0 =ΦH/K(tG/K(infG/K +e +(E(l − 1))) ⊗ ˜ΦK(X)) +=ΦH/K(tG/K(infG/K +e +(E(l − 1)))) ⊗ ΦH/K ◦ ˜ΦK(X) +=ΦH/K(tG/K(infG/K +e +(E(l − 1)))) ⊗ ΦH(X). +For convenience, let TG/K,H/K(−) denote the functor ΦH/K(tG/K(infG/K +e +(−))), and by Proposition +3.1 we have TG/K,H/K(−) = TH/K,H/K(−). If ⟨TG/K,H/K(E(l − 1))⟩ equals to the Bousfield class of +some Johnson-Wilson theory, certainly this would give us an upper bound of BSm(G; H, K). The +idea of the above reduction actually comes from Balmer and Sanders’ computation [6, Proposition +7.1] of Zariski topology of the Balmer spectrum Spc(SH(Z/p)c). They used Hovey-Sadofsky- +Kuhn’s result [20, 24] +⟨TZ/p,Z/p(E(l − 1))8⟩ = ⟨E(l − 2)⟩ +to get BSm(Z/p; Z/p, e) ≤ 1. In fact, BSm(Z/p; Z/p, e) = 1, which means that the determination +of ⟨TG/K,H/K(E(l − 1))⟩ could give us the least upper bound of BSm(G; H, K). If H/K is a finite +abelian p-group, then Theorem 1.11 confirmed that +⟨TG/K,H/K(E(l − 1))⟩ = ⟨E(l − 1 − rankp(H/K))⟩. +In 2019, Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton [7] obtained that when G is a fi- +nite abelian p-group, BSm(G; H, K) is exactly rankp(H/K). In particularly, they did not use the +Bousfield class ⟨TG/K,H/K(E(l − 1))⟩ to determine the upper bound of BSm(G; H, K), but used the +method [32] of derived defect base by recognizing TG/K,H/K(E(l − 1)) as suitable sections of the +structure sheaf on a certain non-connective derived scheme. There must be some beautiful math +living behind such a beautiful result. In order to make this problem more approachable to general +audiences, we give a new approach to determine the upper bound of BSm(G; H, K). +Our new approach is by use of Theorem 6.1, and here is a sketch of the proof for Theorem +6.1. Theorem 6.1 is a generalization of [20, Theorem 1.2]. When trying to generalize [20, The- +orem 1.2], we find that the determination of ⟨TG/K,H/K(E(l − 1))⟩ can be transformed into the +8Actually their construction is tZ/p(infZ/p +e +(−))Z/p, but by Proposition 3.2 and Proposition 3.1, tZ/p(infZ/p +e +(−))Z/p and +TZ/p,Z/p(−) are the same construction. + +12 +Yangyang Ruan +explanation of Balmer and Sanders’ new blue-shift phenomenon [6]. More generally, the deter- +mination of ⟨TG/K,H/K(E(l − 1))⟩ can be transformed into the explanation of general blue-shift +phenomenon 1.2. Observing that if G/K is a finite abelian p-group, then TG/K,H/K(E(l − 1)) +inherits the Landweber exactness of E(l − 1), details see Lemma 5.11, we only have to deter- +mine the periodicity of TG/K,H/K(E(l − 1)) by Hovey’s Theorem 5.10. Here we choose Hovey’s +definition 5.5 of vn-periodicity for TG/K,H/K(E(l − 1)) and find that the determination of the pe- +riodicity of TG/K,H/K(E(l − 1)) is equivalent to the computation of the projective dimension of +π∗(TG/K,H/K(E(l − 1))) as an E(l − 1)∗-module. By homology algebra, the projective dimension +of π∗(TG/K,H/K(E(l − 1))) is measured by the maximal length of a π∗(TG/K,H/K(E(l − 1)))-regular +sequence in the maximal ideal Il−1 = (p, v1, · · · , vl−2) of E(l − 1)∗. Then by Corollary 1.9, finding +some-tuple of pj-series [pj]E(l−1)(x) in π∗(TG/K,H/K(E(l−1))) will give an upper bound of the pro- +jective dimension of π∗(TG/K,H/K(E(l−1))). Given the periodicity of E(l−1), by inductively using +Lemma 6.40, we will get a lower bound of the projective dimension of π∗(TG/K,H/K(E(l − 1))), +details see Lemma 6.40. This is our idea to prove Theorem 1.11 and Theorem 6.1. +There are several significant differences between our new proof and the earlier of [7]. First, +our proof is self-contained, while their proof of [7, Theorem 3.4] is based on a series of work +[31, 32]. Second, our proof is more conceptual in the sense that we have an intuitive idea to +explain general blue-shift phenomenon and successfully achieve it. When G is a non-abelian +group, BSm(G; H, K) is not completely known, our new proof may help to bring some intuition to +this problem. Third, they [7] used derived algebraic geometry and the geometry of the stack of +formal groups to describe the chromatic height shifting behaviour of TG/K,H/K(E(l−1)). However, +our method only need the tool of some-tuple of the pj-series in π∗(TH/K,H/K(E(l − 1))) and some +linear algebra. +3 +The homotopy groups π∗(TA,C(E)) and their maps +Follow the notions of [19, Section 5], in this section we assume that E is a complex-oriented +cohomology theory, particularly p-complete theory with an associated formal group of height n. +The homotopy group of the classical Tate construction tA(infA +e (E))A is computed in [17], and the +homotopy group of generalized Tate spectrum TA,C(E) has already been known to the experts +over years, but there is not any version with enough proving details. In this section, we provide +a detailed proof of Theorem 1.6. The functor TG,N(−) is related to TG,N(−) by the following +proposition. +Proposition 3.1. Let G be a finite p-group or T m = U(1) × · · · × U(1) +������������������������������������������ +m +, and N be its normal +subgroup. Then TG,N(−) = TN,N(−). +Proof. By definition, ΦN(−) = ˜ΦN ◦ resG +N(−), combining with the fact that +resG +N(tG(infG +e (−)) = tN(resG +N ◦infG +e (−) = tN(infN +e (−)), +details see [6, Example 5. 18], we have ΦN(tG(infG +e (−))) = TN,N(−). +□ +First, we recall the definition [26] of the relative geometric N-fixed point functor ˜ΦN(−) : +SH(G) → SH(G/N). For a family F of subgroups of G closed under G-conjugacy, there is a + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +13 +universal space EF characterized by its fixed point data: EF K be contractible if K ∈ F and +empty if K � F . There is a map EF+ → S 0 induced by EF → ∗, and let �EF denote its cofiber. +Then by the long exact sequence of non-equivariant homotopy groups induced by this confiber +sequence, we obtain that �EF K is homotopy equivalent to ∗ if K ∈ F and S 0 if K � F . Therefore +�EF1 ⊗ �EF2 ≃ �E(F1 ∪F1) where ≃ denotes the homotopy equivalence. Let F [N] denote the family +of subgroups of G which do not contain N, then the definition of ˜ΦN(−) is ( �EF [N] ⊗ (−))N. �EG +denotes �EF where F is the family subgroups only containing the trivial subgroup. To compute +π∗(TG,N(E)), we give an equivalent description of π∗(TG,N(E)). +Proposition 3.2. Let G be a finite p-group or T m, and N be its normal subgroup. Let E be a +non-equivariant spectrum. Then +TG,N(E) ≃ ( ˜ΦN(F(EG+, infG +e (E))))G/N and π∗(TG,N(E)) � πG/N +∗ +( ˜ΦN(F(EG+, infG +e (E)))), +where G/N-equivariant homotopy group is defined by a complete G/N-universe in the sense of +Lewis, May and Steinberger [26]. If the family subgroups of G which do not contain N are {e}, +then TG,N(−) = tG(infG +e (−))G. +Proof. Since �EF [N] ⊗ �EG ≃ �EF [N], we have +TG,N(E) =( ˜ΦNtG(infG +e (E)))G/N +=(( �EF [N] ⊗ �EG ⊗ F(EG+, infG +e (E)))N)G/N +≃(( �EF [N] ⊗ F(EG+, infG +e (E)))N)G/N = ( ˜ΦN(F(EG+, infG +e (E))))G/N. +By the adjunction [S n, ( ˜ΦN(F(EG+, infG +e (E))))G/N] � [infG/N +e +(S n), ˜ΦN(F(EG+, infG +e (E)))]G/N, we +identify the homotopy group π∗( ˜ΦN(F(EG+, infG +e (E))))G/N with the G/N-equivariant homotopy +group πG/N +∗ +( ˜ΦN(F(EG+, infG +e (E)))). +If {e} is the family subgroups of G which do not contain N, then �EF [N] = �EG and TG,N(−) = +tG(infG +e (−))G. +□ +Let N be a normal subgroup of G, then the following theorem of Costenoble describes the +behavior of relative geometric N-fixed point functor ˜ΦN(−) on the homotopy group. +Theorem 3.3. (Costenoble [26, Chapter II proposition 9.13.]) Let kG be a ring spectrum and +set kG/N = +˜ΦN(kG). +Then for a finite G/N-CW spectrum X, k∗ +G/N(X) is the localization of +k∗ +G(infG +G/N(X)) obtained by inverting the Euler classes χV ∈ kV +G(S 0) of those representations V +of G such that VN = 0. +From Proposition 3.2 and Theorem 3.3, it follows that to compute π∗(TG,N(E)), we only need to +compute πG +∗ (F(EG+, infG +e (E))), then invert the Euler classes χV ∈ F(EG+, infG +e (E))V(S 0) of those +complex G-representations V such that VN = 0. By the equivariant suspension isomorphism, +we have χV ∈ F(EG+, infG +e (E))V(S 0) � F(EG+, infG +e (E))|V|(S |V|−V), where |V| denote the real +dimension of V. By Theorem 3.3 and the following observation +πG +∗ (F(EG+, infG +e (E))) =π∗(G/G+ ∧ S 0, F(EG+, infG +e (E)))G +=π∗(S 0, F(EG+, infG +e (E))G) +=π∗(BG+, E) = E∗(BG+), +we identify the G-equivariant homotopy group πG +∗ (F(EG+, infG +e (E))) with E∗(BG+). + +14 +Yangyang Ruan +3.1 +The E∗-cohomology of the classifying space of a finite abelian p-group +First we introduce the Weierstrass Preparation Theorem. +Theorem 3.4. (Weierstrass Preparation Theorem [40, 29, 41]) Let R be a graded local commuta- +tive ring, complete in the topology defined by the powers of an ideal m. Suppose +α(x) = +∞ +� +i=0 +aixi ∈ R[[x]] +satisfies α(x) ≡ anxn mod (m, xn+1) with an ∈ R a unit. Then +(i) (Euclidean algorithm) Given f(x) ∈ R[[x]], there exist unique r(x) ∈ R[x] and q(x) ∈ R[[x]] +such that f(x) = r(x) + α(x)q(x) with deg r(x) ≤ n − 1. +(ii) The ring R[[x]]/(α(x)) is a free R-module with basis {1, x, · · · , xn−1}. +(iii) (Factorization) There is a unique factorization α(x) = ε(x)g(x) with ε(x) a unit and g(x) a +monic polynomial of degree n, we call g(x) the Weierstrass polynomial of α(x). +The number n is called the Weierstrass degree of α(x) and denoted by degW α(x). +Recall some basic properties of the associated formal group law F over E∗. +Proposition 3.5. Let E be a p-complete complex-oriented spectrum with an associated formal +group of height n. Let In denote the maximal ideal of E∗ and vn be a unit of E∗. Then for any +integer m, the m-series of F satisfies +(i) [m]E(x) ≡ mx mod (x2); +(ii) [mk]E(x) = [m]E([k]E(x)); +(iii) [p]E(x) = vnxpn mod In; +(iv) [m − k]E(x) = [m]E(x) −F [k]E(x) = ([m]E(x) − [k]E(x)) · ε([m]E(x), [k]E(x)), where +ε([m]E(x), [k]E(x)) is a unit in E∗[[x]]. +Lemma 3.6. Let gj(x) denote the Weierstrass polynomial of [pj]E(x) and gj +1(x) = g1(gj−1 +1 +(x)). +Then gj(x) = gj +1(x). +Proof. Suppose that [p]E(x) = px + a2x2 + · · · + apn−1xpn−1 + vnxpn mod (xpn+1), and we apply +Theorem 3.4 to [p]E(x) ∈ E∗[[x]], then [p]E(x) = ε(x)g1(x) with ε(x) a unit and g1(x) = px + +a2x2 + · · · + apn−1xpn−1 + vnxpn. And we apply this theorem 3.4 to [pj]E(x) ∈ E∗[[x]], by the fact +that [pj]E(x) = [p]E([pj−1]E(x)), then [pj]E(x) = ε j(x)gj(x) with ε j(x) a unit. By the uniqueness +of factorization 3.4 and the fact that gj +1(x) = [pj]E(x) = v1+pn+···+p(j−1)n +n +xpjn mod In, then gj(x) = +gj +1(x). +□ +The following lemma gives the computation of E∗(BA+). + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +15 +Lemma 3.7. Let E be a p-complete complex-oriented spectrum with an associated formal group +of height n. If A is an abelian p-group of form Z/pi1 ⊕ · · · ⊕ Z/pim, then +E∗(BA+) � E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)). +Proof. If A = Z/pj, then there is a fiber sequence: +S 1 → BZ/pj → CP∞ ψpj +→ CP∞. +Note that the Euler class of the Gysin sequence of S 1 → BZ/pj → CP∞ is ψpj,2(x) = [pj]E(x) ∈ +E2(CP∞ ++ ), then we have a long exact sequence: +· · · +� E∗[[x]] +∪[pj]E(x)� E∗+2[[x]] +� E∗+2(BZ/pj ++) +� · · · . +Since [pj]E(x) is not a zero divisor in E∗[[x]], the long exact sequence splits. Therefore, we obtain +E∗(BZ/pj ++) � E∗[[x]]/([pj]E(x)). +As we all know, K¨unneth isomorphism is not always true for product spaces X × Y, but if E- +cohomology of the space X or Y is a finitely generated free module over E∗, the K¨unneth isomor- +phism is true. By Weierstrass Preparation Theorem 3.4, we have an E∗-ring isomorphism +η : E∗[[x]]/([pj]E(x)) � E∗[x]/(gj(x)) +that maps f(x) to r(x), where gj(x) is the Weierstrass polynomial of [pj]E(x), which implies that +E∗[[x]]/([pj]E(x)) is a finite free E∗-module of rank pjn = degW[pj]E(x). This finishes the proof. +□ +Note that E∗(BZ/pj ++) is a Hopf algebra over E∗. And η induces a coalgebra structure on +E∗[x]/(gj(x)): +E∗[[x]]/([pj]E(x)) +µ∗ +BZ/pj +−−−−−→ +E∗[[x]]/([pj]E(x)) ⊗E∗ E∗[[x]]/([pj]E(x)) +η +� +η⊗η +� +E∗[x]/(gj(x)) +η⊗η◦µ∗ +BZ/pj◦η−1 +−−−−−−−−−−−−→ +E∗[x]/(gj(x)) ⊗E∗ E∗[x]/(gj(x)), +then combining with Lemma 3.6, we have +Proposition 3.8. Let E be a p-complete complex-oriented spectrum with an associated formal +group of height n. Then there is an E∗-algebra isomorphism +η : E∗[[x]]/([pj]E(x)) � E∗[x]/(gj +1(x)), +where the coalgebra structure on E∗[x]/(gj +1(x)) is given by the map +η ◦ µ∗ +BZ/pj ◦ η−1 : E∗[x]/(gj +1(x)) → E∗[x]/(gj +1(x)) ⊗E∗ E∗[x]/(gj +1(x)). + +16 +Yangyang Ruan +3.2 +Euler classes and formal groups +In this paper, we always identify Z/pj with the set {0, 1, · · · , pj − 1}. Let ρ w +n : Z/pj → U(1) +denote the complex character that maps h to e +2whπi +pj +for w ∈ Z/pj. Suppose that A has the form +Z/pi1 ⊕ · · · ⊕ Z/pim. By the representation theory of abelian groups [38, Propositon 4.5.1.], +{ρ( w1 +pi1 ,···, wm +pim ) = µU(1) ◦ (ρ w1 +pi1 × · · · × ρ wm +pim ) = ρ w1 +pi1 · · · ρ wm +pim : A → U(1) | (w1, · · · , wm) ∈ A} +formed all irreducible complex representations of Z/pi1 ⊕ · · · ⊕ Z/pim. +Recall the definition [14] of Euler classes for the A-spectrum F(EA+, infA +e (E)). Let V be any +complex A-representation with an inner product, let eV : S 0 → S V send the non-basepoint to 0, +and let χV ∈ F(EA+, infA +e (E))V(S 0) be the image of the unit of F(EA+, infA +e (E))0(S 0) under the +map e∗ +V : F(EA+, infA +e (E))0(S 0) � F(EA+, infA +e (E))V(S V) → F(EA+, infA +e (E))V(S 0). +Since any finite abelian p-group A with rankp(A) = m is isomorphic to a subgroup of T m, we +first show how to specifically identify E∗(BU(1)+) � E∗[[x]] with πU(1) +∗ +(F(EU(1)+, infU(1) +e +(E))). +Let R denote the U(1)-spectrum F(EU(1)+, infU(1) +e +(E)). We may assume that E is a homotopy +commutative ring spectrum, and by [8, Theorem 6.23.] F(EU(1)+, infU(1) +e +(E)) is a homotopy +commutative U(1)-ring spectrum. Firstly, recall the definition [32, Definition 5.1] of a Thom +class µV : S V−|V| → R for V with respect to R, µV is a map of U(1)-spectra such that its canonical +extension to an R-module map +R ⊗ S V−|V| +idR⊗µV +−−−−−→ R ⊗ R +µ +−−−−−→ R +is an equivalence, where µ denotes the multiplication map of the ring spectrum R. Secondly, we +will find the Thom class µV. Since all irreducible complex representations of abelian groups are +complex one-dimensional, we may choose V to be C. For the principal U(1)-bundle C → C → ∗, +we have a Thom space S C, which gives a Thom isomorphism +φC : F(EU(1)+, infU(1) +e +(E))∗(S 0) → F(EU(1)+, infU(1) +e +(E))∗+2(S C), +by the equivariant suspension isomorphism, we can rewrite φC as an isomorphism +πU(1) +∗ +(F(EU(1)+, infU(1) +e +(E))) � πU(1) +∗ +(F(EU(1)+, infU(1) +e +(E)) ⊗ S 2−C). +By [32, Remark 5.2], this Thom isomorphism φC gives rise to such a Thom class µC : S C−2 → +F(EU(1)+, infU(1) +e +(E)) for C with respect to F(EU(1)+, infU(1) +e +(E)). Follow the notions of [13, +Remark 2.2], we also insist that φC(y) = y · µC for all y ∈ F(EU(1)+, infU(1) +e +(E))∗(S 0). Since +χV : S −|V| eV→ S V−|V| µV +→ F(EU(1)+, infU(1) +e +(E)), we have +χC = φC(eC) = eC · µC = e∗ +C(µC). +For the universal principal U(1)-bundle U(1) → EU(1) → BU(1), we have a Thom space +MU(1) ≃ BU(1), which gives a Thom isomorphism ∪x : E∗(BU(1)+) → E∗+2(BU(1)+), and +it corresponds to ·χC under the following identification +F(EU(1)+, infU(1) +e +(E))∗(S 0) +·µC +−−−−−→ +F(EU(1)+, infU(1) +e +(E))∗+2(S C) +� +� +e∗ +C +� +E∗(BU(1)+) +∪x +−−−−−→ F(EU(1)+, infU(1) +e +(E))∗+2(S 0) � E∗+2(BU(1)+). + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +17 +Then x corresponds to χC under the isomorphism between F(EU(1)+, infU(1) +e +(E))∗(S 0) and +E∗(BU(1)+). +Lemma 3.9. Let ρ w +pj be an irreducible complex Z/pj-representation with w ∈ Z/pj. Let ρ# +w +pj be +the map +F(EU(1)+, infU(1) +e +(E))∗(S 0) → F(EZ/pj ++, infZ/pj +e +(E))∗(S 0) +induced by ρ w +pj . Then Bρ∗ +w +pj (x) = [pj]E(x) corresponds to χρ w +pj = ρ# +w +pj (µC) under the isomorphism +between πZ/pj +∗ +(F(EZ/pj ++, infZ/pj +e +(E))) and E∗(BZ/pj ++). +Proof. We take V to be C and identify the following two diagrams. +F(EU(1)+, infU(1) +e +(E))∗(S 0) +·χC +� +ρ#w +pj +� F(EZ/pj ++, infZ/pj +e +(E))∗(S 0) +·ρ#w +pj +(χC) +� +F(EU(1)+, infU(1) +e +(E))∗+2(S 0) +ρ#w +pj � F(EZ/pj ++, infZ/pj +e +(E))∗+2(S 0), +E∗(BU(1)+) +∪x +� +Bρ∗w +pj +� E∗(BZ/pj ++) +∪Bρ∗w +pj +(x) +� +E∗+2(BU(1)+) +Bρ∗+2 +w +pj � E∗+2(BZ/pj ++), +which finishes the proof. +□ +Lemma 3.10. Let A be an abelian p-group of form Z/pi1 ⊕ · · · ⊕ Z/pim and ρ( w1 +pi1 ,···, wm +pim ) be an +irreducible complex A-representation with (w1, · · · , wm) ∈ A. Let ρ# +( w1 +pi1 ,··· , wm +pim ) be the map +F(EU(1)+, infU(1) +e +(E))∗(S 0) → F(EA+, infA +e (E))∗(S 0) +induced by ρ( w1 +pi1 ,···, wm +pim ). Then Bρ∗ +( w1 +pi1 ,··· , wm +pim )(x) = [w1]E(x1) +F · · · +F [wm]E(xm), corresponds to +χρ( w1 +pi1 +,··· , wm +pim ) = ρ# +( w1 +pi1 ,··· , wm +pim )(χC) under the isomorphism between πA +∗ (F(EA+, infA +e (E))) and E∗(BA+). +Proof. Since ρ( w1 +pi1 ,···, wm +pim ) : A → U(1) is the composition map +Z/pi1 ⊕ · · · ⊕ Z/pim +ρ w1 +pi1 +×···×ρ wm +pim +−−−−−−−−−−−→ T m +µm +U(1) +−−−−−→ U(1) +which induces the composition of E∗-algebra homomorphisms +E∗(BU(1)+) +Bµm,∗ +U(1) +−−−−−→ E∗(BT m ++ ) +B(ρ w1 +pi1 +×···×ρ wm +pim +)∗ +−−−−−−−−−−−−−−→ E∗(BA+). +Note that Bµm,∗ +U(1)(x) = x1 +F · · · +F xm, then we have +Bρ∗ +( w1 +pi1 ,··· , wm +pim )(x) = B(ρ w1 +pi1 × · · · × ρ wm +pim )∗ ◦ Bµm,∗ +U(1)(x) += B(ρ w1 +pi1 × · · · × ρ wm +pim )∗(x1 +F · · · +F xm) += [w1]E(x1) +F · · · +F [wm]E(xm). +This finishes the proof. +□ + +18 +Yangyang Ruan +Theorem 3.11. (Lubin and Tate [28]) For each k ∈ Z and each nature number j, there exists a +unique series [k]E(x) ∈ E∗[[x]] such that +[k]E(x) ≡ kx +mod (x2) and [k]E([pj]E(x)) = [pj]E([k]E(x)). +For convenience, we denote [w1]E(x1) +F · · · +F [wm]E(xm) by α(w1,···,wm). +Lemma 3.12. Let j be a nature number and E be a p-complete complex-oriented spectrum with +an associated formal group of height n. If A is a finite abelian p-group of form Z/pi1 ⊕· · ·⊕Z/pim, +then there is a bijection +ω : pjF(E∗(BA+)) → {α(w1,··· ,wm) ∈ E∗(BA+) | (pjw1, · · · , pjwm) = 0, (w1, · · · , wm) ∈ A} +f ∗ �→ ω( f ∗) = f ∗(x). +Proof. First +suppose +that +A += +Z/pi. +For +f ∗ +∈ +pjF(E∗(BZ/pi ++)) += +HomE∗−alg(E∗[[x]]/[pj]E(x), E∗(BZ/pi ++)), we can identify f ∗ with f ∗(x) since f ∗ is an E∗- +ring homomorphism, which means that ω is injective. +Then we have to prove that ω is +well-defined, namely +f ∗(x) ∈ {α(w1,··· ,wm) ∈ E∗(BA+) | (pjw1, · · · , pjwm) = 0, (w1, · · · , wm) ∈ A}. +As f ∗ is a graded E∗-algebra homomorphism and deg x = 2, we have +0 = f ∗([pj]E(x)) = [pj]E( f ∗(x)) ∈ E2(BZ/pi ++) � E2[[x]]/[pi]E(x). +Notice that [pj]E(x) ≡ pjx mod (x2), then the constant term of f ∗(x) must be zero. +Since +f ∗(x) ∈ E2(BZ/pi ++), we may suppose that f ∗(x) ≡ kx mod (x2), and by Lubin and Tate the- +orem 3.11, we have f ∗(x) = [k]E(x). By the property that [n1]E([n2]E(x)) = [n1n2]E(x), we +have [pj]E([k]E(x)) = [kpj]E(x). Then f ∗ ∈ HomE∗−alg(E∗[[x]]/[pj]E(x), E∗(BZ/pi ++)) implies that +f ∗(x) ∈ {[w]E(x) ∈ E2[[x]]/[pi]E(x) | pjw = 0, w ∈ Z/pi}, so θ is well-defined. Note that for each +[w]E(x) ∈ E2[[x]]/[pi]E(x) with pjw = 0, there is a group homomorphism ρw : Z/pi → Z/pj that +maps 1 to w and Bρ∗ +w(x) = [w]E(x), so Bρ∗ +w is an E∗-algebra homomorphism, so ω is surjective. +Therefore, ω is a well-defined bijection. +For A = Z/pi1 ⊕ · · · ⊕ Z/pim, there are group inclusions ιk : Z/pik → A that maps w ∈ Z/pik to +(0, · · · , 0, w, 0, · · · , 0) ∈ Z/pi1 ⊕ · · · ⊕ Z/pik−1 ⊕ Z/pik ⊕ Z/pik−1 ⊕ · · · ⊕ Z/pim. By Lemma 3.7, we +have +E∗(BA+) � E∗[[x1]]/([pi1]E(x1)) ⊗E∗ · · · ⊗E∗ E∗[[xm]]/([pim]E(xm)). +There is an isomorphism: +HomE∗−alg(E∗[[x]]/[pj]E(x), E∗(BA+)) → +m +� +k=1 +HomE∗−alg(E∗[[x]]/[pj]E(x), E∗[[x1]]/([pik]E(xk))) +f ∗ �→ Bι∗ +1 ◦ f ∗ ⊗ · · · ⊗ Bι∗ +m ◦ f ∗ +We can identify f ∗ ∈ HomE∗−alg(E∗[[x]]/[pj]E(x), E∗(BA+)) with f ∗(x) ∈ E2(BA+). Then the rest +proof is similar to the case of A = Z/pi, we omit it here. +□ + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +19 +Lemma 3.13. Let A be a finite abelian p-group. If G is a finite abelian p-group or U(1), then the +map E∗(B(−)) : Hom(A,G) → HomE∗−alg(E∗(BG+), E∗(BA+)) defined by f �→ E∗(B f) = B f ∗ is +an isomorphism of groups. +Proof. By Lemma 3.12, it is easy to check that E∗(B(−)) is a bijection. Then the remaining thing +is to prove that E∗(B(−)) is a homomorphism of groups. Let [BA+, BG+] denote the homotopy +class from BA+ to BG+. Since G is abelian, we have Hom(A,G)/InnG = Hom(A,G). Note that A +is a finite abelian p-group, by Dwyer and Zabrodsky’s Theorem [12] or Notbohm’s Theorem [33], +there is a bijection +B : Hom(A,G) → [BA+, BG+] +ρ �→ Bρ. +For a topological space X, let ∆X denote the diagonal map X → X × X, then for any ρ1, ρ2 ∈ +Hom(A,G), there are products µG ◦(ρ1 ×ρ2)◦∆A and µBG+ ◦(Bρ1 × Bρ2)◦∆BA+. By the functorial +property of B, B preserves the product, namely +B(µG ◦ (ρ1 × ρ2) ◦ ∆A) = µBG+ ◦ (Bρ1 × Bρ2) ◦ ∆BA+. +Similarly, By the functorial property of E∗(−), E∗(−) preserves the product, namely +E∗(µBG+ ◦ (Bρ1 × Bρ2) ◦ ∆BA+) = ∆∗ +BA+ ◦ (Bρ1 × Bρ2)∗ ◦ µ∗ +BG+. +This finishes our proof. +□ +By Lemma 3.12 and Lemma 3.13, we have +Theorem 3.14. Let j be a nature number and E be a p-complete complex-oriented spectrum with +an associated formal group of height n. If A is a finite abelian p-group, then there are group +isomorphisms +pjF(E∗(BA+)) � {α(w1,··· ,wm) ∈ E∗(BA+) | (pjw1, · · · , pjwm) = 0, (w1, · · · , wm) ∈ A} +� Hom(A, Z/pj) � V(pj|A). +Furthermore, +p∞F(E∗(BA+)) � Hom(A, U(1)) � A. +3.3 +Maps between E∗-cohomology of classifying spaces +Let A1 and A2 be two abelian p-groups Z/pi1 ⊕ · · · ⊕ Z/pim and Z/pj1 ⊕ · · · ⊕ Z/pjk. Then any +homomorphism f ∈ Hom(A1, A2) is determined by an integer m×k-matrix F ∈ Mm×k(Z(p)). Since +each nature number i can be identified with a self-map of U(1) of degree i, F can be identified +with a map from T m to T k, and there are two commutative diagrams: +A1 +ρ 1 +pi1 +×···×ρ +1 +pim +−−−−−−−−−−−→ T m +f +� +F +� +A2 +ρ 1 +pj1 +×···×ρ 1 +pjk +−−−−−−−−−−−→ T k, +BA1 +B(ρ 1 +pi1 +×···×ρ +1 +pim +) +−−−−−−−−−−−−−→ BT m +B f +� +BF +� +BA2 +B(ρ 1 +pj1 +×···×ρ 1 +pjk +) +−−−−−−−−−−−−−→ BT k. + +20 +Yangyang Ruan +A1 and A2 are associated with the following two fibrations +T m/A1 � T m −−−−−→ BA1 +B(ρ 1 +pi1 +×···×ρ +1 +pim +) +−−−−−−−−−−−−−→ BT m, T k/A2 � T k −−−−−→ BA2 +B(ρ 1 +pj1 +×···×ρ 1 +pjk +) +−−−−−−−−−−−−−→ BT k. +Lemma 3.15. Let E be a p-complete complex-oriented spectrum with an associated formal group +of height n. Then there is a Leray-Serre spectral sequence of T m → ET m → BT m with the E2- +page Hs(BT m; Et(T m)) � Hs(BT m; Z/p) ⊗ Et(T m) � Z/p[[x1, x2, · · · , xm]] ⊗ ∧E∗[y1, y2, · · · , ym], +and its only nontrivial differential is d2(1 ⊗ yi) = xi for 1 ≤ i ≤ m, which implies that it collapses +at E3-page. +Proof. Since ET m is contractible, then the only possible differential is d2(1 ⊗ yi) = xi for 1 ≤ i ≤ +m. +□ +Lemma 3.16. Let E be a p-complete complex-oriented spectrum with an associated formal group +of height n. Then there is a Leray-Serre spectral sequences of T m → BA1 → BT m with the E2- +page Hs(BT m; Et(T m)) � Hs(BT m; Z/p) ⊗ Et(T m) � Z/p[[x1, x2, · · · , xm]] ⊗ ∧E∗[y1, y2, · · · , ym], +and its only nontrivial differential is d2(1 ⊗ yi) = [pij]E(xj) for 1 ≤ j ≤ m, which implies that it +collapses at E3-page. +Proof. The following commutative diagram +BA1 +B(ρ 1 +pi1 +×···×ρ +1 +pim +) +−−−−−−−−−−−−−→ BT m +� +1BTm +� +ET m +−−−−−→ +BT m +induces a map of Leray-Serre spectral sequences, which gives differentials d2(1 ⊗ yi) = [pij]E(xj) +for 1 ≤ j ≤ m. Then by Lemma 3.7, we conclude that it collapses at E3-page. +□ +Theorem 3.17. Let E be a p-complete complex-oriented spectrum with an associated formal +group of height n. Let A1 and A2 be two abelian p-groups Z/pi1⊕· · ·⊕Z/pim and Z/pj1⊕· · ·⊕Z/pjk. +Then any abelian group homomorphism f ∈ Hom(A1, A2) is determined by an integer m×k-matrix +F ∈ Mm×k(Z(p)), and the homomorphism B f ∗ : E∗(BA2+) → E∗(BA1+) can be identified with the +E3-page map of Leray-Serre spectral sequences for two associated fibrations +T m/A1 � T m −−−−−→ BA1 +B(ρ 1 +pi1 +×···×ρ +1 +pim +) +−−−−−−−−−−−−−→ BT m, T k/A2 � T k −−−−−→ BA2 +B(ρ 1 +pj1 +×···×ρ 1 +pjk +) +−−−−−−−−−−−−−→ BT k. +where the map of these two fibrations is given by the following commutative diagram: +BA1 +B(ρ 1 +pi1 +×···×ρ +1 +pim +) +−−−−−−−−−−−−−→ BT m +B f +� +BF +� +BA2 +B(ρ 1 +pj1 +×···×ρ 1 +pjk +) +−−−−−−−−−−−−−→ BT k. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +21 +3.4 +The homotopy groups π∗(TA,C(E)) +The following lemma determines all complex representations V of A such that VC = 0. +Lemma 3.18. Let A be an abelian group of form Z/pi1 ⊕ · · · ⊕ Z/pim and C be its subgroup +Z/pj1 ⊕ · · · ⊕ Z/pjm with a group inclusion +ϕ : Z/pj1 ⊕ · · · ⊕ Z/pjm → Z/pi1 ⊕ · · · ⊕ Z/pim +(w1, · · · , wk) �→ (pi1−j1w1, · · · , pim−jmwm). +There is a group homomorphism from A/C to A as follows: +φ : Z/pi1−j1 ⊕ · · · ⊕ Z/pim−jm → Z/pi1 ⊕ · · · ⊕ Z/pim +(w1, · · · , wm) �→ (pj1w1, · · · , pjmwm). +Then +{ρ( w1 +pi1 ,···, wm +pim ) = ρ w1 +pi1 · · · ρ wm +pim : A → U(1) | (w1, · · · , wm) ∈ A − imφ(A/C)} +forms all irreducible complex representations V of A such that VC = 0. +Proof. Note that +{ρ( w1 +pi1 ,··· , wm +pim ) : A → U(1) | (w1, · · · , wm) ∈ A} +formed all irreducible complex representations of A. Then for any (u1, · · · , um) ∈ C, we have +ρ( w1 +pi1 ,···, wm +pim )(ϕ(u1, · · · , um)) = ρ w1 +pi1 (pi1−j1u1) · · · wm +pim (pim−jmum) += e +2πi( w1u1 +pj1 +···+ wmum +pjm ) += +ß1 +if pj1|w1, · · · , pjm|wm, +nonconstant Otherwise. +And pj1|w1, · · · , pjm|wm ⇔ (w1, · · · , wm) ∈ imφ(A/C). +□ +Now, we prove Theorem 1.6. +Proof of Theorem 1.6. From Theorem 3.3, it follows that π∗(TA,C(E)) is the localization +of π∗(F(EA+, infA +e (E))) +� +E∗(BA+) obtained by inverting +the Euler classes χV +∈ +F(EA+, infA +e (E))|V|(S |V|−V) of those complex representations V of A such that VC = 0. By Theorem +3.7, we have +E∗(BA+) � E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)). +By Lemma 3.18, we have {ρ( w1 +pi1 ,···, wm +pim ) : A → U(1) | (w1, · · · , wm) ∈ A − imφ(A/C)} forms all +irreducible complex representations V of A such that VC = 0. Each representation ρ( w1 +pi1 ,···, wm +pim ) : +A → U(1) induces a homormorphism Bρ∗ +( w1 +pi1 ,···, wm +pim ) : E∗(BU(1)+) � E∗[[x]] → E∗(BA+), and +by Lemma 3.10, the image Bρ∗ +( w1 +pi1 ,··· , wm +pim )(x) is the Euler class [w1]E(x1) +F · · · +F [wm]E(xm) = +α(w1,··· ,wm). +□ + +22 +Yangyang Ruan +4 +Generalized relations of roots and coefficients of a polynomial +In this section, we prove generalized relations of roots and coefficients of a polynomial, namely +Theorem 1.7. Let R be a commutative ring with 1. Recall that there is a map λ : R[x] → +Pmap(R, R) with λ( f(x)) = [ f(x)] for f(x) ∈ R[x]. Let R[x]n denote the set of polynomials of +degree at most n and λR[x]n denote the map that restricts λ to R[x]n, then what conditions does R +satisfy with such that λR[x]n is injective ? To give a sufficient condition, we take a fresh look at +the equality f(r) = 0 induced by a root r ∈ R of a polynomial map [ f(x)] ∈ Pmap(R, R). Without +loss of generality, we may suppose that f(x) = a0 + a1x + · · · + anxn with a0, a1, · · · , an ∈ R. +f(r) = 0 means that the “R-vector” (a0, a1, · · · , an) is a solution of the homogeneous R-linear +equation x0 + rx1 + · · · + rnxn = 0. Then we need the definition of “R-vector”, R-linear and so on. +4.1 +Basic concepts +Definition 4.1. Let R be a commutative ring with 1 and n be a positive integer. +Let Rn = +{(a1, a2, · · · , an) | ai ∈ R, 1 ≤ i ≤ n}, then for (a1, a2, · · · , an), (b1, b2, · · · , bn) ∈ Rn, +(a1, a2, · · · , an) = (b1, b2, · · · , bn) ⇔ ai = bi(1 ≤ i ≤ n) ∈ R. +Rn has two operations as follows, for (a1, a2, · · · , an), (b1, b2, · · · , bn) ∈ Rn, r ∈ R, then +(i) Vector addition: (a1, a2, · · · , an) + (b1, b2, · · · , bn) = (a1 + b1, a2 + b2, · · · , an + bn); +(ii) Scalar multiplication: r(a1, a2, · · · , an) = (ra1, ra2, · · · , ran). +These two operations on Rn satisfy the following eight rules. For any a, b, c ∈ Rn, r, k ∈ R, +1. a + b = b + a; +2. (a + b) + c = a + (b + c); +3. there is a unique vector 0 = (0, 0, · · · , 0) in Rn such that 0 + a = a + 0 = a, then 0 is called +the zero vector of Rn; +4. for any a = (a1, a2, · · · , an) ∈ Rn, there is a vector −a = (−a1, −a2, · · · , −an) ∈ Rn, called +the negative of a, such that a + (−a) = (−a) + a = 0; +5. 1(a) = a; +6. (kr)a = k(ra); +7. (k + r)a = ka + ra; +8. r(a + b) = ra + rb. +Then Rn is called an n-dimensional R-vector space or R-linear space, and any a ∈ Rn is called +an n-dimensional R-vector. +And we have the notion of subspace. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +23 +Definition 4.2. If a nonempty subset U of Rn satisfies that +(i) a, b ∈ U ⇒ a + b ∈ U; +(ii) a ∈ U, r ∈ R ⇒ ra ∈ U. Then U is called an R-vector subspace of Rn. +Proposition 4.3. Let R be a commutative ring with 1. For t1, t2, · · · , tn ∈ R, if there is a system of +homogeneous R-linear equations + + + + + + + + + + + +x0 + t1x1 + t2 +1x2 + · · · + tn−1 +1 +xn−1 = 0 +x0 + t2x1 + t2 +2x2 + · · · + tn−1 +2 +xn−1 = 0 +... +x0 + tnx1 + t2 +nx2 + · · · + tn−1 +n +xn−1 = 0 +(4.1) +with variables x0, x1, · · · , xn−1. Then the solution of Equations 4.1 is an R-vector subspace of Rn. +4.2 +n-tuple of a polynomial over a commutative ring +Now, we give a sufficient condition such that the solution of Equations 4.1 is unique. +Lemma 4.4. Let R be a commutative ring with 1. For t1, t2, · · · , tn ∈ R, any 1 ≤ i � j ≤ n, +ti − tj is not zero or a zero divisor. If there is a system of homogeneous R-linear equations 4.1 with +variables x0, x1, · · · , xn−1, then the solution of Equations 4.1 is the subspace {0} of Rn. +Proof. For constants c0, c1, · · · , cn−1, d ∈ R, if t is not zero or a zero divisor, then the solutions of +c0x0 + c1x1 + · · · + cn−1xn−1 = d and tc0x0 + tc1x1 + · · · + tcn−1xn−1 = td are the same, that is +c0x0 + c1x1 + · · · + cn−1xn−1 = d ⇔ tc0x0 + tc1x1 + · · · + tcn−1xn−1 = td. +We use Gaussian elimination to solve the R-linear equations: +â1 t1 t2 +1 · · · tn−1 +1 +1 t2 t2 +2 · · · tn−1 +2 +1 t3 t2 +3 · · · tn−1 +3 +... +... +... ... +... +1 tn t2 +n · · · tn−1 +n +ì +→ +â1 +t1 +t2 +1 +· · · +tn−1 +1 +0 t2 − t1 t2 +2 − t2 +1 · · · tn−1 +2 +− tn−1 +1 +0 t3 − t1 t2 +3 − t2 +1 · · · tn−1 +3 +− tn−1 +1 +... +... +... +... +... +0 tn − t1 t2 +n − t2 +1 · · · tn−1 +n +− tn−1 +1 +ì +→ +â1 t1 +t2 +1 +· · · +tn−1 +1 +0 1 t1 + t2 · · · �n−2 +i=0 tn−2−i +1 +ti +2 +0 1 t1 + t3 · · · �n−2 +i=0 tn−2−i +1 +ti +3 +... +... +... +... +... +0 1 t1 + tn · · · �n−2 +i=0 tn−2−i +1 +ti +n +ì +, +then inductively carry out the above process and finally obtain the upper triangular matrix + + + + + + + + +1 t1 +t2 +1 +· · · +tn−1 +1 +0 1 t1 + t2 · · · +�n−2 +i=0 tn−2−i +1 +ti +2 +0 0 +1 +· · · �n−2 +i=1 tn−2−i +1 +�i−1 +j=0 ti−1−j +2 +t j +3 +... +... +... +... +... +0 0 +0 +· · · +1 + + + + + + + + +, +this finishes the proof. +□ + +24 +Yangyang Ruan +Definition 4.5. Let R be a commutative ring with 1. we define an n-tuple {t1, t2, · · · , tn} of R such +that for any 1 ≤ i � j ≤ n, ti − tj is not zero or a zero divisor; if for any 1 ≤ i � j ≤ n, ti − tj +is invertible in R, we call {t1, t2, · · · , tn} an invertible n-tuple of R. Let f(x) be a polynomial over +R, we call {r1, r2, · · · , rn} an n-tuple of f(x) if it is an n-tuple of R and also is a subset of roots of +f(x). +Definition 4.6. Let R be a commutative ring with 1, and d is not zero or a zero divisor in R. For +r ∈ R, we call t is divisible by d if there is an element t′ ∈ R such that t = dt′. +Remark 4.7. Since d is not zero or a zero divisor in R, for t ∈ R, the solution of t = dx in R is +unique. +Proposition 4.8. Let R be a commutative ring with 1. If R has an n-tuple {t1, t2, · · · , tn}, then +λR[x]n−1 is injective. +Proof. For any two polynomial f1(x) � f2(x) ∈ R[x]n−1, without loss of generality we may +suppose that f1(x) = �n−1 +k=0 akxk, f2(x) = �n−1 +k=0 bkxk. +Then f1(x) � f2(x) implies that there +is 1 ≤ k0 ≤ n − 1 such that ak0 − bk0 +� 0. +If λR[x]n−1( f1(x)) = λR[x]n−1( f2(x)), that is +[ f1(y) − f2(y) = ( f1 − f2)(y)] = [0], which implies that ( f1 − f2)(ti) = 0 for any 1 ≤ i ≤ n. Then the +n-dimensional R-vector (a0 − b0, a1 − b1, · · · , an−1 − bn−1) is a solution of Equations 4.1. And by +Lemma 4.4, the solution of Equations 4.1 is {0}. So (a0−b0, a1−b1, · · · , an−1−bn−1) = (0, 0, · · · , 0), +which contradicts to our assumption that ak0 − bk0 � 0. This finishes the proof. +□ +Lemma 4.9. Let R be a commutative ring with 1 and R has an n-tuple {t1, t2, · · · , tn}. Let αi +denote the column n-dimensional R-vector (ti +1, ti +2, · · · , ti +n)T. If 0 ≤ i1 < i2 < · · · < in−1, then +det(α0, αi1, αi2, · · · , αin−1) is divisible by det(α0, α1, · · · , αn−1). +Proof. By directly computation, we have +det(α0, α1, · · · , αn−1) = +� +1≤j m, then by Lemma 4.4 the solution of Equations 4.1 is the sub- +space {0}. Since (a0, a1, · · · , an) is a solution of Equations 4.1, we must have (a0, a1, · · · , an) = 0. +(ii)(iii) If n ≤ m, then by Lemma 4.10, we finish the proof. +□ +Corollary 1.9 and Corollary 1.10 can be easily deduced from Theorem 1.7. +4.3 +Applications of Vanishing ring condition +Follow the notions of [19, Section 5], in this subsection we assume that E is a complex-oriented +cohomology theory, particularly p-complete theory with an associated formal group of height n. +Then there are two ring homomorphism +λ : E∗[[x1]]/([pj]E(x1))[x] → Pmap(E∗[[x1]]/([pj]E(x1)), E∗[[x1]]/([pj]E(x1))), +λ′ : E∗[x1]/(gj(x1))[x] → Pmap(E∗[x1]/(gj(x1)), E∗[x1]/(gj(x1))). +Proposition 4.11. λ is identified with λ′ by the commutative diagram +E∗[[x1]]/([pj]E(x1))[x] +λ +−−−−−→ Pmap(E∗[[x1]]/([pj]E(x1)), E∗[[x1]]/([pj]E(x1))) +� +� +� +� +E∗[x1]/(gj(x1)[x]) +λ′ +−−−−−→ +Pmap(E∗[x1]/(gj(x1)), E∗[x1]/(gj(x1))). +Proposition 4.12. Under the isomorphism +η : E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)) � E∗[x1, · · · , xm]/(gi1(x1), · · · , gim(xm)), +λ is identified with λ′ by the commutative diagram +E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm))[x] +λ +−−−−−−−−−−−→ Pmap(E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim ]E(xm)), E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim ]E(xm)) +� +� +� +� +E∗[x1, · · · , xm]/(gi1(x1), · · · , gim (xm))[x] +λ′ +−−−−−−−−−−−→ +Pmap(E∗[x1, · · · , xm]/(gi1(x1), · · · , gim (xm)), E∗[x1, · · · , xm]/(gi1(x1), · · · , gim (xm))). +Corollary 4.13. +(i) [16, Theorem 1.1] If G is a finite p-group, then tG(infG +e (K(n)))G ≃ ∗; +(ii) [9, Proposition 3.10] Let G be a finite p-group and H be a non-cyclic subgroup, then +ΦHKUG ≃ ∗. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +27 +Proof. (i) By the proof of [16, Theorem 1.1], it suffices to prove that tZ/p(infZ/p +e +(K(n)))Z/p ≃ ∗. +Let f(x) be v−1 +n +[p]K(n)(x) +xpn−1 . Since xpn, ([2]K(n)(x))pn, · · · , ([p−1]K(n)(x))pn are different invertible roots +in π∗(tZ/p(infZ/p +e +(K(n)))Z/p) = π∗(TZ/p,Z/p(K(n))) = K(n)∗((x))/(vnxpn). By using Theorem 1.10, +we have tZ/p(infZ/p +e +(K(n)))Z/p ≃ ∗. +(ii) By the proof of [9, Proposition 3.10], it suffices to prove that ΦZ/p×Z/pKUZ/p×Z/p ≃ ∗. Let +f(x) be (x+1)p−1 +x +. Since the Euler classes x1 − 1, x2 +1 − 1, · · · , xp−1 +1 +− 1, x2 − 1 are different invertible +roots in π∗(ΦZ/p×Z/pKUZ/p×Z/p) = L−1 +Z/p×Z/pZ[x1, x2]/(xp +1 − 1, xp +2 − 1), where the multiplicatively +closed set LZ/p×Z/p is generated by all Euler classes induced by one dimensional complex represen- +tations of Z/p×Z/p. Note that the difference of any two roots has the forms (xm +1 −xn +1) = xn +1(xm−n +1 +−1) +or (x2−xn +1) = xn +1(xp−n +1 +x2−1), since xn +1 is invertible in L−1 +Z/p×Z/pZ[x1, x2]/(xp +1−1, xp +2−1) and xp−n +1 +x2−1 +is the Euler class, we conclude that ΦZ/p×Z/pKUZ/p×Z/p ≃ ∗. +□ +5 +Algebraic periodicity and Landweber exactness +Most of this section are due to Greenlees-Sadofsky [16] and Hovey [21], we just add some +details here. +5.1 +Algebraic periodicity +There are two kinds of definitions of vn-periodic for a p-local and complex-oriented spectrum +E due to Greenlees-Sadofsky [16] and Hovey [21]. They are almost the same, and Hovey’s defi- +nition 5.5 is stronger than Greenlees-Sadofsky’s definition 5.3. In this paper, we choose Hovey’s +definition as our definition of vn-periodic for a p-local and complex-oriented spectrum E. +Recall a finite spectrum X has type n if K(n − 1)∗X = 0 but K(n)∗X � 0. +Lemma 5.1. (Hopkins and Smith [18]) All finite spectrum of type n have the same Bousfield class +and is denoted by F(n). F(n) has a vn self-map and its telescope is denoted by T(n). +Let M(pi0, vi1 +1 , · · · , vin−1 +n−1) be a finite spectrum with +π∗(BP ∧ M(pi0, vi1 +1 , · · · , vin−1 +n−1)) = BP∗/(pi0, vi1 +1 , · · · , vin−1 +n−1). +Such spectra are of type n and are called generalized Moore spectra. M(pi0, vi1 +1 , · · · , vin−1 +n−1) are +guaranteed to exist for sufficiently large multi-indices I = (i0, · · · , in−1) by the periodicity theorem +of Smith [18], written up in [35, Section 6.4]. +We use the notation X∧ +In for the completion of X with respect to the ideal In = (p, v1, · · · , vn−1) ⊂ +BP∗. More precisely, the construction is +X∧ +In = +lim +←− +(i0,i1,··· ,in−1) +(X ∧ M(pi0, vi1 +1 , · · · , vin−1 +n−1)), +(5.1) +where the inverse limit is taken over maps +M(pj0, vj1 +1 , · · · , vjn−1 +n−1) → M(pi0, vi1 +1 , · · · , vin−1 +n−1) + +28 +Yangyang Ruan +commuting with inclusion of the bottom cell. Such maps are easily constructed by courtesy of +the nilpotence theorem of [18] (see for example [18, Proposition 3.7] for existence of these maps +and some uniqueness properties). By [34, Definition 1.4], for any spectrum E there is an E- +localization functor LE : SH(e) → SH(e). The following theorem says that localization with +respect to F(n) is completion at In. +Theorem 5.2. (Hovey[21, Theorem 2.1]) For any spectrum X, the map X +→ +lim +←−−(X ∧ +M(pi0, vi1 +1 , · · · , vin−1 +n−1)) is a F(n)-localization, namely LF(n)X = X∧ +In. +If E is p-local and complex-oriented, then there is a unique map f : BP → E such that +f ∗ : BP∗(CP∞) � BP∗[[xBP]] → E∗(CP∞) � E∗[[xE]] +maps the BP-orientation xBP to the E-orientation xE. And there is a homomorphism +f ∧ 1M(pi0,vi1 +1 ,··· ,vin−1 +n−1 )∗ : π∗(BP ∧ M(pi0, vi1 +1 , · · · , vin−1 +n−1)) → π∗(E ∧ M(pi0, vi1 +1 , · · · , vin−1 +n−1)) +and we still use vi denote f ∧ 1M(pi0,vi1 +1 ,···,v +in−1 +n−1 )∗(vi). +Definition 5.3. (Greenlees-Sadofsky’s vn-periodic, [16, Definition 1.3]) Let E be a p-local and +complex-oriented spectrum, E is called vn-periodic if vn is a unit on the nontrivial spectrum +E ∧ M(pi0, vi1 +1 , · · · , vin−1 +n−1) for sufficiently large multi-indices I = (i0, i1, · · · , in−1). +Remark 5.4. +(i) The above definition is independent of the choice of multi-index I and of the +spectrum M(pi0, vi1 +1 , · · · , vin−1 +n−1). By Theorem 5.2, the equivalent definition of vn-periodic for +E is that vn is a unit on the nontrivial spectrum LF(n)E. +(ii) If a p-local and complex-oriented spectrum E is vn-periodic, then n is unique. +There is another definition of vn-periodic due to Hovey [21]. +Definition 5.5. (Hovey’s vn-periodic, [21]) Let E be a p-local and complex-oriented spectrum. +(i) E is called at most vn-periodic if vn is a unit on E∗/In, by the exactness of +E∗/In +·vn +−−−−−→ E∗/In −−−−−→ E∗/In+1, +which is equivalent to E∗/In+1 = 0. +(ii) E is called at least vn-periodic if E∗/In � 0. +E is called vn-periodic if vn is a unit of E∗/In � 0. +If we say some spectrum is vn-periodic, we mean it in the sense of Hovey’s definition 5.5. +The following proposition says that Hovey’s vn-periodic 5.5 implies that Greenlees-Sadofsky’s +vn-periodic 5.3. +Proposition 5.6. Let E be a p-local and complex-oriented spectrum. If vn is a unit of E∗/In � 0, +then vn is a unit on a nontrivial spectrum E ∧ M(pi0, vi1 +1 , · · · , vin−1 +n−1). + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +29 +Proof. Suppose vn ≡ u mod In for some unit u of E∗/In, then there exists an element t ∈ In such +that vn = u + t. Since u−1 − u−2t + u−3t2 − · · · is a power series that converges in (E∗)∧ +In, vn is a unit +of (E∗)∧ +In. By Theorem 5.2, vn is a unit in π∗(LF(n)E) = (E∗)∧ +In. +Since there exists a generalized Moore spectrum M(pi0, vi1 +1 , · · · , vin−1 +n−1) of type n with large +enough multi-index I = (i0, i1, · · · , in−1), from the construction 5.1 for E, it follows that vn is +a unit in +π∗(E ∧ M(pi0, vi1 +1 , · · · , vin−1 +n−1)) = E∗/(pi0, vi1 +1 , · · · , vin−1 +n−1). +This completes the proof. +□ +5.2 +Landweber exactness +The Brown-Peterson spectrum BP is a ring spectrum with the product map µBP : BP∧BP → BP +and the unit map ηBP : S → BP. E is called a BP-module spectrum if there is a BP-module map +ν : BP ∧ E → E such that the following diagram commute. +BP ∧ BP ∧ E +1BP∧ν +� +µBP∧1E +� BP ∧ E +ν +� +BP ∧ E +ν +� E, +S ∧ E +≃ +� +ηBP∧1E� BP ∧ E +ν +� +E +1E +� E. +A particular good kind of BP-module spectrum is a Landweber exact spectrum [27]. +Proposition 5.7. (The Landweber exact functor [27]) Let F be a formal group law and p a prime, +˜vi the coefficient of xpi in +[p]F(x) = ˜v0x + ˜v1xp + · · · + ˜vixpi + · · · . +If for each i multiplication by ˜vi is monic on Z(p)[˜v1, ˜v2, · · · ]/(˜v0, ˜v1, · · · , ˜vi−1), then F is +Landweber exact and hence gives a cohomology theory E∗(−) = BP∗(−) ⊗BP∗ Z(p)[˜v1, ˜v2, · · · ]. +By Brown representation theorem [2] this defines a spectrum and the spectra arising this way are +called Landweber exact spectra. +Recall a lemma. +Lemma 5.8. (Ravenel [34, Lemma 1.34.]) Let X be a non-equivariant spectrum and f : �d X → +X be a self-map of X with cofibre Y. Let T(X) denote the telescope lim +−−→ +�−id X of f. Then ⟨X⟩ = +⟨T(X)⟩ ∨ ⟨Y⟩. +For two non-equivariant spectra E and F, recall that ⟨F⟩ ≤ ⟨E⟩ if for any spectrum X ∈ SH(e), +E∗X = 0 ⇒ F∗X = 0. The Landweber exact spectrum with the assumption of periodicity deter- +mines its Bousfield class. +Lemma 5.9. Let E be Landweber exact. +(i) If E is at most vn-periodic, then ⟨E⟩ ≤ ⟨E(n)⟩; +(iI) if E is at least vn-periodic, then ⟨E⟩ ≥ ⟨E(n)⟩. + +30 +Yangyang Ruan +Proof. Applying Ravenel’s lemma 5.8 repeatedly using vn-self map 5.1, we get +⟨S 0⟩ = ⟨T(0)⟩ ∨ · · · ∨ ⟨T(n)⟩ ∨ ⟨F(n + 1)⟩. +Smashing with E, we have +⟨E⟩ = ⟨E ∧ T(0)⟩ ∨ · · · ∨ ⟨E ∧ T(n)⟩ ∨ ⟨E ∧ F(n + 1)⟩. +Since E is Landweber exact, E is a BP-module spectrum, so E is a retract of BP ∧ E, then +⟨E⟩ = ⟨BP ∧ E⟩ = ⟨BP ∧ E ∧ T(0)⟩ ∨ · · · ∨ ⟨BP ∧ E ∧ T(n)⟩ ∨ ⟨BP ∧ E ∧ F(n + 1)⟩. +By Hovey’s theorem [21, Theorem 1.9.] that ⟨BP ∧ T(n)⟩ = ⟨K(n)⟩, we have +⟨E⟩ = ⟨E ∧ K(0)⟩ ∨ · · · ∨ ⟨E ∧ K(n)⟩ ∨ ⟨BP ∧ E ∧ F(n + 1)⟩. +If E is at most vn-periodic, then by proposition 5.6, we have E ∧ F(n + 1) = 0 and +⟨E⟩ = ⟨E ∧ K(0)⟩ ∨ · · · ∨ ⟨E ∧ K(n)⟩ ≤ ⟨K(0)⟩ ∨ · · · ∨ ⟨K(n)⟩ = ⟨E(n)⟩. +If E is at least vn-periodic, that is E∗/In � 0, then we get E∗/Ij � 0 for j ≤ n. And by +proposition 5.6, we have E ∧ F( j) � 0 for j ≤ n. Since E is Landweber exact, E∗/Ij → v−1 +j E∗/Ij +is injective, so v−1 +j E∗/Ij � 0 and E ∧T( j) � 0 for j ≤ n. Note that ⟨E ∧T( j)⟩ = ⟨E ∧ K( j)⟩ and for +any F ∈ SH(e), ⟨F ∧ K( j)⟩ is either 0 or ⟨K( j)⟩, then we have ⟨E ∧ K( j)⟩ = ⟨K( j)⟩ for j ≤ n and +⟨E⟩ = ⟨K(0)⟩ ∨ · · · ∨ ⟨K(n)⟩ ∨ ⟨BP ∧ E ∧ F(n + 1)⟩ ≥ ⟨K(0)⟩ ∨ · · · ∨ ⟨K(n)⟩ = ⟨E(n)⟩. +□ +Theorem 5.10. ([21, Corollary 1.12]) If E is a vn-periodic Landweber exact spectrum, then +⟨E⟩ = ⟨E(n)⟩ = ⟨K(0) ∨ · · · ∨ K(n)⟩. +Lemma 5.11. If E is Landweber exact, then TA,C(E) is Landweber exact. +Proof. Note that E∗(BA+) is a finite free module over E∗. Since E is Landweber exact, v0, · · · , vi +form a regular sequence of E∗(BA+) for all p and i. By Theorem 1.6, we know that π∗(TA,C(E)) +is a localization of E∗(BA+). Since the localization will not increase the size of the kernel of +E∗(BA+)/Ii +·vi→ E∗(BA+)/Ii, then v0, · · · , vi form a regular sequence of π∗(TA,C(E)) for all p and i. +This finishes the proof. +□ +6 +Generalized Tate construction lowers Bousfield class +In this section, we prove the following theorem. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +31 +Theorem 6.1. (Generalized Tate construction lowers Bousfield class) Let m be a positive inte- +ger and E be a p-complete, complex oriented spectrum with an associated formal group of height +n. Let A be an abelian p-group of form Z/pi1 ⊕· · ·⊕Z/pim and C be its subgroup Z/pj1 ⊕· · ·⊕Z/pjm +with ik ≤ jk for 1 ≤ k ≤ m. There is a group homomorphism φ from A/C to A as follows: +φ : Z/pi1−j1 ⊕ Z/pi2−j2 ⊕ · · · ⊕ Z/pim−jm → Z/pi1 ⊕ Z/pi2 ⊕ · · · ⊕ Z/pim +(w1, w2, · · · , wm) �→ (pi1−j1w1, pi2−j2w2, · · · , pim−jmwm). +If E is Landweber exact, then +(i) TA,C(E) is Landweber exact; +(ii) TA,C(E) is at least vn−rankp(C)-periodic and at most vn−t-periodic; +(iii) ⟨TA,C(E)⟩ = ⟨E(n − sA,C;E)⟩ for some sA,C;E with t ≤ sA,C;E ≤ rankp(C), When k > n, +E(n − k) = ∗. +where +t = max +j∈N+ ⌈ +logp |V(pj|A)| − logp |V(pj|imφ(A/C))| +j +⌉. +Especially, if A is a finite abelian p-group and C is its direct summand, then blue-shift number +sA,C;E = rankp(C); A = Z/pj, then blue-shift number sA,C;E = 1. However, t does not always +equal rankp(C). For example, A = Z/p2 ⊕ Z/p2 ⊕ Z/p2 and C = Z/p ⊕ Z/p ⊕ Z/p, then t = 2 but +rankp(C) = 3. +The (i) of Theorem 6.1 is proved by Lemma 5.11. By Theorem 5.10, the (i) and (ii) of Theorem +6.1 imply the (iii) of Theorem 6.1. It remains to prove the (ii) of Theorem 6.1. +In Theorem 6.1, E∗ � E∗ is a local ring with the maximal ideal In and E is vn-periodic which is +equivalent to E∗/In+1 = 0 but E∗/In � 0. To determine the periodicity of TA,C(E), we define the +following integer sA,C;E motivated by Definition 5.5. +Definition 6.2. There is an ascending chain of ideals +I−1 = ∅ ⊆ I0 = (0) ⊆ I1 ⊆ · · · ⊆ In+1−q ⊆ · · · ⊆ In+1 = π∗(TA,C(E)), +then sA,C;E is the maximal integer q such that In+1−q = π∗(TA,C(E)) and also is the minimal +integer q such that In−q ⊊ π∗(TA,C(E)). Which is equivalent to +π∗(TA,C(E))/In+1−q = +ß0 +if 0 ≤ q ≤ sA,C;E, +� 0 +if sA,C;E < q. +Remark 6.3. By Theorem 1.6, π∗(TA,C(E)) is an E∗-module. And by Lemma 5.11, TA,C(E) is +Landweber exact. So v0, v1, · · · , vn−sA,C;E form a regular π∗(TA,C(E))-sequence in E∗ and this +sequence is maximal. Then the integer n + 1 − sA,C;E is the depth of π∗(TA,C(E)) or the projective +dimension of π∗(TA,C(E)) as an E∗-module. +There is an easy observation. + +32 +Yangyang Ruan +Lemma 6.4. The blue-shift number +sA,C;E = sA,C;E. +Then the (ii) of Theorem 6.1 is equivalent to t ≤ sA,C;E ≤ rankp(C) where +t = max +j∈N+ ⌈ +logp |V(pj|A)| − logp |V(pj|imφ(A/C))| +j +⌉. +And we divide its proof into three cases: +(1) A = C is an elementary abelian p-group; +(2) A = C is a general abelian p-group; +(3) A is a general abelian p-group and C is its proper subgroup. +Although (1) is a special case of (2), the whole proof for the case (1) is inspiring and the proof +for the upper bound of sA,A;E is different from the corresponding proof for the case (2). For all +above three cases, the key proof lies in the looking for lower bounds of sA,C;E. If we could find +some-tuple of pj-series [pj]E(x) or its Weierstrass polynomial gj(x) (In this section, we do not +distinguish between [pj]E(x) and gj(x)) in π∗(TA,C(E)), then by Corollary 1.10 we get a lower +bound of sA,C;E. +6.1 +Proof for the case (1) A = C is an elementary abelian p-group +Let A be an elementary abelian group with rankp(A) = m. From Proposition 3.1 and Theorem +1.6, it follows that +π∗(TA,A(E)) � L−1 +A E∗[[x1, x2, · · · , xm]]/([p]E(x1), · · · , [p]E(xm)), +where the multiplicatively closed set LA is generated by the set +MA = {α(w1,w2,··· ,wm) | (w1, w2, · · · , wm) ∈ A − {e} = A∗}. +And we have +π∗(TA,A(E))/In+1−q � �L−1 +A,n+1−qE∗/In+1−q[[x1, x2, · · · , xm]]/([p]E(x1), · · · , [p]E(xm)), +where the multiplicatively closed set �LA,n+1−q is mod In+1−q reduction of LA and generated by the +set +‹ +MA,n+1−q = {�α(w1,w2,···,wm) | (w1, w2, · · · , wm) ∈ A∗}. +Note that +[p]E(x) = vn+1−qxpn+1−q + · · · + vnxpn ∈ π∗(TA,A(E))/In+1−q[x]. +Let g1,n+1−q(x) = vn+1−qx+· · ·+vnxpq−1, then [p]E(x) = g1,n+1−q(xpn+1−q) mod In+1−q. The follow- +ing lemma gives a pm-tuple of [p]E(x) in π∗(TA,A(E)) under the assumption that π∗(TA,A(E)) � 0. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +33 +Lemma 6.5. If π∗(TA,A(E)) � 0, then pF(π∗(TA,A(E))) is a pm-tuple of [p]E(x) in π∗(TA,A(E)). +Furthermore, follow the notation in [19, Lemma 6.3], for a, b ∈ π∗(TA,A(E)), we will write a ∼ b if +a = ε · b where ε is a unit in π∗(TA,A(E)), let pF(π∗(TA,A(E)))/ ∼ denote pF(π∗(TA,A(E))) quotient +this equivalent relation, then pF(π∗(TA,A(E)))/ ∼ is an abelian group. +Proof. By Theorem 3.14, we have +pF(π∗(TA,A(E))) � {α(w1,w2,···,wm) ∈ π∗(TA,A(E)) | (w1, w2, · · · , wm) ∈ A}. +To prove that pF(π∗(TA,A(E))) is a |pF(π∗(TA,A(E)))|-tuple of [p]E(x) in π∗(TA,A(E)), we first +check that pF(π∗(TA,A(E))) is a set of roots of [p]E(x). +By Proposition 3.5, we have for +(w1, w2, · · · , wm) ∈ A, (pw1, pw2, · · · , pwm) = 0 and +[p]E(α(w1,w2,··· ,wm)) =[p]E([w1]E(x1) +F [w2]E(x2) +F · · · +F [wm]E(xm)) +=[pw1]E(x1) +F [pw2]E(x2) +F · · · +F [pwm]E(xm) = 0. +Then we check that the difference of any two elements of pF(π∗(TA,A(E))) is not a zero divisor in +in π∗(TA,A(E)). From the formula x −F y = (x − y) · ε(x, y), where x, y ∈ pF(π∗(TA,A(E))), ε(x, y) +is a unit in π∗(TA,A(E)), it follows that +(α(u1,u2,···,um) − α(w1,w2,···,wm)) · ε(α(u1,u2,··· ,um), α(w1,w2,··· ,wm)) +=α(u1,u2,··· ,um) −F α(w1,w2,···,wm) = α(u1−w1,u2−w2,··· ,um−wm), +where ε(α(u1,u2,···,um), α(w1,w2,···,wm)) is a unit in π∗(TA,A(E)). +Since π∗(TA,A(E)) +� +0 and +(u1, u2, · · · , um) � (w1, w2, · · · , wm), α(u1−w1,u2−w2,··· ,um−wm) ∈ LA is not zero or a zero divisor in +π∗(TA,A(E)). So pF(π∗(TA,A(E))) is a pm-tuple of [p]E(x) in π∗(TA,A(E)). +Finally, we give pF(π∗(TA,A(E)))/ ∼ an abelian group structure: +(i) Addition: α(u1,u2,···,um) + α(w1,w2,··· ,wm) ∼ α(u1+w1,u2+w2,···,um+wm); +(ii) Inverse: −α(w1,w2,··· ,wm) ∼ α(−w1,−w2,··· ,−wm). +This completes the proof. +□ +The following lemma gives a pm-tuple of g1,n+1−q(x) in π∗(TA,A(E))/In+1−q under the assump- +tion that π∗(TA,A(E))/In+1−q � 0. +Lemma 6.6. Let pF(π∗(TA,A(E))/In+1−q)pn+1−q denote the subset +{‡ +αpn+1−q +(w1,w2,··· ,wm) ∈ π∗(TA,A(E))/In+1−q | (w1, w2, · · · , wm) ∈ A}. +If π∗(TA,A(E))/In+1−q � 0, then pF(π∗(TA,A(E))/In+1−q)pn+1−q is a pm-tuple of g1,n+1−q(x) in +π∗(TA,A(E))/In+1−q, and pF(π∗(TA,A(E))/In+1−q)pn+1−q/ ∼ is an abelian group. +Proof. Note that +g1,n+1−q(‡ +αpn+1−q +(w1,w2,···,wm)) = [p]E(α(w1,w2,··· ,wm)) +mod In+1−q, + +34 +Yangyang Ruan +so {‡ +αpn+1−q +(w1,w2,··· ,wm) | (w1, w2, · · · , wm) ∈ A} is a set of roots of g1,n+1−q(x) in π∗(TA,A(E))/In+1−q. +For any two different elements ‡ +αpn+1−q +(u1,u2,···,um), ‡ +αpn+1−q +(w1,w2,··· ,wm) ∈ pF(π∗(TA,A(E))/In+1−q)pn+1−q, +we have +0 � ‡ +αpn+1−q +(u1,u2,··· ,um)−‡ +αpn+1−q +(w1,w2,···,wm) = �αpn+1−q +(u1,u2,···,um)−�αpn+1−q +(w1,w2,···,wm) = (�α(u1,u2,··· ,um)−�α(w1,w2,··· ,wm))pn+1−q +for the coefficient Fp. Since π∗(TA,A(E))/In+1−q � 0 and +�α(u1,u2,···,um) − �α(w1,w2,··· ,wm) = ε−1(�α(u1,u2,··· ,um), �α(w1,w2,···,wm)) · �α(u1−w1,u2−w2,···,um−wm) ∈ �LA,q, +(�α(u1,u2,··· ,um) − �α(w1,w2,··· ,wm))pn+1−q is not zero or a zero divisor in π∗(TA,A(E))/In+1−q. Therefore, +pF(π∗(TA,A(E))/In+1−q)pn+1−q is a pm-tuple of g1,n+1−q(x) in π∗(TA,A(E))/In+1−q. +pF(π∗(TA,A(E))/In+1−q)pn+1−q/ ∼ has an abelian group structure: +(i) Addition: �αpn+1−q +(u1,u2,···,um) + �αpn+1−q +(w1,w2,··· ,wm) ∼ �αpn+1−q +(u1+w1,u2+w2,···,um+wm); +(ii) Inverse: −�αpn+1−q +(w1,w2,··· ,wm) ∼ �αpn+1−q +(−w1,−w2,··· ,−wm). +This completes the proof. +□ +For any q ≤ n + 1, there is a surjective map θq : A → pF(π∗(TA,A(E))/In+1−q)pn+1−q that maps +(w1, w2, · · · , wm) to ‡ +αpn+1−q +(w1,w2,··· ,wm), then we have +Lemma 6.7. θq is a bijection if and only if π∗(TA,A(E))/In+1−q � 0. +Proof. ⇒: Since θq is a bijection, then for (u1, u2, · · · , um) � (w1, w2, · · · , wm) ∈ A, +0 � ‡ +αpn+1−q +(u1,u2,···,um) − ‡ +αpn+1−q +(w1,w2,··· ,wm) ∈ π∗(TA,A(E))/In+1−q, +which implies that π∗(TA,A(E))/In+1−q � 0. +⇐: We only have to prove that θq is injective. Since π∗(TA,A(E))/In+1−q � 0, then for any +(w1, w2, · · · , wm) ∈ A∗, 0 � ‡ +αpn+1−q +(w1,w2,···,wm) ∈ �LA,q. So if (u1, u2, · · · , um) � (w1, w2, · · · , wm) ∈ +A, then +‡ +αpn+1−q +(u1,u2,··· ,um)−‡ +αpn+1−q +(w1,w2,···,wm) = (ε−1(�α(u1,u2,···,um), �α(w1,w2,···,wm))·�α(u1−w1,u2−w2,··· ,um−wm))pn+1−q � 0, +thus θq is injective. +□ +When q = n + 1, I0 = (0) and pF(π∗(TA,A(E))/In+1−q)pn+1−q = pF(π∗(TA,A(E))). +Lemma 6.8. pF(π∗(TA,A(E))) is an abelian group and θn+1 is an abelian group homomorphism. +If n < m, then θn+1 is trivial and pF(π∗(TA,A(E))) � e. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +35 +Proof. The group structure of pF(π∗(TA,A(E))) is induced by the formal group law of E, and for +any two elements α(u1,u2,···,um), α(w1,w2,··· ,wm) ∈ pF(π∗(TA,A(E))), their sum is defined by +α(u1,u2,···,um) +F α(w1,w2,··· ,wm) = α(u1+w1,u2+w2,···,um+wm). +Then θn+1 is an abelian group homomorphism. +If n < m, we assume that π∗(TA,A(E)) � 0. +By Lemma 6.7, θn+1 is a bijection and +|pF(π∗(TA,A(E)))| = pm. Then pF(π∗(TA,A(E))) is a pm-tuple of [p]E(x) in π∗(TA,A(E)). Note +that 1 ∈ (p, v1, · · · , vn) and degW[p]E(x) = pn < pm. By Corollary 1.10,we have π∗(TA,A(E)) = 0. +Then θn+1 is trivial and pF(π∗(TA,A(E))) � e. +□ +Corollary 6.9. π∗(TA,A(E))/In+1−q = 0 for q < m + 1, which implies that sA,A;E ≥ m. +Proof. Assume that there exists q0 < m + 1 such that π∗(TA,A(E))/In+1−q0 � 0. By Lemma 6.7, +θq0 is a bijection and |pF(π∗(TA,A(E))/In+1−q0)pn+1−q0| = pm. Then pF(π∗(TA,A(E))/In+1−q0)pn+1−q0 +is a pm-tuple of g1,n+1−q0(x) in π∗(TA,A(E))/In+1−q0. Note that pm > deg g1,n+1−q0(x) = pq0−1 and +1 ∈ (vn+1−q0, · · · , vn). So by Corollary 1.10, we have π∗(TA,A(E))/In+1−q0 = 0. +□ +Although by Corollary 6.9 and the exactness of +π∗(TA,A(E))/In−m +·vn−m +−−−−−→ π∗(TA,A(E))/In−m −−−−−→ π∗(TA,A(E))/In+1−m, +we know that vn−m is a unit in π∗(TA,A(E))/In−m. To achieve our main idea, here we give another +proof of this fact by using Theorem 1.7. Let q = m + 1, we have +Lemma 6.10. Let n ≥ m, then +(i) +vn−m = (−1)pm−1vn +� +(w1,w2,···,wm)∈A∗ +fl +αpn−m +(w1,w2,··· ,wm), +(ii) +0 = (−1)pm−2vn +� +w(1)�w(2)�···�w(pm−2)∈A∗ +fl +αpn−m +w(1)fl +αpn−m +w(2) · · · fl +αpn−m +w(pm−2), +... +(iii) +vn−i = (−1)pm−pm−ivn +� +w(1)�w(2)�···�wpm−i∈A∗ +fl +αpn−m +w(1)fl +αpn−m +w(2) · · · fl +αpn−m +wpm−i, +... +(iv) +0 = −vn +� +(w1,w2,··· ,wm)∈A∗ +fl +αpn−m +(w1,w2,···,wm), +and the right side of the top equality is invertible in π∗(TA,A(E))/In−m. + +36 +Yangyang Ruan +Remark 6.11. Since π∗(TA,A(E))/In−m may be 0, +the fact that vn−m is invertible in +π∗(TA,A(E)))/In−m does not imply that TA,A(E)) is vn−m-periodic, but implies that TA,A(E) is at +most vn−m-periodic. +Proof. If π∗(TA,A(E))/In−m = 0, obviously this is true; if π∗(TA,A(E))/In−m � 0, then by Lemma +6.7, we obtain that θm+1 is a bijection and |pF(π∗(TA,A(E))/In−m)pn−m| = pm. So π∗(TA,A(E))/In−m +has a pm-tuple pF(π∗(TA,A(E))/In−m)pn−m of g1,n−m(x). Then by Theorem 1.7, we have +vn−mx + · · · + vnxpm = vn +� +(w1,w2,···,wm)∈A +(x − fl +αpn−m +(w1,w2,··· ,wm)) ∈ π∗(TA,A(E))/In−m[x]. +□ +To get the upper bound of sA,A;E, we first generalize Ando-Morava-Sadofsky’s theorem [1, +Proposition 2.3] from Z/p to an elementary abelian p-group. +Theorem 6.12. +π∗(TA,A(BP⟨n⟩)) �φ L′−1 +A BP⟨n − m⟩∗[[x1, x2, · · · , xm]], +where φ is the ring isomorphism constructed in the following proof, and the multiplicatively closed +set L′ +A is generated by the set +{φ(α(w1,···,wm)) | α(w1,···,wm) = [w1]BP⟨n⟩(x1) +F · · · +F [wm]BP⟨n⟩(xm), (w1, · · · , wm) ∈ A∗}. +Proof. As similar to Theorem 1.6, replacing E by BP⟨n⟩, we have +π∗(TA,A(BP⟨n⟩))) � L−1 +A BP⟨n⟩∗[[x1, x2, · · · , xm]]/([p]BP⟨n⟩(x1), · · · , [p]BP⟨n⟩(xm)), +where the multiplicatively closed set LA is generated by the set +{α(w1,···,wm) = [w1]BP⟨n⟩(x1) +F · · · +F [wm]BP⟨n⟩(xm) | (w1, · · · , wm) ∈ A∗}. +We always require a ring map to map 1 to 1. First, we construct a ring map +φ : π∗(TA,A(BP⟨n⟩)) → L′−1 +A BP⟨n − m⟩∗[[x1, x2, · · · , xm]], +which send vi to vi (0 ≤ i ≤ n − m), xi to xi (i ≤ m), and send [p]BP⟨n⟩(xi) to 0 for 1 ≤ i ≤ m, +then we have a system of non-homogeneous L′−1 +A BP⟨n − m⟩∗[[x1, x2, · · · , xm]]-linear equations +{φ([p]BP⟨n⟩(xi)) = 0, 1 ≤ i ≤ m}. We view φ([p]BP⟨n⟩(xi)) = 0 as a non-homogeneous linear +equation +xpn−m+1 +i +φ(vn−m+1) + xpn−m+2 +i +φ(vn−m+2) + · · · + xpn +i φ(vn) = −(v0xi + v1xp +i + · · · + vn−mxpn−m +i +) +with variables φ(vn−m+1), φ(vn−m+2), · · · , φ(vn). Since xi is invertible for 1 ≤ i ≤ m, one may use +Gaussian elimination to get the unique solution of φ(vn−m+1), φ(vn−m+2), · · · , φ(vn). Then we define +φ(vi) as the solution of φ(vi) for n − m + 1 ≤ i ≤ n, So φ is a well-defined ring map. There is a map +ϕ : L′−1 +A BP⟨n − m⟩∗[[x1, x2, · · · , xm]] → π∗(TA,A(BP⟨n⟩)) +defined in the obvious way, that becomes an inverse map. +□ + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +37 +Since there is a map:BP⟨n⟩ → v−1 +n BP⟨n⟩ ≃ E(n), by Theorem 6.12, we use the ring isomor- +phism φ to give the following ring isomorphism: +Corollary 6.13. Let A be an elementary abelian p-group with rankp(A) = m. If n ≥ m, then +π∗(TA,A(E))/In−m �φ L′−1 +A E∗[[x1, x2, · · · , xm]]/In−m � L′−1 +A K(n − m)∗[[x1, x2, · · · , xm]], +where φ is the ring isomorphism constructed in the proof of Theorem 6.12, and the multiplicatively +closed set L′ +A is generated by the set +{φ(�α(w1,··· ,wm)) | �α(w1,··· ,wm) = [w1]E(x1) +F · · · +F [wm]E(xm), (w1, · · · , wm) ∈ A∗}. +Note that if n ≥ m, L′−1 +A BP⟨n − m⟩∗[[x1, x2, · · · , xm]] is non-trivial, then by Corollary 6.13, we +have +Corollary 6.14. Let A be an elementary abelian p-group with rankp(A) = m. If n ≥ m, then +π∗(TA,A(E))/In−m � 0. +Remark 6.15. Another way to prove this corollary is by using Lemma 6.40. +By Corollary 6.9 and Corollary 6.14, we have +Theorem 6.16. Let A be a elementary abelian p-group with rankp(A) = m, then sA,A;E = m. +6.2 +Proof for the case (2) A = C is a general abelian p-group +In Subsection 6.1, we devise a powerful tool in the proof for the case (1), which is the +|pF(π∗(TA,A(E)))|-tuple pF(π∗(TA,A(E))) of [p]E(x) in π∗(TA,A(E)). Certainly, this tool can also +be used to explain general blue-shift phenomenon. +More generally, it is natural to consider +|pjF(π∗(TA,A(E)))|-tuple pjF(π∗(TA,A(E))) of [pj]E(x) in π∗(TA,A(E)) for any positive integer j. +Then we could use this tuple of [pj]E(x) to get the solution of some vi, and investigate whether vi +is invertible by the invertible roots of [pj]E(x) in this tuple. Recall that +[p]E(x) = vn+1−qxpn+1−q + · · · + vnxpn ∈ π∗(TA,A(E))/In+1−q[x]. +Then there is a natural problem of how to compute the pj-series [pj]E(x). There is an iteration +formula [pj]E(x) = [p]E([pj−1]E(x)). However, it is too difficult to obtain an accurate formula for +[pj]E(x). This may be one reason why the generalization of previous work to finite abelian groups +is hard. But we can deal with [pj]E(x). The major key insight of our breakthrough is that instead +of trying to obtain an accurate formula of [pj]E(x), it only suffices to compute the leading and the +last terms of [pj]E(x) in E∗/In+1−q[x], as indicated by the method we used in Subsection 6.1. +Without loss of generality, we may suppose that A is Z/pi1 ⊕ Z/pi2 ⊕ · · · ⊕ Z/pim. From Propo- +sition 3.1 and Theorem 1.6, it follows that +π∗(TA,A(E)) � L−1 +A E∗[[x1, x2, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)), +where the multiplicatively closed set LA is generated by the set +MA = {α(w1,w2,··· ,wm) | (w1, w2, · · · , wm) ∈ A∗}. + +38 +Yangyang Ruan +Then for q ≤ n + 1, we have +π∗(TA,A(E))/In+1−q � �L−1 +A,n+1−qE∗/In+1−q[[x1, x2, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)), +where the multiplicatively closed set �LA,n+1−q is mod In+1−q reduction of LA and generated by the +set +‹ +MA,n+1−q = {�α(w1,w2,···,wm) | (w1, w2, · · · , wm) ∈ A∗}. +Lemma 6.17. Let A be a finite abelian p-group. If π∗(TA,A(E)) � 0, then p∞F(π∗(TA,A(E))) is an +|A|-tuple of π∗(TA,A(E)), and p∞F(π∗(TA,A(E)))/ ∼ is an abelian group. +Proof. The proof is similar to the proof of Lemma 6.5. By direct checking of the definition, +we conclude that p∞F(π∗(TA,A(E))) is an |A|-tuple of π∗(TA,A(E)) under the assumption that +π∗(TA,A(E)) � 0. +□ +Lemma 6.18. Let V(pj|A) denote the subgroup {a ∈ A | pja = 0} of A. If π∗(TA,A(E)) � 0, then +pjF(π∗(TA,A(E))) is a |V(pj|A)|-tuple of [pj]E(x) in π∗(TA,A(E)), and pjF(π∗(TA,A(E)))/ ∼ is an +abelian group. +Proof. The proof is similar to the proof of Lemma 6.5. +□ +The following lemma shows the expression of [pj]E(x) in π∗(TA,A(E))/In+1−q. +Lemma 6.19. +[pj]E(x) = v1+pn+1−q+···+p(j−1)(n+1−q) +n+1−q +xpj(n+1−q) + · · · + v1+pn+···+p(j−1)n +n +xpjn ∈ π∗(TA,A(E))/In+1−q[x]. +Proof. Recall that [p]E(x) = vn+1−qxpn+1−q + · · · + vnxpn ∈ E∗/In+1−q[x]. By Proposition 3.5 that +[pj]E(x) = [p]E([pj−1]E(x)), we obtain the leading and the last terms of [pj]E(x) by iteration. +□ +We follow the method used in Subsection 6.1. Let [pj]E(x) = gj,n+1−q(xpj(n+1−q)) ∈ E∗/In+1−q[x], +then by Lemma 3.6 we have gj,n+1−q(x) = gj +1,n+1−q(x) = a1x + · · · + apj(q−1)xpj(q−1). +Lemma 6.20. Let pjF(π∗(TA,A(E))/In+1−q)pj(n+1−q) denote the subset +{·� +αpj(n+1−q) +(w1,w2,··· ,wm) ∈ π∗(TA,A(E))/In+1−q | (pjw1, pjw2, · · · , pjwm) = 0, (w1, w2, · · · , wm) ∈ A}. +If π∗(TA,A(E))/In+1−q � 0, then pjF(π∗(TA,A(E))/In+1−q)pj(n+1−q) is a |V(pj|A)|-tuple of gj +1,n+1−q(x) +in π∗(TA,A(E))/In+1−q, and pjF(π∗(TA,A(E))/In+1−q)pj(n+1−q)/ ∼ is an abelian group. +Proof. The proof is similar to the proof of Lemma 6.6. +□ +There is a surjective map θ j +q +: +V(pj|A) +→ +pjF(π∗(TA,A(E))/In+1−q)pj(n+1−q) that maps +(w1, w2, · · · , wm) to ·� +αpj(n+1−q) +(w1,w2,···,wm). +Lemma 6.21. θ j +q is a bijection if and only if π∗(TA,A(E))/In+1−q � 0. +Proof. The proof is similar to the proof of Lemma 6.7. +□ + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +39 +If π∗(TA,A(E))/In+1−q +� +0, then by Lemma 6.21, θ j +q is a bijection for any j +≥ +1. +Combining with Lemma 6.18, we have |pjF(π∗(TA,A(E))/In+1−q)pj(n+1−q)| = |V(pj|A)|. +Then +pjF(π∗(TA,A(E))/In+1−q)pj(n+1−q) is a |V(pj|A)|-tuple of gj +1,n+1−q(x) in π∗(TA,A(E))/In+1−q. +Lemma 6.22. Let j be any positive integer, then π∗(TA,A(E))/In+1−q = 0 for q < +logp |V(pj|A)| +j ++ 1. +Proof. Assume that there exists j0 and q0 < +logp |V(pj0|A)| +j0 ++ 1 such that π∗(TA,A(E))/In+1−q0 � 0. +By Lemma 6.21, θ j0 +q0 is a bijection and |pj0F(π∗(TA,A(E))/In+1−q0)pj0(n+1−q0)| = |V(pj0|A)|. Then +pj0F(π∗(TA,A(E))/In+1−q0)pj0(n+1−q0) is a |V(pj0|A)|-tuple of gj0 +1,n+1−q0(x) in π∗(TA,A(E))/In+1−q0. Note +that the unit v1+pn+···+p(j0−1)n +n +is the last coefficient of gj0 +1,n+1−q0(x), and q0 < +logp |V(pj0|A)| +j0 ++ 1 implies +that |V(pj0|A)| > deg gj0 +1,n+1−q0(x) = pj0(q0−1). So by Corollary 1.10, we have π∗(TA,A(E))/In+1−q0 = +0, which contradicts to our assumption. This completes the proof. +□ +Recall that A is Z/pi1 ⊕ Z/pi2 ⊕ · · · ⊕ Z/pim, then we have +Lemma 6.23. +⌈ +logp |V(pj|A)| +j +⌉ = +ß= m if 1 ≤ j ≤ min{i1, · · · , im}, +≤ m if j > min{i1, · · · , im}. +Proof. Note that logp |V(p|A)| is exactly the number of Z/p factors in the maximal elementary +abelian subgroup of A, then we have +logp |V(p|A)| = rankp(A) = m. +Since V(pj|A) is a subgroup of A and Z/pj ⊕ · · · ⊕ Z/pj, we obtain that +|V(pj|A)| ≤ pj logp |V(p|A)| and logp |V(pj|A)| ≤ j logp |V(p|A)|, +where the equality holds if and only if 1 ≤ j ≤ min{i1, · · · , im}. Since logp |V(p|A)| is an integer, +we have +⌈ +logp |V(pj|A)| +j +⌉ ≤ logp |V(p|A)|. +This completes the proof. +□ +When q = n + 1, I0 = (0) and pjF(π∗(TA,A(E))/In+1−q)pj(n+1−q) = pjF(π∗(TA,A(E))). +Lemma 6.24. pjF(π∗(TA,A(E))) is an abelian group and θ j +n+1 is an abelian group homomorphism. +If n < m, then θ j +n+1 is trivial and pjF(π∗(TA,A(E))) � e. +Proof. The group structure of pjF(π∗(TA,A(E))) is induced by the formal group law of E, and for +any two elements α(u1,u2,···,um), α(w1,w2,··· ,wm) ∈ pjF(π∗(TA,A(E))), their sum is defined by +α(u1,u2,···,um) +F α(w1,w2,··· ,wm) = α(u1+w1,u2+w2,···,um+wm). +Then θ j +n is an abelian group homomorphism. + +40 +Yangyang Ruan +If n < m, we assume that π∗(TA,A(E)) � 0. +By Lemma 6.21, θ j +n+1 is a bijection and +|pjF(π∗(TA,A(E)))| += +|V(pj|A)|. +Then +pjF(π∗(TA,A(E))) is a |V(pj|A)|-tuple of [pj]E(x) in +π∗(TA,A(E)). Note that +1 ∈ (p, v1, · · · , vn) and degW[p]E(x) = pn < |V(p|A)| = pm. +By Corollary 1.10,we have π∗(TA,A(E)) = 0. Then θ j +n+1 is trivial and pjF(π∗(TA,A(E))) � e. +□ +By Lemma 6.22 and Lemma 6.23, we have +Corollary 6.25. π∗(TA,A(E))/In+1−q = 0 for q < m + 1, which implies that sA,A;E ≥ m. +To achieve our main idea, here we give another proof of the fact that vn−m is a unit in +π∗(TA,A(E))/In−m by using Theorem 1.7. Let q = m + 1, we have +Lemma 6.26. Let n ≥ m. For 1 ≤ j ≤ min{i1, · · · , im}, vn−m is a unit in π∗(TA,A(E))/In−m. +Proof. If π∗(TA,A(E))/In−m = 0, obviously this is true; if π∗(TA,A(E))/In−m � 0, for 1 ≤ j ≤ +min{i1, · · · , im}, V(pj|A) � Z/pj⊕· · ·⊕Z/pj and |V(pj|A)| = pjm. Then by Lemma 6.21, we obtain +that θ j +m+1 is a bijection and | +Â� +pjF(π∗(TA,A(E)))pj(n−m)| = |V(pj|A)| = pjm. So π∗(TA,A(E))/In−m has a +pjm-tuple +Â� +pjF(π∗(TA,A(E)))pj(n−m) of gj +1,n−m(x). Then by Theorem 1.7, we have +v1+pn−m+···+p(j−1)(n−m) +n−m +x+· · ·+v1+pn+···+p(j−1)n +n +xpjm = v1+pn+···+p(j−1)n +n +� +(w1,w2,···,wm)∈V(pj|A) +(x−‡ +αpj(n−m) +(w1,w2,··· ,wm)). +Then +v1+pn−m+···+p(j−1)(n−m) +n−m += (−1)pjmv1+pn+···+p(j−1)n +n +� +(w1,w2,··· ,wm)∈V(pj|A)∗ +‡ +αpj(n−m) +(w1,w2,··· ,wm) ∈ π∗(TA,A(E))/In−m. +□ +By Lemma 6.40, we have +Corollary 6.27. Let A be a finite abelian p-group with rankp(A) = m. +If n ≥ m, then +π∗(TA,A(E))/In−m � 0. +By Corollary 6.25 and Corollary 6.27, we have +Theorem 6.28. Let A be a finite abelian p-group with rankp(A) = m, then sA,A;E = m. +6.3 +Proof for the case (3) A is a general abelian p-group and C is its proper sub- +group. +Without loss of generality, we may suppose that A is Z/pi1 ⊕ Z/pi2 ⊕ · · · ⊕ Z/pim with i1 ≤ i2 ≤ +· · · ≤ im and C is its subgroup Z/pj1 ⊕ Z/pj2 ⊕ · · · ⊕ Z/pjm with a group inclusion +ϕ : Z/pj1 ⊕ Z/pj2 ⊕ · · · ⊕ Z/pjm → Z/pi1 ⊕ Z/pi2 ⊕ · · · ⊕ Z/pim + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +41 +(w1, w2, · · · , wm) �→ (pi1−j1w1, pi2−j2w2, · · · , pim−jmwm), +otherwise we could replace a set of generators of A. There is also a group inclusion from A/C to +A as follows: +φ : Z/pi1−j1 ⊕ · · · ⊕ Z/pim−jm → Z/pi1 ⊕ · · · ⊕ Z/pim +(w1, · · · , wm) �→ (pi1−j1w1, · · · , pim−jmwm). +From Theorem 1.6, it follows that +π∗(TA,C(E)) � L−1 +C E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)), +where the multiplicatively closed set LC is generated by the set +MC = {α(w1,w2,··· ,wm) | (w1, w2, · · · , wm) ∈ A − imφ(A/C)}. +Then +π∗(TA,C(E))/In+1−q � �L−1 +C,n+1−qE∗/In+1−q[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)), +where the multiplicatively closed set �LC,n+1−q is mod In+1−q reduction of LC and generated by the +set +‹ +MC,n+1−q = {�α(w1,w2,···,wm) | (w1, w2, · · · , wm) ∈ A − imφ(A/C)}. +To find tuples of π∗(TA,C(E)), we still focus on the Euler classes α(w1,w2,··· ,wm) for +(w1, w2, · · · , wm) ∈ A. Note that +α(u1,u2,··· ,um) − α(w1,w2,··· ,wm) = α(u1−w1,u2−w2,···,um−wm) · ε−1(α(u1,u2,··· ,um), α(w1,w2,···,wm)), +where ε(α(u1,u2,···,um), α(w1,w2,··· ,wm)) is a unit in π∗(TA,C(E)). If π∗(TA,C(E)) � 0 and (u1 − w1, u2 − +w2, · · · , um − wm) ∈ A − imφ(A/C), then α(u1,u2,··· ,um) − α(w1,w2,··· ,wm) is not a zero divisor in +π∗(TA,C(E)). Since imφ(A/C) is a subgroup of A, A is the disjoint union � +1≤i≤|C| +Ä +ai + imφ(A/C) +ä +of the cosets of imφ(A/C), where {ai ∈ A | 1 ≤ i ≤ |C|} is a complete set of coset representatives +of imφ(A/C) in A. Thus we have +Lemma 6.29. Let A be a finite abelian p-group and C be its subgroup. Let [A : imφ(A/C)] denote +a complete set of coset representatives of imφ(A/C) in A, and S[A:imφ(A/C)] denote the subset +{α(w1,w2,···,wm) ∈ π∗(TA,C(E)) | (w1, w2, · · · , wm) ∈ [A : imφ(A/C)]}. +If π∗(TA,C(E)) � 0, then S[A:imφ(A/C)] is a |C|-tuple of π∗(TA,C(E)). +Lemma 6.30. Let S[A:imφ(A/C)], j denote the subset +{α(w1,w2,··· ,wk) ∈ π∗(TA,C(E)) | (pjw1, pjw2, · · · , pjwm) = 0, (w1, w2, · · · , wm) ∈ [A : imφ(A/C)]}. +If π∗(TA,C(E)) � 0, then S[A:imφ(A/C)], j is an |S[A:imφ(A/C)], j|-tuple of [pj]E(x) in π∗(TA,C(E)). +Proof. This proof is similar to the proof of Lemma 6.18. +□ + +42 +Yangyang Ruan +Lemma 6.31. Let Â� +Spj(n+1−q) +[A:imφ(A/C)], j denote the subset +{·� +αpj(n+1−q) +(w1,w2,···,wm) ∈ π∗(TA,C(E))/In+1−q | (pjw1, pjw2, · · · , pjwm) = 0, (w1, w2, · · · , wm) ∈ [A : imφ(A/C)]}. +If π∗(TA,C(E))/In+1−q +� +0, then Â� +Spj(n+1−q) +[A:imφ(A/C)], j is an | Â� +Spj(n+1−q) +[A:imφ(A/C)], j|-tuple of gj +1,n+1−q(x) in +π∗(TA,C(E))/In+1−q. +Let V(pj|[A : imφ(A/C)]) denote the set +{(w1, w2, · · · , wm) ∈ [A : imφ(A/C)] | (pjw1, pjw2, · · · , pjwm) = 0}, +then there is a surjective map θ j +q +: +V(pj|[A +: +imφ(A/C)]) +→ +Â� +Spj(n+1−q) +[A:imφ(A/C)], j that maps +(w1, w2, · · · , wm) to ·� +αpj(n+1−q) +(w1,w2,···,wm). +Lemma 6.32. θ j +q is a bijection if and only if π∗(TA,C(E))/In+1−q � 0. +Proof. The proof is similar to the proof of Lemma 6.7. +□ +Corollary 6.33. Let A be a finite abelian p-group and C be its proper subgroup. +Let [A : +imφ(A/C)] denote any complete set of coset representatives of imφ(A/C) in A and j be any positive +integer, then π∗(TA,C(E))/In+1−q = 0 for q < +logp |V(pj|[A:imφ(A/C)])| +j ++ 1. +Proof. Assume that there exists a complete set [A : imφ(A/C)]0, an integer j0, and an integer +q0 < +logp |V(pj0|[A:imφ(A/C)]0)| +j0 ++ 1 such that π∗(TA,C(E))/In+1−q0 � 0. By Lemma 6.32, θ j0 +q0 is a +bijection. Then +Â� +Spn+1−q0 +[A:imφ(A/C)]0, j0 is an | Â� +Spn+1−q0 +[A:imφ(A/C)]0, j0|-tuple of gj0 +1,n+1−q0(x) in π∗(TA,C(E))/In+1−q0. +Note that the unit v1+pn+···+p(j0−1)n +n +is the last coefficient of gj0 +1,n+1−q0(x). Since C is a proper subgroup +of A, we have +|V(pj0|[A : imφ(A/C)]0)| > deg gj0 +1,n+1−q0(x) = pj0(q0−1). +So by Corollary 1.10, we have π∗(TA,C(E))/In+1−q0 = 0, which contradicts to our assumption. +This completes the proof. +□ +Note that |V(pj|[A : imφ(A/C)])| depends on the choice of [A : imφ(A/C)]. +Let [A : +imφ(A/C)]max denote a complete set of coset representatives of imφ(A/C) in A such that |V(pj|[A : +imφ(A/C)]max)| is maximal. We first simplify |V(pj|[A : imφ(A/C)]max)| by the following lemma. +Lemma 6.34. Let A be a finite abelian p-group and C be its proper subgroup. Let A′ denote the +minimal direct summand of A that contains C, then +|V(pj|[A : imφ(A/C)]max)| = |V(pj|[A′ : imφ(A′/C)]max)|. +Lemma 6.35. Let A be a finite abelian p-group and C be its direct summand, then +|V(pj|[A : imφ(A/C)]max)| = |V(pj|C)|. + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +43 +Proof. Since A = C ⊕ A/C, then [A : imφ(A/C)] = {ai | 1 ≤ i ≤ |C|} where ai = (ci, a′ +i) for ci ∈ C +and a′ +i ∈ A/C. V(pj|[A : imφ(A/C)]) = {(ci, a′ +i) | 1 ≤ i ≤ |C|, (pjci, pja′ +i) = 0}, we choose a′ +i = 0 +for 1 ≤ i ≤ |C|, then |V(pj|[A : imφ(A/C)]max)| = |V(pj|C)|. +□ +To compute |V(pj|[A : imφ(A/C)]max)|, we need the following lemma. +Lemma 6.36. Let A be a finite abelian p-group and C be its proper subgroup. Then there is an +injection of cosets +� +1≤i≤ +|V(pj|A)| +|V(pj|imφ(A/C))| +Ä +bi + V(pj|imφ(A/C)) +ä +֒→ +� +1≤k≤|C| +Ä +ak + imφ(A/C) +ä +induced by the inclusion V(pj|A) ֒→ A. +Proof. If bi ∈ ak + imφ(A/C), then bi + V(pj|imφ(A/C)) ⊆ ak + imφ(A/C). So it suffices to +prove that for any 1 ≤ k ≤ |C|, ak + imφ(A/C) contains at most one bi for 1 ≤ i ≤ +|V(pj|A)| +|V(pj|imφ(A/C))|. +If ak + imφ(A/C) contains bi1 and bi2 for 1 ≤ i1 � i2 ≤ +|V(pj|A)| +|V(pj|imφ(A/C))|, then there are a′, a′′ ∈ +imφ(A/C) such that bi1 = ak + a′, bi2 = ak + a′′, which follows that bi1 − bi2 = a′ − a′′. Note that +a′ − a′′ ∈ imφ(A/C), then bi1 − bi2 ∈ imφ(A/C). Since +bi1 − bi2 ∈ V(pj|A) − V(pj|imφ(A/C)) = V(pj|A − imφ(A/C)) ⊆ A − imφ(A/C), +this is a contradiction. +□ +By Lemma 6.36 and Lemma 6.34, we have +Corollary 6.37. Let A be a finite abelian p-group and C be its proper subgroup. Let A′ denote the +minimal direct summand of A that contains C, then +|V(pj|[A : imφ(A/C)]max)| = +|V(pj|A)| +|V(pj|imφ(A/C))| = +|V(pj|A′)| +|V(pj|imφ(A′/C))| +and +max +j∈N+ [ +logp |V(pj|[A : imφ(A/C)]max)| +j +] = max +j∈N+ [ +logp |V(pj|A′)| − logp |V(pj|imφ(A′/C))| +j +]. +Remark 6.38. [ +logp |V(pj|[A:imφ(A/C)]max)| +j +] reaches the maximum when j ≤ logp |A|. +By Corollary 6.33 and Corollary 6.37, we have +Corollary 6.39. Let A be a finite abelian p-group and C be its proper subgroup, then +π∗(TA,C(E))/In+1−q = 0 for q < max +j∈N+ +logp |V(pj|A)| − logp |V(pj|imφ(A/C))| +j ++ 1. +Which implies that +sA,C;E ≥ max +j∈N+ ⌈ +logp |V(pj|A)| − logp |V(pj|imφ(A/C))| +j +⌉. + +44 +Yangyang Ruan +Lemma 6.40. Let A be an abelian p-group and C be its subgroup with an inclusion +ϕ : C = Z/pj1 ⊕ Z/pj2 ⊕ · · · ⊕ Z/pjm → A = Z/pi1 ⊕ Z/pi2 ⊕ · · · ⊕ Z/pim +(w1, w2, · · · , wm) �→ (pi1−j1w1, pi2−j2w2, · · · , pim−jmwm). +Let A′ be the subgroup of A with A = A′ ⊕Z/pim and C′ be the subgroup of C with C = C′ ⊕Z/pjm. +If E is Landweber exact and π∗(TA′,C′(E))/In−k � 0, then +(i) π∗(TA,C(E))/In−k−1 � 0 if jm > 0; +(ii) π∗(TA,C(E))/In−k � 0 if jm = 0. +Proof. We first prove the case (i): jm > 0. If E is Landweber exact, then by Lemma 5.11 we +obtain that TA′,C′(E) is Landweber exact. Since π∗(TA′,C′(E))/In−k � 0, by exactness of +0 −−−−−−→ π∗(TA′,C′(E))/In−k−1 +·vn−k−1 +−−−−−−→ π∗(TA′,C′(E))/In−k−1 −−−−−−→ π∗(TA′,C′(E))/In−k −−−−−−→ 0, +we obtain that vn−k−1 is not a unit in π∗(TA′,C′(E))/In−k−1 � 0. By Theorem 1.6, we have +π∗(TA′,C′(E)) � L−1 +C′ E∗(BA′ ++), +where the multiplicatively closed set LC′ is generated by the set +MC′ = {α(w1,··· ,wm) ∈ E∗(BA′ ++) | (w1, · · · , wm) ∈ A′ − imφ(A′/C′)∗)}. +Let L′ +C,i denote the multiplicatively closed set generated by the set +M′ +C,i = {�α(w1,···,wm) ∈ E∗(BA′ ++)[[xm]]/Ii | (w1, · · · , wm) ∈ A − imφ(A/C)∗)}. +Since E is Landweber exact, by a similar proof of Lemma 5.11, we deduce that for each i multi- +plication by vi is monic on L′−1 +C,i E∗(BA′ ++)[[xm]]/Ii. Note that there is an injective homomorphism +L−1 +C′ E∗(BA′ ++) ֒→ L′−1 +C E∗(BA′ ++)[[xm]], +then we have the following commutative diagram +0 −−−−−−−→ +�L−1 +C′,iE∗(BA′ ++)/Ii +·vi +−−−−−−−→ +�L−1 +C′,iE∗(BA′ ++)/Ii +−−−−−−−→ +�L−1 +C′,i+1E∗(BA′ ++)/Ii+1 +−−−−−−−→ 0 +� +� +� +0 −−−−−−−→ L′−1 +C,i E∗(BA′ ++)[[xm]]/Ii +·vi +−−−−−−−→ L′−1 +C,i E∗(BA′ ++)[[xm]]/Ii −−−−−−−→ L′−1 +C,i+1E∗(BA′ ++)[[xm]]/Ii+1 −−−−−−−→ 0, +and deduce that the homomorphism L−1 +C′ E∗(BA′ ++)/Ii → L′−1 +C E∗(BA′ ++)[[xm]]/Ii is injective for each +i. Since π∗(TA′,C′(E))/In−k � 0, we have L′−1 +C,n−kE∗(BA′ ++)[[xm]]/In−k � 0. By exactness of +0 −−−−−−−−→ L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +·vn−k−1 +−−−−−−−−→ L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 −−−−−−−−→ L′−1 +C,n−kE∗(BA′ ++)[[xm]]/In−k −−−−−−−−→ 0, +we +obtain +that +vn−k−1 +is +not +a +unit +in +L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +and +L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +→ +v−1 +n−k−1L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +is +injective. +As + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +45 +[pim]E(xm) is a power series with the invertible leading term v1+pn−k−1+···+p(im−1)(n−k−1) +n−k−1 +xpim(n−k−1) +m +in v−1 +n−k−1L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1, it is a unit. This implies that the map +L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 −−−−−−−→ v−1 +n−k−1L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +·[pim]E(xm) +−−−−−−−−→ v−1 +n−k−1L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +is injective, hence +L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +·[pim]E(xm) +−−−−−−−−→ L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +is also. Using the Gysin sequence of S 1 → BA +B(id×ρ +1 +pim +) +→ +B(A′ × U(1)), we have +E∗(BA+) � E∗(BA′ ++)[[xm]]/([pim]E(xm)) +and +π∗(TA,C(E))/In−k−1 � L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/(In−k−1, [pim]E(xm)). +Then we obtain a short exact sequence: +0 +� L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +·[pim]E(xm) � L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 +� π∗(TA,C(E))/In−k−1 +� 0 . +Now [pim]E(xm) is not a unit in v−1 +n−k−1L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k−1 since its leading coefficient +v1+pn−k−1+···+p(im−1)(n−k−1) +n−k−1 +is not a unit. Therefore π∗(TA,C(E))/In−k−1 � 0. +Now we prove the case (ii): jm = 0, that is C = C′. Since π∗(TA′,C′(E))/In−k � 0, we have +L′−1 +C,n−kE∗(BA′ ++)[[xm]]/In−k � 0. As +π∗(TA,C(E))/In−k � L′−1 +C,n−kE∗(BA′ ++)[[xm]]/(In−k, [pim]E(xm)), +then we obtain a short exact sequence: +L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k +·[pim]E(xm)� L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k +� π∗(TA,C(E))/In−k +� 0 . +Since xm is not invertible in L′−1 +C,n−k−1E∗(BA′ ++)[[xm]]/In−k, which implies that ·[pim]E(xm) is not +surjective, thus π∗(TA,C(E))/In−k � 0. +□ +By inductively using Lemma 6.40, we have +Corollary 6.41. Let A be a finite abelian p-group and C be its proper subgroup. If n ≥ rankp(C), +then π∗(TA,C(E))/In−rankp(C) � 0. +By Corollary 6.39 and Corollary 6.41, we have +Theorem 6.42. Let A be a finite abelian p-group and C be its proper subgroup, then +t ≤ sA,C;E ≤ rankp(C) +where +t = max +j∈N+ ⌈ +logp |V(pj|A)| − logp |V(pj|imφ(A/C))| +j +⌉. +By Lemma 6.35, Lemma 6.23 and Theorem 6.42, we have +Corollary 6.43. Let A be a finite abelian p-group and C be its direct summand, then +sA,C;E = rankp(C). + +46 +Yangyang Ruan +7 +General blue-shift phenomenon for non-abelian cases +Our approach rely heavily on the computation of E∗(BA+) for a finite abelian group A, but +there is no known method to compute E∗(BG+) for a general finite group G. However under some +assumptions, our approach still could obtain partial solution of general blue-shift phenomenon 1.2 +for a non-abelian p-group G. One of the most important problems is how to compute the roots of +[pj]E(x) in π∗(TG,N(E)). This problem is equivalent to how to compute the roots of [pj]E(x) in +E∗(BG+), the equivalence is a consequence of the fact (see Section 3) that π∗(TG,N(E)) is a local- +ization L−1 +N E∗(BG+) of E∗(BG+) with respect to those Euler classes χV ∈ F(EG+, infG +e (E))V(S 0) +of those representations V of G such that VN = 0. As is pointed out in the Introduction that if G +is a non-abelian group, then ψpj +G need not be a homomorphism, so we can not use the functorial +property of B to obtain a self-map of BG. There is a possible way to get around the difficulty. +Inspired by Jackowski-Mcclure-Oliver’s work [23], we regard Bψpj +G as an unstable Adams opera- +tion, which motivates us to give Definition 1.13. There is an equivalent description of the unstable +Adams operation. +Proposition 7.1. Let G be a finite p-group and G′ be the commutator group of G with a quotient +homomorphism ǫ : G → G/G′. Then there is an unstable Adams operation f : BG → BG of +degree p if and only if there is a homomorphism ρ : G → G such that the following diagram +G +ǫ +−−−−−→ G/G′ +ρ +� +ψp +G/G′ +� +G +ǫ +−−−−−→ G/G′ +commutes. +Proof. ⇐: Take f to be Bρ, then by the functorial property of B we obtain that Bρ is an unstable +Adams operation of degree p. +⇒: By Dwyer and Zabrodsky’s Theorem [12] or Notbohm’s Theorem [33], there is a bijection +B : Rep(G,G) = Hom(G,G)/InnG → [BG+, BG+] +ρ �→ Bρ, +and a homomorphism ρ ∈ Rep(G,G) such that f ≃ Bρ. Then by Definition 1.13, we have +Bǫ ◦ f ≃ Bǫ ◦ Bρ = B(ǫ ◦ ρ) ≃ ψp +B(G/G′) ◦ Bǫ = B(ψp +G/G′ ◦ ǫ). +Similarly, there also is a bijection +B : Rep(G,G/G′) = Hom(G,G/G′) → [BG+, B(G/G′)+], +which implies that ǫ ◦ ρ = ψp +G/G′ ◦ ǫ. This finishes the proof. +□ +Let G be a finite p-group and N be its normal subgroup. Here we still choose Hovey’s defi- +nition 5.5 of vn-periodicity for TG,N(E). By Definition 6.2, we find that the determination of the + +General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial +47 +periodicity of TG,N(E) is equivalent to the computation of the projective dimension n + 1 − sG,N;E +of π∗(TG,N(E)) as an E∗-module. +Let A be an abelian group, then each homomorphism from G to A must factor through G/G′ +and we have a bijection +ǫ# +A : Hom(G/G′, A) → Hom(G, A) +ρ �→ ǫ# +A(ρ) = ρ ◦ ǫ. +There is a map Bǫ∗ : E∗(B(G/G′)+) → E∗(BG+) induced by Bǫ. Since ǫ(N) is a subgroup of +G/G′, then the quotient group G/G′/ǫ(N) can be canonically embedded in G/G′ by φ. Note that +{ǫ# +U(1)(ρw) ∈ Hom(G, U(1)) | w ∈ G/G′ − φ(G/G′/ǫ(N))∗} contains all irreducible complex one +dimensional G-representation such that VN = 0. Let [G/G′ : φ(G/G′/ǫ(N))] denote a complete +set of coset representatives of φ(G/G′/ǫ(N)) in G/G′, and S[G/G′:φ(G/G′/ǫ(N))] denote the subset +{Bǫ∗(χρw) ∈ π∗(TG,N(E)) | w ∈ [G/G′ : φ(G/G′/ǫ(N))]}. +Lemma 7.2. Let S[G/G′:φ(G/G′/ǫ(N))], j denote the subset +{Bǫ∗(χρw) ∈ π∗(TG,N(E)) | pjw = 0, w ∈ [G/G′ : φ(G/G′/ǫ(N))]}. +If Conjecture 1.14 is true and π∗(TG,N(E)) � 0, then S[G/G′:φ(G/G′/ǫ(N))], j is an |S[G/G′:φ(G/G′/ǫ(N))], j|- +tuple of [pj]E(x) in π∗(TG,N(E)). +Proof. If Conjecture 1.14 is true, then there is an unstable Adams operation f : BG → BG of +degree p and E2( f(−)) = [p]E(−) : E2(BG+) → E2(BG+). Let f j = f ◦ f j−1, then for any +Bǫ∗(χρw) ∈ S[G/G′:φ(G/G′/ǫ(N))], j, we have +[pj]E(Bǫ∗(χρw)) = E∗( f j)(Bǫ∗(χρw)) = Bǫ∗(ψpj,∗ +B(G/G′)(χρw)) = Bǫ∗([pj]E(χρw)) = Bǫ∗(χρpjw) = 0. +So S[G/G′:φ(G/G′/ǫ(N))], j is a set of roots of [pj]E(x) in π∗(TG,N(E)). +For any w, u ∈ [G/G′ : φ(G/G′/ǫ(N))], we have +Bǫ∗(χρw) − Bǫ∗(χρu) = Bǫ∗(χρw − χρu) = Bǫ∗(χρw−u · ε−1(χρw, χρu)) = Bǫ∗(χρw−u) · Bǫ∗(ε−1(χρw, χρu)) +where ε(χρw, χρu) is a unit in π∗(TG/G′,ǫ(N)(E)). +Since Bǫ∗ is a ring homomorphism, +Bǫ∗(ε−1(χρw, χρu)) is a unit in π∗(TG,N(E)). If π∗(TG,N(E)) � 0, then Bǫ∗(χρw−u) ∈ LN, so the +difference of any two elements in S[G/G′:φ(G/G′/ǫ(N))], j is not a zero divisor in π∗(TG,N(E)). This +completes the proof. +□ +Theorem 7.3. Let E be a p-complete, complex oriented spectrum with an associated formal group +of height n. Let G be a finite p-group and N be its normal subgroup. Let [G/G′ : φ(G/G′/ǫ(N))] +denote any complete set of coset representatives of φ(G/G′/ǫ(N)) in G/G′ and j be any positive in- +teger. If Conjecture 1.14 is true, then π∗(TG,N(E))/In+1−q = 0 for q < +logp |V(pj|[G/G′:φ(G/G′/ǫ(N))])| +j ++ +1. From Lemma 6.34, it follows that +π∗(TG,N(E))/In+1−q = 0 for q < max +j∈N+ +logp |V(pj|G/G′)| − logp |V +Ä +pj|imφ(G/G′/ǫ(N)) +ä +| +j ++ 1, + +48 +Yangyang Ruan +which implies that +sG,N;E = sG,N;E ≥ max +j∈N+ ⌈ +logp |V(pj|G/G′)| − logp |V +Ä +pj|imφ(G/G′/ǫ(N)) +ä +| +j +⌉. +References +[1] Ando, M., Morava, J., Sadofsky, H.: Completions of Z/(p)-Tate cohomology of periodic +spectra. Geom. 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Graduate Texts in Mathematics, +29, Springer, NewYork, 1960. +Yangyang Ruan +Institute of Mathematics +Academy of Mathematics and Systems Science, Chinese Academy of Sciences +No. 55, Zhongguancun East Road, Haidian District, Beijing, 100190 +P.R.China +E-mails: ruanyy@amss.ac.cn + diff --git a/ttE4T4oBgHgl3EQfWAzx/content/tmp_files/load_file.txt b/ttE4T4oBgHgl3EQfWAzx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ec5de80ca9f10fe20dff9853f1dd928fb416bc5 --- /dev/null +++ b/ttE4T4oBgHgl3EQfWAzx/content/tmp_files/load_file.txt @@ -0,0 +1,1826 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf,len=1825 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='05030v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='AT] 12 Jan 2023 General Blue-Shift Phenomenon and Generalized Relations of Roots and Coefficients of a Polynomial Yangyang Ruan, ruanyy@amss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='cn Institute of Mathematics, Chinese Academy of Sciences Beijing 100190, China Abstract In chromatic homotopy theory, there is a well-known conjecture called blue- shift phenomenon (BSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Recently, Balmer-Sanders (Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=', 208(2017), 283-326) and Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton (Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=', 216(2019), 215- 240) showed that a new BSP is closely related to the Zariski topology of Balmer spectrum of the category of compact genuine A-spectra for a finite abelian group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To unify these two BSP to one framework, we propose a general blue-shift phenomenon (GBSP) in this paper and have a new idea to explain it in a more conceptual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To carry out our idea, we use the roots of p j-series of formal group law of a complex oriented spectrum E in the homotopy group of the generalized Tate spectrum of E originally due to the seminal pa- per of Hopkins-Kuhn-Ravenel (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=', 13(2000), 553-594).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' This motivates us to go further to study the relation of roots and coefficients of a polynomial in a com- mutative ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And we propose a notion called n-tuple of a polynomial in R to obtain generalized relations of roots and coefficients of this polynomial in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' These generalized relations have a broad application prospect in reducing the relations of R, especially they play an extremely important role in explaining GBSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By taking this brand-new approach, we successfully achieve our idea of the explanation of GBSP for some abelian cases, and obtain that the generalized Tate construction lowers Bousfield class along with many Tate vanishing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' This strengthens and extends previous theorems of Balmer-Sanders (In- vent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=', 208(2017), 283-326) and Ando-Morava-Sadofsky (Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=', 2(1998), 145-174).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Though our approach could only recover Barthel-Hausmann-Naumann-Nikolaus- Noel-Stapleton (Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=', 216(2019), 215-240), it seems more accessible to deal with GBSP for non-abelian cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Besides, our approach greatly simplifies the original proof of Bonventre-Guillou-Stapleton (arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='03797), which showed that its applications are not restricted to GBSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Thus our approach deserves more applications and further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 1 Introduction In chromatic homotopy theory, blue-shift is a well-known phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The study of this phenomenon is a widely concerned and extremely active area in algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Roughly 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Primary 55N22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Secondary 55N20, 55P42, 55P91, 55Q10, 55R40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' General blue-shift phenomenon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Generalized Tate construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Equivariant stable homo- topy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Bousfield class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Relation of roots and coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' n-tuple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' pj-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 1 2 Yangyang Ruan speaking, for a finite group G, applying the categorical G-fixed point functor (−)G for the classical Tate construction tG(infG e (E)) (see Section 2 for details) of a non-equivariant vn-periodic1 spec- trum E, one obtains a new spectrum tG(infG e (E))G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The blue-shift results obtained by far abounds, we summarise various blue-shift phenomena into the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (Classical blue-shift phenomenon) tG(infG e (E))G is vn−sG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E-periodic for some positive integer sG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To make Tate vanishing results fit into this framework, especially when sG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E > n, the vn−sG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E-periodic ring spectrum denotes the contractible spectrum ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' We call sG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E blue-shift number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' As far as we know, classical blue-shift phenomenon was discovered by Davis and Mahowald [11] in 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' They found that if G is a cyclic group of order 2, denoted by Z/2, then the construc- tion tZ/2(infZ/2 e (−))Z/2 maps the v1-periodic 2-local ring spectrum bu, which denotes the connected complex K-theory, to a v0-periodic spectrum K(Z2) which denotes the Eilenberg-Maclane spec- trum for 2-adic integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And they conjectured an extended result in which bu is replaced by the spectrum BP⟨n⟩ of [22] and K(Z2) is replaced by BP⟨n − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let K(n) denote the n-th Morava K- theory, then in 1994 Greenlees and Sadofsky [16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1] found that tG(infG e (K(n)))G ≃ ∗ for any p-group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In 1996, Hovey and Sadofsky [20] discovered that when G = Z/p, E is vn-periodic and Landweber exact2, blue-shift number sZ/p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E is 1 for any prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In 1998, Ando-Morava- Sadofsky [1] confirmed that Davis and Mahowald’s conjecture is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let T(n) be the telescope of any vn-self map of a complex of type n3, then in 2004 Kuhn [24] proved that tG(infG e (T(n)))G ≃ ∗ for any p-group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For a finite group G, let SH(G) denote the G-equivariant stable homotopy category and SH(G)c denote its full subcategory that consists of all compact objects of SH(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In 2017, Balmer and Sanders [6] showed that classical blue-shift phenomenon, namely Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1, for G = Z/p is closely related to the Zariski topology of Balmer spectrum Spc(SH(Z/p)c) of SH(Z/p)c, which is a Z/p-equivariant analog of Devinatz-Hopkins-Smith’s work [10, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To compute the Zariski topology of Balmer spectrum Spc(SH(G)c), they proposed a new construc- tion that replaces the functor (−)G in the construction tG(infG e (−))G of classical blue-shift phe- nomenon by the geometric fixed point functor ΦG(−), hence a new blue-shift phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In 2019, Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton [7] obtained the Zariski topol- ogy of Spc(SH(A)c) for an abelian group A by studying this new blue-shift phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To unify classical blue-shift phenomenon and new blue-shift phenomenon to one framework, we propose a general blue-shift phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To be precise, let N be a normal subgroup of G and ˜ΦN be the relative geometric N-fixed point functor from SH(G) to SH(G/N), then we consider a more general functor ( ˜ΦN(tG(infG e (−))))G/N which is obtained by replacing (−)G in the construc- tion tG(infG e (−))G by the functor ( ˜ΦN(−))G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For convenience, let TG,N(−) denote the functor ( ˜ΦN(tG(infG e (−))))G/N : SH(e) → SH(e) from non-equivariant spectra to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In this paper, we call TG,N(−) the generalized Tate construction for non-equivariant spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And for a non- equivariant spectrum E, we call TG,N(E) the generalized Tate spectrum of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then the general blue-shift phenomenon can be stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 1Usually vn-periodic means that vn is a unit in the homotopy ring π∗(E), but in this paper, we choose a less restrictive definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5 due to Hovey [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 2Details see [27] or Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 3Details see Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 3 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (General blue-shift phenomenon) The functor TG,N(−) maps a vn-periodic spectrum E to a vn−sG,N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E-periodic spectrum TG,N(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In other words, this generalized Tate con- struction reduces chromatic periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (i) When N = G, TG,N(−) is the construction ΦG(tG(infG e (−))) in Balmer and Sanders’s new blue-shift phenomenon, details see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) When the family subgroups of G which do not contain N are {e}, one special case is that G = Z/pj and N = Z/p, TG,N(−) is the construction tG(infG e (−))G in classical blue-shift phenomenon, details see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The goal of this paper is to study this general blue-shift phenomenon, namely Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And our main idea to explain this phenomenon is that since the homotopy group π∗(TG,N(E)) of generalized Tate spectrum TG,N(E) is a graded ring, it must be isomorphic to a quotient of a free graded ring by some relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And we may reduce these relations like solving equations to obtain vn−sG,N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E, then we need to prove the solution of vn−sG,N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E is invertible in π∗(TG,N(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Inspired by Hopkins-Kuhn-Ravenel’s work [19], we use the roots of pj-series of formal group law of E in π∗(TG,N(E)) to carry out our main idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And we follow their assumption that E is a p-complete and complex oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Recall that a ring spectrum E is complex oriented if there exists an element x ∈ E2(CP∞) such that the image i∗(x) of the map i∗ : E2(CP∞) → E2(CP1) induced by i : S 2 � CP1 ֒→ CP∞ is the canonical generator of E2(S 2) � π0E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Such a class x is called a complex orientation of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The complex orientated E with the multiplication map µCP∞ : CP∞ ×CP∞ → CP∞ gives an associated formal group law F over E∗: x1 +F x2 = F(x1, x2) = µ∗ CP∞(x) ∈ E∗(CP∞ × CP∞) = E∗[[x1, x2]], where “[[]]” denotes the formal power series ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For any integer m, the m-series of F is the formal power series [m]E(x) = x +F x +F · · · +F x �������������������������������������������� m ∈ E∗[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let vn denote the coefficient of xpn in [p]E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Say that F (i) has height ≥ n if vi = 0 for i < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) has height exactly n if it has height ≥ n and vn ∈ E∗ is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' When localized at p, such formal group laws are classified by height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By using the Gysin sequence of S 1 → BZ/pj → CP∞ and the fact that [pj]E(x) is not a zero divisor in E∗[[x]], one obtains that E∗(BZ/pj) � E∗[[x]]/([pj]E(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Besides, E∗(BZ/pj) is a Hopf algebra over E∗ where the coalgebra structure is induced by the multiplication map µBZ/pj : BZ/pj × BZ/pj → BZ/pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To compute the roots of [pj]E(x) in a graded E∗-algebra4, we recall a definition due to Hopkins-Kuhn-Ravenel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' ([19, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=']) Let R be a graded E∗-algebra and j be a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then the set of E∗-algebra homomorphisms HomE∗−alg(E∗[[x]]/([pj]E(x)), R), denoted by pjF(R), forms a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 4In [19], a graded E∗-algebra means that a graded Hopf algebra over E∗, and we follow their notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For a graded E∗-algebra R, a root of [pj]E(x) in R is an element r ∈ R such that [pj]E(r) = 0 in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 4 Yangyang Ruan Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In other words, f ∗ ∈ HomE∗−alg(E∗[[x]]/([pj]E(x)), R) is an E∗-ring homomorphism so that there is a one-one correspondence between f ∗ and its image f ∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If we identify f ∗ with its image f ∗(x), since f ∗([pj]E(x)) = [pj]E( f ∗(x)) = 0, then f ∗ is viewed as a root of [pj]E(x) in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And pjF(R) is viewed as a set of roots of [pj]E(x) in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If π∗(TG,N(E)) has an E∗-algebra structure, then by Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5 pjF(π∗(TG,N(E))) is viewed as a set of roots of [pj]E(x) in π∗(TG,N(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By simplifying the construction TG,N(−), we identify the homotopy group π∗(TG,N(E)) with the G/N-equivariant homotopy group πG/N ∗ ( ˜ΦN(F(EG, infG e (E)))) of a G/N-spectrum ˜ΦN(F(EG, infG e (E)), details see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Combining with Costenoble’s Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3, we identify πG/N ∗ ( ˜ΦN(F(EG, infG e (E)))) with L−1 N E∗(BG), where the multiplicatively closed set LN is generated by the set MN = {χV ∈ E∗(BG) | V is any complex G-representation such that VN = 0} of Euler classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The work of [19] is one of the most important and profound results in the study of the generalized cohomology of BG, and they showed that if G is an abelian group, E∗(BG) can be computed and represented by a beautiful E∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' However, it is regrettable that by far, there exsits no method to compute E∗(BG) for a general non-abelian group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' One of the difficulties might lie in the fact that BG may not be an H-space for a non-abelian group G, in which case E∗(BG) may not possess a coalgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The E∗-algebra structure is critical, so we take G to be an abelian group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since BG is homotopy equivalent to the classifying space of the p- Sylow group of G after localizing at p for a prime p, so without loss of generality we always work p-locally and assume that A is an abelian p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Here we take N to be a subgroup C of A, and obtain the homotopy group π∗(TA,C(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (The homotopy group of generalized Tate spectrum TA,C(E)) Let m be a positive integer and E be a p-complete, complex oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let A be an abelian p-group of form Z/pi1 ⊕ · · · ⊕ Z/pim and C be its subgroup Z/pj1 ⊕ · · ⊕ Z/pjm with jk ≤ ik for 1 ≤ k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' There is a group homomorphism φ5 from A/C to A as follows: φ : Z/pi1−j1 ⊕ Z/pi2−j2 ⊕ · · · ⊕ Z/pim−jm → Z/pi1 ⊕ Z/pi2 ⊕ · · · ⊕ Z/pim (w1, w2, · · · , wm) �→ (pi1−j1w1, pi2−j2w2, · · · , pim−jmwm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then π∗(TA,C(E)) � L−1 C E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)), where the multiplicatively closed set LC is generated by the set MC = {α(w1,···,wm) = [w1]E(x1) +F · · · +F [wm]E(xm) ∈ E∗(BA) | (w1, · · · , wm) ∈ A − imφ(A/C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' As pjF(π∗(TA,C(E))) is well-defined, then by Weierstrass Preparation Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4, we have an E∗-algebra isomorphism η : E∗[[x]]/([pj]E(x)) → E∗[x]/(gj(x)) 5To describe the multiplicatively closed set LC, the group homomorphism φ : A/C → A arises, details see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 5 where gj(x) is the Weierstrass polynomial of [pj]E(x), which identifies the power series [pj]E(x) with the polynomial gj(x) and their corresponding roots in π∗(TA,C(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To determinate the peri- odicity of TA,C(E), we study the relation of roots and coefficients of gj(x) in π∗(TA,C(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1 and f(x) be a polynomial over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' A polynomial f(x) in R[x] can viewed as a polynomial map from R to R, which maps r ∈ R to f(r) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' We denote the set of such polynomial maps by Pmap(R, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To be more precise, Pmap(R, R) is the quotient R[x]/ ∼, let [ f(x)] denote the equivalent class of f(x): f(x) ∼ g(x) if for any r ∈ R, f(r) = g(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' There is a map λ : R[x] → Pmap(R, R) with λ( f(x)) = [ f(x)] for f(x) ∈ R[x], what conditions does R satisfy with such that λ is injective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To serve our purpose here, we restrict ourself to a narrow version of this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R[x]n denote the set of polynomials of degree at most n and λR[x]n denote the map that restricts λ to R[x]n, then the question now is what condition does R satisfy with so that λR[x]n is injective ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' A sufficient condition is that R has a set S in which the difference of any two elements is not a zero divisor, and we call such S an |S |-tuple of R, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then if S also is a subset of roots of a polynomial f(x) over R, we call such S an |S |-tuple of f(x) in R, see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And by using these two notions, we generalize the relation of roots and coefficients of a polynomial over a commutative ring and obtain Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (Generalized relations of roots and coefficients of a polynomial) Let R be a commutative ring with 1 and f(x) = a0 + a1x + · · · + amxm be a polynomial over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Suppose that R has an n-tuple {r1, r2, · · · , rn} of f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (i) If n > m, then ai = 0 in R for 0 ≤ i ≤ m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) if n = m, then ai = (−1)nan � 1≤k1�k2�···�kn−i≤n rk1rk2 · · · rkn−i in R for 0 ≤ i ≤ n − 1 and f(x) = an n � i=1 (x − ri);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (iii) if n ≤ m, then ai = det(α0,··· ,αi−1,β,αi+1,··· ,αn−1) det(α0,α1,··· ,αn−1) in R for 0 ≤ i ≤ n − 1, where αi denotes the column R-vector (ri 1, ri 2, · · · , ri n)T for 0 ≤ i ≤ n − 1 and β denote the column R-vector (− �m i=n airi 1, − �m i=n airi 2, · · · , − �m i=n airi n)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (i) It is impossible for a nonzero polynomial over a field to have the number of roots more than its degree, whereas it is possible for a nonzero polynomial over a commu- tative ring, such as the nonzero polynomial x2 over Z[x1, x2]/(x2 1, x2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) To some extent, this theorem is a generalization of polynomial factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' It is easy to see that the first two cases of this theorem imply that f(x) has a polynomial factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The third case just showed that if n ≤ m, one can obtain a factorization f(x) = an �n i=1(x − ri) in R[x]/(am−n+1, am−n+2, · · · , am).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If R has a set S in which the difference of any two elements is invertible in R, we call such S an invertible |S |-tuple of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The first corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7 shows that generalized relations of roots and coefficients of a polynomial can be viewed in some sense as polynomial interpolation over a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 6 Yangyang Ruan Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1 and f(x) = a0+a1x+· · ·+amxm be a polynomial over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If R has an invertible n-tuple {r1, r2, · · · , rn} of f(x), then f(x) = n � j=1 � 1≤i≤n,i�j x − ri rj − ri (− m � i1=n ai1ri1 j ), when m < n, − �m i1=n ai1ri1 j denotes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The other corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7 gives a sufficient yet useful condition to guarantee the vanishment of a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (Vanishing ring condition) Let f(x) = a0 + a1x + · · · + amxm be a polynomial over a commutative ring R with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' R has an n-tuple {r1, r2, · · · , rn} of f(x) under the assumption that R � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (i) If n > m and 1 belongs to the ideal (a0, a1, · · · , an) of R, then R = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) if n ≤ m and 1 belongs to the ideal (a0 − det(β,α1,α2,··· ,αn−1) det(α0,α1,··· ,αn−1) , a1 − det(α0,β,α2··· ,αn−1) det(α0,α1,···,αn−1) , · · · , an − det(α0,···,αi−1,β,αi+1,···,αn−1) det(α0,α1,···,αn−1) ) of R, then R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The usefulness of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='10 can be seen in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='13 which includes a proof of Tate vanishing result [16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1] of Morava K-theory and a proof of ΦHKUG ≃ ∗ [9, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='10] for a p-group G and a non-cyclic subgroup H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And our method greatly simplifies those original proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Studying the relation of roots and coefficients of a polynomial in R has a broad application prospect in reducing the relations of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The most common situation is that one obtains some relations of R, like an n-tuple {r1, r2, · · · , rn} of f(x), then dedecates to reduce these relations to get a desired relation, like the solution of ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' A useful application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7 in dealing with practical mathematical problems is the explanation of general blue-shift phenomenon which is motivated by computing the Zariski topology of Balmer spectrum Spc(SH(G)c), details see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For a finite abelian p-group A, let rankp(A) denote the number of Z/p factors in the maximal elementary abelian subgroup of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let ⟨E⟩ denote Bousfield class of E and E(k) denote k-th Johnson-Wilson theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By using pjF(π∗(TA,C(E))) and generalized relations of roots and coeffi- cients of gj(x) in π∗(TA,C(E)), we have a partial answer of general blue-shift phenomenon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2 for abelian cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E be a p-complete, complex oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let A be a finite abelian p-group and C be its direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If E is Landweber exact, then (i) TA,C(E) is Landweber exact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) TA,C(E) is vn−rankp(C)-periodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (iii) ⟨TA,C(E)⟩ = ⟨E(n − rankp(C))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' When k > n, E(n − k) = ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (i) By [21, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='12], the assumption on E implies that ⟨E⟩ = ⟨E(n)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 7 (ii) When A = C = Z/p and E = E(n), this theorem implies the corresponding case of [20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2], and gives an upper bound of BSm(Z/p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Z/p, e), that is BSm(Z/p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Z/p, e) ≤ 16, which implies [6, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (iii) When A = C = (Z/p)k and E is the n-th Morava E-theory En, this theorem implies [36, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (iv) Note that ⟨TA,A(E(n))⟩ = ⟨E(n − rankp(A))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If A = C = H/K is an abelian p-group, then this theorem gives an upper bound of BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K), that is BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K) ≤ rankp(H/K)7, which implies [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let G be a finite p-group and N be its normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To answer general blue-shift phe- nomenon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2 for non-abelian cases, one of the most important problems that we have to deal with is how to compute the roots of [pj]E(x) in π∗(TG,N(E)), and this problem is equivalent to how to compute the roots of [pj]E(x) in E∗(BG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If G is an abelian group, we could define a homomor- phism ψpj G : G → G g �→ gpj, and by the functorial property of the classifying space functor B, we have Bψpj G = ψpj BG and ψpj,∗ BG : E∗(BG) → E∗(BG) is an E∗-algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then the generators of the kernel of ψpj,2 BG : E2(BG) → E2(BG) are roots of [pj]E(x) in E∗(BG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' But if G is a non-abelian group, then ψpj G need not be a homomorphism, so we can not use the functorial property of B to obtain a self-map of BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Inspired by Jackowski-Mcclure-Oliver’s work [23], we regard Bψpj G as an unstable Adams operation, which motivates us to give the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let G be a finite p-group and G′ be the commutator group {aba−1b−1 | a, b ∈ G} of G with a quotient homomorphism ǫ : G → G/G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' A self-map f : BG → BG is called an unstable Adams operation of degree p if the following diagram BG Bǫ −−−−−→ B(G/G′) f \uf8e6\uf8e6� ψp B(G/G′) \uf8e6\uf8e6� BG Bǫ −−−−−→ B(G/G′) commutes up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let G be a finite p-group and E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then there is an unstable Adams operation f : BG → BG of degree p and E2( f(−)) = [p]E(−) : E2(BG) → E2(BG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 6Details see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 7Details see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 8 Yangyang Ruan For a real number r, let ⌈r⌉ denote the least integer of no less than r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For a finite abelian group A, let V(pj|A) denote the subgroup {a ∈ A | pja = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since ǫ(N) is a subgroup of G/G′, then the quotient group G/G′/ǫ(N) can be canonically embedded in G/G′ by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3) Let E be a p-complete, complex oriented spectrum with an as- sociated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let G be a finite p-group and N be its normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='14 is true, then sG,N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='E ≥ max j∈N+ ⌈ logp |V(pj|G/G′)| − logp |V Ä pj|imφ(G/G′/ǫ(N)) ä | j ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The motivation from the computation of the Zariski topol- ogy of Balmer spectrum of the G-equivariant stable homotopy category can be found in Section 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In Section 3, we compute the homotopy group of generalized Tate spectrum TA,C(E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In Section 4, we generalize the relation of roots and coefficients of a polynomial in a commutative ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In Sec- tion 5, we recall the definition of algebraic periodicity and Landweber exactness for a spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11 is a corollary of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1, we give a detailed proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 in Section 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In Section 7, we provide a possible way to deal with general blue-shift phenomenon for non-abelian cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Acknowledgement: Firstly, I thank professor Kuhn Nicholas John for introducing me the prob- lem of computing Balmer spectrum in an International Workshop on Algebraic Topology at Fudan University in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Secondly, I thank professor Stefan Schwede for teaching me lots of knowledge about the G-equivariant stable homotopy category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Thirdly, as most our work is based on my PhD thesis [39], I thank professor Xu An Zhao for his carefully reading my PhD thesis and making me correct some vague arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Finally, I thank Long Huang and Ran Wang for carefully reading my draft and suggesting lots of improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 2 Motivation from the computation of the Zariski topology of Balmer spectrum Spc(SH(G)c) Our work is motivated by computing the Zariski topology of Balmer spectrum, this leads us to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' so let us illustrate why Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11 can be applied to compute the Balmer spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' SH(G)c has a symmetric monoidal structure whose tensor product and unit object are the smash product of G-spectra and G-sphere spectrum S G respectively, which make it resembles a commu- tative ring with a unit, so one could introduce the method of algebraic geometry and define “prime ideal” and “spectrum” for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In 2005, Blamer [4] defined the spectrum Spc(SH(G)c), which is similar to the spectrum of a commutative ring with a unit, of SH(G)c as a set of all proper “prime ideals” with Zarisiki topology, and now this spectrum is called Balmer spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' When G is the trivial group e, SH(G) is the classical stable homotopy category SH(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Hopkins and Smith [18] classified all thick subcategories of SH(e)c by using the work of Ravenel [34] and Mitchell [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In other words, they got the Balmer spectrum Spc(SH(e)c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let K(0) and K(∞) denote the rational and mod p Eilenberg-Maclane spectra K(Q), K(Z/p) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then all proper “prime ideals” General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 9 of SH(e)c are of the following form Cp,m = {X ∈ SH(e)c | K(m − 1)∗(X) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For each p, there is a descending chain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1) Cp,1 ⊋ Cp,2 ⊋ · · · ⊋ Cp,∞ due to [34, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The topology space Spc(SH(e)c) can be described by the following diagram, C2,∞ C3,∞ · · Cp,∞ · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' C2,n+1 C3,n+1 · · Cp,n+1 · · C2,n C3,n · · Cp,n · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' C2,2 ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ C3,2 ▼ ▼ ▼ ▼ · · Cp,2 ❧❧❧❧❧❧❧ · · C0,1 where the line between any two points denotes that there is an inclusion relation between the two “prime ideals”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The computation of Spc(SH(e)c) is one of the main tools used in applications of the nilpotence theorem of Devinatz, Hopkins and Smith [10, 18] to global questions in stable homotopy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Strickland [37] tried to generalize the non-equivariant case to the G-equivariant case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For any subgroup H of a finite group G, Strickland used the geometric H-fixed point functor ΦH(−) : SH(G) → SH(e) which resembles a “ring homomorphsim” to pull back Cp,m to obtain “prime ideals” in SH(G)c, then got the G-equivariant “prime ideals” PG(H, p, m) = (ΦH)−1(Cp,m) = {X ∈ SH(G)c | K(m − 1)∗ΦH(X) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In 2017, Balmer and Sanders [6, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='14] confirmed that all G-equivariant proper “prime ideals” of SH(G)c are obtained by this way, which means that they determined the set structure of Balmer spectrum Spc(SH(G)c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To compute the Zarisiki topology of Spc(SH(G)c), it suffices to give an equivalent condition for any two “prime ideals” PG(K, q, l), PG(H, p, m) of SH(G)c to have an inclusion relation PG(K, q, l) ⊆ PG(H, p, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Balmer and Sanders [6, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='12 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4] obtained two necessary conditions for the inclusion: one is p = q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' the other is that K is a subgroup of H up to G-conjucate, which is denoted by K ≤G H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Therefore, the determination of Zariski topology of Spc(SH(G)c) can be reduced to the computation of the following number BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K) := min{l − m = i ∈ Z| PG(K, p, l) ⊆ PG(H, p, m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 10 Yangyang Ruan There is an observation that l ≥ m which is due to Kuhn and Lloyd [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' It suffices to prove that for each l < m, there is a finite G-spectrum X such that X ∈ PG(K, p, l) and X � PG(H, p, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By Mitchell’s work [30], there is a non-equivariant finite spectrum Y such that Y ∈ Cp,m but Y � Cp,m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then we take X to be a G-spectrum Y with the trivial G-action, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To determine BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K), some intuition for the relation PG(K, p, l) ⊆ PG(H, p, m) would be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' From the descending chain 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 and the fact that ΦK(X) ∈ SH(e)c, it follows that K(m − 1) ⊗ ΦK(X) = 0 ⇔ m−1 � i=0 K(i) ⊗ ΦK(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To transform the above equation into a more convenient form, we recall Bousfield’s [3] definition of a non-equivariant spectrum E, ⟨E⟩ denotes the equivalence class of E: E ∼ F if for any spectrum X ∈ SH(e), E∗X = 0 ⇔ F∗X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And ⟨E⟩ is called Bousfield class of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Due to Ravenel [34, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' ], the Bousfield class ⟨�n i=0 K(i)⟩ equals to the Bousfield class ⟨E(n)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then we have for X ∈ SH(G)c, m−1 � i=0 K(i) ⊗ ΦK(X) = 0 ⇔ E(m − 1) ⊗ ΦK(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Thus for X ∈ SH(G)c, K(m − 1) ⊗ ΦK(X) = 0 ⇔ E(m − 1) ⊗ ΦK(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' PG(K, p, l) ⊆ PG(H, p, m) is equivalent to the fact that for X ∈ SH(G)c, E(l − 1)∗ΦK(X) = 0 implies E(m − 1)∗ΦH(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The inclusion H ֒→ G provides a restriction functor resG H : SH(G) → SH(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Assume that K ⊴ G, the surjective homomorphism G → G/K induces an inflation functor infG G/K : SH(G/K) → SH(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let ˜ΦK be the relative geometric K-fixed point functor from SH(G) to SH(G/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By [26, Chapter II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' §9], we have resG/K e ˜ΦK � ΦK and 0 = E(l − 1) ⊗ ΦK(X) = E(l − 1) ⊗ resG/K e ˜ΦK(X) = resG/K e (infG/K e (E(l − 1)) ⊗ ˜ΦK(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let G/K+ denote the disjoint union of G/K and a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By [5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 Theorem], we get resG/K e (−) � G/K+ ⊗ (−) and 0 = resG/K e (infG/K e (E(l − 1)) ⊗ ˜ΦK(X)) = G/K+ ⊗ infG/K e (E(l − 1)) ⊗ ˜ΦK(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E(G/K) denote the Milnor construction, which is an infinite join G/K ∗ G/K ∗ · · · ∗ G/K, for the group G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then 0 = E(G/K)+ ⊗ infG/K e (E(l − 1)) ⊗ ˜ΦK(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let �E(G/K) be the unreduced suspension of E(G/K) with one of the cone points as basepoint, then we have 0 =F( �E(G/K), ΣE(G/K)+ ⊗ infG/K e (E(l − 1)) ⊗ ˜ΦK(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2) General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 11 By [15, Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5], we have F( �EG, ΣEG+ ⊗ −) � F(EG+, −) ⊗ �EG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' tG(kG) := F(EG+, kG)⊗ �EG is so-called classical Tate construction in the sense of Greenlees and May [14] for a G-spectrum kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Assume that K ⊴ H, we apply geometric H/K-fixed point functor ΦH/K(−) to Formula 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since ΦH/K(−) preserves weak equivalences, we obtain 0 = ΦH/K(tG/K(infG/K e (E(l − 1)) ⊗ ˜ΦK(X))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Note that for X ∈ SH(G), Y ∈ SH(G)c, tG(X) ⊗ Y � tG(X ⊗ Y) (details see [6, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='8]), we have 0 = ΦH/K(tG/K(infG/K e (E(l − 1))) ⊗ ˜ΦK(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' From the facts that for any G/K-spectra X and Y, ΦH/K(X ⊗ Y) = ΦH/K(X) ⊗ ΦH/K(Y), and ΦH/K ◦ ˜ΦK � ΦH, it follows that 0 =ΦH/K(tG/K(infG/K e (E(l − 1))) ⊗ ˜ΦK(X)) =ΦH/K(tG/K(infG/K e (E(l − 1)))) ⊗ ΦH/K ◦ ˜ΦK(X) =ΦH/K(tG/K(infG/K e (E(l − 1)))) ⊗ ΦH(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For convenience, let TG/K,H/K(−) denote the functor ΦH/K(tG/K(infG/K e (−))), and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 we have TG/K,H/K(−) = TH/K,H/K(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If ⟨TG/K,H/K(E(l − 1))⟩ equals to the Bousfield class of some Johnson-Wilson theory, certainly this would give us an upper bound of BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The idea of the above reduction actually comes from Balmer and Sanders’ computation [6, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1] of Zariski topology of the Balmer spectrum Spc(SH(Z/p)c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' They used Hovey-Sadofsky- Kuhn’s result [20, 24] ⟨TZ/p,Z/p(E(l − 1))8⟩ = ⟨E(l − 2)⟩ to get BSm(Z/p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Z/p, e) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In fact, BSm(Z/p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Z/p, e) = 1, which means that the determination of ⟨TG/K,H/K(E(l − 1))⟩ could give us the least upper bound of BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If H/K is a finite abelian p-group, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11 confirmed that ⟨TG/K,H/K(E(l − 1))⟩ = ⟨E(l − 1 − rankp(H/K))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In 2019, Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton [7] obtained that when G is a fi- nite abelian p-group, BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K) is exactly rankp(H/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In particularly, they did not use the Bousfield class ⟨TG/K,H/K(E(l − 1))⟩ to determine the upper bound of BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K), but used the method [32] of derived defect base by recognizing TG/K,H/K(E(l − 1)) as suitable sections of the structure sheaf on a certain non-connective derived scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' There must be some beautiful math living behind such a beautiful result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In order to make this problem more approachable to general audiences, we give a new approach to determine the upper bound of BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Our new approach is by use of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1, and here is a sketch of the proof for Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 is a generalization of [20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' When trying to generalize [20, The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2], we find that the determination of ⟨TG/K,H/K(E(l − 1))⟩ can be transformed into the 8Actually their construction is tZ/p(infZ/p e (−))Z/p, but by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1, tZ/p(infZ/p e (−))Z/p and TZ/p,Z/p(−) are the same construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 12 Yangyang Ruan explanation of Balmer and Sanders’ new blue-shift phenomenon [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' More generally, the deter- mination of ⟨TG/K,H/K(E(l − 1))⟩ can be transformed into the explanation of general blue-shift phenomenon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Observing that if G/K is a finite abelian p-group, then TG/K,H/K(E(l − 1)) inherits the Landweber exactness of E(l − 1), details see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11, we only have to deter- mine the periodicity of TG/K,H/K(E(l − 1)) by Hovey’s Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Here we choose Hovey’s definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5 of vn-periodicity for TG/K,H/K(E(l − 1)) and find that the determination of the pe- riodicity of TG/K,H/K(E(l − 1)) is equivalent to the computation of the projective dimension of π∗(TG/K,H/K(E(l − 1))) as an E(l − 1)∗-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By homology algebra, the projective dimension of π∗(TG/K,H/K(E(l − 1))) is measured by the maximal length of a π∗(TG/K,H/K(E(l − 1)))-regular sequence in the maximal ideal Il−1 = (p, v1, · · · , vl−2) of E(l − 1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then by Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='9, finding some-tuple of pj-series [pj]E(l−1)(x) in π∗(TG/K,H/K(E(l−1))) will give an upper bound of the pro- jective dimension of π∗(TG/K,H/K(E(l−1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Given the periodicity of E(l−1), by inductively using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='40, we will get a lower bound of the projective dimension of π∗(TG/K,H/K(E(l − 1))), details see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' This is our idea to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' There are several significant differences between our new proof and the earlier of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' First, our proof is self-contained, while their proof of [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4] is based on a series of work [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Second, our proof is more conceptual in the sense that we have an intuitive idea to explain general blue-shift phenomenon and successfully achieve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' When G is a non-abelian group, BSm(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' H, K) is not completely known, our new proof may help to bring some intuition to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Third, they [7] used derived algebraic geometry and the geometry of the stack of formal groups to describe the chromatic height shifting behaviour of TG/K,H/K(E(l−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' However, our method only need the tool of some-tuple of the pj-series in π∗(TH/K,H/K(E(l − 1))) and some linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 3 The homotopy groups π∗(TA,C(E)) and their maps Follow the notions of [19, Section 5], in this section we assume that E is a complex-oriented cohomology theory, particularly p-complete theory with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The homotopy group of the classical Tate construction tA(infA e (E))A is computed in [17], and the homotopy group of generalized Tate spectrum TA,C(E) has already been known to the experts over years, but there is not any version with enough proving details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' In this section, we provide a detailed proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The functor TG,N(−) is related to TG,N(−) by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let G be a finite p-group or T m = U(1) × · · · × U(1) ������������������������������������������ m , and N be its normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then TG,N(−) = TN,N(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By definition, ΦN(−) = ˜ΦN ◦ resG N(−), combining with the fact that resG N(tG(infG e (−)) = tN(resG N ◦infG e (−) = tN(infN e (−)), details see [6, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 18], we have ΦN(tG(infG e (−))) = TN,N(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ First, we recall the definition [26] of the relative geometric N-fixed point functor ˜ΦN(−) : SH(G) → SH(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For a family F of subgroups of G closed under G-conjugacy, there is a General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 13 universal space EF characterized by its fixed point data: EF K be contractible if K ∈ F and empty if K � F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' There is a map EF+ → S 0 induced by EF → ∗, and let �EF denote its cofiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then by the long exact sequence of non-equivariant homotopy groups induced by this confiber sequence, we obtain that �EF K is homotopy equivalent to ∗ if K ∈ F and S 0 if K � F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Therefore �EF1 ⊗ �EF2 ≃ �E(F1 ∪F1) where ≃ denotes the homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let F [N] denote the family of subgroups of G which do not contain N, then the definition of ˜ΦN(−) is ( �EF [N] ⊗ (−))N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' �EG denotes �EF where F is the family subgroups only containing the trivial subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To compute π∗(TG,N(E)), we give an equivalent description of π∗(TG,N(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let G be a finite p-group or T m, and N be its normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E be a non-equivariant spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then TG,N(E) ≃ ( ˜ΦN(F(EG+, infG e (E))))G/N and π∗(TG,N(E)) � πG/N ∗ ( ˜ΦN(F(EG+, infG e (E)))), where G/N-equivariant homotopy group is defined by a complete G/N-universe in the sense of Lewis, May and Steinberger [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If the family subgroups of G which do not contain N are {e}, then TG,N(−) = tG(infG e (−))G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since �EF [N] ⊗ �EG ≃ �EF [N], we have TG,N(E) =( ˜ΦNtG(infG e (E)))G/N =(( �EF [N] ⊗ �EG ⊗ F(EG+, infG e (E)))N)G/N ≃(( �EF [N] ⊗ F(EG+, infG e (E)))N)G/N = ( ˜ΦN(F(EG+, infG e (E))))G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By the adjunction [S n, ( ˜ΦN(F(EG+, infG e (E))))G/N] � [infG/N e (S n), ˜ΦN(F(EG+, infG e (E)))]G/N, we identify the homotopy group π∗( ˜ΦN(F(EG+, infG e (E))))G/N with the G/N-equivariant homotopy group πG/N ∗ ( ˜ΦN(F(EG+, infG e (E)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If {e} is the family subgroups of G which do not contain N, then �EF [N] = �EG and TG,N(−) = tG(infG e (−))G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ Let N be a normal subgroup of G, then the following theorem of Costenoble describes the behavior of relative geometric N-fixed point functor ˜ΦN(−) on the homotopy group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (Costenoble [26, Chapter II proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=']) Let kG be a ring spectrum and set kG/N = ˜ΦN(kG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then for a finite G/N-CW spectrum X, k∗ G/N(X) is the localization of k∗ G(infG G/N(X)) obtained by inverting the Euler classes χV ∈ kV G(S 0) of those representations V of G such that VN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3, it follows that to compute π∗(TG,N(E)), we only need to compute πG ∗ (F(EG+, infG e (E))), then invert the Euler classes χV ∈ F(EG+, infG e (E))V(S 0) of those complex G-representations V such that VN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By the equivariant suspension isomorphism, we have χV ∈ F(EG+, infG e (E))V(S 0) � F(EG+, infG e (E))|V|(S |V|−V), where |V| denote the real dimension of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3 and the following observation πG ∗ (F(EG+, infG e (E))) =π∗(G/G+ ∧ S 0, F(EG+, infG e (E)))G =π∗(S 0, F(EG+, infG e (E))G) =π∗(BG+, E) = E∗(BG+), we identify the G-equivariant homotopy group πG ∗ (F(EG+, infG e (E))) with E∗(BG+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 14 Yangyang Ruan 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 The E∗-cohomology of the classifying space of a finite abelian p-group First we introduce the Weierstrass Preparation Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (Weierstrass Preparation Theorem [40, 29, 41]) Let R be a graded local commuta- tive ring, complete in the topology defined by the powers of an ideal m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Suppose α(x) = ∞ � i=0 aixi ∈ R[[x]] satisfies α(x) ≡ anxn mod (m, xn+1) with an ∈ R a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then (i) (Euclidean algorithm) Given f(x) ∈ R[[x]], there exist unique r(x) ∈ R[x] and q(x) ∈ R[[x]] such that f(x) = r(x) + α(x)q(x) with deg r(x) ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) The ring R[[x]]/(α(x)) is a free R-module with basis {1, x, · · · , xn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (iii) (Factorization) There is a unique factorization α(x) = ε(x)g(x) with ε(x) a unit and g(x) a monic polynomial of degree n, we call g(x) the Weierstrass polynomial of α(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The number n is called the Weierstrass degree of α(x) and denoted by degW α(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Recall some basic properties of the associated formal group law F over E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let In denote the maximal ideal of E∗ and vn be a unit of E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then for any integer m, the m-series of F satisfies (i) [m]E(x) ≡ mx mod (x2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) [mk]E(x) = [m]E([k]E(x));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (iii) [p]E(x) = vnxpn mod In;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (iv) [m − k]E(x) = [m]E(x) −F [k]E(x) = ([m]E(x) − [k]E(x)) · ε([m]E(x), [k]E(x)), where ε([m]E(x), [k]E(x)) is a unit in E∗[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let gj(x) denote the Weierstrass polynomial of [pj]E(x) and gj 1(x) = g1(gj−1 1 (x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then gj(x) = gj 1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Suppose that [p]E(x) = px + a2x2 + · · · + apn−1xpn−1 + vnxpn mod (xpn+1), and we apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4 to [p]E(x) ∈ E∗[[x]], then [p]E(x) = ε(x)g1(x) with ε(x) a unit and g1(x) = px + a2x2 + · · · + apn−1xpn−1 + vnxpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And we apply this theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4 to [pj]E(x) ∈ E∗[[x]], by the fact that [pj]E(x) = [p]E([pj−1]E(x)), then [pj]E(x) = ε j(x)gj(x) with ε j(x) a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By the uniqueness of factorization 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4 and the fact that gj 1(x) = [pj]E(x) = v1+pn+···+p(j−1)n n xpjn mod In, then gj(x) = gj 1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ The following lemma gives the computation of E∗(BA+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If A is an abelian p-group of form Z/pi1 ⊕ · · · ⊕ Z/pim, then E∗(BA+) � E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If A = Z/pj, then there is a fiber sequence: S 1 → BZ/pj → CP∞ ψpj → CP∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Note that the Euler class of the Gysin sequence of S 1 → BZ/pj → CP∞ is ψpj,2(x) = [pj]E(x) ∈ E2(CP∞ + ), then we have a long exact sequence: · · � E∗[[x]] ∪[pj]E(x)� E∗+2[[x]] � E∗+2(BZ/pj +) � · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since [pj]E(x) is not a zero divisor in E∗[[x]], the long exact sequence splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Therefore, we obtain E∗(BZ/pj +) � E∗[[x]]/([pj]E(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' As we all know, K¨unneth isomorphism is not always true for product spaces X × Y, but if E- cohomology of the space X or Y is a finitely generated free module over E∗, the K¨unneth isomor- phism is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By Weierstrass Preparation Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4, we have an E∗-ring isomorphism η : E∗[[x]]/([pj]E(x)) � E∗[x]/(gj(x)) that maps f(x) to r(x), where gj(x) is the Weierstrass polynomial of [pj]E(x), which implies that E∗[[x]]/([pj]E(x)) is a finite free E∗-module of rank pjn = degW[pj]E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ Note that E∗(BZ/pj +) is a Hopf algebra over E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And η induces a coalgebra structure on E∗[x]/(gj(x)): E∗[[x]]/([pj]E(x)) µ∗ BZ/pj −−−−−→ E∗[[x]]/([pj]E(x)) ⊗E∗ E∗[[x]]/([pj]E(x)) η \uf8e6\uf8e6� η⊗η \uf8e6\uf8e6� E∗[x]/(gj(x)) η⊗η◦µ∗ BZ/pj◦η−1 −−−−−−−−−−−−→ E∗[x]/(gj(x)) ⊗E∗ E∗[x]/(gj(x)), then combining with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='6, we have Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then there is an E∗-algebra isomorphism η : E∗[[x]]/([pj]E(x)) � E∗[x]/(gj 1(x)), where the coalgebra structure on E∗[x]/(gj 1(x)) is given by the map η ◦ µ∗ BZ/pj ◦ η−1 : E∗[x]/(gj 1(x)) → E∗[x]/(gj 1(x)) ⊗E∗ E∗[x]/(gj 1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 16 Yangyang Ruan 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2 Euler classes and formal groups In this paper, we always identify Z/pj with the set {0, 1, · · · , pj − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let ρ w n : Z/pj → U(1) denote the complex character that maps h to e 2whπi pj for w ∈ Z/pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Suppose that A has the form Z/pi1 ⊕ · · · ⊕ Z/pim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By the representation theory of abelian groups [38, Propositon 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' ], {ρ( w1 pi1 ,···, wm pim ) = µU(1) ◦ (ρ w1 pi1 × · · · × ρ wm pim ) = ρ w1 pi1 · · · ρ wm pim : A → U(1) | (w1, · · · , wm) ∈ A} formed all irreducible complex representations of Z/pi1 ⊕ · · · ⊕ Z/pim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Recall the definition [14] of Euler classes for the A-spectrum F(EA+, infA e (E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let V be any complex A-representation with an inner product, let eV : S 0 → S V send the non-basepoint to 0, and let χV ∈ F(EA+, infA e (E))V(S 0) be the image of the unit of F(EA+, infA e (E))0(S 0) under the map e∗ V : F(EA+, infA e (E))0(S 0) � F(EA+, infA e (E))V(S V) → F(EA+, infA e (E))V(S 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since any finite abelian p-group A with rankp(A) = m is isomorphic to a subgroup of T m, we first show how to specifically identify E∗(BU(1)+) � E∗[[x]] with πU(1) ∗ (F(EU(1)+, infU(1) e (E))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R denote the U(1)-spectrum F(EU(1)+, infU(1) e (E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' We may assume that E is a homotopy commutative ring spectrum, and by [8, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='] F(EU(1)+, infU(1) e (E)) is a homotopy commutative U(1)-ring spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Firstly, recall the definition [32, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1] of a Thom class µV : S V−|V| → R for V with respect to R, µV is a map of U(1)-spectra such that its canonical extension to an R-module map R ⊗ S V−|V| idR⊗µV −−−−−→ R ⊗ R µ −−−−−→ R is an equivalence, where µ denotes the multiplication map of the ring spectrum R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Secondly, we will find the Thom class µV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since all irreducible complex representations of abelian groups are complex one-dimensional, we may choose V to be C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For the principal U(1)-bundle C → C → ∗, we have a Thom space S C, which gives a Thom isomorphism φC : F(EU(1)+, infU(1) e (E))∗(S 0) → F(EU(1)+, infU(1) e (E))∗+2(S C), by the equivariant suspension isomorphism, we can rewrite φC as an isomorphism πU(1) ∗ (F(EU(1)+, infU(1) e (E))) � πU(1) ∗ (F(EU(1)+, infU(1) e (E)) ⊗ S 2−C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By [32, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2], this Thom isomorphism φC gives rise to such a Thom class µC : S C−2 → F(EU(1)+, infU(1) e (E)) for C with respect to F(EU(1)+, infU(1) e (E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Follow the notions of [13, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2], we also insist that φC(y) = y · µC for all y ∈ F(EU(1)+, infU(1) e (E))∗(S 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since χV : S −|V| eV→ S V−|V| µV → F(EU(1)+, infU(1) e (E)), we have χC = φC(eC) = eC · µC = e∗ C(µC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For the universal principal U(1)-bundle U(1) → EU(1) → BU(1), we have a Thom space MU(1) ≃ BU(1), which gives a Thom isomorphism ∪x : E∗(BU(1)+) → E∗+2(BU(1)+), and it corresponds to ·χC under the following identification F(EU(1)+, infU(1) e (E))∗(S 0) µC −−−−−→ F(EU(1)+, infU(1) e (E))∗+2(S C) � \uf8e6\uf8e6� e∗ C \uf8e6\uf8e6� E∗(BU(1)+) ∪x −−−−−→ F(EU(1)+, infU(1) e (E))∗+2(S 0) � E∗+2(BU(1)+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 17 Then x corresponds to χC under the isomorphism between F(EU(1)+, infU(1) e (E))∗(S 0) and E∗(BU(1)+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let ρ w pj be an irreducible complex Z/pj-representation with w ∈ Z/pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let ρ# w pj be the map F(EU(1)+, infU(1) e (E))∗(S 0) → F(EZ/pj +, infZ/pj e (E))∗(S 0) induced by ρ w pj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then Bρ∗ w pj (x) = [pj]E(x) corresponds to χρ w pj = ρ# w pj (µC) under the isomorphism between πZ/pj ∗ (F(EZ/pj +, infZ/pj e (E))) and E∗(BZ/pj +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' We take V to be C and identify the following two diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' F(EU(1)+, infU(1) e (E))∗(S 0) χC � ρ#w pj � F(EZ/pj +, infZ/pj e (E))∗(S 0) ρ#w pj (χC) � F(EU(1)+, infU(1) e (E))∗+2(S 0) ρ#w pj � F(EZ/pj +, infZ/pj e (E))∗+2(S 0), E∗(BU(1)+) ∪x � Bρ∗w pj � E∗(BZ/pj +) ∪Bρ∗w pj (x) � E∗+2(BU(1)+) Bρ∗+2 w pj � E∗+2(BZ/pj +), which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let A be an abelian p-group of form Z/pi1 ⊕ · · · ⊕ Z/pim and ρ( w1 pi1 ,···, wm pim ) be an irreducible complex A-representation with (w1, · · · , wm) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let ρ# ( w1 pi1 ,··· , wm pim ) be the map F(EU(1)+, infU(1) e (E))∗(S 0) → F(EA+, infA e (E))∗(S 0) induced by ρ( w1 pi1 ,···, wm pim ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then Bρ∗ ( w1 pi1 ,··· , wm pim )(x) = [w1]E(x1) +F · · · +F [wm]E(xm), corresponds to χρ( w1 pi1 ,··· , wm pim ) = ρ# ( w1 pi1 ,··· , wm pim )(χC) under the isomorphism between πA ∗ (F(EA+, infA e (E))) and E∗(BA+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since ρ( w1 pi1 ,···, wm pim ) : A → U(1) is the composition map Z/pi1 ⊕ · · · ⊕ Z/pim ρ w1 pi1 ×···×ρ wm pim −−−−−−−−−−−→ T m µm U(1) −−−−−→ U(1) which induces the composition of E∗-algebra homomorphisms E∗(BU(1)+) Bµm,∗ U(1) −−−−−→ E∗(BT m + ) B(ρ w1 pi1 ×···×ρ wm pim )∗ −−−−−−−−−−−−−−→ E∗(BA+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Note that Bµm,∗ U(1)(x) = x1 +F · · · +F xm, then we have Bρ∗ ( w1 pi1 ,··· , wm pim )(x) = B(ρ w1 pi1 × · · · × ρ wm pim )∗ ◦ Bµm,∗ U(1)(x) = B(ρ w1 pi1 × · · · × ρ wm pim )∗(x1 +F · · · +F xm) = [w1]E(x1) +F · · · +F [wm]E(xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ 18 Yangyang Ruan Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (Lubin and Tate [28]) For each k ∈ Z and each nature number j, there exists a unique series [k]E(x) ∈ E∗[[x]] such that [k]E(x) ≡ kx mod (x2) and [k]E([pj]E(x)) = [pj]E([k]E(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For convenience, we denote [w1]E(x1) +F · · · +F [wm]E(xm) by α(w1,···,wm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let j be a nature number and E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If A is a finite abelian p-group of form Z/pi1 ⊕· · ·⊕Z/pim, then there is a bijection ω : pjF(E∗(BA+)) → {α(w1,··· ,wm) ∈ E∗(BA+) | (pjw1, · · · , pjwm) = 0, (w1, · · · , wm) ∈ A} f ∗ �→ ω( f ∗) = f ∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' First suppose that A = Z/pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For f ∗ ∈ pjF(E∗(BZ/pi +)) = HomE∗−alg(E∗[[x]]/[pj]E(x), E∗(BZ/pi +)), we can identify f ∗ with f ∗(x) since f ∗ is an E∗- ring homomorphism, which means that ω is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then we have to prove that ω is well-defined, namely f ∗(x) ∈ {α(w1,··· ,wm) ∈ E∗(BA+) | (pjw1, · · · , pjwm) = 0, (w1, · · · , wm) ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' As f ∗ is a graded E∗-algebra homomorphism and deg x = 2, we have 0 = f ∗([pj]E(x)) = [pj]E( f ∗(x)) ∈ E2(BZ/pi +) � E2[[x]]/[pi]E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Notice that [pj]E(x) ≡ pjx mod (x2), then the constant term of f ∗(x) must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since f ∗(x) ∈ E2(BZ/pi +), we may suppose that f ∗(x) ≡ kx mod (x2), and by Lubin and Tate the- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='11, we have f ∗(x) = [k]E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By the property that [n1]E([n2]E(x)) = [n1n2]E(x), we have [pj]E([k]E(x)) = [kpj]E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then f ∗ ∈ HomE∗−alg(E∗[[x]]/[pj]E(x), E∗(BZ/pi +)) implies that f ∗(x) ∈ {[w]E(x) ∈ E2[[x]]/[pi]E(x) | pjw = 0, w ∈ Z/pi}, so θ is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Note that for each [w]E(x) ∈ E2[[x]]/[pi]E(x) with pjw = 0, there is a group homomorphism ρw : Z/pi → Z/pj that maps 1 to w and Bρ∗ w(x) = [w]E(x), so Bρ∗ w is an E∗-algebra homomorphism, so ω is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Therefore, ω is a well-defined bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For A = Z/pi1 ⊕ · · · ⊕ Z/pim, there are group inclusions ιk : Z/pik → A that maps w ∈ Z/pik to (0, · · · , 0, w, 0, · · · , 0) ∈ Z/pi1 ⊕ · · · ⊕ Z/pik−1 ⊕ Z/pik ⊕ Z/pik−1 ⊕ · · · ⊕ Z/pim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7, we have E∗(BA+) � E∗[[x1]]/([pi1]E(x1)) ⊗E∗ · · · ⊗E∗ E∗[[xm]]/([pim]E(xm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' There is an isomorphism: HomE∗−alg(E∗[[x]]/[pj]E(x), E∗(BA+)) → m � k=1 HomE∗−alg(E∗[[x]]/[pj]E(x), E∗[[x1]]/([pik]E(xk))) f ∗ �→ Bι∗ 1 ◦ f ∗ ⊗ · · · ⊗ Bι∗ m ◦ f ∗ We can identify f ∗ ∈ HomE∗−alg(E∗[[x]]/[pj]E(x), E∗(BA+)) with f ∗(x) ∈ E2(BA+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then the rest proof is similar to the case of A = Z/pi, we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 19 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let A be a finite abelian p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If G is a finite abelian p-group or U(1), then the map E∗(B(−)) : Hom(A,G) → HomE∗−alg(E∗(BG+), E∗(BA+)) defined by f �→ E∗(B f) = B f ∗ is an isomorphism of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='12, it is easy to check that E∗(B(−)) is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then the remaining thing is to prove that E∗(B(−)) is a homomorphism of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let [BA+, BG+] denote the homotopy class from BA+ to BG+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since G is abelian, we have Hom(A,G)/InnG = Hom(A,G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Note that A is a finite abelian p-group, by Dwyer and Zabrodsky’s Theorem [12] or Notbohm’s Theorem [33], there is a bijection B : Hom(A,G) → [BA+, BG+] ρ �→ Bρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For a topological space X, let ∆X denote the diagonal map X → X × X, then for any ρ1, ρ2 ∈ Hom(A,G), there are products µG ◦(ρ1 ×ρ2)◦∆A and µBG+ ◦(Bρ1 × Bρ2)◦∆BA+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By the functorial property of B, B preserves the product, namely B(µG ◦ (ρ1 × ρ2) ◦ ∆A) = µBG+ ◦ (Bρ1 × Bρ2) ◦ ∆BA+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Similarly, By the functorial property of E∗(−), E∗(−) preserves the product, namely E∗(µBG+ ◦ (Bρ1 × Bρ2) ◦ ∆BA+) = ∆∗ BA+ ◦ (Bρ1 × Bρ2)∗ ◦ µ∗ BG+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' This finishes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='12 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='13, we have Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let j be a nature number and E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If A is a finite abelian p-group, then there are group isomorphisms pjF(E∗(BA+)) � {α(w1,··· ,wm) ∈ E∗(BA+) | (pjw1, · · · , pjwm) = 0, (w1, · · · , wm) ∈ A} � Hom(A, Z/pj) � V(pj|A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Furthermore, p∞F(E∗(BA+)) � Hom(A, U(1)) � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3 Maps between E∗-cohomology of classifying spaces Let A1 and A2 be two abelian p-groups Z/pi1 ⊕ · · · ⊕ Z/pim and Z/pj1 ⊕ · · · ⊕ Z/pjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then any homomorphism f ∈ Hom(A1, A2) is determined by an integer m×k-matrix F ∈ Mm×k(Z(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since each nature number i can be identified with a self-map of U(1) of degree i, F can be identified with a map from T m to T k, and there are two commutative diagrams: A1 ρ 1 pi1 ×···×ρ 1 pim −−−−−−−−−−−→ T m f \uf8e6\uf8e6� F \uf8e6\uf8e6� A2 ρ 1 pj1 ×···×ρ 1 pjk −−−−−−−−−−−→ T k, BA1 B(ρ 1 pi1 ×···×ρ 1 pim ) −−−−−−−−−−−−−→ BT m B f \uf8e6\uf8e6� BF \uf8e6\uf8e6� BA2 B(ρ 1 pj1 ×···×ρ 1 pjk ) −−−−−−−−−−−−−→ BT k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 20 Yangyang Ruan A1 and A2 are associated with the following two fibrations T m/A1 � T m −−−−−→ BA1 B(ρ 1 pi1 ×···×ρ 1 pim ) −−−−−−−−−−−−−→ BT m, T k/A2 � T k −−−−−→ BA2 B(ρ 1 pj1 ×···×ρ 1 pjk ) −−−−−−−−−−−−−→ BT k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then there is a Leray-Serre spectral sequence of T m → ET m → BT m with the E2- page Hs(BT m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Et(T m)) � Hs(BT m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Z/p) ⊗ Et(T m) � Z/p[[x1, x2, · · · , xm]] ⊗ ∧E∗[y1, y2, · · · , ym], and its only nontrivial differential is d2(1 ⊗ yi) = xi for 1 ≤ i ≤ m, which implies that it collapses at E3-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since ET m is contractible, then the only possible differential is d2(1 ⊗ yi) = xi for 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then there is a Leray-Serre spectral sequences of T m → BA1 → BT m with the E2- page Hs(BT m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Et(T m)) � Hs(BT m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Z/p) ⊗ Et(T m) � Z/p[[x1, x2, · · · , xm]] ⊗ ∧E∗[y1, y2, · · · , ym], and its only nontrivial differential is d2(1 ⊗ yi) = [pij]E(xj) for 1 ≤ j ≤ m, which implies that it collapses at E3-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' The following commutative diagram BA1 B(ρ 1 pi1 ×···×ρ 1 pim ) −−−−−−−−−−−−−→ BT m \uf8e6\uf8e6� 1BTm \uf8e6\uf8e6� ET m −−−−−→ BT m induces a map of Leray-Serre spectral sequences, which gives differentials d2(1 ⊗ yi) = [pij]E(xj) for 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7, we conclude that it collapses at E3-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let E be a p-complete complex-oriented spectrum with an associated formal group of height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let A1 and A2 be two abelian p-groups Z/pi1⊕· · ·⊕Z/pim and Z/pj1⊕· · ·⊕Z/pjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then any abelian group homomorphism f ∈ Hom(A1, A2) is determined by an integer m×k-matrix F ∈ Mm×k(Z(p)), and the homomorphism B f ∗ : E∗(BA2+) → E∗(BA1+) can be identified with the E3-page map of Leray-Serre spectral sequences for two associated fibrations T m/A1 � T m −−−−−→ BA1 B(ρ 1 pi1 ×···×ρ 1 pim ) −−−−−−−−−−−−−→ BT m, T k/A2 � T k −−−−−→ BA2 B(ρ 1 pj1 ×···×ρ 1 pjk ) −−−−−−−−−−−−−→ BT k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' where the map of these two fibrations is given by the following commutative diagram: BA1 B(ρ 1 pi1 ×···×ρ 1 pim ) −−−−−−−−−−−−−→ BT m B f \uf8e6\uf8e6� BF \uf8e6\uf8e6� BA2 B(ρ 1 pj1 ×···×ρ 1 pjk ) −−−−−−−−−−−−−→ BT k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4 The homotopy groups π∗(TA,C(E)) The following lemma determines all complex representations V of A such that VC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let A be an abelian group of form Z/pi1 ⊕ · · · ⊕ Z/pim and C be its subgroup Z/pj1 ⊕ · · · ⊕ Z/pjm with a group inclusion ϕ : Z/pj1 ⊕ · · · ⊕ Z/pjm → Z/pi1 ⊕ · · · ⊕ Z/pim (w1, · · · , wk) �→ (pi1−j1w1, · · · , pim−jmwm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' There is a group homomorphism from A/C to A as follows: φ : Z/pi1−j1 ⊕ · · · ⊕ Z/pim−jm → Z/pi1 ⊕ · · · ⊕ Z/pim (w1, · · · , wm) �→ (pj1w1, · · · , pjmwm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then {ρ( w1 pi1 ,···, wm pim ) = ρ w1 pi1 · · · ρ wm pim : A → U(1) | (w1, · · · , wm) ∈ A − imφ(A/C)} forms all irreducible complex representations V of A such that VC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Note that {ρ( w1 pi1 ,··· , wm pim ) : A → U(1) | (w1, · · · , wm) ∈ A} formed all irreducible complex representations of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then for any (u1, · · · , um) ∈ C, we have ρ( w1 pi1 ,···, wm pim )(ϕ(u1, · · · , um)) = ρ w1 pi1 (pi1−j1u1) · · · wm pim (pim−jmum) = e 2πi( w1u1 pj1 +···+ wmum pjm ) = ß1 if pj1|w1, · · · , pjm|wm, nonconstant Otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And pj1|w1, · · · , pjm|wm ⇔ (w1, · · · , wm) ∈ imφ(A/C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ Now, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3, it follows that π∗(TA,C(E)) is the localization of π∗(F(EA+, infA e (E))) � E∗(BA+) obtained by inverting the Euler classes χV ∈ F(EA+, infA e (E))|V|(S |V|−V) of those complex representations V of A such that VC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7, we have E∗(BA+) � E∗[[x1, · · · , xm]]/([pi1]E(x1), · · · , [pim]E(xm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='18, we have {ρ( w1 pi1 ,···, wm pim ) : A → U(1) | (w1, · · · , wm) ∈ A − imφ(A/C)} forms all irreducible complex representations V of A such that VC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Each representation ρ( w1 pi1 ,···, wm pim ) : A → U(1) induces a homormorphism Bρ∗ ( w1 pi1 ,···, wm pim ) : E∗(BU(1)+) � E∗[[x]] → E∗(BA+), and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='10, the image Bρ∗ ( w1 pi1 ,··· , wm pim )(x) is the Euler class [w1]E(x1) +F · · · +F [wm]E(xm) = α(w1,··· ,wm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ 22 Yangyang Ruan 4 Generalized relations of roots and coefficients of a polynomial In this section, we prove generalized relations of roots and coefficients of a polynomial, namely Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Recall that there is a map λ : R[x] → Pmap(R, R) with λ( f(x)) = [ f(x)] for f(x) ∈ R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R[x]n denote the set of polynomials of degree at most n and λR[x]n denote the map that restricts λ to R[x]n, then what conditions does R satisfy with such that λR[x]n is injective ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' To give a sufficient condition, we take a fresh look at the equality f(r) = 0 induced by a root r ∈ R of a polynomial map [ f(x)] ∈ Pmap(R, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Without loss of generality, we may suppose that f(x) = a0 + a1x + · · · + anxn with a0, a1, · · · , an ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' f(r) = 0 means that the “R-vector” (a0, a1, · · · , an) is a solution of the homogeneous R-linear equation x0 + rx1 + · · · + rnxn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then we need the definition of “R-vector”, R-linear and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 Basic concepts Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1 and n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let Rn = {(a1, a2, · · · , an) | ai ∈ R, 1 ≤ i ≤ n}, then for (a1, a2, · · · , an), (b1, b2, · · · , bn) ∈ Rn, (a1, a2, · · · , an) = (b1, b2, · · · , bn) ⇔ ai = bi(1 ≤ i ≤ n) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Rn has two operations as follows, for (a1, a2, · · · , an), (b1, b2, · · · , bn) ∈ Rn, r ∈ R, then (i) Vector addition: (a1, a2, · · · , an) + (b1, b2, · · · , bn) = (a1 + b1, a2 + b2, · · · , an + bn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) Scalar multiplication: r(a1, a2, · · · , an) = (ra1, ra2, · · · , ran).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' These two operations on Rn satisfy the following eight rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For any a, b, c ∈ Rn, r, k ∈ R, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' a + b = b + a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (a + b) + c = a + (b + c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' there is a unique vector 0 = (0, 0, · · · , 0) in Rn such that 0 + a = a + 0 = a, then 0 is called the zero vector of Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' for any a = (a1, a2, · · · , an) ∈ Rn, there is a vector −a = (−a1, −a2, · · · , −an) ∈ Rn, called the negative of a, such that a + (−a) = (−a) + a = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 1(a) = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (kr)a = k(ra);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (k + r)a = ka + ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' r(a + b) = ra + rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then Rn is called an n-dimensional R-vector space or R-linear space, and any a ∈ Rn is called an n-dimensional R-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And we have the notion of subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' General blue-shift phenomenon and generalized relations of roots and coefficients of a polynomial 23 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If a nonempty subset U of Rn satisfies that (i) a, b ∈ U ⇒ a + b ∈ U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' (ii) a ∈ U, r ∈ R ⇒ ra ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then U is called an R-vector subspace of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For t1, t2, · · · , tn ∈ R, if there is a system of homogeneous R-linear equations \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 x0 + t1x1 + t2 1x2 + · · · + tn−1 1 xn−1 = 0 x0 + t2x1 + t2 2x2 + · · · + tn−1 2 xn−1 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' x0 + tnx1 + t2 nx2 + · · · + tn−1 n xn−1 = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1) with variables x0, x1, · · · , xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then the solution of Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 is an R-vector subspace of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='2 n-tuple of a polynomial over a commutative ring Now, we give a sufficient condition such that the solution of Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For t1, t2, · · · , tn ∈ R, any 1 ≤ i � j ≤ n, ti − tj is not zero or a zero divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If there is a system of homogeneous R-linear equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 with variables x0, x1, · · · , xn−1, then the solution of Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 is the subspace {0} of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For constants c0, c1, · · · , cn−1, d ∈ R, if t is not zero or a zero divisor, then the solutions of c0x0 + c1x1 + · · · + cn−1xn−1 = d and tc0x0 + tc1x1 + · · · + tcn−1xn−1 = td are the same, that is c0x0 + c1x1 + · · · + cn−1xn−1 = d ⇔ tc0x0 + tc1x1 + · · · + tcn−1xn−1 = td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' We use Gaussian elimination to solve the R-linear equations: â1 t1 t2 1 · · · tn−1 1 1 t2 t2 2 · · · tn−1 2 1 t3 t2 3 · · · tn−1 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 1 tn t2 n · · · tn−1 n ì → â1 t1 t2 1 · · tn−1 1 0 t2 − t1 t2 2 − t2 1 · · · tn−1 2 − tn−1 1 0 t3 − t1 t2 3 − t2 1 · · · tn−1 3 − tn−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 0 tn − t1 t2 n − t2 1 · · · tn−1 n − tn−1 1 ì → â1 t1 t2 1 · · tn−1 1 0 1 t1 + t2 · · · �n−2 i=0 tn−2−i 1 ti 2 0 1 t1 + t3 · · · �n−2 i=0 tn−2−i 1 ti 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 0 1 t1 + tn · · · �n−2 i=0 tn−2−i 1 ti n ì , then inductively carry out the above process and finally obtain the upper triangular matrix \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 t1 t2 1 · · tn−1 1 0 1 t1 + t2 · · · �n−2 i=0 tn−2−i 1 ti 2 0 0 1 · · �n−2 i=1 tn−2−i 1 �i−1 j=0 ti−1−j 2 t j 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' 0 0 0 · · 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , this finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ 24 Yangyang Ruan Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' we define an n-tuple {t1, t2, · · · , tn} of R such that for any 1 ≤ i � j ≤ n, ti − tj is not zero or a zero divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' if for any 1 ≤ i � j ≤ n, ti − tj is invertible in R, we call {t1, t2, · · · , tn} an invertible n-tuple of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let f(x) be a polynomial over R, we call {r1, r2, · · · , rn} an n-tuple of f(x) if it is an n-tuple of R and also is a subset of roots of f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1, and d is not zero or a zero divisor in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For r ∈ R, we call t is divisible by d if there is an element t′ ∈ R such that t = dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Since d is not zero or a zero divisor in R, for t ∈ R, the solution of t = dx in R is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If R has an n-tuple {t1, t2, · · · , tn}, then λR[x]n−1 is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' For any two polynomial f1(x) � f2(x) ∈ R[x]n−1, without loss of generality we may suppose that f1(x) = �n−1 k=0 akxk, f2(x) = �n−1 k=0 bkxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then f1(x) � f2(x) implies that there is 1 ≤ k0 ≤ n − 1 such that ak0 − bk0 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If λR[x]n−1( f1(x)) = λR[x]n−1( f2(x)), that is [ f1(y) − f2(y) = ( f1 − f2)(y)] = [0], which implies that ( f1 − f2)(ti) = 0 for any 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Then the n-dimensional R-vector (a0 − b0, a1 − b1, · · · , an−1 − bn−1) is a solution of Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' And by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='4, the solution of Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='1 is {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' So (a0−b0, a1−b1, · · · , an−1−bn−1) = (0, 0, · · · , 0), which contradicts to our assumption that ak0 − bk0 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let R be a commutative ring with 1 and R has an n-tuple {t1, t2, · · · , tn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Let αi denote the column n-dimensional R-vector (ti 1, ti 2, · · · , ti n)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' If 0 ≤ i1 < i2 < · · · < in−1, then det(α0, αi1, αi2, · · · , αin−1) is divisible by det(α0, α1, · · · , αn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE4T4oBgHgl3EQfWAzx/content/2301.05030v1.pdf'} +page_content=' By directly computation, we have det(α0, α1, · · · , αn−1) = � 1≤j 0. +(H4) The map t �→ H(x, y, +√ +t) is convex for each x, y ∈ RN. +(H5) +� ∞ +a +� +t +H(t) +� +s +N−s dt = ∞ and +� b +0 +� +t +H(t) +� +s +N−s dt < ∞, for some a, b > 0. +Due to the presence of the Choquard type non linearity, Problem (1.1) is known +as a Choquard equation. One of the main tool to deal with such type of equations +is Hardy-Littlewood-Sobolev [22] inequality which is stated below. +Proposition 1.1. [22] Let t1, t2 > 1 and 0 < λ < N with 1/t1 + 1/t2 + λ/N = 2, +f ∈ Lt1(RN) and g ∈ Lt2(RN). Then there exist a sharp constant C independent +of f and g such that +���� +� +RN +� +RN +f(x)g(y) +|x − y|λ dxdy +���� ≤ C∥f∥Lt1(RN)∥g∥Lt2(RN). +Choquard type of equations have been studied extensively in the literature, we +refer to [28] for the physical interpretation and survey of such type of equations. For + +GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION +3 +some existence results involving Choquard type equations, we refer to the works +of Moroz-Schaftingen [26, 27] (Laplace operator), Avenia-Siciliano-Squassina [6], +Mukherjee-Sreenadh [29](fractional Laplace operator), Patrizia-Xiang-Zhang [31] +(p-Laplacian), Xie-Wang-Zhang [37] ((p, q)-Laplacian) and Pucci-Xiang-Zhang [33], +Belchior-Bueno-Miyagaki-Pereira [11] (fractional p-Laplacian). +Alves-R˘adulescu-Tavares [4] discussed the generalized choquard problem in Orlicz- +Sobolev spaces. +Problem (1.1) involves the potential term which vanishes at infinity, such type +of equations studied widely by many researchers. In 2013, Alves-Souto [5] proved +the existence result for the equation +(1.2) +− ∆u + V (x)u = K(x)f(u) in RN, +where N ≥ 3. They assumed that V, K : RN → (0, ∞) are continuous functions +and satisfy the following conditions: +(K′ +1) K ∈ L∞(RN) and if {An} is a sequence of Borel sets such that sup +n |An| < ∞ +then +lim +s→∞ +� +An∩Bs(0)c K(x) = 0 uniformly in n ∈ N. +(K′ +2) One of the following condition is true: +(K21) +K +V ∈ L∞(RN). +(K22) +K(x) +[V (x)] +2∗−p +2∗−2 +→ 0 as |x| → ∞ for some p ∈ (2, 2∗). +If V, K satisfies (K′ +1) − (K′ +2) then we say (V, K) ∈ K. +Further, Chen-Yuan [13], considered the problem: +(1.3) +− ∆u + V (x)u = +�� +RN +K(y)F(u(y)) +|x − y|λ +dy +� +K(x)f(u(x)) in RN, +where they assumed that (V, K) ∈ K but the conditions (K′ +1) and (K22) is replaced +by the conditions (K1) and (K23), respectively. (K1) and (K23) are as follows: +(K1) K ∈ L∞(RN) and if {An} is a sequence of Borel sets such that sup +n |An| < ∞ +then +lim +s→∞ +� +An∩Bs(0)c |K(x)| +2N +2N−λ = 0 uniformly in n ∈ N. +(K23) |K(x)| +2N +2N−λ +[V (x)] +2∗−p +2∗−2 +→ 0 as |x| → ∞ for some p ∈ (2, 2∗). +In this sequence, Li-Teng-Wu [21] studied the following fractional Schr¨odinger +equation: +(1.4) +(−∆)su + V (x)u = |u|2∗ +s−2u + λK(x)f(u) in RN, +where λ > 0, s ∈ (0, 1), 2∗ +s = +2N +N−2s and (−∆)s is the fractional Laplace operator of +order s. They assumed that (V, K) ∈ K but in the condition (K22), 2∗ is replaced +by 2∗ +s. +Luo-Li-Li [24], considered the fractional choquard equation: +(1.5) +(−∆)su + V (x)u = +�� +RN +K(y)F(u(y)) +|x − y|λ +dy +� +K(x)f(u(x)) in RN, + +4 +SHILPA GUPTA AND GAURAV DWIVEDI +in which they assumed that the conditions (K1) and +(K24) |K(x)| +2N +2N−λ +[V (x)] +2∗s −p +2∗s −2 +→ 0 as |x| → ∞ for some p ∈ (2, 2∗ +s) +are satisfied. +After that, many researches studied the nonlinear equations involving vanishing +potential with different type of operators and different conditions on the non linear- +ity, we refer to, Deng-Li-Shuai [15] (p-Laplace operator), Perera-Squassina-Yang [32] +(fractional p-Laplacian), Isernia [19] (Fractional p&q-Laplacian), Isernia-Repovˇs +[20] (Double-phase operator). Recently, Silva-Souto [35] developed the existence +result for generalized Schr¨odinger equation in Orlicz-Sobolev spaces. +Existence result for Choquard type equations with vanishing potential has been +obtained by Chen-Yuan [13], Alves-Figueiredo-Yang [3] (for Laplace operator), +Albuquerque-Silva-Sousa [14](fractional coupled Choquard-type systems). +In this paper, we assume that V, K : RN → (0, ∞) are continuous functions and +satisfies (K1). Moreover, we assume that +(K2) V, K satisfies one of the following conditions: +(K2a) +K +V ∈ L∞(RN). +(K2b) |K(x)| +2N +2N−λ +L(x) +→ 0 as |x| → ∞, where L(x) = min +t>0 +� +V (x) H(x,x,t) +Ψ(x,x,t) +� +· +Inspired by the mentioned research works, in this article, we study (1.1) via varia- +tional techniques. The main novelties of this paper are as follows: +• To introduce the homogeneous fractional Musielak-Sobolev spaces and in- +vestigate their properties which are needed to study the Problem (1.1). +Provide the characterization of these spaces which is written in the form of +Theorem 2.13. +• To prove the suitable version of Hardy-Littlewood-Sobolev inequality for +Lebesque Musielak spaces. +To the best of our knowledge, this is the first paper which proves the existence +result for generalized fractional Laplace operator with the vanishing potential to- +gether with Choquard type non linearity. +We assume that f : R → R is continuous and satisfies the following conditions: +(f1) There exist a generalized N-function Ψ : RN × RN × R → [0, ∞) and +ψ1l, ψ2l ∈ (h2, h∗ +1) such that ψ1 ≤ ψ(x,y,t)|t|2 +Ψ(x,y,t) +≤ ψ2, ∀(x, y) ∈ RN × RN and +t ̸= 0 and +lim +t→0 +f(t) +ψ(x, x, t)t = 0, ∀ x ∈ RN, +where Ψ(x, y, t) = +� |t| +0 ψ(x, y, r)r dr and +2N +2N−λ = l. +(f2) lim +t→∞ +F(t) +(H∗(x, x, t))1/l = 0, ∀ x ∈ RN, where H∗ is define in (2.2). +(f3) For i ∈ {1, 2}, lim +t→∞ +f(t) +(H∗(x, x, t)) +b−1 +h∗ +i += 0, ∀ x ∈ RN for some bl ∈ (h2, h∗ +1). +(f4) There exist σ > h2/2 such that +0 < σF(t) = σ +� t +0 +f(s)ds ≤ 2tf(t), + +GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION +5 +for all t > 0, x ∈ RN. +This article is organized as follows: We discuss the definition and properties of +Lebesque Musielak spaces and fractional Musielak-Sobolev spaces in Section 2. The +functional setup needed to prove our result is provided Section 3. We also state our +main results in Section 3. Section 4 deals with the proof of Theorem 2.13. Section +5 deals with the proof of Theorem 3.3. Finally, in Section 6, we prove Theorem 3.4. +2. Musielak spaces and Functional Setting +Let Ω ⊆ RN be any open set. Define, +H(x, y, t) = +� |t| +0 +h(x, y, s)s ds, +where h : Ω × Ω × [0, ∞) → [0, ∞). +Recall that, H(x, y, t) : Ω × Ω × R → [0, ∞) is called a generalized N-function if +it satisfies the following conditions: +(1) H is continuous, even and convex function of t. +(2) H(x, y, t) = 0 if and only if t = 0, ∀x, y ∈ Ω. +(3) lim +t→0 +H(x,y,t) +t += 0 and lim +t→∞ +H(x,y,t) +t += ∞, ∀x, y ∈ Ω. +For any generalized N-function A : Ω × Ω × R → [0, ∞), we define the function +ax : Ω × [0, ∞) → [0, ∞) such that +ax(x, t) = a(x, x, t) ∀(x, t) ∈ Ω × [0, ∞) +and +Ax(x, t) = +� |t| +0 +ax(x, s)s ds. +We say that a generalized N-function H, satisfies the weak ∆2-condition if there +exists C > 0 and a non-negative function k ∈ L1(Ω) such that +H(x, y, 2t) ≤ CH(x, y, t) + k(x) ∀(x × y × t) ∈ Ω × Ω × [0, ∞). +If k = 0, then H is said to satisfy ∆2-condition. Throughout this paper, we assume +that H is a generalized N-function which satisfy ∆2-condition. +Next, we define the complementary function � +H corresponding to generalized +N-function H as +� +H(x, y, t) = +� |t| +0 +�h(x, y, s)s ds, +where �h is defined as �h(x, y, t) = sup{s : h(x, y, s)s ≤ t} ∀(x, y, t) ∈ Ω × Ω × [0, ∞). +Moreover, the function H and its complementary function � +H satisfy the following +Young’s inequality [23, Proposition 2.1]: +s1s2 ≤ H(x, y, s1) + � +H(x, y, s2) ∀x, y ∈ Ω, s1, s2 > 0. +The Lebesque-Musielak space LHx(Ω) is defined as: +LHx(Ω) = +� +u : Ω → R is measurable +���� +� +Ω +Hx (x, τ|u|) dx < ∞, for some τ > 0 +� +· +LHx(Ω) is a normed space [30] with the Luxemburg norm +∥u∥LHx(Ω) = inf +� +τ > 0 +���� +� +Ω +Hx (x, τ|u|) dx ≤ 1 +� +· + +6 +SHILPA GUPTA AND GAURAV DWIVEDI +Theorem 2.1. [30] The space LHx(Ω) is separable and reflexive Banach space. +Theorem 2.2. [1] The space Cc(Ω) is dense in LHx(Ω). +Moreover, C∞ +c (Ω) is +dense in LHx(Ω). +Proposition 2.3. [1] Let H and � +H be complimentary N-functions. Then, for any +u ∈ LHx(Ω) and v ∈ L � +Hx(Ω), we have +���� +� +Ω +uv dx +���� ≤ 2∥u∥LHx(Ω)∥v∥L � +Hx(Ω). +Lemma 2.4. [1] Let v ∈ L � +Hx(Ω). Then +(2.1) +Gv(u) = +� +Ω +u(x)v(x)dx +is a bounded linear functional on LHx(Ω), i.e., Gv ∈ (LHx(Ω))∗. +Also, every +bounded linear functional in LHx(Ω) is of the form (2.1) for some v ∈ L � +Hx(Ω). +Moreover, (LHx(Ω))∗ is isomorphic to L � +Hx(Ω). +Remark 2.5. By Lemma 2.4, we have +∥v∥L � +Hx(Ω) = ∥Gv∥(LHx(Ω))∗ = +sup +∥u∥LHx (Ω)≤1 +����� +� +Ω +u(x)v(x)dx +���� +� +· +For a given generalized N-function H and s ∈ (0, 1), fractional Musielak-Sobolev +space is denoted by W s,H(Ω) and is defined as +W s,H(Ω) = +� +u ∈ LHx(Ω) : +� +Ω +� +Ω +H +� +x, y, τ|u(x) − u(y)| +|x − y|s +� +dx dy +|x − y|N < ∞, for some τ > 0 +� +. +W s,H(Ω) is a normed space with the norm +∥u∥ = ∥u∥LHx(Ω) + [u]s,H, +where +[u]s,H = inf +� +τ > 0 +���� +� +Ω +� +Ω +H +� +x, y, |u(x) − u(y)| +τ|x − y|s +� +dx dy +|x − y|N ≤ 1 +� +· +We define the Lebesque-Musielak space LH(dµ) as: +LH(dµ) = +� +u : Ω × Ω → R is measurable +���� +� +Ω +� +Ω +H (x, y, τ|u(x, y)|) dµ < ∞, for some τ > 0 +� +, +where dµ = +dxdy +|x − y|N is a measure on the set Ω × Ω. +Remark 2.6. [u]s,H is finite if and only if (u(x) − u(y)) +|x − y|s +∈ LH(dµ) and [u]s,H = +���� +u(x) − u(y) +|x − y|s +���� +LH(dµ) +· +Theorem 2.7. [7] W s,H(Ω) is a separable and reflexive Banach space. +The space W s,H +0 +(Ω) is defined as +W s,H +0 +(Ω) = {u ∈ W s,H(RN) : u = 0 a.e. in RN\Ω}. +Next, we state the generalized Poincar´e’s inequality: + +GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION +7 +Theorem 2.8. [7] Let Ω be a bounded open subset of +RN and 0 < s < 1. Then +there exist a positive constant c > 0 such that +∥u∥LHx(Ω) ≤ c[u]s,H, +∀ u ∈ W. +This implies that, [·]s,H is the norm on W s,H +0 +(Ω), which is equivalent to the norm +∥ · ∥. +For a given generalized N-function H : RN × RN × R → [0, ∞), we define the +Sobolev conjugate function H∗ : RN × R → [0, ∞) as: +(2.2) +H∗(x, t) = Hx(x, G−1(t)), ∀ t ≥ 0, +where +G(x, t) = +�� t +0 +� +r +Hx(x, r) +� +s +N−S +dr +� N−s +N +, ∀ t ≥ 0. +One can verify that H∗ is a generalized N-function. +Theorem 2.9. Let s ∈ (0, 1) and H be any generalized N-function satisfying (H5). +Then the embedding W s,H(RN) ֒→ LH∗(RN) is continuous. +Moreover, in this +embedding the space LH∗(RN) is optimal among all the Musielak spaces. +Proof. The proof is similar to the proof of [2, Theorem 6.1]. We omit the details. +□ +Proposition 2.10. Let H be any generalized N-function satisfying (H2) − (H3). +Assume that u ∈ LHx(RN). Then, we have +(1) min +� +ρh1, ρh2� +Hx(x, t) ≤ Hx(x, ρt) ≤ max +� +ρh1, ρh2� +Hx(x, t), ∀ρ, t > 0 +(2) min +� +∥u∥h1 +LHx(RN), ∥u∥h2 +LHx(RN) +� +≤ +� +RN Hx(x, |u|)dx ≤ max +� +∥u∥h1 +LHx(RN), ∥u∥h2 +LHx(RN) +� +· +Proof. Proof of (1) is similar to the proof of [16, Lemma 2.1]. By (H3) and Propo- +sition 2.10, we have +(2.3) +b1 min +� +ρh1, ρh2� +≤ Hx(x, ρ) ≤ b2 max +� +ρh1, ρh2� +∀ρ > 0. +Hence, (2.3) and the definition of norm implies (2). +□ +Let H be any generalized N-function satisfying (H2) − (H3). Then we define +weighted Lebesque-Musielak space LHx +V (RN) as: +LHx +V (RN) = +� +u : RN → R is measurable +���� +� +RN V (x)Hx (x, τ|u|) dx < ∞, for some τ > 0 +� +· +LHx +V (RN) is a normed space [30] with the Luxemburg norm +∥u∥V,H = inf +� +τ > 0 +���� +� +RN V (x)Hx (x, τ|u|) dx ≤ 1 +� +· +Corollary 2.11. Let H be any generalized N-function satisfying (H2) − (H3). +Assume that u ∈ LHx +V (RN). Then, we have +min +� +∥u∥h1 +V,H, ∥u∥h2 +V,H +� +≤ +� +RN V (x)Hx(x, |u|)dx ≤ max +� +∥u∥h1 +V,H, ∥u∥h2 +V,H +� +· +Proposition 2.12. [25, Lemma 4.3] Let H be any generalized N-function satisfying +(H2). Assume that u ∈ H∗ +x(RN) and ρ, t ≥ 0. Then, we have +(1) min +� +ρh∗ +1, ρh∗ +2� +H∗ +x(x, t) ≤ H∗ +x(x, ρt) ≤ max +� +ρh∗ +1, ρh∗ +2� +H∗ +x(x, t), + +8 +SHILPA GUPTA AND GAURAV DWIVEDI +(2) min +� +∥u∥h∗ +1 +LH∗x(RN), ∥u∥h∗ +2 +LH∗x(RN) +� +≤ +� +RN H∗ +x(x, |u|)dx ≤ max +� +∥u∥h∗ +1 +LH∗x(RN), ∥u∥h∗ +2 +LH∗x(RN) +� +, +where, h∗ +1 = +Nh1 +N−sh1 and h∗ +2 = +Nh2 +N−sh2 · +2.1. Homogeneous fractional Musielak-Sobolev space. Fractional Musielak- +Sobolev spaces are not sufficient to study the Problem (1.1), as inf V (x) can be zero. +In this section, we introduce the suitable space to study Problem (1.1) which we +called homogeneous fractional Musielak-Sobolev space and investigate their prop- +erties. +One can be verify that the space C∞ +c (RN) is normed space with the norm [·]s,H. +However, the normed space (C∞ +c (RN), [·]s,H) is not complete. Further, we define +the completion Ds,H(RN) of (C∞ +c (RN), [·]s,H) in the standard way. More precisely, +Ds,H(RN) = +� +[un] : {un} ⊆ C∞ +c (RN) is a Cauchy sequence under the norm [·]s,H +� +, +where [un] is the equivalence class of the Cauchy sequence {un} with the equivalence +relation ′ ∼′ +s,H, which is defined as {un} ∼s,H {vn} iff lim +n→∞[un − vn]s,H = 0. +Ds,H(RN) is the Banach space with the norm ∥[un]∥Ds,H(RN) = lim +n→∞[un]s,H. +Next, we define the characterization of the normed space (Ds,H(RN), ∥·∥Ds,H(RN)). +Consider the space +˚ +W s,H(RN) = +� +u ∈ LH∗ +x(RN) : [u]s,H < ∞ +� +. +˚ +W s,H(RN) is a normed space with the norm ∥u∥ ˚ +W s,H(RN) = [u]s,H. +Theorem 2.13. Let H be a generalized N-function and s ∈ (0, 1). Then C∞ +c (RN) +is the dense subspace of ˚ +W s,H(RN). Moreover, there exist an linear isomorphism +between ˚ +W s,H(RN) and Ds,H(RN). In other words, the space Ds,H(RN) can be +identified as ˚ +W s,H(RN) and ∥ · ∥Ds,H(RN ) = ∥ · ∥ ˚ +W s,H(RN ) = [·]s,H. +We provide a proof of the Theorem 2.13 in Section 4. +Due to the presence of potential term V in the Problem (1.1), we consider the +following weighted space: +W = +� +u ∈ Ds,H(RN) : +� +RN V (x)Hx (x, |u|) dx < ∞ +� +which is a normed space with the norm +∥u∥W = ∥u∥Ds,H(RN) + ∥u∥V,H· +For the shake of simplicity, we denote ∥ · ∥W as ∥ · ∥. +Next, we have the following lemma from the definition of the space W. +Lemma 2.14. The space W is compactly embedded in LH +loc(RN). Also, W is con- +tinuously embedded in LH∗ +x(RN). +Next, we will state some results which are used to prove our main result. Define +the function m : ˚ +W s,H(RN) → R as +m(u) = +� +RN +� +RN H +� +x, y, |u(x) − u(y)| +|x − y|s +� +dx dy +|x − y|N · +Proposition 2.15. [7] For all u ∈ ˚ +W s,H(RN) we have +(1) If [u]s,H > 1 then [u]h− +s,H ≤ m(u) ≤ [u]h+ +s,H. + +GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION +9 +(2) If [u]s,H < 1 then [u]h+ +s,H ≤ m(u) ≤ [u]h− +s,H. +In particular, m(u) = 1 iff [u]s,H = 1. +Moreover, if {un} ⊂ ˚ +W s,H(RN) then +∥un∥ → 0 iff m(un) → 0. +Theorem 2.16. The space W is continuously embedded in LP +Q(RN), where Q(x) = +|K(x)|l and p : RN × RN × [0, ∞) → [0, ∞), P(x, y, t) = +� |t| +0 p(x, y, r)r dr is a +generalized N-function such that +p1 ≤ p(x, y, |t|)|t|2 +P(x, y, |t|) +≤ p2, ∀(x, y) ∈ RN × RN and t ̸= 0 +for some p1, p2 ∈ (h2, h∗ +1)· +Proof. The proof is similar to [35, Lemma 5.1]. +□ +As we have discussed, the Hardy-Littlewood-Sobolev inequality is the primary +tool for dealing with the Choquard type non linearity in the context of variational +methods. So far, we do not have Hardy-Littlewood-Sobolev inequality for Lebesque +Musielak spaces. By taking advantage of the condition (H2) and using Proposition +1.1, we prove and use the following version of Hardy-Littlewood-Sobolev inequality +in Lebesque Musielak spaces. +Proposition 2.17. For any u ∈ W, we have K(x)F(u(x)) ∈ Ll(RN). Moreover, +for all ǫ > 0 there exist cǫ > 0 such that +���� +� +RN +� +RN +K(x)K(y)F(u(x))F(u(y)) +|x − y|λ +dxdy +���� +≤ C1 max +� +ǫ2∥u∥2ψ1 + c2 +ǫ∥u∥2h∗ +1/l, ǫ2∥u∥2ψ2 + c2 +ǫ∥u∥2h∗ +2/l� +≤ C2(∥u∥2ψ1 + ∥u∥2h∗ +1/l + ∥u∥2ψ2 + ∥u∥2h∗ +2/l) +for some C1, C2 > 0. +Proof. Let u ∈ W. It follows, from (f1) − (f2) that, for all ǫ > 0 there exist cǫ > 0 +such that +|F(t)| ≤ ǫΨx(x, t) + cǫ(H∗ +x(x, t))1/l, ∀(x, t) ∈ RN × R. +By Proposition 2.10, we have +� +RN|K(x)F(u(x))|ldx ≤ 2l−1 +� +RN(ǫ(Ψx(x, u(x)))l + cl +ǫH∗ +x(x, u(x)))dx +≤ c1ǫl max +� +∥u∥ψ1l +Lψ1l(RN), ∥u∥ψ2l +Lψ2l(RN ) +� ++ c2cl +ǫ max +� +∥u∥h∗ +1 +LH∗x(RN), ∥u∥h∗ +2 +LH∗x(RN) +� +, +for some c1, c2 > 0. Further, by Theorem 2.16 and Lemma 2.14, one gets +(2.4) +� +RN |K(x)F(u(x))|ldx ≤ c3ǫl max +� +∥u∥ψ1l, ∥u∥ψ2l� ++c4cl +ǫ max +� +∥u∥h∗ +1, ∥u∥h∗ +2 +� +< ∞, +which implies, K(x)F(u(x)) ∈ Ll(RN), for some c3, c4 > 0. +By Proposition 1.1 and (2.4), we get +���� +� +RN +� +RN +K(x)K(y)F(u(x))F(u(y)) +|x − y|λ +dxdy +���� +≤ +� +c3ǫl max +� +∥u∥ψ1l, ∥u∥ψ2l� ++ c4cl +ǫ max +� +∥u∥h∗ +1, ∥u∥h∗ +2 +��2/l + +10 +SHILPA GUPTA AND GAURAV DWIVEDI +≤ C1 max +� +ǫ2∥u∥2ψ1 + c2 +ǫ∥u∥2h∗ +1/l, ǫ2∥u∥2ψ2 + c2 +ǫ∥u∥2h∗ +2/l� +≤ C2(∥u∥2ψ1 + ∥u∥2h∗ +1/l + ∥u∥2ψ2 + ∥u∥2h∗ +2/l) +for some C1, C2 > 0. +□ +3. Functional Setting +First, we define a weak solution to (1.1) and the corresponding energy functional. +Definition 1. We say that u ∈ W is a weak solution of (1.1) if the following holds: +� +RN +� +RN h +� +x, y, |u(x) − u(y)| +|x − y|s +� (u(x) − u(y))(v(x) − v(y)) +|x − y|N+2s +dx dy ++ +� +RN V (x)hx(x, |u|)uvdx = +� +RN +� +RN +K(x)K(y)F(u(x))f(u(y))v(y) +|x − y|λ +dxdy, ∀v ∈ W. +Thus, the energy functional I : W → R corresponding to (1.1) is given by +I(u) = +� +RN +� +RN H +�|u(x) − u(y)| +|x − y|s +� +dx dy +|x − y|N + +� +RN V (x)Hx(x, |u|)dx +− 1 +2 +� +RN +� +RN +K(x)K(y)F(u(x))F(u(y)) +|x − y|λ +dxdy. +It can be seen that I is well defined by Proposition 2.17, C1 [4, Lemma 3.2] and +the derivative of I at any point u ∈ W is given by +I′(u)(v) = +� +RN +� +RN h +� +x, y, |u(x) − u(y)| +|x − y|s +� (u(x) − u(y))(v(x) − v(y)) +|x − y|N+2s +dx dy ++ +� +RN V (x)hx(x, |u|)uvdx − +� +RN +� +RN +K(x)K(y)F(u(x))f(u(y))v(y) +|x − y|λ +dxdy, ∀v ∈ W. +Moreover, the critical points of I are the weak solutions to (1.1). +Let J : W → R such that +J(u) = +� +RN +� +RN H +� +x, y, |u(x) − u(y)| +|x − y|s +� +dx dy +|x − y|N + +� +RN V (x)Hx(x, |u|)dx. +Remark 3.1. The functional J is convex, since H is convex. +Consequently, J +is weakly lower semicontinuous, i.e., if {un} ⇀ u in W s,H +0 +(RN) then J(u) ≤ +lim inf +n→∞ J(un). +Lemma 3.2. Suppose that the function t �→ H(x, y, +√ +t) is convex for each x, y ∈ +RN. Moreover, we assume that the sequence {un} converges weakly to u in W and +lim sup +n→∞ ⟨J′(un), un − u⟩ ≤ 0. +Then {un} converges strongly to u in W. +Proof. The proof is similar to [9, Lemma 4.9]. +□ +The main existence result of this paper is as follows: +Theorem 3.3. Suppose that the conditions (f1)−(f4), (K1)−(K2) and (H1)−(H5) +are satisfied. Then the Problem (1.1) has a nontrivial weak solution. + +GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION +11 +To prove the existence of ground state solution, we need the following additional +assumption on f: +(GS) The map t �→ +f(t) +t|t| +h2 +2 −2 is strictly increasing for t > 0. +Theorem 3.4. If (f1) − (f4), (GS), (K1) − (K2) and (H1) − (H5) are satisfied, +then the solution obtained through Theorem 3.3 is a ground state solution. +4. Proof of the Theorem 2.13 +Proof. We present the proof of the theorem in three steps: +Step 1: In this step, we will prove that C∞ +c (RN) is dense in ˚ +W s,H(RN), i.e., for +any u ∈ ˚ +W s,H(RN) there exist a sequence in (C∞ +c (RN), [·]s,H) which converges to +u in ˚ +W s,H(RN). +Let ρ ∈ C∞ +c (RN) be the standard mollifier with support inside B1(0). Define, +ρǫ(x) = ǫ−nρ +� x +ǫ +� +. It can be seen that ρǫ(x) ∈ C∞ +c (RN), +� +RN ρǫ(x)dx = 1 and +support of ρǫ belongs to Bǫ(0). +Let u ∈ ˚ +W s,H(RN) be any arbitrary element. Then uǫ = ρǫ ∗ u ∈ C∞(RN). +Next, we claim that [uǫ − u]s,H → 0 as ǫ → 0. +By using Proposition 2.3, Remarks 2.5, 2.6 and the properties of mollifiers, we +have +[uǫ − u]s,H = +���� +(uǫ(x) − u(x)) − (uǫ(y) − u(y)) +|x − y|s +���� +LH(dµ) += +sup +∥v∥ +L � +H(dµ)≤1 +����� +� +RN +� +RN +(uǫ(x) − u(x)) − (uǫ(y) − u(y)) +|x − y|s +v(x, y)dµ +���� +� +≤ +sup +∥v∥ +L � +H(dµ)≤1 +2∥v∥L � +H(dµ) +�� +|ξ|<1 +ρ(ξ)dξ +���� +(u(x − ǫξ) − u(y − ǫξ)) − (u(x) − u(y)) +|x − y|s +���� +LH(dµ) +� += 2 +� +|ξ|<1 +ρ(ξ) +���� +(u(x − ǫξ) − u(y − ǫξ)) − (u(x) − u(y)) +|x − y|s +���� +LH(dµ) +dξ. +As we know that, w(x, y) = |u(x) − u(y)| +|x − y|s +∈ LH(dµ) and C∞ +c (dµ) is dense in +LH(dµ), hence, for a given ǫ > 0 there exist k(x, y) ∈ C∞ +c (dµ) such that ∥w − +k∥LH(dµ) ≤ ǫ +4. Further, we have +∥w(x − ǫξ, y − ǫξ) − k(x − ǫξ, y − ǫξ)∥LH(dµ) ≤ ǫ +4 +and +∥K(x − ǫξ, y − ǫξ) − k(x, y)∥LH(dµ) ≤ ǫ +4 +for sufficiently small ǫ and for all |ξ| ≤ 1. Therefore, we get [uǫ − u]s,H ≤ ǫ. As ǫ +was arbitrary, we get [uǫ − u]s,H → 0 as ǫ → 0. We conclude the claim by using +Theorem 2.2. +Step 2: Let {un} ⊆ (C∞ +c (RN), [·]s,H) be a Cauchy sequence. +Claim: There exist u ∈ ˚ +W s,H(RN) such that un → u in ˚ +W s,H(RN). +By Theorem 2.9, we have + +12 +SHILPA GUPTA AND GAURAV DWIVEDI +∥un∥LH∗x(RN) ≤ c[un]s,H < ∞, ∀n. +Hence, {un} ⊆ LH∗ +x(RN) and {un} is Cauchy sequence in LH∗ +x(RN). As we know +that LH∗ +x(RN) is a Banach space, thus there exist u ∈ LH∗ +x(RN) such that un → u +in LH∗ +x(RN). This implies that, un(x) → u(x) a.e. in RN. By the continuity of H, +we have H +� +x, y, |un(x) − un(y)| +|x − y|s +� +→ H +� +x, y, |u(x) − u(y)| +|x − y|s +� +a.e. in RN. +Thanks to the Fatou’s lemma, +� +RN +� +RN H +� +x, y, |u(x) − u(y)| +|x − y|s +� +dµ ≤ lim inf +n→∞ +� +RN +� +RN H +� +x, y, |un(x) − un(y)| +|x − y|s +� +dµ < ∞, +which implies u ∈ ˚ +W s,H(RN). +Next, we will prove that [un − u]s,H → 0 as n → ∞. +As {un} ⊆ (C∞ +c (RN), [·]s,H), we have +� +RN +� +RN H +� +x, y, |un(x) − un(y)| +|x − y|s +� +dµ = +� +RN +� +RN H +� +x, y, |un(x) − un(y)| +|x − y|s +� +dx dy +|x − y|N < ∞ +for each n. Thus |un(x) − un(y)| +|x − y|s +∈ LH(dµ). +Let +zn(x, y) = |un(x) − un(y)| +|x − y|s +· +It can be also seen that {zn(x, y)} ia a Cauchy sequence in LH(dµ). As LH(dµ) is +a Banach space, there exist z(x, y) ∈ LH(dµ) such that zn → z in LH(dµ). Further, +by uniqueness of the limit, we have z(x, y) = |u(x) − u(y)| +|x − y|s +· Hence, [un − u]s,H → 0 +as n → ∞, which proves our claim. +Step 3: Let [un] ∈ Ds,H(RN), i.e. [un] is an equivalence class of the Cauchy +sequence {un} ⊆ (C∞ +c (RN), [·]s,H). By Step 2, there exist u ∈ ˚ +W s,H(RN) such that +[un − u]s,H → 0 as n → ∞. Define a function, k : Ds,H(RN) → ˚ +W s,H(RN) such +that k([un]) = u. It can be see that k is well defined one-one, onto and isometry +by Step 1 and Step 2, which completes the proof the theorem. +□ +5. Proof of the Theorem 3.3 +To prove our main result, we first establish a series of lemmas. +Lemma 5.1. There exist positive real numbers α and ρ such that +I(u) ≥ α, +∀u ∈ W : ∥u∥ = ρ. + +GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION +13 +Proof. By using the Corollary 2.11 and Proposition 2.15, we have +I(u) = +� +RN +� +RN H +� +x, y, |u(x) − u(y)| +|x − y|s +� +dx dy +|x − y|N + +� +RN V (x)Hx(x, |u|)dx +− 1 +2 +� +RN +� +RN +K(x)K(y)F(u(x))F(u(y)) +|x − y|λ +dxdy +≥ min +� +[u]h1 +s,H, [u]h2 +s,H +� ++ min +� +∥u∥h1 +V,H, ∥u∥h2 +V,H +� +− 1 +2 +� +RN +� +RN +K(x)K(y)F(u(x))F(u(y)) +|x − y|λ +dxdy· +If ∥u∥ < 1, Proposition 2.17 implies +I(u) ≥ ∥u∥h2 − (C1ǫ2∥u∥2ψ1 + C1c2 +ǫ∥u∥2h∗ +1/l) +≥ ∥u∥h2 +� +1 − +C1ǫ2 +∥u∥h2−2ψ1 +� +− C1c2 +ǫ∥u∥2h∗ +1/l. +We conclude the result by choosing ρ and ǫ sufficiently small and using the fact +that (2h∗ +1/l) > h2. +□ +Lemma 5.2. There exist ν0 ∈ W and β > 0 such that +I(ν0) < 0 and ∥ν0∥ > β. +Proof. By (f4), there exist m1, m2 > 0 such that +(5.1) +F(s) ≥ m1sσ − m2, +∀ s ∈ [0, ∞). +Let u ∈ W\{0} and u ≥ 0 with compact support K ⊆ RN. For t > 1, by Corollary +2.11 and Proposition 2.15, we have +I(tu) = +� +RN +� +RN H +� +x, y, |tu(x) − tu(y)| +|x − y|s +� +dx dy +|x − y|N + +� +RN V (x)Hx(x, |tu|)dx +− 1 +2 +� +K +� +K +K(x)K(y)F(tu(x))F(tu(y)) +|x − y|λ +dxdy +≤ th2 � +max +� +[u]h1 +s,H, [u]h2 +s,H +� ++ max +� +∥u∥h1 +V,H, ∥u∥h2 +V,H +�� +− 1 +2 +� +K +� +K +K(x)K(y)(m1tσ(u(x))σ − m2)(m1tσ(u(y))σ − m2) +|x − y|λ +dxdy +this implies that I(tu) → −∞ as t → ∞, since 2σ > h2. Now, by setting ν0 = tu +for sufficiently large t, we get the desired result. +□ +By Lemmas 5.1 and 5.2, the geometric conditions of the mountain pass theorem +are satisfied for the functional I. +Hence, by the version of the mountain pass +theorem without (PS) condition, there exists a sequence {un} ⊆ W such that +I(un) → cM and I′(un) → 0 as n → ∞, where +0 < cM = inf +γ∈Γ max +t∈[0,1] I(γ(t)) > 0, +and +Γ = {γ ∈ C([0, 1], W) : γ(0) = 0, γ(1) < 0}· +Lemma 5.3. The (PS)cM sequence is bounded in W. Moreover, there exist u ∈ W +such that, up to a subsequence, we have un ⇀ u weakly in W. + +14 +SHILPA GUPTA AND GAURAV DWIVEDI +Proof. Since {un} is a (PS)cM sequence of I, we have I(un) → cM and I′(un) → 0 +as n → ∞, i.e., +� +RN +� +RN H +�|un(x) − un(y)| +|x − y|s +� +dx dy +|x − y|N + +� +RN V (x)Hx(x, |un|)dx +(5.2) +− 1 +2 +� +RN +� +RN +K(x)K(y)F(un(x))F(un(y)) +|x − y|λ +dxdy = cM + δn, +where δn → 0 as n → ∞ and +���� +� +RN +� +RN h +� +x, y, |un(x) − un(y)| +|x − y|s +� (un(x) − un(y))(v(x) − v(y)) +|x − y|N+2s +dx dy ++ +� +RN V (x)hx(x, |un|)unvdx − +� +RN +� +RN +K(x)K(y)F(un(x))f(un(y))v(y) +|x − y|λ +dxdy +���� ≤ εn∥v∥, +(5.3) +∀v ∈ W, where εn → 0 as n → ∞. On taking v = un, by (5.2), (5.3) and using +(f4), we obtain +�� +RN +� +RN H +� +x, y, |un(x) − un(y)| +|x − y|s +� +dx dy +|x − y|N +− 1 +σ +� +RN +� +RN h +� +x, y, |un(x) − un(y)| +|x − y|s +� (un(x) − un(y))2 +|x − y|N+2s +dxdy +� ++ +� +RN (V (x)Hx(x, |u|) − 1 +σ V (x)hx(x, |u|)u2)dx ≤ c5(1 + ∥un∥), +for some c5 > 0. It follows from (H2) that +� +1 − h2 +σ +� � +RN +� +RN H +� +x, y, |un(x) − un(y)| +|x − y|s +� +dx dy +|x − y|N ++ +� +1 − h2 +σ +� � +RN V (x)Hx(x, |un|)dx ≤ c5(1 + ∥un∥). +If ∥un∥ > 1, by Corollary 2.11 and Proposition 2.15, we have +� +1 − h2 +σ +� +([un]h1 +s,H + ∥un∥h1 +V,H) ≤ c5(1 + ∥un∥) +� +1 − h2 +σ +� +∥un∥h1 ≤ c5(1 + ∥un∥). +Consequently, ∥un∥ ≤ c6 for some c6 > 0. Thus {un} is bounded in W. As W is a +reflexive Banach space, ∃ u ∈ W such that up to a subsequence, we have un ⇀ u +weakly in W. +□ +Lemma 5.4. Let {un} is bounded in W such that un ⇀ u weakly in W. Then +lim +n→∞ +� +RN |K(x)f(un(x))(un(x) − u(x))|ldx = 0. +Proof. Let {un} is bounded in W such that un ⇀ u weakly in W. By Lemma 2.14, +we have un(x) → u(x) a.e. x ∈ RN. + +GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION +15 +Define Q(x) = |K(x)|l. It follows from (f1) and (f3) that, for all ǫ > 0 there +exist t0, t1, cǫ > 0 such that +(5.4) +f(t) ≤ ǫ +� +ψx(x, t)t + (H∗ +x(x, t)) +b−1 +h∗ +i +� ++ cǫ(H∗ +x(x, t)) +b−1 +h∗ +i χ[t0,t1](t), ∀(x, t) ∈ RN × R. +Further, by (5.4), we have +K(x)f(un(x)) ≤ ǫK(x) +� +ψx(x, un(x))un(x) + (H∗ +x(x, un(x))) +b−1 +h∗ +i +� ++ cǫK(x)(H∗ +x(x, un(x))) +b−1 +h∗ +i χ[t0,t1](un(x)), ∀(x, t) ∈ RN × R. +Consider, +� +RN|K(x)f(un(x))(un(x) − u(x))|ldx +≤ 2l−1ǫl +� +RN Q(x) +���� +� +ψx(x, un(x))un(x) + (H∗ +x(x, un(x))) +b−1 +h∗ +i +� +(un(x) − u(x)) +���� +l +dx ++ 2l−1cl +ǫ +� +RN Q(x) +����(H∗ +x(x, un(x))) +b−1 +h∗ +i χ[t0,t1](un(x))(un(x) − u(x)) +���� +l +dx. +(5.5) +Now, define the set An = {x ∈ RN : |un(x)| ≥ t0}. Thus, (H3) and definition of +H∗ implies +c7|An| ≤ +� +An +H∗ +x(x, t0)dx ≤ +� +An +H∗ +x(x, un(x))dx ≤ +� +RN H∗ +x(x, un(x))dx < c8, +since {un} is bounded in W, for some c7, c8 > 0. +Therefore, we have sup +n∈N +|An| < ∞. Using (K1), we get +lim +d→∞ +� +An∩Bd(0)c |K(x)| +2N +2N−λ dx = 0 uniformly in n ∈ N +consequently, for a given ǫ > 0 there exist d0 > 0 such that +(5.6) +� +An∩Bd0 (0)c |K(x)| +2N +2N−λ dx < ǫ +b +(b−1) +for each n. +Using H¨older’s inequality and Proposition 2.12, we have +� +Bd0 (0)c Q(x) +����(H∗ +x(x, un(x))) +b−1 +h∗ +i χ[t0,t1](un(x))(un(x) − u(x)) +���� +l +dx +≤ c9 +� +An∩Bd0(0)c Q(x)|un(x)|(b−1)lχ[t0,t1](|un(x)|)|un(x) − u(x)|ldx +≤ c9 max +i∈{1,2} + + + +�� +An∩Bd0(0)c Q(x)|un(x)|blχ[t0,t1](un(x))dx +� (b−1) +b +�� +An∩Bd0(0)c Q(x)|un(x) − u(x)|bldx +� 1 +b  + + + +16 +SHILPA GUPTA AND GAURAV DWIVEDI +≤ c10t(b−1)l +1 +�� +An∩Bd0(0)c Q(x) +� (b−1) +b +for some c9, c10 > 0. +Further, by (5.6), we obtain +(5.7) +� +Bd0 (0)c Q(x) +����(H∗ +x(x, un(x))) +b−1 +h∗ +i χ[t0,t1](un(x))(un(x) − u(x)) +���� +l +dx ≤ c11ǫ, +for some c11 > 0. +By Propositions 2.10, 2.12, Lemma 2.14, Theorem 2.16 and H¨older’s inequality, +we have +� +Bd0(0)c Q(x) +���� +� +ψx(x, un(x))un(x) + (H∗ +x(x, un(x))) +b−1 +h∗ +i +� +(un(x) − u(x)) +���� +l +dx +≤ c12 max +i∈{1,2} +�� +RN Q(x) +� +|un(x)|(ψi−1)l + |un(x)|(b−1)l� +|un(x) − u(x)|ldx +� +≤ c12 max +i∈{1,2} + + + +�� +RN Q(x)|un(x)|ψildx +� (ψi−1) +ψi +�� +RN Q(x)|un(x) − u(x)|ψildx +� 1 +ψi + + + ++ c12 max +i∈{1,2} + + + +�� +RN Q(x)|un(x)|bldx +� (b−1) +b +�� +RN Q(x)|un(x) − u(x)|bldx +� 1 +b + + + +≤ c13 max +i∈{1,2} +� +∥un(x) − u(x)∥l +Lψil +Q (RN) + ∥un(x) − u(x)∥l +Lbl +Q(RN) +� +for some c12, c13 > 0. +Therefore, by Theorem 2.16, we obtain +(5.8) +� +Bd0(0)c Q(x) +���� +� +ψy(x, un(x))un(x) + (H∗ +x(x, un(x))) +b−1 +h∗ +i +� +(un(x) − u(x)) +���� +l +dx ≤ c14, +for some c14 > 0. +Consequently, (5.5), (5.7) and (5.8) implies +� +Bd0 (0)c |K(x)f(un(x))(un(x) − u(x))|ldx ≤ 2l−1ǫlc14 + 2l−1cl +ǫc11ǫ → 0 +as ǫ was arbitrary. +On the other side, by (f3) and Strauss compactness lemma [12, Theorem A.I], +we have +lim +n→∞ +� +Bd0(0) +|K(x)f(un(x))(un(x) − u(x))|ldx = 0, +which completes the proof. +□ +Proof of the Theorem 3.3. Both the geometric conditions of the mountain +pass theorem follow from Lemmas 5.1 and 5.2. Next, we will prove that the func- +tional I satisfies the (PS)cM condition. +Let {un} ⊆ W be any Palais-Smale sequence, i.e., I(un) → cM and I′(un) → 0 +in dual space of W. By Lemma 5.3, we conclude that {un} is bounded in W and + +GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION +17 +un ⇀ u weakly in W. As a consequence, I′(un)(un − u) → 0 as n → ∞, i.e., +(5.9) +J′(un)(un − u) − +� +RN +� +RN +K(x)K(y)F(un(x))f(un(y))(un(y) − u(y)) +|x − y|λ +dxdy → 0 +as n → 0. Next, we claim that that +(5.10) +� +RN +� +RN +K(x)K(y)F(un(x))f(un(y))(un(y) − u(y)) +|x − y|λ +dxdy → 0 as n → 0. +Let |K(y)|l = Q(y). By (2.4), we have +(5.11) +∥K(x)F(un(x))∥Ll(RN) ≤ c15, +for some c15 > 0 since {un} is bounded in W. By Lemma 5.4, we have +(5.12) +lim +n→∞ +� +RN |K(x)f(un(x))(un(x) − u(x))|ldx = 0. +By (5.11), (5.12) and Proposition 1.1, the claim in the (5.10) is proved. Hence, by +Lemma 3.2, we have un → u. Thus, (PS)cM condition is satisfied for the functional +I. +Hence, by the mountain pass theorem, there exists critical point uM of I with +level cM, i.e., I′(uM) = 0 and I(uM) = cM > 0. Thus, uM is the non-trivial weak +solution of the Problem (1.1). +□ +6. Ground State Solution +In this section, we prove that the solution obtained through Theorem 3.3 is a +ground state solution. Let us recall the definition of a ground state solution: +Definition 2. A weak solution u0 of (1.1) is called a ground state solution if it has +the least energy, i.e., we say, the solution u0 is ground state solution of (1.1) if +(6.1) +I(u0) = r = inf +u∈S I(u), +where S is the set of all critical points of the functional I. +To prove that the solution obtained in Theorem 3.3 is a ground state solution, +we use the minimization method, in particular, Nehari manifold method. We define +(6.2) +ℵ = {u ∈ W\{0}|I′(u)u = 0} and b = inf +u∈ℵ I(u). +The set ℵ is called the Nehari manifold. It can be observed that S ⊆ ℵ. The +key idea of this method is to search for a non-trivial critical point of I in ℵ instead +of the whole space W. To know more about this method, one can refer to [36]. +The existence of a ground state solution is proved by many researchers, we refer to, +[11, 13, 24, 25, 26, 27] and reference therein. +For u ∈ W, define the function, hu : [0, ∞) → R such that hu(t) = I(tu), i.e., +hu(t) = +� +RN +� +RN H +� +x, y, |tu(x) − tu(y)| +|x − y|s +� +dx dy +|x − y|N + +� +RN V (x)Hx(x, |tu(x)|)dx +− 1 +2 +� +RN +� +RN +K(x)K(y)F(tu(x))F(tu(y)) +|x − y|λ +dxdy. +Lemma 6.1. Let (f1) − (f4), (GS), (K1) − (K2) and (H1) − (H5) hold. If u ∈ +W\{0}, then there exists unique tu > 0 such that tuu ∈ ℵ. Moreover, max +t∈[0,∞] hu(t) = +hu(tu) = I(utu). + +18 +SHILPA GUPTA AND GAURAV DWIVEDI +Proof. We observe that h′ +u(t) = 0 if and only if tu ∈ ℵ. Lemma 5.1 and Lemma 5.2 +imply that hu(t) > 0 for sufficiently small t and hu(t) < 0 for sufficiently large t. +Thus, ∃ tu ∈ (0, ∞) such that +max +t∈[0,∞] hu(t) = hu(tu) = I(utu). +Consequently, +h′ +u(tu) = 0 and tuu ∈ ℵ. Next, we will prove the uniqueness of tu. If t is the critical +point of hu, then we have +h′ +u(t) = +� +RN +� +RN h +� +x, y, |tu(x) − tu(y)| +|x − y|s +� (tu(x) − tu(y))2 +t|x − y|N+2s +dx dy ++ +� +RN +V (x)hx(x, |tu(x)|)(tu(x))2dx +t +− +� +RN +� +RN +K(x)K(y)F(u(x))f(tu(y))u(y) +|x − y|λ +dxdy = 0, +which implies that +� +RN +� +RN h +� +x, y, |tu(x) − tu(y)| +|x − y|s +� (tu(x) − tu(y))2 +th2|x − y|N+2s dx dy + +� +RN +V (x)hx(x, |tu(x)|)(tu(x))2dx +th2 += +� +RN +� +RN +K(x)K(y)F(tu(x))f(tu(y))tu(y) +th2|x − y|λ +dxdy. +(6.3) +On proceeding as [34, Lemma 4.3], one can check that the right hand side of (6.3) +is decreasing for t > 0. Consider, +� +RN +� +RN +K(x)K(y)F(tu(x))f(tu(y))tu(y) +th2|x − y|λ +dxdy += +� +RN K(y) +�� +RN +K(x)F(tu(x))dx +|x − y|λ +� f(tu(y))tu(y) +th2 +dy += +� +RN K(y) +�� +RN +K(x)F(tu(x))dx +t +h2 +2 |x − y|λ +� +f(tu(y))|u(y)| +h2 +2 +|tu(y)| +h2 +2 −2(tu(y)) +dy +which implies the left hand side of (6.3) is increasing strictly for t > 0 by (GS) and +(f4). Therefore, tu is a unique critical point of hu. +□ +Proof of the Theorem 3.4 It is enough to prove that, cM = b = r, where b +and r as are defined in (6.1) and (6.2), respectively. +By using the fact that S ⊆ ℵ, we have b ≤ r. Also, it can be seen r ≤ cM. It will +be sufficient to prove that b ≥ cM. +If v ∈ ℵ, then h′ +v(1) = 0. By Lemma 6.1, we have max +t∈[0,∞] hv(t) = hv(1) = I(v). +Choose a function γ : [0, 1] → W such that γ(t) = tt0v, where t0 > 0 such that +I(t0v) < 0, which implies that γ ∈ Γ. Therefore, we have cM ≤ max +t∈[0,1] I(γ(t)) = +max +t∈[0,1] I(tt0v) ≤ max +t≥0 I(tv) = I(v), which is true for every element v ∈ ℵ. 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Lett., 135 (2023), 108418. +Shilpa Gupta +Department of Mathematics +Birla Institute of Technology and Science Pilani +Pilani Campus, Vidya Vihar +Pilani, Jhunjhunu +Rajasthan, India - 333031 +Email address: p20180442@pilani.bits-pilani.ac.in; shilpagupta890@gmail.com +Gaurav Dwivedi +Department of Mathematics +Birla Institute of Technology and Science Pilani +Pilani Campus, Vidya Vihar +Pilani, Jhunjhunu +Rajasthan, India - 333031 +Email address: gaurav.dwivedi@pilani.bits-pilani.ac.in + diff --git a/wdE3T4oBgHgl3EQfOgk_/content/tmp_files/load_file.txt b/wdE3T4oBgHgl3EQfOgk_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c1175802958b9e1823857b145ff97758f293919 --- /dev/null +++ b/wdE3T4oBgHgl3EQfOgk_/content/tmp_files/load_file.txt @@ -0,0 +1,777 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf,len=776 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='04393v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='AP] 11 Jan 2023 GROUND STATE SOLUTION FOR A GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION WITH VANISHING POTENTIAL IN HOMOGENEOUS FRACTIONAL MUSIELAK SOBOLEV SPACES SHILPA GUPTA AND GAURAV DWIVEDI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' This paper aims to establish the existence of a weak solution for the following problem: (−∆)s Hu(x)+V (x)h(x, x, |u|)u(x) = �� RN K(y)F (u(y)) |x − y|λ dy � K(x)f(u(x)) in RN, where N ≥ 1, s ∈ (0, 1), λ ∈ (0, N), H(x, y, t) = � |t| 0 h(x, y, r)r dr, h : RN × RN × [0, ∞) → [0, ∞) is a generalized N-function and (−∆)s H is a generalized fractional Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The functions V, K : RN → (0, ∞), non-linear function f : R → R are continuous and F (t) = � t 0 f(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' First, we introduce the homogeneous fractional Musielak-Sobolev space and investigate their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' After that, we pose the given problem in that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' To establish our existence results, we prove and use the suitable version of Hardy-Littlewood-Sobolev inequality for Lebesque Musielak spaces together with variational technique based on the mountain pass theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We also prove the existence of a ground state solution by the method of Nehari manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Introduction This paper aims to establish the existence of a weak solution to the following problem: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) (−∆)s Hu(x) + V (x)h(x, x, |u|)u(x) = �� RN K(y)F(u(y)) |x − y|λ dy � K(x)f(u(x)) in RN, where N ≥ 1, s ∈ (0, 1), λ ∈ (0, N), H(x, y, t) = � |t| 0 h(x, y, r)r dr, and h : RN × RN × [0, ∞) → [0, ∞) is a generalized N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The functions V, K : RN → (0, ∞), non-linear function f : R → R are continuous and F(t) = � t 0 f(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The operator (−∆)s H is called the generalized fractional Laplace operator and is defined as: (−∆)s Hu(x) = 2 lim ǫ→0 � RN\\Bǫ(x) h � x, y, |u(x) − u(y)| |x − y|s � u(x) − u(y) |x − y|s dy |x − y|N+s · 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' 35J20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' 35J62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Variational methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Choquard Equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Fractonal Musielak Sobolev spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Vanishing Potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Hardy-Littlewood-Sobolev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' 1 2 SHILPA GUPTA AND GAURAV DWIVEDI The operator (−∆)s H generalizes fractional p-Laplace operator, fractional (p, q)- Laplace operator, weighted fractional Laplace operator and fractional double-phase operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' More specifically, if we replace H by tp, tp+tq, a(x)tp and tp+a(x)tq, then (−∆)s H reduces to the fractional p-Laplacian, fractional (p, q)-Laplacian, weighted fractional Laplace operator and fractional double-phase operator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The existence results for problems of type (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) are examined in fractional Orlicz- Sobolev spaces when H(x, y, t) is independent of x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In this context, we quote the work of Bahrouni-Ounaies [9], Bahrouni-Ounaies-Tavares [10], Missaoui-Ounaies [25], and Silva-Carvalho-Albuquerque-Bahrouni [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In the case, when H(x, y, t) depends on all x, y and t, the existence of a solution for the problems of the type (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) is studied in fractional Musielak-Sobolev spaces (see Section 2, for the definitions and properties of fractional Musielak-Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=') The study of Musielak spaces started in the mid-1970s with the work of Musielak [30] and Hudzik [17, 18], where the authors provide the general framework for Musielak spaces in terms of modular function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Recently, Azroul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [7] devel- oped the idea of fractional Musielak-Sobolev spaces, which are the generalization of fractional Orlicz-Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In the same paper, authors also obtained the existence result for the non local problem � (−∆)s Hu(x) + h (x, x, |u|) u = f(x, u) in Ω, u = 0 in x ∈ RN\\Ω, where Ω ⊆ RN is a bounded and smooth domain, N ≥ 1, 0 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In the paper [8], authors proved some embedding and extension results for fractional Musielak- Sobolev spaces and established the existence result for the following non local prob- lem � (−∆)s1 Hu(x) + (−∆)s2 Hu(x) = f(x, u) in Ω, u = 0 in RN\\Ω, where Ω ⊆ RN is a bounded and smooth domain, N ≥ 1, 0 < s1 ≤ s2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In this article, we consider the following assumptions on the functions H and h : (H1) h(x, y, ·) ∈ C1 in (0, ∞), ∀x, y ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (H2) h1 ≤ h(x,y,|t|)|t|2 H(x,y,|t|) ≤ h2 < N for all x, y ∈ RN and t ̸= 0 for some 1 ≤ h1 < h2 < h∗ 1, where h∗ 1 = Nh1 N−sh1 ≤ h∗ 2 = Nh2 N−sh2 · (H3) inf x,y∈RN H(x, y, 1) = b1 and sup x,y∈RN H(x, y, 1) = b2 for some b1, b2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (H4) The map t �→ H(x, y, √ t) is convex for each x, y ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (H5) � ∞ a � t H(t) � s N−s dt = ∞ and � b 0 � t H(t) � s N−s dt < ∞, for some a, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Due to the presence of the Choquard type non linearity, Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) is known as a Choquard equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' One of the main tool to deal with such type of equations is Hardy-Littlewood-Sobolev [22] inequality which is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [22] Let t1, t2 > 1 and 0 < λ < N with 1/t1 + 1/t2 + λ/N = 2, f ∈ Lt1(RN) and g ∈ Lt2(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then there exist a sharp constant C independent of f and g such that ���� � RN � RN f(x)g(y) |x − y|λ dxdy ���� ≤ C∥f∥Lt1(RN)∥g∥Lt2(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Choquard type of equations have been studied extensively in the literature, we refer to [28] for the physical interpretation and survey of such type of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' For GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 3 some existence results involving Choquard type equations, we refer to the works of Moroz-Schaftingen [26, 27] (Laplace operator), Avenia-Siciliano-Squassina [6], Mukherjee-Sreenadh [29](fractional Laplace operator), Patrizia-Xiang-Zhang [31] (p-Laplacian), Xie-Wang-Zhang [37] ((p, q)-Laplacian) and Pucci-Xiang-Zhang [33], Belchior-Bueno-Miyagaki-Pereira [11] (fractional p-Laplacian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Alves-R˘adulescu-Tavares [4] discussed the generalized choquard problem in Orlicz- Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) involves the potential term which vanishes at infinity, such type of equations studied widely by many researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In 2013, Alves-Souto [5] proved the existence result for the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2) − ∆u + V (x)u = K(x)f(u) in RN, where N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' They assumed that V, K : RN → (0, ∞) are continuous functions and satisfy the following conditions: (K′ 1) K ∈ L∞(RN) and if {An} is a sequence of Borel sets such that sup n |An| < ∞ then lim s→∞ � An∩Bs(0)c K(x) = 0 uniformly in n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (K′ 2) One of the following condition is true: (K21) K V ∈ L∞(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (K22) K(x) [V (x)] 2∗−p 2∗−2 → 0 as |x| → ∞ for some p ∈ (2, 2∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' If V, K satisfies (K′ 1) − (K′ 2) then we say (V, K) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Further, Chen-Yuan [13], considered the problem: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3) − ∆u + V (x)u = �� RN K(y)F(u(y)) |x − y|λ dy � K(x)f(u(x)) in RN, where they assumed that (V, K) ∈ K but the conditions (K′ 1) and (K22) is replaced by the conditions (K1) and (K23), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (K1) and (K23) are as follows: (K1) K ∈ L∞(RN) and if {An} is a sequence of Borel sets such that sup n |An| < ∞ then lim s→∞ � An∩Bs(0)c |K(x)| 2N 2N−λ = 0 uniformly in n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (K23) |K(x)| 2N 2N−λ [V (x)] 2∗−p 2∗−2 → 0 as |x| → ∞ for some p ∈ (2, 2∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In this sequence, Li-Teng-Wu [21] studied the following fractional Schr¨odinger equation: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4) (−∆)su + V (x)u = |u|2∗ s−2u + λK(x)f(u) in RN, where λ > 0, s ∈ (0, 1), 2∗ s = 2N N−2s and (−∆)s is the fractional Laplace operator of order s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' They assumed that (V, K) ∈ K but in the condition (K22), 2∗ is replaced by 2∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Luo-Li-Li [24], considered the fractional choquard equation: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='5) (−∆)su + V (x)u = �� RN K(y)F(u(y)) |x − y|λ dy � K(x)f(u(x)) in RN, 4 SHILPA GUPTA AND GAURAV DWIVEDI in which they assumed that the conditions (K1) and (K24) |K(x)| 2N 2N−λ [V (x)] 2∗s −p 2∗s −2 → 0 as |x| → ∞ for some p ∈ (2, 2∗ s) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' After that, many researches studied the nonlinear equations involving vanishing potential with different type of operators and different conditions on the non linear- ity, we refer to, Deng-Li-Shuai [15] (p-Laplace operator), Perera-Squassina-Yang [32] (fractional p-Laplacian), Isernia [19] (Fractional p&q-Laplacian), Isernia-Repovˇs [20] (Double-phase operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Recently, Silva-Souto [35] developed the existence result for generalized Schr¨odinger equation in Orlicz-Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Existence result for Choquard type equations with vanishing potential has been obtained by Chen-Yuan [13], Alves-Figueiredo-Yang [3] (for Laplace operator), Albuquerque-Silva-Sousa [14](fractional coupled Choquard-type systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In this paper, we assume that V, K : RN → (0, ∞) are continuous functions and satisfies (K1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, we assume that (K2) V, K satisfies one of the following conditions: (K2a) K V ∈ L∞(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (K2b) |K(x)| 2N 2N−λ L(x) → 0 as |x| → ∞, where L(x) = min t>0 � V (x) H(x,x,t) Ψ(x,x,t) � Inspired by the mentioned research works, in this article, we study (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) via varia- tional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The main novelties of this paper are as follows: To introduce the homogeneous fractional Musielak-Sobolev spaces and in- vestigate their properties which are needed to study the Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Provide the characterization of these spaces which is written in the form of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' To prove the suitable version of Hardy-Littlewood-Sobolev inequality for Lebesque Musielak spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' To the best of our knowledge, this is the first paper which proves the existence result for generalized fractional Laplace operator with the vanishing potential to- gether with Choquard type non linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We assume that f : R → R is continuous and satisfies the following conditions: (f1) There exist a generalized N-function Ψ : RN × RN × R → [0, ∞) and ψ1l, ψ2l ∈ (h2, h∗ 1) such that ψ1 ≤ ψ(x,y,t)|t|2 Ψ(x,y,t) ≤ ψ2, ∀(x, y) ∈ RN × RN and t ̸= 0 and lim t→0 f(t) ψ(x, x, t)t = 0, ∀ x ∈ RN, where Ψ(x, y, t) = � |t| 0 ψ(x, y, r)r dr and 2N 2N−λ = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (f2) lim t→∞ F(t) (H∗(x, x, t))1/l = 0, ∀ x ∈ RN, where H∗ is define in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (f3) For i ∈ {1, 2}, lim t→∞ f(t) (H∗(x, x, t)) b−1 h∗ i = 0, ∀ x ∈ RN for some bl ∈ (h2, h∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (f4) There exist σ > h2/2 such that 0 < σF(t) = σ � t 0 f(s)ds ≤ 2tf(t), GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 5 for all t > 0, x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' This article is organized as follows: We discuss the definition and properties of Lebesque Musielak spaces and fractional Musielak-Sobolev spaces in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The functional setup needed to prove our result is provided Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We also state our main results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Section 4 deals with the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Section 5 deals with the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Finally, in Section 6, we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Musielak spaces and Functional Setting Let Ω ⊆ RN be any open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Define, H(x, y, t) = � |t| 0 h(x, y, s)s ds, where h : Ω × Ω × [0, ∞) → [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Recall that, H(x, y, t) : Ω × Ω × R → [0, ∞) is called a generalized N-function if it satisfies the following conditions: (1) H is continuous, even and convex function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (2) H(x, y, t) = 0 if and only if t = 0, ∀x, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (3) lim t→0 H(x,y,t) t = 0 and lim t→∞ H(x,y,t) t = ∞, ∀x, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' For any generalized N-function A : Ω × Ω × R → [0, ∞), we define the function ax : Ω × [0, ∞) → [0, ∞) such that ax(x, t) = a(x, x, t) ∀(x, t) ∈ Ω × [0, ∞) and Ax(x, t) = � |t| 0 ax(x, s)s ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We say that a generalized N-function H, satisfies the weak ∆2-condition if there exists C > 0 and a non-negative function k ∈ L1(Ω) such that H(x, y, 2t) ≤ CH(x, y, t) + k(x) ∀(x × y × t) ∈ Ω × Ω × [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' If k = 0, then H is said to satisfy ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Throughout this paper, we assume that H is a generalized N-function which satisfy ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we define the complementary function � H corresponding to generalized N-function H as � H(x, y, t) = � |t| 0 �h(x, y, s)s ds, where �h is defined as �h(x, y, t) = sup{s : h(x, y, s)s ≤ t} ∀(x, y, t) ∈ Ω × Ω × [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, the function H and its complementary function � H satisfy the following Young’s inequality [23, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1]: s1s2 ≤ H(x, y, s1) + � H(x, y, s2) ∀x, y ∈ Ω, s1, s2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The Lebesque-Musielak space LHx(Ω) is defined as: LHx(Ω) = � u : Ω → R is measurable ���� � Ω Hx (x, τ|u|) dx < ∞, for some τ > 0 � LHx(Ω) is a normed space [30] with the Luxemburg norm ∥u∥LHx(Ω) = inf � τ > 0 ���� � Ω Hx (x, τ|u|) dx ≤ 1 � 6 SHILPA GUPTA AND GAURAV DWIVEDI Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [30] The space LHx(Ω) is separable and reflexive Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [1] The space Cc(Ω) is dense in LHx(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, C∞ c (Ω) is dense in LHx(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [1] Let H and � H be complimentary N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then, for any u ∈ LHx(Ω) and v ∈ L � Hx(Ω), we have ���� � Ω uv dx ���� ≤ 2∥u∥LHx(Ω)∥v∥L � Hx(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [1] Let v ∈ L � Hx(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) Gv(u) = � Ω u(x)v(x)dx is a bounded linear functional on LHx(Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', Gv ∈ (LHx(Ω))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Also, every bounded linear functional in LHx(Ω) is of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) for some v ∈ L � Hx(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, (LHx(Ω))∗ is isomorphic to L � Hx(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4, we have ∥v∥L � Hx(Ω) = ∥Gv∥(LHx(Ω))∗ = sup ∥u∥LHx (Ω)≤1 ����� � Ω u(x)v(x)dx ���� � For a given generalized N-function H and s ∈ (0, 1), fractional Musielak-Sobolev space is denoted by W s,H(Ω) and is defined as W s,H(Ω) = � u ∈ LHx(Ω) : � Ω � Ω H � x, y, τ|u(x) − u(y)| |x − y|s � dx dy |x − y|N < ∞, for some τ > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' W s,H(Ω) is a normed space with the norm ∥u∥ = ∥u∥LHx(Ω) + [u]s,H, where [u]s,H = inf � τ > 0 ���� � Ω � Ω H � x, y, |u(x) − u(y)| τ|x − y|s � dx dy |x − y|N ≤ 1 � We define the Lebesque-Musielak space LH(dµ) as: LH(dµ) = � u : Ω × Ω → R is measurable ���� � Ω � Ω H (x, y, τ|u(x, y)|) dµ < ∞, for some τ > 0 � , where dµ = dxdy |x − y|N is a measure on the set Ω × Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [u]s,H is finite if and only if (u(x) − u(y)) |x − y|s ∈ LH(dµ) and [u]s,H = ���� u(x) − u(y) |x − y|s ���� LH(dµ) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [7] W s,H(Ω) is a separable and reflexive Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The space W s,H 0 (Ω) is defined as W s,H 0 (Ω) = {u ∈ W s,H(RN) : u = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' in RN\\Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we state the generalized Poincar´e’s inequality: GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 7 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [7] Let Ω be a bounded open subset of RN and 0 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then there exist a positive constant c > 0 such that ∥u∥LHx(Ω) ≤ c[u]s,H, ∀ u ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' This implies that, [·]s,H is the norm on W s,H 0 (Ω), which is equivalent to the norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' For a given generalized N-function H : RN × RN × R → [0, ∞), we define the Sobolev conjugate function H∗ : RN × R → [0, ∞) as: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2) H∗(x, t) = Hx(x, G−1(t)), ∀ t ≥ 0, where G(x, t) = �� t 0 � r Hx(x, r) � s N−S dr � N−s N , ∀ t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' One can verify that H∗ is a generalized N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let s ∈ (0, 1) and H be any generalized N-function satisfying (H5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then the embedding W s,H(RN) ֒→ LH∗(RN) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, in this embedding the space LH∗(RN) is optimal among all the Musielak spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The proof is similar to the proof of [2, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let H be any generalized N-function satisfying (H2) − (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Assume that u ∈ LHx(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then, we have (1) min � ρh1, ρh2� Hx(x, t) ≤ Hx(x, ρt) ≤ max � ρh1, ρh2� Hx(x, t), ∀ρ, t > 0 (2) min � ∥u∥h1 LHx(RN), ∥u∥h2 LHx(RN) � ≤ � RN Hx(x, |u|)dx ≤ max � ∥u∥h1 LHx(RN), ∥u∥h2 LHx(RN) � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proof of (1) is similar to the proof of [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By (H3) and Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='10, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3) b1 min � ρh1, ρh2� ≤ Hx(x, ρ) ≤ b2 max � ρh1, ρh2� ∀ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Hence, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3) and the definition of norm implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ Let H be any generalized N-function satisfying (H2) − (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then we define weighted Lebesque-Musielak space LHx V (RN) as: LHx V (RN) = � u : RN → R is measurable ���� � RN V (x)Hx (x, τ|u|) dx < ∞, for some τ > 0 � LHx V (RN) is a normed space [30] with the Luxemburg norm ∥u∥V,H = inf � τ > 0 ���� � RN V (x)Hx (x, τ|u|) dx ≤ 1 � Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let H be any generalized N-function satisfying (H2) − (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Assume that u ∈ LHx V (RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then, we have min � ∥u∥h1 V,H, ∥u∥h2 V,H � ≤ � RN V (x)Hx(x, |u|)dx ≤ max � ∥u∥h1 V,H, ∥u∥h2 V,H � Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [25, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3] Let H be any generalized N-function satisfying (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Assume that u ∈ H∗ x(RN) and ρ, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then, we have (1) min � ρh∗ 1, ρh∗ 2� H∗ x(x, t) ≤ H∗ x(x, ρt) ≤ max � ρh∗ 1, ρh∗ 2� H∗ x(x, t), 8 SHILPA GUPTA AND GAURAV DWIVEDI (2) min � ∥u∥h∗ 1 LH∗x(RN), ∥u∥h∗ 2 LH∗x(RN) � ≤ � RN H∗ x(x, |u|)dx ≤ max � ∥u∥h∗ 1 LH∗x(RN), ∥u∥h∗ 2 LH∗x(RN) � , where, h∗ 1 = Nh1 N−sh1 and h∗ 2 = Nh2 N−sh2 · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Homogeneous fractional Musielak-Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Fractional Musielak- Sobolev spaces are not sufficient to study the Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1), as inf V (x) can be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In this section, we introduce the suitable space to study Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) which we called homogeneous fractional Musielak-Sobolev space and investigate their prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' One can be verify that the space C∞ c (RN) is normed space with the norm [·]s,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' However, the normed space (C∞ c (RN), [·]s,H) is not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Further, we define the completion Ds,H(RN) of (C∞ c (RN), [·]s,H) in the standard way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' More precisely, Ds,H(RN) = � [un] : {un} ⊆ C∞ c (RN) is a Cauchy sequence under the norm [·]s,H � , where [un] is the equivalence class of the Cauchy sequence {un} with the equivalence relation ′ ∼′ s,H, which is defined as {un} ∼s,H {vn} iff lim n→∞[un − vn]s,H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Ds,H(RN) is the Banach space with the norm ∥[un]∥Ds,H(RN) = lim n→∞[un]s,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we define the characterization of the normed space (Ds,H(RN), ∥·∥Ds,H(RN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Consider the space ˚ W s,H(RN) = � u ∈ LH∗ x(RN) : [u]s,H < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' ˚ W s,H(RN) is a normed space with the norm ∥u∥ ˚ W s,H(RN) = [u]s,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let H be a generalized N-function and s ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then C∞ c (RN) is the dense subspace of ˚ W s,H(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, there exist an linear isomorphism between ˚ W s,H(RN) and Ds,H(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In other words, the space Ds,H(RN) can be identified as ˚ W s,H(RN) and ∥ · ∥Ds,H(RN ) = ∥ · ∥ ˚ W s,H(RN ) = [·]s,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We provide a proof of the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='13 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Due to the presence of potential term V in the Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1), we consider the following weighted space: W = � u ∈ Ds,H(RN) : � RN V (x)Hx (x, |u|) dx < ∞ � which is a normed space with the norm ∥u∥W = ∥u∥Ds,H(RN) + ∥u∥V,H· For the shake of simplicity, we denote ∥ · ∥W as ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we have the following lemma from the definition of the space W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The space W is compactly embedded in LH loc(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Also, W is con- tinuously embedded in LH∗ x(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we will state some results which are used to prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Define the function m : ˚ W s,H(RN) → R as m(u) = � RN � RN H � x, y, |u(x) − u(y)| |x − y|s � dx dy |x − y|N · Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [7] For all u ∈ ˚ W s,H(RN) we have (1) If [u]s,H > 1 then [u]h− s,H ≤ m(u) ≤ [u]h+ s,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 9 (2) If [u]s,H < 1 then [u]h+ s,H ≤ m(u) ≤ [u]h− s,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' In particular, m(u) = 1 iff [u]s,H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, if {un} ⊂ ˚ W s,H(RN) then ∥un∥ → 0 iff m(un) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The space W is continuously embedded in LP Q(RN), where Q(x) = |K(x)|l and p : RN × RN × [0, ∞) → [0, ∞), P(x, y, t) = � |t| 0 p(x, y, r)r dr is a generalized N-function such that p1 ≤ p(x, y, |t|)|t|2 P(x, y, |t|) ≤ p2, ∀(x, y) ∈ RN × RN and t ̸= 0 for some p1, p2 ∈ (h2, h∗ 1)· Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The proof is similar to [35, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ As we have discussed, the Hardy-Littlewood-Sobolev inequality is the primary tool for dealing with the Choquard type non linearity in the context of variational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' So far, we do not have Hardy-Littlewood-Sobolev inequality for Lebesque Musielak spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By taking advantage of the condition (H2) and using Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1, we prove and use the following version of Hardy-Littlewood-Sobolev inequality in Lebesque Musielak spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' For any u ∈ W, we have K(x)F(u(x)) ∈ Ll(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, for all ǫ > 0 there exist cǫ > 0 such that ���� � RN � RN K(x)K(y)F(u(x))F(u(y)) |x − y|λ dxdy ���� ≤ C1 max � ǫ2∥u∥2ψ1 + c2 ǫ∥u∥2h∗ 1/l, ǫ2∥u∥2ψ2 + c2 ǫ∥u∥2h∗ 2/l� ≤ C2(∥u∥2ψ1 + ∥u∥2h∗ 1/l + ∥u∥2ψ2 + ∥u∥2h∗ 2/l) for some C1, C2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let u ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' It follows, from (f1) − (f2) that, for all ǫ > 0 there exist cǫ > 0 such that |F(t)| ≤ ǫΨx(x, t) + cǫ(H∗ x(x, t))1/l, ∀(x, t) ∈ RN × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='10, we have � RN|K(x)F(u(x))|ldx ≤ 2l−1 � RN(ǫ(Ψx(x, u(x)))l + cl ǫH∗ x(x, u(x)))dx ≤ c1ǫl max � ∥u∥ψ1l Lψ1l(RN), ∥u∥ψ2l Lψ2l(RN ) � + c2cl ǫ max � ∥u∥h∗ 1 LH∗x(RN), ∥u∥h∗ 2 LH∗x(RN) � , for some c1, c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Further, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='16 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='14, one gets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4) � RN |K(x)F(u(x))|ldx ≤ c3ǫl max � ∥u∥ψ1l, ∥u∥ψ2l� +c4cl ǫ max � ∥u∥h∗ 1, ∥u∥h∗ 2 � < ∞, which implies, K(x)F(u(x)) ∈ Ll(RN), for some c3, c4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4), we get ���� � RN � RN K(x)K(y)F(u(x))F(u(y)) |x − y|λ dxdy ���� ≤ � c3ǫl max � ∥u∥ψ1l, ∥u∥ψ2l� + c4cl ǫ max � ∥u∥h∗ 1, ∥u∥h∗ 2 ��2/l 10 SHILPA GUPTA AND GAURAV DWIVEDI ≤ C1 max � ǫ2∥u∥2ψ1 + c2 ǫ∥u∥2h∗ 1/l, ǫ2∥u∥2ψ2 + c2 ǫ∥u∥2h∗ 2/l� ≤ C2(∥u∥2ψ1 + ∥u∥2h∗ 1/l + ∥u∥2ψ2 + ∥u∥2h∗ 2/l) for some C1, C2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Functional Setting First, we define a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) and the corresponding energy functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We say that u ∈ W is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) if the following holds: � RN � RN h � x, y, |u(x) − u(y)| |x − y|s � (u(x) − u(y))(v(x) − v(y)) |x − y|N+2s dx dy + � RN V (x)hx(x, |u|)uvdx = � RN � RN K(x)K(y)F(u(x))f(u(y))v(y) |x − y|λ dxdy, ∀v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Thus, the energy functional I : W → R corresponding to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) is given by I(u) = � RN � RN H �|u(x) − u(y)| |x − y|s � dx dy |x − y|N + � RN V (x)Hx(x, |u|)dx − 1 2 � RN � RN K(x)K(y)F(u(x))F(u(y)) |x − y|λ dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' It can be seen that I is well defined by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='17, C1 [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2] and the derivative of I at any point u ∈ W is given by I′(u)(v) = � RN � RN h � x, y, |u(x) − u(y)| |x − y|s � (u(x) − u(y))(v(x) − v(y)) |x − y|N+2s dx dy + � RN V (x)hx(x, |u|)uvdx − � RN � RN K(x)K(y)F(u(x))f(u(y))v(y) |x − y|λ dxdy, ∀v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, the critical points of I are the weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let J : W → R such that J(u) = � RN � RN H � x, y, |u(x) − u(y)| |x − y|s � dx dy |x − y|N + � RN V (x)Hx(x, |u|)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The functional J is convex, since H is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Consequently, J is weakly lower semicontinuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', if {un} ⇀ u in W s,H 0 (RN) then J(u) ≤ lim inf n→∞ J(un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Suppose that the function t �→ H(x, y, √ t) is convex for each x, y ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, we assume that the sequence {un} converges weakly to u in W and lim sup n→∞ ⟨J′(un), un − u⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then {un} converges strongly to u in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The proof is similar to [9, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ The main existence result of this paper is as follows: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Suppose that the conditions (f1)−(f4), (K1)−(K2) and (H1)−(H5) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then the Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) has a nontrivial weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 11 To prove the existence of ground state solution, we need the following additional assumption on f: (GS) The map t �→ f(t) t|t| h2 2 −2 is strictly increasing for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' If (f1) − (f4), (GS), (K1) − (K2) and (H1) − (H5) are satisfied, then the solution obtained through Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3 is a ground state solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proof of the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We present the proof of the theorem in three steps: Step 1: In this step, we will prove that C∞ c (RN) is dense in ˚ W s,H(RN), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', for any u ∈ ˚ W s,H(RN) there exist a sequence in (C∞ c (RN), [·]s,H) which converges to u in ˚ W s,H(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let ρ ∈ C∞ c (RN) be the standard mollifier with support inside B1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Define, ρǫ(x) = ǫ−nρ � x ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' It can be seen that ρǫ(x) ∈ C∞ c (RN), � RN ρǫ(x)dx = 1 and support of ρǫ belongs to Bǫ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let u ∈ ˚ W s,H(RN) be any arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then uǫ = ρǫ ∗ u ∈ C∞(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we claim that [uǫ − u]s,H → 0 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3, Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='6 and the properties of mollifiers, we have [uǫ − u]s,H = ���� (uǫ(x) − u(x)) − (uǫ(y) − u(y)) |x − y|s ���� LH(dµ) = sup ∥v∥ L � H(dµ)≤1 ����� � RN � RN (uǫ(x) − u(x)) − (uǫ(y) − u(y)) |x − y|s v(x, y)dµ ���� � ≤ sup ∥v∥ L � H(dµ)≤1 2∥v∥L � H(dµ) �� |ξ|<1 ρ(ξ)dξ ���� (u(x − ǫξ) − u(y − ǫξ)) − (u(x) − u(y)) |x − y|s ���� LH(dµ) � = 2 � |ξ|<1 ρ(ξ) ���� (u(x − ǫξ) − u(y − ǫξ)) − (u(x) − u(y)) |x − y|s ���� LH(dµ) dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' As we know that, w(x, y) = |u(x) − u(y)| |x − y|s ∈ LH(dµ) and C∞ c (dµ) is dense in LH(dµ), hence, for a given ǫ > 0 there exist k(x, y) ∈ C∞ c (dµ) such that ∥w − k∥LH(dµ) ≤ ǫ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Further, we have ∥w(x − ǫξ, y − ǫξ) − k(x − ǫξ, y − ǫξ)∥LH(dµ) ≤ ǫ 4 and ∥K(x − ǫξ, y − ǫξ) − k(x, y)∥LH(dµ) ≤ ǫ 4 for sufficiently small ǫ and for all |ξ| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Therefore, we get [uǫ − u]s,H ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' As ǫ was arbitrary, we get [uǫ − u]s,H → 0 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We conclude the claim by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Step 2: Let {un} ⊆ (C∞ c (RN), [·]s,H) be a Cauchy sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Claim: There exist u ∈ ˚ W s,H(RN) such that un → u in ˚ W s,H(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='9, we have 12 SHILPA GUPTA AND GAURAV DWIVEDI ∥un∥LH∗x(RN) ≤ c[un]s,H < ∞, ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Hence, {un} ⊆ LH∗ x(RN) and {un} is Cauchy sequence in LH∗ x(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' As we know that LH∗ x(RN) is a Banach space, thus there exist u ∈ LH∗ x(RN) such that un → u in LH∗ x(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' This implies that, un(x) → u(x) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By the continuity of H, we have H � x, y, |un(x) − un(y)| |x − y|s � → H � x, y, |u(x) − u(y)| |x − y|s � a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Thanks to the Fatou’s lemma, � RN � RN H � x, y, |u(x) − u(y)| |x − y|s � dµ ≤ lim inf n→∞ � RN � RN H � x, y, |un(x) − un(y)| |x − y|s � dµ < ∞, which implies u ∈ ˚ W s,H(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we will prove that [un − u]s,H → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' As {un} ⊆ (C∞ c (RN), [·]s,H), we have � RN � RN H � x, y, |un(x) − un(y)| |x − y|s � dµ = � RN � RN H � x, y, |un(x) − un(y)| |x − y|s � dx dy |x − y|N < ∞ for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Thus |un(x) − un(y)| |x − y|s ∈ LH(dµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let zn(x, y) = |un(x) − un(y)| |x − y|s It can be also seen that {zn(x, y)} ia a Cauchy sequence in LH(dµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' As LH(dµ) is a Banach space, there exist z(x, y) ∈ LH(dµ) such that zn → z in LH(dµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Further, by uniqueness of the limit, we have z(x, y) = |u(x) − u(y)| |x − y|s Hence, [un − u]s,H → 0 as n → ∞, which proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Step 3: Let [un] ∈ Ds,H(RN), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [un] is an equivalence class of the Cauchy sequence {un} ⊆ (C∞ c (RN), [·]s,H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Step 2, there exist u ∈ ˚ W s,H(RN) such that [un − u]s,H → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Define a function, k : Ds,H(RN) → ˚ W s,H(RN) such that k([un]) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' It can be see that k is well defined one-one, onto and isometry by Step 1 and Step 2, which completes the proof the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proof of the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3 To prove our main result, we first establish a series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' There exist positive real numbers α and ρ such that I(u) ≥ α, ∀u ∈ W : ∥u∥ = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By using the Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='11 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='15, we have I(u) = � RN � RN H � x, y, |u(x) − u(y)| |x − y|s � dx dy |x − y|N + � RN V (x)Hx(x, |u|)dx − 1 2 � RN � RN K(x)K(y)F(u(x))F(u(y)) |x − y|λ dxdy ≥ min � [u]h1 s,H, [u]h2 s,H � + min � ∥u∥h1 V,H, ∥u∥h2 V,H � − 1 2 � RN � RN K(x)K(y)F(u(x))F(u(y)) |x − y|λ dxdy· If ∥u∥ < 1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='17 implies I(u) ≥ ∥u∥h2 − (C1ǫ2∥u∥2ψ1 + C1c2 ǫ∥u∥2h∗ 1/l) ≥ ∥u∥h2 � 1 − C1ǫ2 ∥u∥h2−2ψ1 � − C1c2 ǫ∥u∥2h∗ 1/l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We conclude the result by choosing ρ and ǫ sufficiently small and using the fact that (2h∗ 1/l) > h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' There exist ν0 ∈ W and β > 0 such that I(ν0) < 0 and ∥ν0∥ > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By (f4), there exist m1, m2 > 0 such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) F(s) ≥ m1sσ − m2, ∀ s ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let u ∈ W\\{0} and u ≥ 0 with compact support K ⊆ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' For t > 1, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='11 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='15, we have I(tu) = � RN � RN H � x, y, |tu(x) − tu(y)| |x − y|s � dx dy |x − y|N + � RN V (x)Hx(x, |tu|)dx − 1 2 � K � K K(x)K(y)F(tu(x))F(tu(y)) |x − y|λ dxdy ≤ th2 � max � [u]h1 s,H, [u]h2 s,H � + max � ∥u∥h1 V,H, ∥u∥h2 V,H �� − 1 2 � K � K K(x)K(y)(m1tσ(u(x))σ − m2)(m1tσ(u(y))σ − m2) |x − y|λ dxdy this implies that I(tu) → −∞ as t → ∞, since 2σ > h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Now, by setting ν0 = tu for sufficiently large t, we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ By Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2, the geometric conditions of the mountain pass theorem are satisfied for the functional I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Hence, by the version of the mountain pass theorem without (PS) condition, there exists a sequence {un} ⊆ W such that I(un) → cM and I′(un) → 0 as n → ∞, where 0 < cM = inf γ∈Γ max t∈[0,1] I(γ(t)) > 0, and Γ = {γ ∈ C([0, 1], W) : γ(0) = 0, γ(1) < 0}· Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The (PS)cM sequence is bounded in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, there exist u ∈ W such that, up to a subsequence, we have un ⇀ u weakly in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' 14 SHILPA GUPTA AND GAURAV DWIVEDI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Since {un} is a (PS)cM sequence of I, we have I(un) → cM and I′(un) → 0 as n → ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', � RN � RN H �|un(x) − un(y)| |x − y|s � dx dy |x − y|N + � RN V (x)Hx(x, |un|)dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2) − 1 2 � RN � RN K(x)K(y)F(un(x))F(un(y)) |x − y|λ dxdy = cM + δn, where δn → 0 as n → ∞ and ���� � RN � RN h � x, y, |un(x) − un(y)| |x − y|s � (un(x) − un(y))(v(x) − v(y)) |x − y|N+2s dx dy + � RN V (x)hx(x, |un|)unvdx − � RN � RN K(x)K(y)F(un(x))f(un(y))v(y) |x − y|λ dxdy ���� ≤ εn∥v∥, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3) ∀v ∈ W, where εn → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' On taking v = un, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3) and using (f4), we obtain �� RN � RN H � x, y, |un(x) − un(y)| |x − y|s � dx dy |x − y|N − 1 σ � RN � RN h � x, y, |un(x) − un(y)| |x − y|s � (un(x) − un(y))2 |x − y|N+2s dxdy � + � RN (V (x)Hx(x, |u|) − 1 σ V (x)hx(x, |u|)u2)dx ≤ c5(1 + ∥un∥), for some c5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' It follows from (H2) that � 1 − h2 σ � � RN � RN H � x, y, |un(x) − un(y)| |x − y|s � dx dy |x − y|N + � 1 − h2 σ � � RN V (x)Hx(x, |un|)dx ≤ c5(1 + ∥un∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' If ∥un∥ > 1, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='11 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='15, we have � 1 − h2 σ � ([un]h1 s,H + ∥un∥h1 V,H) ≤ c5(1 + ∥un∥) � 1 − h2 σ � ∥un∥h1 ≤ c5(1 + ∥un∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Consequently, ∥un∥ ≤ c6 for some c6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Thus {un} is bounded in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' As W is a reflexive Banach space, ∃ u ∈ W such that up to a subsequence, we have un ⇀ u weakly in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let {un} is bounded in W such that un ⇀ u weakly in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Then lim n→∞ � RN |K(x)f(un(x))(un(x) − u(x))|ldx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let {un} is bounded in W such that un ⇀ u weakly in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='14, we have un(x) → u(x) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 15 Define Q(x) = |K(x)|l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' It follows from (f1) and (f3) that, for all ǫ > 0 there exist t0, t1, cǫ > 0 such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4) f(t) ≤ ǫ � ψx(x, t)t + (H∗ x(x, t)) b−1 h∗ i � + cǫ(H∗ x(x, t)) b−1 h∗ i χ[t0,t1](t), ∀(x, t) ∈ RN × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Further, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4), we have K(x)f(un(x)) ≤ ǫK(x) � ψx(x, un(x))un(x) + (H∗ x(x, un(x))) b−1 h∗ i � + cǫK(x)(H∗ x(x, un(x))) b−1 h∗ i χ[t0,t1](un(x)), ∀(x, t) ∈ RN × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Consider, � RN|K(x)f(un(x))(un(x) − u(x))|ldx ≤ 2l−1ǫl � RN Q(x) ���� � ψx(x, un(x))un(x) + (H∗ x(x, un(x))) b−1 h∗ i � (un(x) − u(x)) ���� l dx + 2l−1cl ǫ � RN Q(x) ����(H∗ x(x, un(x))) b−1 h∗ i χ[t0,t1](un(x))(un(x) − u(x)) ���� l dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='5) Now, define the set An = {x ∈ RN : |un(x)| ≥ t0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Thus, (H3) and definition of H∗ implies c7|An| ≤ � An H∗ x(x, t0)dx ≤ � An H∗ x(x, un(x))dx ≤ � RN H∗ x(x, un(x))dx < c8, since {un} is bounded in W, for some c7, c8 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Therefore, we have sup n∈N |An| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Using (K1), we get lim d→∞ � An∩Bd(0)c |K(x)| 2N 2N−λ dx = 0 uniformly in n ∈ N consequently, for a given ǫ > 0 there exist d0 > 0 such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='6) � An∩Bd0 (0)c |K(x)| 2N 2N−λ dx < ǫ b (b−1) for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Using H¨older’s inequality and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='12, we have � Bd0 (0)c Q(x) ����(H∗ x(x, un(x))) b−1 h∗ i χ[t0,t1](un(x))(un(x) − u(x)) ���� l dx ≤ c9 � An∩Bd0(0)c Q(x)|un(x)|(b−1)lχ[t0,t1](|un(x)|)|un(x) − u(x)|ldx ≤ c9 max i∈{1,2} \uf8f1 \uf8f2 \uf8f3 �� An∩Bd0(0)c Q(x)|un(x)|blχ[t0,t1](un(x))dx � (b−1) b �� An∩Bd0(0)c Q(x)|un(x) − u(x)|bldx � 1 b \uf8fc \uf8fd \uf8fe 16 SHILPA GUPTA AND GAURAV DWIVEDI ≤ c10t(b−1)l 1 �� An∩Bd0(0)c Q(x) � (b−1) b for some c9, c10 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Further, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='6), we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='7) � Bd0 (0)c Q(x) ����(H∗ x(x, un(x))) b−1 h∗ i χ[t0,t1](un(x))(un(x) − u(x)) ���� l dx ≤ c11ǫ, for some c11 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='10, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='16 and H¨older’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' we have � Bd0(0)c Q(x) ���� � ψx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' un(x))un(x) + (H∗ x(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' un(x))) b−1 h∗ i � (un(x) − u(x)) ���� l dx ≤ c12 max i∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2} �� RN Q(x) � |un(x)|(ψi−1)l + |un(x)|(b−1)l� |un(x) − u(x)|ldx � ≤ c12 max i∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2} \uf8f1 \uf8f2 \uf8f3 �� RN Q(x)|un(x)|ψildx � (ψi−1) ψi �� RN Q(x)|un(x) − u(x)|ψildx � 1 ψi \uf8fc \uf8fd \uf8fe + c12 max i∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2} \uf8f1 \uf8f2 \uf8f3 �� RN Q(x)|un(x)|bldx � (b−1) b �� RN Q(x)|un(x) − u(x)|bldx � 1 b \uf8fc \uf8fd \uf8fe ≤ c13 max i∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2} � ∥un(x) − u(x)∥l Lψil Q (RN) + ∥un(x) − u(x)∥l Lbl Q(RN) � for some c12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' c13 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Therefore, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='16, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='8) � Bd0(0)c Q(x) ���� � ψy(x, un(x))un(x) + (H∗ x(x, un(x))) b−1 h∗ i � (un(x) − u(x)) ���� l dx ≤ c14, for some c14 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Consequently, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='8) implies � Bd0 (0)c |K(x)f(un(x))(un(x) − u(x))|ldx ≤ 2l−1ǫlc14 + 2l−1cl ǫc11ǫ → 0 as ǫ was arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' On the other side, by (f3) and Strauss compactness lemma [12, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='I], we have lim n→∞ � Bd0(0) |K(x)f(un(x))(un(x) − u(x))|ldx = 0, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ Proof of the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Both the geometric conditions of the mountain pass theorem follow from Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we will prove that the func- tional I satisfies the (PS)cM condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let {un} ⊆ W be any Palais-Smale sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', I(un) → cM and I′(un) → 0 in dual space of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3, we conclude that {un} is bounded in W and GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 17 un ⇀ u weakly in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' As a consequence, I′(un)(un − u) → 0 as n → ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='9) J′(un)(un − u) − � RN � RN K(x)K(y)F(un(x))f(un(y))(un(y) − u(y)) |x − y|λ dxdy → 0 as n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we claim that that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='10) � RN � RN K(x)K(y)F(un(x))f(un(y))(un(y) − u(y)) |x − y|λ dxdy → 0 as n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let |K(y)|l = Q(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4), we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='11) ∥K(x)F(un(x))∥Ll(RN) ≤ c15, for some c15 > 0 since {un} is bounded in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4, we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='12) lim n→∞ � RN |K(x)f(un(x))(un(x) − u(x))|ldx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='12) and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1, the claim in the (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='10) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2, we have un → u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Thus, (PS)cM condition is satisfied for the functional I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Hence, by the mountain pass theorem, there exists critical point uM of I with level cM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', I′(uM) = 0 and I(uM) = cM > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Thus, uM is the non-trivial weak solution of the Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Ground State Solution In this section, we prove that the solution obtained through Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3 is a ground state solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let us recall the definition of a ground state solution: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' A weak solution u0 of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) is called a ground state solution if it has the least energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', we say, the solution u0 is ground state solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) if (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) I(u0) = r = inf u∈S I(u), where S is the set of all critical points of the functional I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' To prove that the solution obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3 is a ground state solution, we use the minimization method, in particular, Nehari manifold method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We define (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2) ℵ = {u ∈ W\\{0}|I′(u)u = 0} and b = inf u∈ℵ I(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The set ℵ is called the Nehari manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' It can be observed that S ⊆ ℵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The key idea of this method is to search for a non-trivial critical point of I in ℵ instead of the whole space W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' To know more about this method, one can refer to [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' The existence of a ground state solution is proved by many researchers, we refer to, [11, 13, 24, 25, 26, 27] and reference therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' For u ∈ W, define the function, hu : [0, ∞) → R such that hu(t) = I(tu), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', hu(t) = � RN � RN H � x, y, |tu(x) − tu(y)| |x − y|s � dx dy |x − y|N + � RN V (x)Hx(x, |tu(x)|)dx − 1 2 � RN � RN K(x)K(y)F(tu(x))F(tu(y)) |x − y|λ dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Let (f1) − (f4), (GS), (K1) − (K2) and (H1) − (H5) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' If u ∈ W\\{0}, then there exists unique tu > 0 such that tuu ∈ ℵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Moreover, max t∈[0,∞] hu(t) = hu(tu) = I(utu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' 18 SHILPA GUPTA AND GAURAV DWIVEDI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' We observe that h′ u(t) = 0 if and only if tu ∈ ℵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2 imply that hu(t) > 0 for sufficiently small t and hu(t) < 0 for sufficiently large t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Thus, ∃ tu ∈ (0, ∞) such that max t∈[0,∞] hu(t) = hu(tu) = I(utu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Consequently, h′ u(tu) = 0 and tuu ∈ ℵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Next, we will prove the uniqueness of tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' If t is the critical point of hu, then we have h′ u(t) = � RN � RN h � x, y, |tu(x) − tu(y)| |x − y|s � (tu(x) − tu(y))2 t|x − y|N+2s dx dy + � RN V (x)hx(x, |tu(x)|)(tu(x))2dx t − � RN � RN K(x)K(y)F(u(x))f(tu(y))u(y) |x − y|λ dxdy = 0, which implies that � RN � RN h � x, y, |tu(x) − tu(y)| |x − y|s � (tu(x) − tu(y))2 th2|x − y|N+2s dx dy + � RN V (x)hx(x, |tu(x)|)(tu(x))2dx th2 = � RN � RN K(x)K(y)F(tu(x))f(tu(y))tu(y) th2|x − y|λ dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3) On proceeding as [34, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3], one can check that the right hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3) is decreasing for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Consider, � RN � RN K(x)K(y)F(tu(x))f(tu(y))tu(y) th2|x − y|λ dxdy = � RN K(y) �� RN K(x)F(tu(x))dx |x − y|λ � f(tu(y))tu(y) th2 dy = � RN K(y) �� RN K(x)F(tu(x))dx t h2 2 |x − y|λ � f(tu(y))|u(y)| h2 2 |tu(y)| h2 2 −2(tu(y)) dy which implies the left hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='3) is increasing strictly for t > 0 by (GS) and (f4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Therefore, tu is a unique critical point of hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ Proof of the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='4 It is enough to prove that, cM = b = r, where b and r as are defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By using the fact that S ⊆ ℵ, we have b ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Also, it can be seen r ≤ cM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' It will be sufficient to prove that b ≥ cM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' If v ∈ ℵ, then h′ v(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='1, we have max t∈[0,∞] hv(t) = hv(1) = I(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Choose a function γ : [0, 1] → W such that γ(t) = tt0v, where t0 > 0 such that I(t0v) < 0, which implies that γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Therefore, we have cM ≤ max t∈[0,1] I(γ(t)) = max t∈[0,1] I(tt0v) ≤ max t≥0 I(tv) = I(v), which is true for every element v ∈ ℵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Hence, b ≥ cM, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' □ Acknowledgement Gaurav Dwivedi acknowledges the funding by Science and Engineering Research Board, India, under the grant CRG/2020/002087.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Conflict of Interest This work does not have any conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' GENERALIZED CHOQUARD SCHR ¨ODINGER EQUATION 19 References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Press, Somerville, (2010), 597–632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' [37] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Xie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Zhang, Existence of solutions for the (p, q)-Laplacian equation with nonlocal Choquard reaction, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=', 135 (2023), 108418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' Shilpa Gupta Department of Mathematics Birla Institute of Technology and Science Pilani Pilani Campus, Vidya Vihar Pilani, Jhunjhunu Rajasthan, India - 333031 Email address: p20180442@pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='bits-pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content=' shilpagupta890@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='com Gaurav Dwivedi Department of Mathematics Birla Institute of Technology and Science Pilani Pilani Campus, Vidya Vihar Pilani, Jhunjhunu Rajasthan, India - 333031 Email address: gaurav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='dwivedi@pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='bits-pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} +page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQfOgk_/content/2301.04393v1.pdf'} diff --git a/x9AzT4oBgHgl3EQfCfrw/content/tmp_files/2301.00962v1.pdf.txt b/x9AzT4oBgHgl3EQfCfrw/content/tmp_files/2301.00962v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea63ad6cd18dd3052b59a8bddc3fb6909a4e2d5b --- /dev/null +++ b/x9AzT4oBgHgl3EQfCfrw/content/tmp_files/2301.00962v1.pdf.txt @@ -0,0 +1,1200 @@ +The derived moduli stack of logarithmic flat connections +Francis Bischoff∗ +Abstract +We give an explicit finite-dimensional model for the derived moduli stack of flat connections on +Ck with logarithmic singularities along a weighted homogeneous Saito free divisor. We investigate in +detail the case of plane curves of the form xp = yq and relate the moduli spaces to the Grothendieck- +Springer resolution. We also discuss the shifted Poisson geometry of these moduli spaces. Namely, +we conjecture that the map restricting a logarithmic connection to the complement of the divisor +admits a shifted coisotropic structure and we construct a shifted Poisson structure on the formal +neighbourhood of a canonical connection in the case of plane curves xp = yq. +Contents +1 +Introduction +1 +2 +Homogeneous free divisors and logarithmic flat connections +4 +3 +Finite dimensional model +5 +4 +Proof of Theorem 3.5 +8 +5 +Plane curves xp − yq +12 +1 +Introduction +Let D ⊂ Ck be a hypersurface cut out by a reduced holomorphic function f. In [30] Saito considers +the subsheaf, usually denoted TCk(−logD), of holomorphic vector fields on Ck which preserve the +ideal generated by f. In general, it is coherent and closed under the Lie bracket, but may fail to +be locally free. In fact, Saito provides a very explicit criterion for determining whether the sheaf is +locally free. When it is, D is said to be a free divisor and TCk(−logD), known as the logarithmic +tangent bundle, defines a Lie algebroid. Examples of free divisors include smooth hypersurfaces, +plane curves and simple normal crossings. In general, D may be highly singular. +Let G be a connected complex reductive group with Lie algebra g and assume that D is a free +divisor which is homogeneous under a given C∗-action on Ck with the property that all its weights +are strictly positive. In this paper, we are interested in studying the moduli space of TCk(−logD)- +representations on principal G-bundles, also known as logarithmic flat connections. +There is a +standard way of defining this moduli space as the Maurer-Cartan locus of an infinite-dimensional +differential graded Lie algebra (dgla) LD,g which is associated to D and g. Let Ω1 +Ck(logD) denote the +logarithmic cotangent bundle, which is the dual to TCk(−logD), and let Ω● +Ck(logD) = ∧●Ω1 +Ck(logD) +be the exterior algebra. This defines a commutative differential graded algebra when equipped with +the Lie algebroid differential d. Then LD,g = Ω● +Ck(logD) ⊗ g inherits the structure of a dgla. The +Maurer-Cartan locus of this dgla is defined to be the following set +MC(LD,g) = {ω ∈ L1 +D,g ∣ dω + 1 +2[ω,ω] = 0}. +∗Exeter College and Mathematical Institute, University of Oxford; francis.bischoff@maths.ox.ac.uk +1 +arXiv:2301.00962v1 [math.AG] 3 Jan 2023 + +Here, ω ∈ Ω1 +Ck(logD) ⊗ g is a Lie algebra valued 1-form, and it defines the following connection +∇ = d + ω, which has a logarithmic singularity along D. It’s curvature is given by the following +expression +F(ω) = dω + 1 +2[ω,ω]. +The degree 0 component of the dgla is L0 +D,g = Map(Ck,g), which is the Lie algebra of the infinite +dimensional gauge group G = Map(Ck,G). This group acts on the Maurer-Cartan locus, giving +the correct equivalence between flat connections. As a result, the moduli space of flat logarithmic +connections is defined to be the stack quotient +[MC(LD,g)/G]. +Although this construction involves infinite dimensional spaces, in [4] we provide a purely finite +dimensional model. More precisely, we show that the category of logarithmic flat connections with +fixed residue data is equivalent to the stack quotient of an affine algebraic variety by the action of +an algebraic group. +The purpose of the present paper is to provide a derived enhancement of the moduli stack. There +are several different approaches to derived geometry in the literature, such as [16, 12, 19, 31, 28]. +In this paper, we have opted to go with the notion of bundles of curved dgla’s, which requires +relatively little technology and is sufficient for our purposes. Let us recall the definition from [2]. +Definition 1.1. A bundle of curved differential graded Lie algebras over a variety M consists of a +graded vector bundle L● starting in degree 2, which is equipped with the following data +1. a section F ∈ Γ(M,L2), +2. a degree 1 bundle map δ ∶ L● → L●[1] +3. a smoothly varying graded Lie bracket [−,−] on the fibres of L●, +satisfying the following conditions +1. the Bianchi identity δF = 0, +2. δ2 = [F,−], +3. δ is a graded derivation of the bracket [−,−]. +If [M/G] is a stack, defined by the data of a Lie groupoid G over M, then we define a bundle of +curved dgla’s over the stack to be such a bundle over M, which is equipped with an equivariant +action of G preserving the data (F,δ,[−,−]). We will use the respective terminology of derived +manifolds and derived stacks to refer to this data. +There is a standard way of constructing a derived manifold from the data of a dgla, and we +can apply it to the case of LD,g. Namely, we take the base to be M = L1 +D,g and take the bundle +L● to be trivial with fibre given by the truncation L●≥2 +D,g. The section is given by the curvature F, +and the bracket is simply the constant one inherited from LD,g. The bundle map δ varies over M, +and above a point ω ∈ L1 +D,g, it is given by the twisted differential δω = d + [ω,−]. Let us denote the +resulting derived manifold MD,g. It can be further upgraded to a derived stack by noting that the +gauge group G lifts to an action on L●≥2 +D,g via the adjoint representation. +A derived manifold (or stack) has an underlying classical truncation π0(M), defined as the +vanishing locus of the section F. In the example under consideration, the classical truncation is +given by the Maurer-Cartan locus, and hence +π0([MD,g/G]) = [MC(LD,g)/G]. +2 + +For this reason, we say that [MD,g/G] is the derived moduli stack of flat logarithmic G-connections. +The main result of this paper is a finite-dimensional model of this derived stack. +Here is a brief description of this result. +Given an element A ∈ g, we consider the infinite +dimensional derived moduli stack [MD,g(A)/G] of G-connections whose ‘residue’ is conjugate to +A. Details of this are given in Section 2. Let A = S + N0 be the Jordan decomposition, where +S is semisimple and N0 is nilpotent. In Section 3 we construct a finite dimensional dgla (U0,δS) +associated to S, with corresponding derived stack [US/Aut(S)]. This is interpreted as a certain +sub-moduli space of flat connections on the fibre f −1(1). Then, associated to the element A, we +construct a derived substack [W(A)/Aut(S)] of the shifted tangent bundle T[−1][US/Aut(S)]. +The main result is Theorem 3.5, which states that there is an equivalence of derived stacks +q ∶ [W(A)/Aut(S)] → [MD,g(A)/G]. +By this we mean that q induces an equivalence between the groupoids of solutions to the MC +equation, and given any solution w, the derivative dqw is a quasi-isomorphism of tangent complexes. +In Section 5 we turn to the case of a plane curve defined by the function f = xp − yq. This +is the simplest case above k = 1, and already it exhibits interesting behaviour. We construct an +explicit derived stack [Q(A)/PS] from the data of a parabolic subgroup PS of the centralizer of +exp( 2πi +pq S) and a representation H1(U0). In Theorem 5.7 we show that, given a condition on the +eigenvalues of adS, the derived stack [Q(A)/PS] is equivalent to [W(A)/Aut(S)]. For general S, +the moduli space can have extra components and we illustrate this in Example 5.9. The derived +stack [Q(A)/PS] can be interpreted in terms of spaces showing up in geometric representation +theory, such as the Grothendieck-Springer resolution. +Hence Theorem 5.7 can be viewed as a +higher dimensional generalization of Boalch’s description in [5] of the moduli space of logarithmic +connections on the disc. +Speculations about Poisson geometry +Going back to the work of Atiyah-Bott [1] and Goldman [15], we know that the moduli space of flat +connections on a closed Riemann surface admits a symplectic structure. If the surface is punctured, +then the moduli space admits a Poisson structure, whose symplectic leaves are obtained by fixing +boundary conditions at the punctures [14]. +This picture has since been generalized in several +directions, including to the case of flat connections with singularities [6, 8, 7]. More recently, the +moduli space of local systems on higher dimensional manifolds has been studied using tools from +derived algebraic geometry. For a compact oriented manifold M of dimension d, the moduli space +of local systems LocG(M) is a derived stack equipped with a shifted symplectic structure of degree +2−d [25]. If M has a boundary ∂M = N, then LocG(N) has a 3−d-shifted symplectic structure, and +the restriction map r ∶ LocG(M) → LocG(N) has a Lagrangian structure [9], inducing on LocG(M) +a 2 − d-shifted Poisson structure [22]. We wish to generalize this picture to the case of logarithmic +flat connections in higher dimensions. +In the above setting of a map f ∶ Ck → C, the inverse image of the unit circle f −1(S1) is a +manifold of dimension 2k−1, usually with boundary, and so LocG(f −1(S1)) has a Poisson structure +of degree 3 − 2k. Given a logarithmic flat connection, we can restrict it to f −1(S1) and take its +holonomy. This should define a map +r ∶ [W(A)/Aut(S)] → LocG(f −1(S1)). +(1.1) +This map was studied by Boalch [5] in the special case of k = 1, where f −1(S1) = S1. In this case +LocG(S1) = G/G has a 1-shifted symplectic structure, and the work of Boalch (suitably interpreted +by [29]) shows that r has a Lagrangian structure. In higher dimensions we conjecture that the map +can be equipped with a shifted coisotropic structure in the sense of [21, 22]. +3 + +Conjecture 1.2. The map r can be naturally equipped with a coisotropic structure. +In order to avoid the analytic issues that arise in taking the holonomy, it may be preferable to +replace LocG(f −1(S1)) with a moduli space of flat connections on the complement of D. +In recent work [23, 24], Pantev and To¨en studied the moduli spaces of local systems and flat +connections on non-compact algebraic varieties. They constructed shifted Poisson structures and +explained how to obtain the symplectic leaves by imposing suitable boundary conditions at infinity. +Conjecture 1.2 may be viewed as providing another source of boundary conditions for the moduli +spaces associated to f −1(Ck ∖ D). We hope that it may also be used in conjunction with their +results, for example by considering the map r in the presence of additional boundary conditions at +the boundary of the fibres of f. +One implication of the conjecture is that the moduli spaces [W(A)/Aut(S)] should admit +2(1 − k)-shifted Poisson structures. In Theorem 5.14 we provide evidence for the conjecture by +constructing a −2-shifted Poisson structure on the formal neighbourhood of a special connection +in the case of plane curves xp = yq. Our construction is somewhat ad hoc, but it makes use of an +invariant inner product on the Lie algebra g, as well as the intersection pairing on the cohomology +of the curve f −1(1). We have also not checked that our shifted Poisson structure fits into the +formalism developed by [10]. We hope to address all these issues in future work. +Acknowledgements. I would like to thank Elliot Cheung for pointing me to the paper [2]. +2 +Homogeneous free divisors and logarithmic flat connec- +tions +Assume that the given C∗ action on Ck has strictly positive weights. It is generated infinitesimally +by an Euler vector field +E = +k +∑ +i=1 +nizi∂zi, +where ni ∈ Z>0 are positive integers. This vector field defines a weight grading on the holomorphic +functions OCk (and more generally tensor fields) on Ck, such that the coordinate function zi has +weight ni. This grading will play an important role. Because of our assumption, each weight space +is finite-dimensional over C. We also assume that the function f defining D is homogeneous of +weight r: E(f) = rf. +The C∗ action determines an action Lie algebroid C⋉Ck which is generated by the Euler vector +field. Because E is a section of TCk(−logD), there is an induced Lie algebroid morphism +i ∶ C ⋉ Ck → TCk(−logD), +(λ,z) ↦ λEz. +The logarithmic 1-form dlogf = df +f is a closed section of Ω1 +Ck(logD). Hence, it determines a Lie +algebroid morphism +π ∶ TCk(−logD) → C, +V ↦ 1 +rf V (f), +where C is considered as an abelian Lie algebra. The composition p = π ○ i ∶ C ⋉ Ck → C is given by +projection to the first factor. This has a section +j ∶ C → C ⋉ Ck, +λ ↦ (λ,0), +4 + +which is also a Lie algebroid morphism. Altogether, we have the following diagram of Lie algebroids: +C ⋉ Ck +C +TCk(−logD) +i +π +p +j +Each Lie algebroid determines a differential graded Lie algebra, whose Maurer-Cartan locus con- +sists of flat algebroid connections. Furthermore, each morphism of Lie algebroids determines a +pullback morphism between dgla’s, and as a result, a pullback morphism between categories of +representations, or more generally, derived moduli stacks of flat connections. This gives rise to the +following diagram of (infinite-dimensional) derived stacks: +[(OCk ⊗ g)/G] +[g/G] +[MD,g/G] +i∗ +π∗ +p∗ +j∗ +In this diagram, [g/G] is the moduli stack of g-representations of C. It is the stack quotient +corresponding to the adjoint action of G on its Lie algebra. [(OCk ⊗ g)/G] is the moduli stack of +g-representations of C⋉Cr. In both cases the derived structure is trivial because the Lie algebroids +have rank 1. +Now fix an element A ∈ g, let OA ⊂ g be its adjoint orbit, and let GA ⊆ G be its centralizer +subgroup. This determines a substack [OA/G] ⊂ [g/G] which is Morita equivalent to BGA. The +preimage [MD,g(A)/G] ∶= (j∗i∗)−1(BGA) is the derived stack of logarithmic flat connections ω +whose ‘residue’ j∗i∗(ω) lies in OA. More precisely, the base of the derived manifold MD,g(A) is +given by +M(A) = {ω ∈ Ω1 +Ck(logD) ⊗ g ∣ j∗i∗(ω) ∈ OA}, +with the bundle of curved dgla’s restricted from MD,g. The action of G preserves M(A). +3 +Finite dimensional model +Let A = S + N0 be the Jordan decomposition of A, where S is semisimple, N0 is nilpotent, and +[S,N0] = 0. In this section we will construct a finite dimensional model for [MD,g(A)/G]. +The dgla LD,g +We start by analysing the structure of the dgla LD,g. Being constructed from the cdga Ω● +Ck(logD) +and the Lie algebra g, LD,g inherits their derivations. The basic ones are as follows: +• the Lie algebroid differential d, which has degree +1 and squares to 0, +• the interior multiplication with the Euler vector field ιE, which has degree −1 and squares to +0, +• the adjoint action of S, adS, which has degree 0. +5 + +By taking commutator brackets we arrive at further derivations, such as LE = [ιE,d], the Lie +derivative with respect to E, which is a derivation of degree 0. We can also wedge any derivation +by a differential form to obtain a new derivation. Let α0 = 1 +rdlogf, which is a closed logarithmic +1-form. Then α0adS is a degree +1 derivation which squares to 0. Among the 5 derivations just +described, almost all of them commute. The only two non-vanishing commutator brackets are the +following: +[ιE,d] = LE, +[ιE,α0adS] = adS. +The second bracket follows as a consequence of the identity ιE(α0) = 1. We are primarily interested +in studying the dgla structure arising from +δS = d + α0adS, +which is a degree +1 derivation that squares to 0. We are also interested in the following degree 0 +derivation +LS ∶= [ιE,δS] = LE + adS. +This operator is diagonalisable in the sense that any element β ∈ LD,g has a Taylor series expansion +β = ∑ +u +βu +where each term satisfies LS(βu) = uβu. Indeed, the operator adS is diagonalizable on g with +finitely many eigenvalues since S is semisimple. The eigenspaces of LE are the weight spaces. We +noted earlier that the weight degrees of holomorphic functions are strictly positive integers, and +that each weight space is finite dimensional. As an operator on Ω● +Ck(logD), the eigenvalues of LE +may not be positive, but they are integers which are bounded below. Hence, the eigenvalues of LS +have the form ui + Z≥0, for finitely many complex numbers ui. +Let LD,g,u denote the u-eigenspace, and note that it is finite-dimensional. Because LS is a +derivation, the Lie bracket respects this decomposition: +[−,−] ∶ LD,g,u × LD,g,v → LD,g,u+v. +In particular, LD,g,0 is a finite-dimensional Lie subalgebra. The derivations δS and ιE commute +with LS, and hence preserve its eigenspaces. In particular, they restrict to LD,g,0. +Now introduce the degree 0 derivation P = α0ιE. This derivation satisfies P 2 = P, and hence +induces a decomposition LD,g = ker(P) ⊕ im(P). Let U = ker(P) and let I = im(P). With respect +to the bracket, U is a subalgebra and I is an abelian ideal. +Lemma 3.1. The derivation ιE vanishes on U. For every degree i it defines an isomorphism +ιE ∶ Ii → U i−1, +with inverse given by multiplication by α0. Therefore, as a graded Lie algebra, LD,g is isomorphic +to U ⋉ U[−1], where U acts on U[−1] via the adjoint action. +Proof. It is clear that ker(ιE) ⊆ U = ker(P). For the opposite inclusion, suppose that P(x) = 0. +Then ιE(x) is in the kernel of multiplication by α0. Since α0 is a non-vanishing algebroid 1-form, +ιE(x) must be of the form α0 ∧ y. But then +0 = ι2 +E(x) = ιE(α0 ∧ y) = y − α0 ∧ ιE(y), +which implies that ιE(x) = 0, as required. The image of ιE is contained in U since ι2 +E = 0. To see +surjectivity, we can explicitely construct the inverse as mulitplication by α0. Given x ∈ U, check +6 + +that α0 ∧ x = P(α0 ∧ x) ∈ I. Hence α0∧ ∶ U i−1 → Ii. Then for x ∈ U, we have ιE(α0 ∧ x) = x, and for +P(y) ∈ I we have α0 ∧ ιEP(y) = P 2(y) = P(y). +Now define the isomorphism Ξ ∶ LD,g → U ⋉ U[−1] by the following formula in degree i: +U i ⊕ Ii → U i ⊕ U i−1, +(x,y) ↦ (x,(−1)iιE(y)). +This preserves Lie brackets. +The commutator [P,LS] = 0. Therefore, the two operators can be simultaneously diagonalized. +In particular, we have the decomposition LD,g,0 = U0 ⊕ I0. The results of the previous lemma +remain true for this subalgebra. Next, we have [P,δS] = α0LS. If we re-write this as the following +identity +PδS = α0LS + δSP +then we can deduce that δS preserves I. Indeed, applying this identity to an element of the form +x = P(y), we obtain +PδS(x) = α0LSP(y) + δSP 2(y) = α0PLS(y) + δSP(y) = δS(x). +On the other hand, the differential δS does not preserve U. But by applying the identity to an +element x ∈ U, we compute that the ‘off-diagonal’ term is given by PδS(x) = α0LS(x). This term +vanishes when we restrict to the subalgebra LD,g,0. Hence, we obtain the following corollary. +Corollary 3.2. The subalgebra U0 is preserved by δS, and there is an isomorphism of dgla’s +(LD,g,0,δS) ≅ (U0,δS) ⋉ (U0,δS)[−1]. +Proof. On the subspace LD,g,0 we have [ιE,δS] = 0. This implies that the morphism Ξ from Lemma +3.1 is a chain map. +The gauge group Aut(S) +Viewing S ∈ g as a representation of C, we can pull it back to obtain a representation p∗S of +C ⋉ Ck. Let Aut(S) be the subgroup of the gauge group G consisting of gauge transformations +which preserve p∗S: +Aut(S) = {g ∈ G ∣ g ∗ p∗S = p∗S}. +It is a finite-dimensional algebraic group whose Lie algebra is L0 +D,g,0. We recall the description of +its Levi decomposition which was given in [4]. The automorphism group of j∗p∗S = S is GS, the +centralizer subgroup of S in G, which is reductive. The pullback functor j∗ defines a homomorphism +j∗ ∶ Aut(S) → GS, +g ↦ g(0), +and the pullback functor p∗ defines a splitting. The kernel of j∗, denoted Aut0(S), is the unipotent +radical. Hence the isomorphism Aut(S) ≅ Aut0(S) ⋊ GS provides the Levi decomposition. +Define the following gauge action of Aut(S) on L1 +D,g,0: +g ∗ x = gxg−1 − δS(g)g−1, +where δS(g)g−1 = dgg−1 + α0(S − gSg−1). +Lemma 3.3. The gauge action of Aut(S) is well-defined. In terms of the decomposition L1 +D,g,0 = +U 1 +0 ⊕ I1 +0 it is given by +g ∗ (x,y) = (gxg−1 − δS(g)g−1,gyg−1), +where x ∈ U 1 +0 and y ∈ I1 +0. +Furthermore, Aut(S) acts on L●≥2 +D,g,0 by conjugation, preserving the +decomposition U0 ⊕ I0 and the Lie bracket. +7 + +Proof. A computation shows that LS(g ∗ x) = g(LSx)g−1 for x ∈ L1 +D,g, showing that L1 +D,g,0 is +preserved. Similarly, LS(gxg−1) = g(LSx)g−1 for x ∈ Lj +D,g, showing that the conjugation action +preserves L●≥2 +D,g,0. Next, for x ∈ LD,g,0 we have P(gxg−1) = gP(x)g−1, implying that the conjugation +also preserves U0 and I0. Finally, +P(δS(g)g−1) = α0(LE(g)g−1 + S − gSg−1), +which vanishes for g ∈ Aut(S). Hence δS(g)g−1 ∈ U 1 +0 . +The finite-dimensional derived stack +Given the finite dimensional dgla LD,g,0 we obtain a derived manifold WS. The base manifold +is the vector space WS = L1 +D,g,0, the bundle of curved dgla’s is the trivial bundle WS × L●≥2 +D,g,0, +the curvature section is given by the standard formula FS(w) = δS(w) + 1 +2[w,w], and the twisted +differential δ is given by +δS,w = δS + [w,−], +for w ∈ WS. +Furthermore, Lemma 3.3 gives an equivariant action of Aut(S) on WS × L●≥2 +D,g,0, +preserving the bracket. It is also straightforward to check that this action preserves FS and δ. +Hence, we obtain a derived stack [WS/Aut(S)]. +U0 is a sub-dgla of LD,g,0, which is preserved by the action of Aut(S). Hence, it gives rise to +a derived substack [US/Aut(S)] of [WS/Aut(S)]. Furthermore, since I0 is an ideal of LD,g,0, we +also get a projection morphism [WS/Aut(S)] → [US/Aut(S)]. +Proposition 3.4. The derived stack [WS/Aut(S)] is isomorphic to the shifted tangent bundle +T[−1][US/Aut(S)]. +Proof. This follows from Corollary 3.2 and Lemma 3.3. +We are actually interested in a substack of [WS/Aut(S)] which is determined by the element +A = S + N0. Recall that the image of Aut(S) under j∗ is GS, the centralizer of S. This implies +that for any element ω ∈ WS, the image j∗i∗(ω) ∈ gS = Lie(GS). We will require that this element +be contained in GS ∗ N0, the adjoint orbit of N0 in gS. Namely, define +W(A) = {ω ∈ WS ∣ j∗i∗(ω) ∈ GS ∗ N0}. +Let W(A) be the derived manifold obtained by pulling back the bundle of curved dgla’s from WS +to W(A). The action of Aut(S) restricts to an action on this sub-manifold. Hence, we obtain a +derived stack [W(A)/Aut(S)]. +Theorem 3.5. [W(A)/Aut(S)] is equivalent to [MD,g(A)/G], the derived stack of logarithmic +flat connections whose residue lies in the adjoint orbit OA of A. +4 +Proof of Theorem 3.5 +In this section we will give the proof of the equivalence between [W(A)/Aut(S)] and [MD,g(A)/G]. +There is a natural morphism +q ∶ [W(A)/Aut(S)] → [MD,g(A)/G], +which we describe as follows: +8 + +1. The map on the base manifolds is given by the following formula +q ∶ W(A) → M(A), +ω ↦ α0S + ω. +2. The map on bundles of curved dgla is given by the inclusion L●≥2 +D,g,0 → L●≥2 +D,g. +3. The group Aut(S) includes into G as a subgroup, and the map q is equivariant. +In order to show that q is an equivalence, we must show two things. First, there is an underlying +functor between the classical groupoids: +π0(q) ∶ Aut(S) ⋉ MC(W(A)) → G ⋉ MC(MD,g(A)). +We need to show that this is an equivalence of categories. This is implied by [4, Theorem 5.5] and +the following lemma. +Lemma 4.1. Let ω ∈ W(A). Then ιE(ω) is nilpotent. +Proof. For ω ∈ L1 +D,g,0, we have ιE(ω) ∈ U 0 +0 = Lie(Aut(S)). If ω ∈ W(A) we have in addition that +j∗ιE(ω) ∈ GS ∗ N0, and so is nilpotent. Let ιE(ω) = Bs + Bn be the Jordan decomposition, where +Bs is semisimple and Bn is nilpotent. Then j∗(Bs) = 0, so that Bs ∈ Lie(Aut0(S)). But since +Aut0(S) is unipotent, this implies that Bs = 0, and hence ιE(ω) is nilpotent. +Second, a derived stack has a tangent complex at every point of its MC locus, and the map q +induces a chain map between the tangent complexes: +dqw ∶ Tw[W(A)/Aut(S)] → Tq(w)[MD,g(A)/G]. +We need to show that this is a quasi-isomorphism at each point of the MC locus. We will do this +by first constructing an explicit homotopy at the special point q(0) (which is generally not in our +space), and then apply the homological perturbation lemma to obtain the quasi-isomorphism at all +points. +The homotopy +Let a ∶ LD,g,0 → LD,g be the inclusion and let b ∶ LD,g → LD,g,0 be the projection to the degree 0 +component. Both a and b are chain maps with respect to δS, but in general only a preserves the +Lie bracket. Furthermore, b ○ a = idLD,g,0. +Recall that a given element β ∈ LD,g has a Taylor expansion in the eigenvalues of LS: +β = ∑ +u +βu, +where each term satisfies LS(βu) = uβu. As we saw, the eigenvalues have the form ui + Z≥0 for +finitely many complex numbers ui. For this reason the series +β′ = ∑ +u≠0 +1 +uβu +converges to a well-defined element of LD,g. We use this to define the following degree −1 operator +h ∶ Li +D,g → Li−1 +D,g, +∑ +u +βu ↦ ιE(∑ +u≠0 +1 +uβu). +The following lemma results from straightforward computation. It has the upshot that a defines +a quasi-isomorphism of dgla’s from (LD,g,0,δS) to (LD,g,δS). +9 + +Lemma 4.2. The operator h defines a homotopy between ab and idLD,g. In other words, it satisfies +[δS,h] = idLD,g − ab. +Furthermore, it satisfies the ‘side conditions’ h ○ a = 0, b ○ h = 0 and h2 = 0. Finally, it vanishes on +U and sends Ii to U i−1. +The perturbation +We will now perturb the differential δS and show that a continues to define a quasi-isomorphism. +This is achieved by using the perturbation lemma [13]. +Let w ∈ W(A) satisfy the Maurer-Cartan equation δS(w) + 1 +2[w,w] = 0 and consider the per- +turbed differential δS,w = δS + [w,−]. +This is a differential on LD,g, and we want an induced +perturbation of the homotopy data (a,b,h,δS) of the previous section. +Lemma 4.3. The endomorphism adw ○ h of LD,g is nilpotent. +Proof. The element w can be decomposed as w = γ + α0N, where γ ∈ U 1 +0 and N ∈ U 0 +0 . By Lemma +4.1, N is nilpotent. Recall from Lemma 4.2 that h vanishes on U and its image is contained in U. +Furthermore, since U is a subalgebra of LD,g, it is preserved by adγ. As a result h ○ adγ ○ h = 0. +Hence, it suffices to show that the operator α0adN ○ h is nilpotent. +Now note that adN and multiplication by α0 commute. Since N ∈ U 0 +0 , adN also commutes with +h. This implies that +(h ○ α0adN)k = (adN)k ○ ˜hk, +where ˜h is the operator ˜h(β) = h(α0 ∧ β). +But this will vanish for large enough k since N is +nilpotent. +The upshot of this lemma is that we can now define the following perturbed maps: +h′ = +∞ +∑ +p=0 +(−hadw)ph, +δ′ = δS + +∞ +∑ +p=0 +b(−adwh)padwa, +a′ = +∞ +∑ +p=0 +(−hadw)pa, +b′ = +∞ +∑ +p=0 +b(−adwh)p. +The perturbation lemma says that δ′ defines a differential on LD,g,0, that a′ and b′ define chain +maps between (LD,g,0,δ′) and (LD,g,δS,w), and that the following equations are satisfied: +b′ ○ a′ = idLD,g,0, +[δS,w,h′] = idLD,g − a′ ○ b′, +h′ ○ a′ = 0, +b′ ○ h′ = 0, +h′ ○ h′ = 0. +The following lemma identifies the perturbations. +Lemma 4.4. The perturbations are given by +a′ = a, +b′ = b, +δ′ = δS,w. +In particular, the inclusion a ∶ (LD,g,0,δS,w) → (LD,g,δS,w) is a quasi-isomorphism. Furthermore, +h′ vanishes on U and sends I to U. +10 + +Proof. The element w ∈ L1 +D,g,0 and so adw restricts to LD,g,0 and commutes with both a and b. +As a result of this and the side conditions of Lemma 4.2, we have that hadwa = 0 and badwh = 0. +Plugging this into the definitions of the deformed maps gives +a′ = a − ∑ +p≥0 +(−hadw)p(hadwa) = a, +δ′ = δS + badwa − ∑ +p≥0 +b(−adwh)p−1adw(hadwa) = δS,w, +b′ = b − ∑ +p≥0 +(badwh)(−adwh)p = b. +The statement about h′ follows from Lemma 4.2 and the fact that each term in the definition of h′ +starts and ends with h. +The quasi-isomorphism of tangent complexes +Consider a point w ∈ MC(W(A)). It has the form w = γ + α0N, where γ ∈ U 1 +0 , N ∈ U 0 +0 , and +j∗i∗(w) = N(0) ∈ GS ∗ N0. It has corresponding point q(w) ∈ MC(MD,g(A)). In this section we +will describe the morphism of tangent complexes dqw ∶ Tw[W(A)/Aut(S)] → Tq(w)[MD,g(A)/G] +and show that it is a quasi-isomorphism. +We start by describing the tangent complexes. First, the tangent complex of [MD,g(A)/G] is +given as follows: +Tq(w)[MD,g(A)/G] = L0 +D,g → Tq(w)M(A) → L2 +D,g → L3 +D,g → ... +Note that the first term is L0 +D,g = Lie(G), and the second term is the subspace +Tq(w)M(A) = {v ∈ L1 +D,g ∣ j∗i∗(v) ∈ T(S+N(0))OA}, +where we use the fact that j∗i∗(q(w)) = S + N(0). The first map is the derivative of the gauge +action, and a computation shows that it is equal to −δS,w. The minus sign is due to the fact that we +are making the gauge group act on the left. For simplicity we will replace this by δS,w, since it does +not affect the cohomology. The second map is the derivative of the curvature dF, and a calculation +shows that it is given by δS,w. Finally, all higher maps are given by δq(w) = δS,w. Therefore, the +tangent complex is a subcomplex of (LD,g,δS,w). +The tangent complex of [W(A)/Aut(S)] has a similar descriptions. It is given by +Tw[W(A)/Aut(S)] = L0 +D,g,0 → TwW(A) → L2 +D,g,0 → L2 +D,g,0 → ... +As above, L0 +D,g,0 = Lie(Aut(S)) and the second term is the subspace +TwW(A) = {v ∈ L1 +D,g,0 ∣ j∗i∗(v) ∈ TN(0)(GS ∗ N0)}. +Again all maps are given by δS,w (the first map has a minus sign, which we remove for simplicity). +Hence, the tangent complex is a subcomplex of (LD,g,0,δS,w). +The map dqw is easily seen to coincide with a. Therefore, in order to prove that dqw is a quasi- +isomorphism, it suffices to show that the homotopy data (a,b,h′,δS,w) restricts to the tangent +complexes. +Lemma 4.5. The maps (a,b,h′,δS,w) restrict to Tq(w)[MD,g(A)/G] and Tw[W(A)/Aut(S)]. +Therefore, dqw defines a quasi-isomorphism. +11 + +Proof. Since the complexes are modified in degree 1, it suffices to restrict our attention to degrees +0,1,2. The above description of the tangent complexes and dqw immediately implies that a and δS,w +restrict. To check that h′ restricts, we only need to show that h′(L2 +D,g) is contained in Tq(w)M(A). +But this follows because, by Lemma 4.4, the image of h′ is contained in U. +For the map b, consider a point β ∈ Tq(w)M(A). This can be expanded as β = ∑u βu, where +each term satisfies LS(βu) = uβu. +By definition b(β) = β0. +Hence, we need to check that if +j∗i∗(β) ∈ T(S+N(0))OA, then j∗i∗(β0) ∈ TN(0)(GS ∗ N0). These tangent spaces have the following +descriptions +T(S+N(0))OA = Im(adS+N(0) ∶ g → g), +TN(0)(GS ∗ N0) = Im(adN(0) ∶ gS → gS). +Now using the eigenvector expansion, we have +j∗i∗(β) = ∑ +u +j∗i∗(βu) = adS+N(0)(Z), +for some Z ∈ g. One can check that each term in the summand satisfies adS(j∗i∗(βu)) = uj∗i∗(βu). +Since adS ∶ g → g is diagonalizable, we can decompose Z into eigenvectors as well: Z = ∑u Zu. +And since adS+N(0) commutes with adS, it preserves the eigenspaces. Hence, we can match up the +eigenvectors to get +j∗i∗β0 = adS+N(0)(Z0) = adN(0)(Z0), +where Z0 ∈ gS. +5 +Plane curves xp − yq +In this section we give a detailed study of the case of plane curves. Consider +f = xp − yq ∶ C2 → C, +where p and q are relatively prime positive integers satisfying p < q. This function is weighted +homogeneous of degree qp for the Euler vector field E = qx∂x + py∂y, which defines the weight +grading on coordinates ∣x∣ = q and ∣y∣ = p. The logarithmic tangent bundle is generated by the +vector fields E and V = qyq−1∂x + pxp−1∂y, which satisfy [E,V ] = (qp − p − q)V . Let w0 = qp − p − q. +The logarithmic 1-form α0 = +1 +qpdlogf pairs with V to give 0. Therefore, it can be completed to +a dual basis α0,β of the logarithmic cotangent bundle. +The form α0 is closed, and β satisfies +dβ = (p + q − qp)α0 ∧ β. +Cohomology of V +Let Ow denote the subspace of polynomial functions with weight w with respect to E. Note that +any integer w ∈ Z has a unique decomposition w = aq + bp + cqp, where a,b,c ∈ Z, 0 ≤ a < p and +0 ≤ b < q. This decomposition provides a useful way of indexing the weights because of the following +lemma. +Lemma 5.1. Let w = aq + bp + cqp, with the above restrictions on a,b,c. The dimension of Ow is +max(c,−1) + 1, and a basis is given by +xa+cpyb,xa+(c−1)pyb+q,...,xayb+cq. +The vector field V has weight w0, and hence it defines a map +V ∶ Ow → Ow+w0. +12 + +Lemma 5.2. The kernel of V is C[f]. +Proof. A calculation shows that V (f c) = 0. Conversely, let g ∈ ker(V ). Because V is homogeneous, +it suffices to consider the case where g is homogeneous of weight w > 0. The equation V (g) = 0 +implies that ∂xg = pxp−1h and ∂yg = −qyq−1h, for a common polynomial h. Therefore, +wg = E(g) = qx∂xg + py∂yg = qp(xp − yq)h, +so that g = qp +w fh. Hence h is a function of weight w − qp and it lies in the kernel of V . The result +now follows by induction on the weight. +The Jacobian ideal of f is generated by xp−1 and yq−1. Let C = C[x,y]/(xp−1,yq−1), considered +as a C-vector space. It has a natural basis of monomials xayb, where 0 ≤ a < p − 1 and 0 ≤ b < q − 1. +Using this basis, C is naturally graded by weight, and there is a weight preserving injective linear +map C → C[x,y]. Consider the graded polynomial ring C[f], where f has degree pq. Then C[x,y] +is a graded C[f]-module and there is a morphism of graded C[f]-modules +C[f] ⊗C C → C[x,y]. +The action of V on C[x,y] is C[f]-linear, so that the cokernel coker(V ) is also a C[f]-module. +Post-composing with the quotient projection, we obtain the morphism +C[f] ⊗C C → coker(V ). +Lemma 5.3. The morphism C[f] ⊗C C → coker(V ) is an isomorphism of graded C[f]-modules. +Proof. Since V is homogeneous it suffices to consider a single weight at a time: we consider the +cokernel of the map V ∶ Ow−w0 → Ow. Let w = aq + bp + cqp, where 0 ≤ a < p, 0 ≤ b < q and c ≥ 0, +so that Ow has dimension c + 1. Then w − w0 = (a + 1)q + (b + 1)p + (c − 1)qp. If a < p − 1 and +b < q − 1 then Ow−w0 has dimension c. Furthermore V is injective because w − w0 is not a multiple +of qp. Hence coker(V )w is 1-dimensional. If a = p − 1 and b < q − 1, then w − w0 = (b + 1)p + cqp, so +Ow−w0 has dimension c + 1, V is injective, and hence coker(V )w = 0. The same argument applies +to the case a < p − 1 and b = q − 1. The only remaining case is a = p − 1 and b = q − 1. In this case +w − w0 = (c + 1)qp and Ow−w0 has dimension c + 2. But now V has a 1-dimensional kernel and so +coker(V )w = 0. +The upshot is that the cokernel is non-zero precisely when a < p − 1 and b < q − 1, in which case +it is 1-dimensional. These dimensions match with the dimensions of C[f] ⊗C C. Hence it suffices +for us to prove that f cxayb is not in the image of V . We will do this by proving that the following +map +M ∶ C ⊕ Ow−w0 → Ow, +(λ,g) ↦ λf cxayb + V (g) +is represented by a matrix with positive determinant, using the bases of Lemma 5.1. Applying V +to the element xa+1+ipyb+1+jq yields +p(1 + b + jq)xa+(i+1)pyb+jq + q(1 + a + ip)xa+ipyb+(j+1)q. +The salient thing to note is that the basis elements are consecutive and the coefficients are positive. +Hence V is represented by a (c +1)×c matrix such that column i has positive entries in rows i and +i+1 and 0 for the remaining rows. Using the binomial theorem, f cxayb = ∑c +k=0(−1)kxa+(c−k)pyb+kq. +The salient point here is that the terms are non-zero with alternating signs. These give the entries +of the first column of the matrix M. Computing the determinant of M using the Laplace expansion +along the first column shows that it is positive. +13 + +The dgla (U0,δS) +Now we choose a Lie algebra g and a semisimple element S ∈ g. +This induces an eigenspace +decomposition of the Lie algebra +g = ⊕ +λ +gλ, +where gλ is the eigenspace of adS with eigenvalue λ. We will use the following convention: if λ is +not an eigenvalue of adS, then gλ = 0. Note that the decomposition is preserved by the bracket: +[gλ,gµ] ⊆ gλ+µ. +The dgla (U0,δS) has terms in degrees 0 and 1. They are given by +U 0 +0 = ⊕ +w≥0 +Ow ⊗ g−w, +U 1 +0 = ⊕ +w≥0 +Ow ⊗ gw0−wβ, +with δS given by applying V . We will sometimes drop β from the notation. +Applying Lemmas 5.2 and 5.3 we obtain the following description of the cohomology of (U0,δS). +Corollary 5.4. The cohomology of (U0,δS) is given as follows +H0(U0) = ⊕ +c≥0 +f cg−cpq, +H1(U0) ≅ ⊕ +w≥0 +(C[f] ⊗ C)w ⊗ gw0−w. +Furthermore, the graded Lie algebra H●(U0) with zero differential naturally embeds into (U0,δS) +as a quasi-isomorphic sub-dgla. +Let a ∈ g be a real semisimple element. Recall from [5] that this determines a parabolic subgroup +of G +P(a) = {g ∈ G ∣ lim +z→0zagz−a exists in G along any ray}, +where z ∈ C and za = exp(log(z)a). Decomposing S into real and imaginary parts, S = a + ib, we +can define the following subgroup of G +PS ∶= CG(e +−2πi +qp S) ∩ P(−a +qp ). +In this definition CG(e +−2πi +qp S) is the centralizer of e +−2πi +qp S in G. It is reductive but possibly discon- +nected. Let CS denote the connected component of the identity. The group PS is the parabolic +subgroup of CG(e +−2πi +qp S) (or CS) determined by the element −a +qp and it is connected. The reductive +quotient of PS is GS, the centralizer of S in G. Denote the quotient map χ ∶ PS → GS. +Lemma 5.5. The group PS embeds into Aut(S) as the subgroup integrating H0(U0). The gauge +action of PS preserves H1(U0) ⊂ U 1 +0 and is linear. Hence, we have a Lie subgroupoid +PS ⋉ H1(U0) ⊆ Aut(S) ⋉ U 1 +0 . +Proof. Let C ⋉ C2 be the Lie algebroid generated by the action of E and let C ⋉ C be the Lie +algebroid generated by the action of z∂z. The following defines a Lie algebroid morphism +f ∶ C ⋉ C2 → C ⋉ C, +(λ,x,y) ↦ (pqλ,f(x,y)), +and under this map, the logarithmic connection d + 1 +qpSdlogz pulls back to p∗S. As a result, the +pullback defines an embedding of automorphism groups from Aut(d + +1 +qpSdlogz) → Aut(S). In +[3, Proposition 3.4] it is shown that restricting an automorphism to 1 ∈ C defines an embedding +of Aut(d + +1 +qpSdlogz) into G which identifies it with PS. Furthermore, the Lie algebra of PS is +identified with ⊕c≥0 zcg−cpq, and under the pullback, this is sent isomorphically to H0(U0). Finally, +since the action of H0(U0) preserves H1(U0) and is linear, the same is true of PS. +14 + +Given the semisimple element S ∈ g, we say that it is large enough if all the positive integer +eigenvalues of adS are strictly greater than w0. +Proposition 5.6. The inclusion PS ⋉ H1(U0) ⊆ Aut(S) ⋉ U 1 +0 is a Morita equivalence if S is large +enough. +Proof. First, because of the assumption on S and the fact that Ow0 = 0 (see Lemma 5.1), the vector +space U 1 +0 has the following form +U 1 +0 = ⊕ +w>w0 +Ow ⊗ gw0−wβ. +We now proceed in several steps. +1. Claim: The subspace H1(U0) intersects every orbit of Aut(S). Given γ ∈ U 1 +0 , we need to +find an element of Aut(S) which sends γ into H1(U0). +We do this iteratively following +the usual proof of the normal form for ODEs with Fuchsian singularities. First, we expand +γ = ∑w>w0 γw, where γw ∈ Ow ⊗ gw0−wβ. Given a weight w′ > w0, let u ∈ Ow′−w0 ⊗ gw0−w′, and +consider the action of eu ∈ Aut(S) on γ: +eu ∗ γ = euγe−u − V (eu)e−uβ. +We claim that γ is modified in weights w′ and higher. Indeed, expanding we get +V (eu)e−u = V (u) − V (u)u + 1 +2V (u2) + ... +The first term has weight w′. All other terms have higher weights since w′−w0 > 0. Expanding +the term euγwe−u gives +exp(adu)γw = γw + [u,γw] + 1 +2[u,[u,γw]] + ... +The second term has weight w′ − w0 + w > w′, since w > w0. Note that the action on weight +w′ is given by γw′ ↦ γw′ − V (u)β. +By Lemma 5.3, the element γw′ can be decomposed as +γw′ = f cxayb ⊗ Xβ + V (u)β, +where f cxayb ⊗ X ∈ (C[f] ⊗ C)w′ ⊗ gw0−w′ and u ∈ Ow′−w0 ⊗ gw0−w′. +Then (eu ∗ γ)w′ = +γw′ − V (u)β ∈ H1(U0). +Now starting with the lowest weight w′ > w0, we iteratively act on γ by elements eu ∈ Aut(S) +so that the terms up to level w′ lie in H1(U0). This will terminate after finitely many steps +since U 1 +0 is finite-dimensional. The result is an element of H1(U0). +2. Claim: The inclusion functor PS ⋉ H1(U0) → Aut(S) ⋉ U 1 +0 is fully-faithful. We need to show +that given g ∈ Aut(S) and γ ∈ H1(U0), if g ∗ γ ∈ H1(U0), then g ∈ PS. Recall the Levi +decomposition Aut(S) ≅ Aut0(S) ⋊ GS, and note that GS ⊆ PS. It therefore suffices to work +under the assumption that g ∈ Aut0(S). Since such a g is unipotent, it has the form g = eu, for +u ∈ ⊕w>0 Ow ⊗ g−w. Expanding in weights, u = ∑w≥w1 uw, where w1 > 0 is the lowest weight. +From the above expressions for eu ∗ γ, we see that the lowest weight for which γ is modified +is w0 + w1. The corresponding term is given by +(eu ∗ γ)w0+w1 = γw0+w1 + V (uw1). +Since eu ∗ γ ∈ H1(U0), we must have V (uw1) = 0, implying that uw1 ∈ H0(U0) and euw1 ∈ PS. +Let ˜γ = euw1 ∗γ ∈ H1(U0), so that g ∗γ = (eue−uw1 )∗ ˜γ. Using the Baker-Campbell-Hausdorff +formula and the fact that w1 > 0, we see that eue−uw1 = ev, where the lowest weight of v is +strictly greater than w1. Hence the result follows by induction on w1. +15 + +3. Let γ ∈ H1(U0). Claim: The inclusion (H●(U0),adγ) → (U ● +0,δS +adγ) is a quasi-isomorphism. +This follows by the homological perturbation lemma [13]. +Let a ∶ H●(U0) → U ● +0 be the +inclusion. By Corollary 5.4, this is a quasi-isomorphism with respect to the 0 differential on +the domain and δS on the codomain. We have the decomposition U 1 +0 = H1(U0) ⊕ Im(δS). +Let C be a complement to H0(U0), so that U 0 +0 = H0(U0) ⊕ C. It is possible to choose this +complement compatible with the weight decomposition. Using the decomposition we define +the projection b ∶ U ● +0 → H●(U0). The restriction δS∣C ∶ C → Im(δS) is an isomorphism, and +the inverse defines a map h ∶ U 1 +0 → U 0 +0 which has weight −w0. These maps satisfy b ○ a = id, +id − a ○ b = [δS,h], as well as the side conditions h ○ a = 0, b ○ h = 0 and h ○ h = 0. +Now consider the map adγ ∶ U 0 +0 → U 1 +0 which will serve as a perturbation. Note that it restricts +to a map H0(U0) → H1(U0). Expanding in the weights, γ = ∑w>w0 γw. Since the weight of h +is −w0, it follows that adγ ○h is an endomorphism of U 1 +0 which raises the weight of an element +by at least 1. It follows that adγ ○h is nilpotent, allowing us to apply the perturbation lemma. +Using the formulas appearing above Lemma 4.4, we see that a remains unperturbed, δS is +deformed to δS + adγ and the zero differential is deformed to adγ. +The moduli stack [W(A)/Aut(S)] +Now we choose an element A = S + N0 ∈ g, which we write using the Jordan decomposition. This +determines the derived stack [W(A)/Aut(S)]. The base of the derived manifold is +W(A) = {Cβ + Nα0 ∈ U 1 +0 ⊕ U 0 +0 α0 ∣ N(0) ∈ GS ∗ N0}. +The bundle of curved dgla’s is the trivial bundle W(A) × U 1 +0 α0 with trivial dgla structure. The +curvature section is given by F(Cβ + Nα0) = (V (N) + [C,N])β ∧ α0. Applying Lemma 5.5 and +Proposition 5.6 we can construct a smaller model for this derived stack. +By Lemma 5.5, the vector space H1(U0) is a linear representation of PS. +The Lie algebra +Lie(PS) is likewise a representation, and the subspace dχ−1(GS ∗ N0) is preserved by this action +(recall that χ ∶ PS → GS is the projection to the reductive quotient). Let +Q(A) = H1(U0) × dχ−1(GS ∗ N0), +equipped with the action of PS. The infinitesimal action of PS on H1(U0) defines a PS-equivariant +map +FS ∶ Q(A) → H1(U0), +(C,N) ↦ [C,N]. +Viewing this as a section of the bundle Q(A) × H1(U0) we get a derived manifold Q(A), which +represents the derived vanishing locus of FS. This defines the derived stack +[Q(A)/PS]. +By Corollary 5.4 and Lemma 5.5 this maps into [W(A)/Aut(S)]. +Theorem 5.7. There is a map of derived stacks +i ∶ [Q(A)/PS] → [W(A)/Aut(S)]. +For points (0,N) ∈ Q(A) the derivative di(0,N) is a quasi-isomorphism of tangent complexes. Fur- +thermore, if S is large enough, then i is an equivalence. +16 + +Proof. To begin, assume that S is large enough, so that PS ⋉ H1(U0) ⊆ Aut(S) ⋉ U 1 +0 is a Morita +equivalence by Proposition 5.6. Now let Cβ + Nα0 ∈ MC(W(A)). Claim: If C ∈ H1(U0), then +(C,N) ∈ MC(Q(A)). Indeed, observe that F(Cβ + Nα0) = (δS(N) + adCβ(N)) ∧ α0, and that +(H●(U0),adCβ) → (U ● +0,δS + adCβ) +(5.1) +is a quasi-isomorphism. It follows that N ∈ H0(U0), and therefore that the claim is verified. It is +straightforward to deduce from this claim that the induced morphism +PS ⋉ MC(Q(A)) → Aut(A) ⋉ MC(W(A)) +is an equivalence of categories. Now given a point (C,N) ∈ MC(Q(A)), we need to show that +the morphism of tangent complexes di(C,N) ∶ T(C,N)[Q(A)/PS] → T(C,N)[W(A)/Aut(S)] is a +quasi-isomorphism. The differentials on these tangent complexes have the form d + adNα0. By an +argument involving the perturbation lemma, it suffices to prove that di(C,N) is a quasi-isomorphism +with respect to the differentials d. But for these differentials, di(C,N) is a direct sum of Equation +5.1 and a shift of a subspace. Therefore, it is a quasi-isomorphism. Note that by Corollary 5.4, +Equation 5.1 is quasi-isomorphism when C = 0 even if S is not large enough. +Remark 5.8. Connections of the form (0,N) ∈ Q(A) are pullbacks by f of connections on C with +a logarithmic pole at the origin. +The condition in Theorem 5.7 that S is large enough is necessary. In the following Example we +see that the moduli space can have extra components when the condition is not satisfied. +Example 5.9. Let f = x2 − y5, let g = gl3, and let A = S be a diagonal matrix with entries 0,1 and +11. Note that S is not large enough since 1 is a positive eigenvalue of adS which is smaller than +w0 = 3. The subalgebra gS consists of the diagonal matrices and +U 0 +0 = gS ⊕ spanC(x2E23,y5E23,xy3E13), +U 1 +0 = spanC(yE21,y2E12,xy4E23,x2y2E13,y7E13), +where Eij are the elementary matrices. On the other hand, the δS cohomology is given by +H0(U0) = gS ⊕ spanC(fE23), +H1(U0) = spanC(yE21,y2E12,y2fE13). +An arbitrary element of W(A) has the form Cβ + Nα0, where +C = C21yE21 + C12y2E12 + C23xy4E23 + (C(1) +13 x2y2 + C(2) +13 y7)E13, +N = N13xy3E13 + (N (1) +23 f + N (2) +23 (x2 + y5))E23. +The Maurer-Cartan equation consists of the following coupled system of equations: +20N (2) +23 = −C21N13 +5N13 = −C12(N (2) +23 − N (1) +23 ) +6N13 = −C12(N (2) +23 + N (1) +23 ). +Adding the last two equations and substituting the result into the first yields (110−C21C12)N (2) +23 = 0. +Assume first that C21C12 ≠ 110. Then the equations simplify to N (2) +23 = N13 = C12N (1) +23 = 0, and +N ∈ H0(U0). The result is a 5-dimensional variety Ma with 2 irreducible components. Next, assume +that C21C12 = 110. Then we can solve the equations to obtain N13 = −2C12N (1) +23 and N (2) +23 = 11N (1) +23 . +Hence, the result is a smooth irreducible 5-dimensional variety Mb. +17 + +An element of Aut(S) has the form +g = +⎛ +⎜ +⎝ +u +0 +λxy3 +0 +v +ax2 + by5 +0 +0 +w +⎞ +⎟ +⎠ +, +and it acts on C = (C21,C12,C23,C(1) +13 ,C(2) +13 ) and N = (N13,N (1) +23 ,N (2) +23 ) in the following way: +g ∗ C = (v +uC21, u +v C12, v +wC23 − λv +uwC21 − 10(a + b) +w +, u +wC(1) +13 − au +vwC12 − 6λ +w , u +wC(2) +13 − bu +vwC12 − 5λ +w ) +g ∗ N = ( u +wN13, v +wN (1) +23 , v +wN (2) +23 ). +There are two things we can immediately note. First, by looking at the action on N, we can see that +there are orbits in Mb which do not intersect Q(A). Hence, [Mb/Aut(S)] is an extra component of +the moduli space. Second, the product C21C12 is invariant under the action and if we assume that +C21C12 ≠ 110, then it is always possible to send C into H1(U0) (i.e. C23 = 0 and C(1) +13 + C(2) +13 = 0). +The subgroup of Aut(S) which stabilizes this locus is PS, and hence [Ma/Aut(S)] is contained in +[MC(Q(A))/PS]. +∎ +Relation to the Grothendieck-Springer resolution +Recall that PS ⊆ CS is a parabolic subgroup with Levi factor GS. Denote their Lie algebras pS, cS +and gS, respectively. Let P denote the partial flag variety, defined as the set of parabolic subalgebras +of cS which are conjugate to pS. It is isomorphic to CS/PS. Consider the incidence variety +˜c = {(x,p) ∈ cS × P ∣ x ∈ p}, +which is isomorphic to (CS × pS)/PS via the map which sends (g,x) ∈ CS × pS to the pair +(Adg(x),Adg(pS)) ∈ ˜c. When PS is a Borel subgroup, ˜c is the Grothendieck-Springer resolution +(see e.g. [11] for details). The element N0 ∈ gS defines an adjoint orbit O in the Levi quotient of +every parabolic p ∈ P [5, Lemma 1]. This lets us define the following subspace of ˜c: +˜cN0 = {(x,p) ∈ ˜c ∣ dχ(x) ∈ O}, +where dχ denotes the projection to the Levi quotient. The space Q(A) is a PS-equivariant vector +bundle over dχ−1(GS ∗ N0). Hence π ∶ EA = (CS × Q(A))/PS → ˜cN0 is a CS-equivariant vector +bundle and the map FS gives rise to an equivariant section σS of π∗(EA) → EA. In this way, we +obtain a derived stack [EA/CS] which is equivalent to [Q(A)/PS]. +Example 5.10. Let G = GLn and let A = S = pqD, where D is a diagonal matrix with distinct +integer eigenvalues. In this case, S is large enough, CS = G and PS = B is a Borel subgroup. Hence +P is the flag variety Fln. Since N0 = 0, ˜g0 is the Springer resolution, which is isomorphic to T ∗Fln. +We use Corollary 5.4 to compute the cohomology space H1(U0). The weights w showing up in the +decomposition have the form w = (p − 1)q + (q − 1)p + cqp. But (C[f] ⊗ C)w = 0 in this case, so that +H1(U0) = 0. Hence, the moduli space [W(A)/Aut(S)] is equivalent to the quotient stack +[T ∗Fln/GLn]. +Note that the connections corresponding to the points of T ∗Fln are all pulled-back from logarithmic +connections on C, where a similar classification is given by [5, Theorem A]. +∎ +18 + +Example 5.11. Let G = GLn and A = S = qp +r D, where D is a diagonal integer matrix with distinct +eigenvalues and r is a positive integer. +In this case, the eigenvalues of e +−2πi +qp S are rth roots of +unity, implying that CS = ∏r−1 +i=0 GLmi, with the factors indexed by the roots of unity. Since the +eigenvalues of S are assumed to be distinct, PS = ∏r−1 +i=0 Bmi is a product of Borels. Therefore, +P ≅ ∏r−1 +i=0 Flmi is a product of flag varieties and ˜c0 ≅ ∏r−1 +i=0 T ∗Flmi is the product of their cotangent +bundles. Writing the eigenvalues of D as rk+u, with 0 ≤ u < r, the eigenvalues of adS have the form +qp(k1 − k2) + qp +r (u1 − u2). We can ensure that S is large enough by restricting the possible values +of ki. For example, this is guaranteed if k1 − k2 ≠ 0,±1 for all pairs of eigenvalues. By Corollary +5.4, the weights w showing up in the decomposition of H1(U0) have the form +w = qp(1 + k2 − k1) + qp +r (u2 − u1) − p − q. +As in Example 5.10, we must have u2 ≠ u1 in order to get a non-zero contribution. +Now we specialise to GL4, with r = pq and S = D a diagonal matrix with entries +pqk1, +pqk2, +pqk3 + p + q, +pqk4 + p + q, +such that k1 ≪ k2 ≪ k3 ≪ k4. Then CS = GL2 × GL2 and P ≅ P1 × P1. Let T1 and T2 be the +tautological rank 2 vector bundles over P1 × P1 (corresponding respectively to the first and sec- +ond factors). They are both trivial, but are equipped with tautological line subbundle bundles +Li with respective degrees (−1,0) and (0,−1). The cotangent bundle ˜c0 can be identified with +Hom(T1/L1,L1) ⊕ Hom(T2/L2,L2), or alternatively, the nilpotent filtration-preserving endomor- +phisms of T1 and T2. +The weights in the decomposition of H1(U0) have the form qp(1 + kj − ki), for j = 3,4 and +i = 1,2. Hence, H1(U0) can be identified with the subspace of gl4 consisting of the upper right 2×2 +block. Therefore, +EA = Hom(T1/L1,L1) ⊕ Hom(T2/L2,L2) ⊕ Hom(T2,T1), +with section σS(a,b,c) = c ○ b − a ○ c. +∎ +Tangent Lie bialgebra and shifted Poisson geometry +Let A = S ∈ g be a semisimple element and consider the moduli space [W(S)/Aut(S)]. There +is a distinguished point 0 ∈ MC(W(S)) corresponding to the connection 1-form α0S. +In this +section we focus our attention on a formal neighbourhood of this point in the moduli space and +sketch the construction of a −2-shifted Poisson structure on this neighbourhood. According to +the fundamental principal of derived deformation theory (see e.g. +[17, 20, 27, 18]) this formal +neighbourhood is encoded by the shifted tangent complex T0[W(S)/Aut(S)][−1], equipped with +its structure as a differential graded Lie algebra. By Theorem 5.7 this dgla is quasi-isomorphic to +T0[Q(A)/PS][−1] = H0(U0) → H1(U0) ⊕ H0(U0)+ → H1(U0), +where the differential is 0 and H0(U0)+ = ⊕c>0 f cg−cpq. +By [26, Proposition 1.5], a −3-shifted +Lie bialgebra structure on T0[Q(A)/PS][−1] gives rise to a −2-shifted Poisson structure on the +formal neighbourhood. Hence, our strategy is to construct a Lie bialgebra structure on the tangent +complex by embedding it into H●(U0) ⋉ H●(U0)[−1], and then to realize this larger Lie algebra as +a Lagrangian inside a −3-shifted Manin triple. +We construct the Manin triple in stages, starting with the following input data: +• Choose an invariant inner product k on the reductive Lie algebra g. This induces a perfect +pairing between the eigenspaces gλ and g−λ. +19 + +• The C-vector space C = C[x,y]/(xp−1,yq−1) is canonically isomorphic to the degree 1 coho- +mology of f −1(1), which is a curve of genus g = 1 +2(p − 1)(q − 1) with a single puncture. The +isomorphism is given as follows: +C → H1(f −1(1)), +xayb ↦ (xaybβ)∣f −1(1). +By pulling back the intersection pairing on the curve, we obtain a symplectic form I on C. +Up to a scaling constant, it is given by the following formula +I(xayb,xa′yb′) = δ(a + a′ + 2 − p)δ(b + b′ + 2 − q) +aq + bp − w0 +, +where δ is the delta function which evaluates to 1 at 0 and 0 otherwise. We can extend this +to a C[f]-linear pairing on C[f] ⊗C C. +Now consider the Lie algebra c = ⊕c∈Z f cg−cpq. This has two distinguished subalgebras: first +b = H0(U0), where c ≥ 0, and second b−, where c ≤ 0. Note that c can be viewed as a subalgebra of +g and so inherits the pairing k. This defines a perfect pairing between b and b−. +Next, define the following vector space +K = ⊕ +c∈Z +⊕ +0≤a≤p−2 +0≤b≤q−2 +f cxaybgw0−cpq−aq−bp. +This decomposes as K = n+⊕h⊕n− according to whether c is positive, zero, or negative, respectively. +Note that H1(U0) = n+ ⊕ h. Viewing both c and K as subspaces of the Lie algebra C[x,y] ⊗ g, we +can show that K is a representation of c. By combining the symplectic form I on C with the Lie +bracket on g, we define a symmetric c-equivariant map ω ∶ S2(K) → c as follows +ω(f c1xa1yb1X1,f c2xa2yb2X2) = f c1+c2I(xa1yb1,xa2yb2)[X1,X2]. +Post-composing this with the natural projection to b− defines a map µ ∶ S2(K) → b−. This is b- +equivariant, where b− is a b-representation by using the inner product to identify it with b∗. With +respect to the b action on K, both n+ and n+⊕h are sub-representations. Hence the quotient h = n+⊕h +n+ +is naturally a b-representation, and µ descends to define a b-equivariant map ν ∶ S2(h) → b−. We +now define a graded Lie algebra +L = b ⊕ (K ⊕ h)[−1] ⊕ b−[−2]. +The bracket is constructed from the bracket on b, the b-action on the other summands, the sym- +metric pairing µ on K and the symmetric pairing −ν on h. We set the bracket between K and h to +be zero. +Lemma 5.12. The vector space L with the bracket described above defines a graded Lie algebra. +Let p+ = b ⊕ (n+ ⊕ h)[−1] and let p− = (n− ⊕ h)[−1] ⊕ b−[−2]. There are morphisms +p+ → L, +(b,n,h) ↦ (b,n + h,h,0) +p− → L, +(n,h,u) ↦ (0,n + h,−h,u). +These embed p± as complementary subalgebras of L. Furthermore, p+ is isomorphic to H●(U0). +Next we construct an inner product on L. By combining the inner product k on g with the +symplectic pairing I on C, we define a skew symmetric pairing Ω ∶ ∧2K → C as follows +Ω(f c1xa1yb1X1,f c2xa2yb2X2) = f c1+c2I(xa1yb1,xa2yb2)k(X1,X2). +20 + +This pairing is non-degenerate, restricts to a non-degenerate pairing on h and defines a perfect +pairing between n±. We can now define a non-degenerate graded symmetric bilinear pairing +B ∶ S2(L) → C[−2]. +More precisely, given y = (b1,k1,h1,u1),z = (b2,k2,h2,u2) ∈ L, we set +B(y,z) = k(b1,u2) + k(u1,b2) + Ω(k1,k2) − Ω(h1,h2). +Lemma 5.13. The pairing B is a non-degenerate invariant inner product on L. Furthermore, p± +are complementary Lagrangian subalgebras. In other words, (L,p+,p−) is a −2-shifted Manin triple. +Now let C[ϵ] be the cdga generated by a degree +1 variable ϵ and let tr ∶ C[ϵ] → C[−1] be the +trace map defined by sending the element a + bϵ to b. Tensoring with C[ϵ] defines a new triple of +graded Lie algebras (C[ϵ]⊗L,C[ϵ]⊗p+,C[ϵ]⊗p−) and the bilinear form B extends in the following +way +B ∶ S2(C[ϵ] ⊗ L) → C[−3], +(fy,gz) ↦ (−1)∣y∣∣g∣tr(fg)B(y,z). +In this way, we obtain a −3-shifted Manin triple. Note that C[ϵ]⊗p+ ≅ H●(U0)⋉H●(U0)[−1]. The +tangent complex T0[Q(A)/PS][−1] is isomorphic to the subalgebra b ⊕ (n+ ⊕ h) ⊕ b>0ϵ ⊕ (n+ ⊕ h)ϵ, +where b>0 consists of the elements f cX with c > 0. Let M be the direct sum of this subalgebra with +C[ϵ] ⊗ p−. Then M is a coisotropic subalgebra of L and M ⊥ = g0 ⊂ b− ⊂ M is an isotropic ideal. +It follows that (M/M −1,T0[Q(A)/PS][−1],C[ϵ] ⊗ p−/M ⊥) is a −3-shifted Manin triple. Therefore, +by [26, Lemma 1.3], the tangent complex obtains a −3-shifted Lie bialgebra structure. +Theorem 5.14. The tangent complex T0[Q(A)/PS][−1] admits the structure of a −3-shifted Lie +bialgebra. Therefore, the formal neighbourhood of 0 ∈ [W(S)/Aut(S)] admits a −2-shifted Poisson +structure. +References +[1] M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfaces. Philos. Trans. +Roy. Soc. London Ser. A, 308(1505):523–615, 1983. +[2] Kai Behrend, Ionut Ciocan-Fontanine, Junho Hwang, and Michael Rose. The derived moduli +space of stable sheaves. 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Soc., 193(902):x+224, 2008. +22 + diff --git a/x9AzT4oBgHgl3EQfCfrw/content/tmp_files/load_file.txt b/x9AzT4oBgHgl3EQfCfrw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbfe54f7bc75663a5027ef98ee41923a28a8c174 --- /dev/null +++ b/x9AzT4oBgHgl3EQfCfrw/content/tmp_files/load_file.txt @@ -0,0 +1,896 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf,len=895 +page_content='The derived moduli stack of logarithmic flat connections Francis Bischoff∗ Abstract We give an explicit finite-dimensional model for the derived moduli stack of flat connections on Ck with logarithmic singularities along a weighted homogeneous Saito free divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We investigate in detail the case of plane curves of the form xp = yq and relate the moduli spaces to the Grothendieck- Springer resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We also discuss the shifted Poisson geometry of these moduli spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Namely, we conjecture that the map restricting a logarithmic connection to the complement of the divisor admits a shifted coisotropic structure and we construct a shifted Poisson structure on the formal neighbourhood of a canonical connection in the case of plane curves xp = yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Homogeneous free divisors and logarithmic flat connections 4 3 Finite dimensional model 5 4 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5 8 5 Plane curves xp − yq 12 1 Introduction Let D ⊂ Ck be a hypersurface cut out by a reduced holomorphic function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In [30] Saito considers the subsheaf, usually denoted TCk(−logD), of holomorphic vector fields on Ck which preserve the ideal generated by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In general, it is coherent and closed under the Lie bracket, but may fail to be locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In fact, Saito provides a very explicit criterion for determining whether the sheaf is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' When it is, D is said to be a free divisor and TCk(−logD), known as the logarithmic tangent bundle, defines a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Examples of free divisors include smooth hypersurfaces, plane curves and simple normal crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In general, D may be highly singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let G be a connected complex reductive group with Lie algebra g and assume that D is a free divisor which is homogeneous under a given C∗-action on Ck with the property that all its weights are strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this paper, we are interested in studying the moduli space of TCk(−logD)- representations on principal G-bundles, also known as logarithmic flat connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' There is a standard way of defining this moduli space as the Maurer-Cartan locus of an infinite-dimensional differential graded Lie algebra (dgla) LD,g which is associated to D and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let Ω1 Ck(logD) denote the logarithmic cotangent bundle, which is the dual to TCk(−logD), and let Ω● Ck(logD) = ∧●Ω1 Ck(logD) be the exterior algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This defines a commutative differential graded algebra when equipped with the Lie algebroid differential d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then LD,g = Ω● Ck(logD) ⊗ g inherits the structure of a dgla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The Maurer-Cartan locus of this dgla is defined to be the following set MC(LD,g) = {ω ∈ L1 D,g ∣ dω + 1 2[ω,ω] = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' ∗Exeter College and Mathematical Institute, University of Oxford;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' francis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='bischoff@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='uk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='00962v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='AG] 3 Jan 2023 Here, ω ∈ Ω1 Ck(logD) ⊗ g is a Lie algebra valued 1-form, and it defines the following connection ∇ = d + ω, which has a logarithmic singularity along D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It’s curvature is given by the following expression F(ω) = dω + 1 2[ω,ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The degree 0 component of the dgla is L0 D,g = Map(Ck,g), which is the Lie algebra of the infinite dimensional gauge group G = Map(Ck,G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This group acts on the Maurer-Cartan locus, giving the correct equivalence between flat connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' As a result, the moduli space of flat logarithmic connections is defined to be the stack quotient [MC(LD,g)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Although this construction involves infinite dimensional spaces, in [4] we provide a purely finite dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' More precisely, we show that the category of logarithmic flat connections with fixed residue data is equivalent to the stack quotient of an affine algebraic variety by the action of an algebraic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The purpose of the present paper is to provide a derived enhancement of the moduli stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' There are several different approaches to derived geometry in the literature, such as [16, 12, 19, 31, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this paper, we have opted to go with the notion of bundles of curved dgla’s, which requires relatively little technology and is sufficient for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let us recall the definition from [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' A bundle of curved differential graded Lie algebras over a variety M consists of a graded vector bundle L● starting in degree 2, which is equipped with the following data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' a section F ∈ Γ(M,L2), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' a degree 1 bundle map δ ∶ L● → L●[1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' a smoothly varying graded Lie bracket [−,−] on the fibres of L●, satisfying the following conditions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' the Bianchi identity δF = 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' δ2 = [F,−], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' δ is a graded derivation of the bracket [−,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' If [M/G] is a stack, defined by the data of a Lie groupoid G over M, then we define a bundle of curved dgla’s over the stack to be such a bundle over M, which is equipped with an equivariant action of G preserving the data (F,δ,[−,−]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We will use the respective terminology of derived manifolds and derived stacks to refer to this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' There is a standard way of constructing a derived manifold from the data of a dgla, and we can apply it to the case of LD,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Namely, we take the base to be M = L1 D,g and take the bundle L● to be trivial with fibre given by the truncation L●≥2 D,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The section is given by the curvature F, and the bracket is simply the constant one inherited from LD,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The bundle map δ varies over M, and above a point ω ∈ L1 D,g, it is given by the twisted differential δω = d + [ω,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let us denote the resulting derived manifold MD,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It can be further upgraded to a derived stack by noting that the gauge group G lifts to an action on L●≥2 D,g via the adjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' A derived manifold (or stack) has an underlying classical truncation π0(M), defined as the vanishing locus of the section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In the example under consideration, the classical truncation is given by the Maurer-Cartan locus, and hence π0([MD,g/G]) = [MC(LD,g)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 2 For this reason, we say that [MD,g/G] is the derived moduli stack of flat logarithmic G-connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The main result of this paper is a finite-dimensional model of this derived stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Here is a brief description of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Given an element A ∈ g, we consider the infinite dimensional derived moduli stack [MD,g(A)/G] of G-connections whose ‘residue’ is conjugate to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Details of this are given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let A = S + N0 be the Jordan decomposition, where S is semisimple and N0 is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In Section 3 we construct a finite dimensional dgla (U0,δS) associated to S, with corresponding derived stack [US/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This is interpreted as a certain sub-moduli space of flat connections on the fibre f −1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then, associated to the element A, we construct a derived substack [W(A)/Aut(S)] of the shifted tangent bundle T[−1][US/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The main result is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5, which states that there is an equivalence of derived stacks q ∶ [W(A)/Aut(S)] → [MD,g(A)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By this we mean that q induces an equivalence between the groupoids of solutions to the MC equation, and given any solution w, the derivative dqw is a quasi-isomorphism of tangent complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In Section 5 we turn to the case of a plane curve defined by the function f = xp − yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This is the simplest case above k = 1, and already it exhibits interesting behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We construct an explicit derived stack [Q(A)/PS] from the data of a parabolic subgroup PS of the centralizer of exp( 2πi pq S) and a representation H1(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='7 we show that, given a condition on the eigenvalues of adS, the derived stack [Q(A)/PS] is equivalent to [W(A)/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For general S, the moduli space can have extra components and we illustrate this in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The derived stack [Q(A)/PS] can be interpreted in terms of spaces showing up in geometric representation theory, such as the Grothendieck-Springer resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='7 can be viewed as a higher dimensional generalization of Boalch’s description in [5] of the moduli space of logarithmic connections on the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Speculations about Poisson geometry Going back to the work of Atiyah-Bott [1] and Goldman [15], we know that the moduli space of flat connections on a closed Riemann surface admits a symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' If the surface is punctured, then the moduli space admits a Poisson structure, whose symplectic leaves are obtained by fixing boundary conditions at the punctures [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This picture has since been generalized in several directions, including to the case of flat connections with singularities [6, 8, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' More recently, the moduli space of local systems on higher dimensional manifolds has been studied using tools from derived algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For a compact oriented manifold M of dimension d, the moduli space of local systems LocG(M) is a derived stack equipped with a shifted symplectic structure of degree 2−d [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' If M has a boundary ∂M = N, then LocG(N) has a 3−d-shifted symplectic structure, and the restriction map r ∶ LocG(M) → LocG(N) has a Lagrangian structure [9], inducing on LocG(M) a 2 − d-shifted Poisson structure [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We wish to generalize this picture to the case of logarithmic flat connections in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In the above setting of a map f ∶ Ck → C, the inverse image of the unit circle f −1(S1) is a manifold of dimension 2k−1, usually with boundary, and so LocG(f −1(S1)) has a Poisson structure of degree 3 − 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Given a logarithmic flat connection, we can restrict it to f −1(S1) and take its holonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This should define a map r ∶ [W(A)/Aut(S)] → LocG(f −1(S1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1) This map was studied by Boalch [5] in the special case of k = 1, where f −1(S1) = S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this case LocG(S1) = G/G has a 1-shifted symplectic structure, and the work of Boalch (suitably interpreted by [29]) shows that r has a Lagrangian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In higher dimensions we conjecture that the map can be equipped with a shifted coisotropic structure in the sense of [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 3 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The map r can be naturally equipped with a coisotropic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In order to avoid the analytic issues that arise in taking the holonomy, it may be preferable to replace LocG(f −1(S1)) with a moduli space of flat connections on the complement of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In recent work [23, 24], Pantev and To¨en studied the moduli spaces of local systems and flat connections on non-compact algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' They constructed shifted Poisson structures and explained how to obtain the symplectic leaves by imposing suitable boundary conditions at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2 may be viewed as providing another source of boundary conditions for the moduli spaces associated to f −1(Ck ∖ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We hope that it may also be used in conjunction with their results, for example by considering the map r in the presence of additional boundary conditions at the boundary of the fibres of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' One implication of the conjecture is that the moduli spaces [W(A)/Aut(S)] should admit 2(1 − k)-shifted Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='14 we provide evidence for the conjecture by constructing a −2-shifted Poisson structure on the formal neighbourhood of a special connection in the case of plane curves xp = yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Our construction is somewhat ad hoc, but it makes use of an invariant inner product on the Lie algebra g, as well as the intersection pairing on the cohomology of the curve f −1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We have also not checked that our shifted Poisson structure fits into the formalism developed by [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We hope to address all these issues in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' I would like to thank Elliot Cheung for pointing me to the paper [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 2 Homogeneous free divisors and logarithmic flat connec- tions Assume that the given C∗ action on Ck has strictly positive weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is generated infinitesimally by an Euler vector field E = k ∑ i=1 nizi∂zi, where ni ∈ Z>0 are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This vector field defines a weight grading on the holomorphic functions OCk (and more generally tensor fields) on Ck, such that the coordinate function zi has weight ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This grading will play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Because of our assumption, each weight space is finite-dimensional over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We also assume that the function f defining D is homogeneous of weight r: E(f) = rf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The C∗ action determines an action Lie algebroid C⋉Ck which is generated by the Euler vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Because E is a section of TCk(−logD), there is an induced Lie algebroid morphism i ∶ C ⋉ Ck → TCk(−logD), (λ,z) ↦ λEz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The logarithmic 1-form dlogf = df f is a closed section of Ω1 Ck(logD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, it determines a Lie algebroid morphism π ∶ TCk(−logD) → C, V ↦ 1 rf V (f), where C is considered as an abelian Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The composition p = π ○ i ∶ C ⋉ Ck → C is given by projection to the first factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This has a section j ∶ C → C ⋉ Ck, λ ↦ (λ,0), 4 which is also a Lie algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Altogether, we have the following diagram of Lie algebroids: C ⋉ Ck C TCk(−logD) i π p j Each Lie algebroid determines a differential graded Lie algebra, whose Maurer-Cartan locus con- sists of flat algebroid connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, each morphism of Lie algebroids determines a pullback morphism between dgla’s, and as a result, a pullback morphism between categories of representations, or more generally, derived moduli stacks of flat connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This gives rise to the following diagram of (infinite-dimensional) derived stacks: [(OCk ⊗ g)/G] [g/G] [MD,g/G] i∗ π∗ p∗ j∗ In this diagram, [g/G] is the moduli stack of g-representations of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is the stack quotient corresponding to the adjoint action of G on its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' [(OCk ⊗ g)/G] is the moduli stack of g-representations of C⋉Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In both cases the derived structure is trivial because the Lie algebroids have rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now fix an element A ∈ g, let OA ⊂ g be its adjoint orbit, and let GA ⊆ G be its centralizer subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This determines a substack [OA/G] ⊂ [g/G] which is Morita equivalent to BGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The preimage [MD,g(A)/G] ∶= (j∗i∗)−1(BGA) is the derived stack of logarithmic flat connections ω whose ‘residue’ j∗i∗(ω) lies in OA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' More precisely, the base of the derived manifold MD,g(A) is given by M(A) = {ω ∈ Ω1 Ck(logD) ⊗ g ∣ j∗i∗(ω) ∈ OA}, with the bundle of curved dgla’s restricted from MD,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The action of G preserves M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 3 Finite dimensional model Let A = S + N0 be the Jordan decomposition of A, where S is semisimple, N0 is nilpotent, and [S,N0] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this section we will construct a finite dimensional model for [MD,g(A)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The dgla LD,g We start by analysing the structure of the dgla LD,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Being constructed from the cdga Ω● Ck(logD) and the Lie algebra g, LD,g inherits their derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The basic ones are as follows: the Lie algebroid differential d, which has degree +1 and squares to 0, the interior multiplication with the Euler vector field ιE, which has degree −1 and squares to 0, the adjoint action of S, adS, which has degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 5 By taking commutator brackets we arrive at further derivations, such as LE = [ιE,d], the Lie derivative with respect to E, which is a derivation of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We can also wedge any derivation by a differential form to obtain a new derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let α0 = 1 rdlogf, which is a closed logarithmic 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then α0adS is a degree +1 derivation which squares to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Among the 5 derivations just described, almost all of them commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The only two non-vanishing commutator brackets are the following: [ιE,d] = LE, [ιE,α0adS] = adS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The second bracket follows as a consequence of the identity ιE(α0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We are primarily interested in studying the dgla structure arising from δS = d + α0adS, which is a degree +1 derivation that squares to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We are also interested in the following degree 0 derivation LS ∶= [ιE,δS] = LE + adS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This operator is diagonalisable in the sense that any element β ∈ LD,g has a Taylor series expansion β = ∑ u βu where each term satisfies LS(βu) = uβu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Indeed, the operator adS is diagonalizable on g with finitely many eigenvalues since S is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The eigenspaces of LE are the weight spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We noted earlier that the weight degrees of holomorphic functions are strictly positive integers, and that each weight space is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' As an operator on Ω● Ck(logD), the eigenvalues of LE may not be positive, but they are integers which are bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, the eigenvalues of LS have the form ui + Z≥0, for finitely many complex numbers ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let LD,g,u denote the u-eigenspace, and note that it is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Because LS is a derivation, the Lie bracket respects this decomposition: [−,−] ∶ LD,g,u × LD,g,v → LD,g,u+v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In particular, LD,g,0 is a finite-dimensional Lie subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The derivations δS and ιE commute with LS, and hence preserve its eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In particular, they restrict to LD,g,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now introduce the degree 0 derivation P = α0ιE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This derivation satisfies P 2 = P, and hence induces a decomposition LD,g = ker(P) ⊕ im(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let U = ker(P) and let I = im(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' With respect to the bracket, U is a subalgebra and I is an abelian ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The derivation ιE vanishes on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For every degree i it defines an isomorphism ιE ∶ Ii → U i−1, with inverse given by multiplication by α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, as a graded Lie algebra, LD,g is isomorphic to U ⋉ U[−1], where U acts on U[−1] via the adjoint action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is clear that ker(ιE) ⊆ U = ker(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For the opposite inclusion, suppose that P(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then ιE(x) is in the kernel of multiplication by α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since α0 is a non-vanishing algebroid 1-form, ιE(x) must be of the form α0 ∧ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' But then 0 = ι2 E(x) = ιE(α0 ∧ y) = y − α0 ∧ ιE(y), which implies that ιE(x) = 0, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The image of ιE is contained in U since ι2 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' To see surjectivity, we can explicitely construct the inverse as mulitplication by α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Given x ∈ U, check 6 that α0 ∧ x = P(α0 ∧ x) ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence α0∧ ∶ U i−1 → Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then for x ∈ U, we have ιE(α0 ∧ x) = x, and for P(y) ∈ I we have α0 ∧ ιEP(y) = P 2(y) = P(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now define the isomorphism Ξ ∶ LD,g → U ⋉ U[−1] by the following formula in degree i: U i ⊕ Ii → U i ⊕ U i−1, (x,y) ↦ (x,(−1)iιE(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This preserves Lie brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The commutator [P,LS] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, the two operators can be simultaneously diagonalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In particular, we have the decomposition LD,g,0 = U0 ⊕ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The results of the previous lemma remain true for this subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Next, we have [P,δS] = α0LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' If we re-write this as the following identity PδS = α0LS + δSP then we can deduce that δS preserves I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Indeed, applying this identity to an element of the form x = P(y), we obtain PδS(x) = α0LSP(y) + δSP 2(y) = α0PLS(y) + δSP(y) = δS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' On the other hand, the differential δS does not preserve U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' But by applying the identity to an element x ∈ U, we compute that the ‘off-diagonal’ term is given by PδS(x) = α0LS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This term vanishes when we restrict to the subalgebra LD,g,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, we obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The subalgebra U0 is preserved by δS, and there is an isomorphism of dgla’s (LD,g,0,δS) ≅ (U0,δS) ⋉ (U0,δS)[−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' On the subspace LD,g,0 we have [ιE,δS] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This implies that the morphism Ξ from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1 is a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The gauge group Aut(S) Viewing S ∈ g as a representation of C, we can pull it back to obtain a representation p∗S of C ⋉ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let Aut(S) be the subgroup of the gauge group G consisting of gauge transformations which preserve p∗S: Aut(S) = {g ∈ G ∣ g ∗ p∗S = p∗S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is a finite-dimensional algebraic group whose Lie algebra is L0 D,g,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We recall the description of its Levi decomposition which was given in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The automorphism group of j∗p∗S = S is GS, the centralizer subgroup of S in G, which is reductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The pullback functor j∗ defines a homomorphism j∗ ∶ Aut(S) → GS, g ↦ g(0), and the pullback functor p∗ defines a splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The kernel of j∗, denoted Aut0(S), is the unipotent radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence the isomorphism Aut(S) ≅ Aut0(S) ⋊ GS provides the Levi decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Define the following gauge action of Aut(S) on L1 D,g,0: g ∗ x = gxg−1 − δS(g)g−1, where δS(g)g−1 = dgg−1 + α0(S − gSg−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The gauge action of Aut(S) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In terms of the decomposition L1 D,g,0 = U 1 0 ⊕ I1 0 it is given by g ∗ (x,y) = (gxg−1 − δS(g)g−1,gyg−1), where x ∈ U 1 0 and y ∈ I1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, Aut(S) acts on L●≥2 D,g,0 by conjugation, preserving the decomposition U0 ⊕ I0 and the Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' A computation shows that LS(g ∗ x) = g(LSx)g−1 for x ∈ L1 D,g, showing that L1 D,g,0 is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Similarly, LS(gxg−1) = g(LSx)g−1 for x ∈ Lj D,g, showing that the conjugation action preserves L●≥2 D,g,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Next, for x ∈ LD,g,0 we have P(gxg−1) = gP(x)g−1, implying that the conjugation also preserves U0 and I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Finally, P(δS(g)g−1) = α0(LE(g)g−1 + S − gSg−1), which vanishes for g ∈ Aut(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence δS(g)g−1 ∈ U 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The finite-dimensional derived stack Given the finite dimensional dgla LD,g,0 we obtain a derived manifold WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The base manifold is the vector space WS = L1 D,g,0, the bundle of curved dgla’s is the trivial bundle WS × L●≥2 D,g,0, the curvature section is given by the standard formula FS(w) = δS(w) + 1 2[w,w], and the twisted differential δ is given by δS,w = δS + [w,−], for w ∈ WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='3 gives an equivariant action of Aut(S) on WS × L●≥2 D,g,0, preserving the bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is also straightforward to check that this action preserves FS and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, we obtain a derived stack [WS/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' U0 is a sub-dgla of LD,g,0, which is preserved by the action of Aut(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, it gives rise to a derived substack [US/Aut(S)] of [WS/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, since I0 is an ideal of LD,g,0, we also get a projection morphism [WS/Aut(S)] → [US/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The derived stack [WS/Aut(S)] is isomorphic to the shifted tangent bundle T[−1][US/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We are actually interested in a substack of [WS/Aut(S)] which is determined by the element A = S + N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Recall that the image of Aut(S) under j∗ is GS, the centralizer of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This implies that for any element ω ∈ WS, the image j∗i∗(ω) ∈ gS = Lie(GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We will require that this element be contained in GS ∗ N0, the adjoint orbit of N0 in gS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Namely, define W(A) = {ω ∈ WS ∣ j∗i∗(ω) ∈ GS ∗ N0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let W(A) be the derived manifold obtained by pulling back the bundle of curved dgla’s from WS to W(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The action of Aut(S) restricts to an action on this sub-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, we obtain a derived stack [W(A)/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' [W(A)/Aut(S)] is equivalent to [MD,g(A)/G], the derived stack of logarithmic flat connections whose residue lies in the adjoint orbit OA of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 4 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5 In this section we will give the proof of the equivalence between [W(A)/Aut(S)] and [MD,g(A)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' There is a natural morphism q ∶ [W(A)/Aut(S)] → [MD,g(A)/G], which we describe as follows: 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The map on the base manifolds is given by the following formula q ∶ W(A) → M(A), ω ↦ α0S + ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The map on bundles of curved dgla is given by the inclusion L●≥2 D,g,0 → L●≥2 D,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The group Aut(S) includes into G as a subgroup, and the map q is equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In order to show that q is an equivalence, we must show two things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' First, there is an underlying functor between the classical groupoids: π0(q) ∶ Aut(S) ⋉ MC(W(A)) → G ⋉ MC(MD,g(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We need to show that this is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This is implied by [4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5] and the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let ω ∈ W(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then ιE(ω) is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For ω ∈ L1 D,g,0, we have ιE(ω) ∈ U 0 0 = Lie(Aut(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' If ω ∈ W(A) we have in addition that j∗ιE(ω) ∈ GS ∗ N0, and so is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let ιE(ω) = Bs + Bn be the Jordan decomposition, where Bs is semisimple and Bn is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then j∗(Bs) = 0, so that Bs ∈ Lie(Aut0(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' But since Aut0(S) is unipotent, this implies that Bs = 0, and hence ιE(ω) is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Second, a derived stack has a tangent complex at every point of its MC locus, and the map q induces a chain map between the tangent complexes: dqw ∶ Tw[W(A)/Aut(S)] → Tq(w)[MD,g(A)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We need to show that this is a quasi-isomorphism at each point of the MC locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We will do this by first constructing an explicit homotopy at the special point q(0) (which is generally not in our space), and then apply the homological perturbation lemma to obtain the quasi-isomorphism at all points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The homotopy Let a ∶ LD,g,0 → LD,g be the inclusion and let b ∶ LD,g → LD,g,0 be the projection to the degree 0 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Both a and b are chain maps with respect to δS, but in general only a preserves the Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, b ○ a = idLD,g,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Recall that a given element β ∈ LD,g has a Taylor expansion in the eigenvalues of LS: β = ∑ u βu, where each term satisfies LS(βu) = uβu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' As we saw, the eigenvalues have the form ui + Z≥0 for finitely many complex numbers ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For this reason the series β′ = ∑ u≠0 1 uβu converges to a well-defined element of LD,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We use this to define the following degree −1 operator h ∶ Li D,g → Li−1 D,g, ∑ u βu ↦ ιE(∑ u≠0 1 uβu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The following lemma results from straightforward computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It has the upshot that a defines a quasi-isomorphism of dgla’s from (LD,g,0,δS) to (LD,g,δS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 9 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The operator h defines a homotopy between ab and idLD,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In other words, it satisfies [δS,h] = idLD,g − ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, it satisfies the ‘side conditions’ h ○ a = 0, b ○ h = 0 and h2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Finally, it vanishes on U and sends Ii to U i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The perturbation We will now perturb the differential δS and show that a continues to define a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This is achieved by using the perturbation lemma [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let w ∈ W(A) satisfy the Maurer-Cartan equation δS(w) + 1 2[w,w] = 0 and consider the per- turbed differential δS,w = δS + [w,−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This is a differential on LD,g, and we want an induced perturbation of the homotopy data (a,b,h,δS) of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The endomorphism adw ○ h of LD,g is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The element w can be decomposed as w = γ + α0N, where γ ∈ U 1 0 and N ∈ U 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1, N is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Recall from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2 that h vanishes on U and its image is contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, since U is a subalgebra of LD,g, it is preserved by adγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' As a result h ○ adγ ○ h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, it suffices to show that the operator α0adN ○ h is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now note that adN and multiplication by α0 commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since N ∈ U 0 0 , adN also commutes with h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This implies that (h ○ α0adN)k = (adN)k ○ ˜hk, where ˜h is the operator ˜h(β) = h(α0 ∧ β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' But this will vanish for large enough k since N is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The upshot of this lemma is that we can now define the following perturbed maps: h′ = ∞ ∑ p=0 (−hadw)ph, δ′ = δS + ∞ ∑ p=0 b(−adwh)padwa, a′ = ∞ ∑ p=0 (−hadw)pa, b′ = ∞ ∑ p=0 b(−adwh)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The perturbation lemma says that δ′ defines a differential on LD,g,0, that a′ and b′ define chain maps between (LD,g,0,δ′) and (LD,g,δS,w), and that the following equations are satisfied: b′ ○ a′ = idLD,g,0, [δS,w,h′] = idLD,g − a′ ○ b′, h′ ○ a′ = 0, b′ ○ h′ = 0, h′ ○ h′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The following lemma identifies the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The perturbations are given by a′ = a, b′ = b, δ′ = δS,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In particular, the inclusion a ∶ (LD,g,0,δS,w) → (LD,g,δS,w) is a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, h′ vanishes on U and sends I to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The element w ∈ L1 D,g,0 and so adw restricts to LD,g,0 and commutes with both a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' As a result of this and the side conditions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2, we have that hadwa = 0 and badwh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Plugging this into the definitions of the deformed maps gives a′ = a − ∑ p≥0 (−hadw)p(hadwa) = a, δ′ = δS + badwa − ∑ p≥0 b(−adwh)p−1adw(hadwa) = δS,w, b′ = b − ∑ p≥0 (badwh)(−adwh)p = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The statement about h′ follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2 and the fact that each term in the definition of h′ starts and ends with h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The quasi-isomorphism of tangent complexes Consider a point w ∈ MC(W(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It has the form w = γ + α0N, where γ ∈ U 1 0 , N ∈ U 0 0 , and j∗i∗(w) = N(0) ∈ GS ∗ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It has corresponding point q(w) ∈ MC(MD,g(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this section we will describe the morphism of tangent complexes dqw ∶ Tw[W(A)/Aut(S)] → Tq(w)[MD,g(A)/G] and show that it is a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We start by describing the tangent complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' First, the tangent complex of [MD,g(A)/G] is given as follows: Tq(w)[MD,g(A)/G] = L0 D,g → Tq(w)M(A) → L2 D,g → L3 D,g → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that the first term is L0 D,g = Lie(G), and the second term is the subspace Tq(w)M(A) = {v ∈ L1 D,g ∣ j∗i∗(v) ∈ T(S+N(0))OA}, where we use the fact that j∗i∗(q(w)) = S + N(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The first map is the derivative of the gauge action, and a computation shows that it is equal to −δS,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The minus sign is due to the fact that we are making the gauge group act on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For simplicity we will replace this by δS,w, since it does not affect the cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The second map is the derivative of the curvature dF, and a calculation shows that it is given by δS,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Finally, all higher maps are given by δq(w) = δS,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, the tangent complex is a subcomplex of (LD,g,δS,w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The tangent complex of [W(A)/Aut(S)] has a similar descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is given by Tw[W(A)/Aut(S)] = L0 D,g,0 → TwW(A) → L2 D,g,0 → L2 D,g,0 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' As above, L0 D,g,0 = Lie(Aut(S)) and the second term is the subspace TwW(A) = {v ∈ L1 D,g,0 ∣ j∗i∗(v) ∈ TN(0)(GS ∗ N0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Again all maps are given by δS,w (the first map has a minus sign, which we remove for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, the tangent complex is a subcomplex of (LD,g,0,δS,w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The map dqw is easily seen to coincide with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, in order to prove that dqw is a quasi- isomorphism, it suffices to show that the homotopy data (a,b,h′,δS,w) restricts to the tangent complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The maps (a,b,h′,δS,w) restrict to Tq(w)[MD,g(A)/G] and Tw[W(A)/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, dqw defines a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since the complexes are modified in degree 1, it suffices to restrict our attention to degrees 0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The above description of the tangent complexes and dqw immediately implies that a and δS,w restrict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' To check that h′ restricts, we only need to show that h′(L2 D,g) is contained in Tq(w)M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' But this follows because, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4, the image of h′ is contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For the map b, consider a point β ∈ Tq(w)M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This can be expanded as β = ∑u βu, where each term satisfies LS(βu) = uβu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By definition b(β) = β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, we need to check that if j∗i∗(β) ∈ T(S+N(0))OA, then j∗i∗(β0) ∈ TN(0)(GS ∗ N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' These tangent spaces have the following descriptions T(S+N(0))OA = Im(adS+N(0) ∶ g → g), TN(0)(GS ∗ N0) = Im(adN(0) ∶ gS → gS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now using the eigenvector expansion, we have j∗i∗(β) = ∑ u j∗i∗(βu) = adS+N(0)(Z), for some Z ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' One can check that each term in the summand satisfies adS(j∗i∗(βu)) = uj∗i∗(βu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since adS ∶ g → g is diagonalizable, we can decompose Z into eigenvectors as well: Z = ∑u Zu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' And since adS+N(0) commutes with adS, it preserves the eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, we can match up the eigenvectors to get j∗i∗β0 = adS+N(0)(Z0) = adN(0)(Z0), where Z0 ∈ gS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 5 Plane curves xp − yq In this section we give a detailed study of the case of plane curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Consider f = xp − yq ∶ C2 → C, where p and q are relatively prime positive integers satisfying p < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This function is weighted homogeneous of degree qp for the Euler vector field E = qx∂x + py∂y, which defines the weight grading on coordinates ∣x∣ = q and ∣y∣ = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The logarithmic tangent bundle is generated by the vector fields E and V = qyq−1∂x + pxp−1∂y, which satisfy [E,V ] = (qp − p − q)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let w0 = qp − p − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The logarithmic 1-form α0 = 1 qpdlogf pairs with V to give 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, it can be completed to a dual basis α0,β of the logarithmic cotangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The form α0 is closed, and β satisfies dβ = (p + q − qp)α0 ∧ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Cohomology of V Let Ow denote the subspace of polynomial functions with weight w with respect to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that any integer w ∈ Z has a unique decomposition w = aq + bp + cqp, where a,b,c ∈ Z, 0 ≤ a < p and 0 ≤ b < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This decomposition provides a useful way of indexing the weights because of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let w = aq + bp + cqp, with the above restrictions on a,b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The dimension of Ow is max(c,−1) + 1, and a basis is given by xa+cpyb,xa+(c−1)pyb+q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=',xayb+cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The vector field V has weight w0, and hence it defines a map V ∶ Ow → Ow+w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 12 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The kernel of V is C[f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' A calculation shows that V (f c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Conversely, let g ∈ ker(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Because V is homogeneous, it suffices to consider the case where g is homogeneous of weight w > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The equation V (g) = 0 implies that ∂xg = pxp−1h and ∂yg = −qyq−1h, for a common polynomial h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, wg = E(g) = qx∂xg + py∂yg = qp(xp − yq)h, so that g = qp w fh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence h is a function of weight w − qp and it lies in the kernel of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The result now follows by induction on the weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The Jacobian ideal of f is generated by xp−1 and yq−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let C = C[x,y]/(xp−1,yq−1), considered as a C-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It has a natural basis of monomials xayb, where 0 ≤ a < p − 1 and 0 ≤ b < q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Using this basis, C is naturally graded by weight, and there is a weight preserving injective linear map C → C[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Consider the graded polynomial ring C[f], where f has degree pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then C[x,y] is a graded C[f]-module and there is a morphism of graded C[f]-modules C[f] ⊗C C → C[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The action of V on C[x,y] is C[f]-linear, so that the cokernel coker(V ) is also a C[f]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Post-composing with the quotient projection, we obtain the morphism C[f] ⊗C C → coker(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The morphism C[f] ⊗C C → coker(V ) is an isomorphism of graded C[f]-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since V is homogeneous it suffices to consider a single weight at a time: we consider the cokernel of the map V ∶ Ow−w0 → Ow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let w = aq + bp + cqp, where 0 ≤ a < p, 0 ≤ b < q and c ≥ 0, so that Ow has dimension c + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then w − w0 = (a + 1)q + (b + 1)p + (c − 1)qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' If a < p − 1 and b < q − 1 then Ow−w0 has dimension c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore V is injective because w − w0 is not a multiple of qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence coker(V )w is 1-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' If a = p − 1 and b < q − 1, then w − w0 = (b + 1)p + cqp, so Ow−w0 has dimension c + 1, V is injective, and hence coker(V )w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The same argument applies to the case a < p − 1 and b = q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The only remaining case is a = p − 1 and b = q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this case w − w0 = (c + 1)qp and Ow−w0 has dimension c + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' But now V has a 1-dimensional kernel and so coker(V )w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The upshot is that the cokernel is non-zero precisely when a < p − 1 and b < q − 1, in which case it is 1-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' These dimensions match with the dimensions of C[f] ⊗C C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence it suffices for us to prove that f cxayb is not in the image of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We will do this by proving that the following map M ∶ C ⊕ Ow−w0 → Ow, (λ,g) ↦ λf cxayb + V (g) is represented by a matrix with positive determinant, using the bases of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Applying V to the element xa+1+ipyb+1+jq yields p(1 + b + jq)xa+(i+1)pyb+jq + q(1 + a + ip)xa+ipyb+(j+1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The salient thing to note is that the basis elements are consecutive and the coefficients are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence V is represented by a (c +1)×c matrix such that column i has positive entries in rows i and i+1 and 0 for the remaining rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Using the binomial theorem, f cxayb = ∑c k=0(−1)kxa+(c−k)pyb+kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The salient point here is that the terms are non-zero with alternating signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' These give the entries of the first column of the matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Computing the determinant of M using the Laplace expansion along the first column shows that it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 13 The dgla (U0,δS) Now we choose a Lie algebra g and a semisimple element S ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This induces an eigenspace decomposition of the Lie algebra g = ⊕ λ gλ, where gλ is the eigenspace of adS with eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We will use the following convention: if λ is not an eigenvalue of adS, then gλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that the decomposition is preserved by the bracket: [gλ,gµ] ⊆ gλ+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The dgla (U0,δS) has terms in degrees 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' They are given by U 0 0 = ⊕ w≥0 Ow ⊗ g−w, U 1 0 = ⊕ w≥0 Ow ⊗ gw0−wβ, with δS given by applying V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We will sometimes drop β from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Applying Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='3 we obtain the following description of the cohomology of (U0,δS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The cohomology of (U0,δS) is given as follows H0(U0) = ⊕ c≥0 f cg−cpq, H1(U0) ≅ ⊕ w≥0 (C[f] ⊗ C)w ⊗ gw0−w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, the graded Lie algebra H●(U0) with zero differential naturally embeds into (U0,δS) as a quasi-isomorphic sub-dgla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let a ∈ g be a real semisimple element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Recall from [5] that this determines a parabolic subgroup of G P(a) = {g ∈ G ∣ lim z→0zagz−a exists in G along any ray}, where z ∈ C and za = exp(log(z)a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Decomposing S into real and imaginary parts, S = a + ib, we can define the following subgroup of G PS ∶= CG(e −2πi qp S) ∩ P(−a qp ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this definition CG(e −2πi qp S) is the centralizer of e −2πi qp S in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is reductive but possibly discon- nected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let CS denote the connected component of the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The group PS is the parabolic subgroup of CG(e −2πi qp S) (or CS) determined by the element −a qp and it is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The reductive quotient of PS is GS, the centralizer of S in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Denote the quotient map χ ∶ PS → GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The group PS embeds into Aut(S) as the subgroup integrating H0(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The gauge action of PS preserves H1(U0) ⊂ U 1 0 and is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, we have a Lie subgroupoid PS ⋉ H1(U0) ⊆ Aut(S) ⋉ U 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let C ⋉ C2 be the Lie algebroid generated by the action of E and let C ⋉ C be the Lie algebroid generated by the action of z∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The following defines a Lie algebroid morphism f ∶ C ⋉ C2 → C ⋉ C, (λ,x,y) ↦ (pqλ,f(x,y)), and under this map, the logarithmic connection d + 1 qpSdlogz pulls back to p∗S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' As a result, the pullback defines an embedding of automorphism groups from Aut(d + 1 qpSdlogz) → Aut(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In [3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4] it is shown that restricting an automorphism to 1 ∈ C defines an embedding of Aut(d + 1 qpSdlogz) into G which identifies it with PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, the Lie algebra of PS is identified with ⊕c≥0 zcg−cpq, and under the pullback, this is sent isomorphically to H0(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Finally, since the action of H0(U0) preserves H1(U0) and is linear, the same is true of PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 14 Given the semisimple element S ∈ g, we say that it is large enough if all the positive integer eigenvalues of adS are strictly greater than w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The inclusion PS ⋉ H1(U0) ⊆ Aut(S) ⋉ U 1 0 is a Morita equivalence if S is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' First, because of the assumption on S and the fact that Ow0 = 0 (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1), the vector space U 1 0 has the following form U 1 0 = ⊕ w>w0 Ow ⊗ gw0−wβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We now proceed in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Claim: The subspace H1(U0) intersects every orbit of Aut(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Given γ ∈ U 1 0 , we need to find an element of Aut(S) which sends γ into H1(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We do this iteratively following the usual proof of the normal form for ODEs with Fuchsian singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' First, we expand γ = ∑w>w0 γw, where γw ∈ Ow ⊗ gw0−wβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Given a weight w′ > w0, let u ∈ Ow′−w0 ⊗ gw0−w′, and consider the action of eu ∈ Aut(S) on γ: eu ∗ γ = euγe−u − V (eu)e−uβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We claim that γ is modified in weights w′ and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Indeed, expanding we get V (eu)e−u = V (u) − V (u)u + 1 2V (u2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The first term has weight w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' All other terms have higher weights since w′−w0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Expanding the term euγwe−u gives exp(adu)γw = γw + [u,γw] + 1 2[u,[u,γw]] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The second term has weight w′ − w0 + w > w′, since w > w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that the action on weight w′ is given by γw′ ↦ γw′ − V (u)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='3, the element γw′ can be decomposed as γw′ = f cxayb ⊗ Xβ + V (u)β, where f cxayb ⊗ X ∈ (C[f] ⊗ C)w′ ⊗ gw0−w′ and u ∈ Ow′−w0 ⊗ gw0−w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then (eu ∗ γ)w′ = γw′ − V (u)β ∈ H1(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now starting with the lowest weight w′ > w0, we iteratively act on γ by elements eu ∈ Aut(S) so that the terms up to level w′ lie in H1(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This will terminate after finitely many steps since U 1 0 is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The result is an element of H1(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Claim: The inclusion functor PS ⋉ H1(U0) → Aut(S) ⋉ U 1 0 is fully-faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We need to show that given g ∈ Aut(S) and γ ∈ H1(U0), if g ∗ γ ∈ H1(U0), then g ∈ PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Recall the Levi decomposition Aut(S) ≅ Aut0(S) ⋊ GS, and note that GS ⊆ PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It therefore suffices to work under the assumption that g ∈ Aut0(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since such a g is unipotent, it has the form g = eu, for u ∈ ⊕w>0 Ow ⊗ g−w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Expanding in weights, u = ∑w≥w1 uw, where w1 > 0 is the lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' From the above expressions for eu ∗ γ, we see that the lowest weight for which γ is modified is w0 + w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The corresponding term is given by (eu ∗ γ)w0+w1 = γw0+w1 + V (uw1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since eu ∗ γ ∈ H1(U0), we must have V (uw1) = 0, implying that uw1 ∈ H0(U0) and euw1 ∈ PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let ˜γ = euw1 ∗γ ∈ H1(U0), so that g ∗γ = (eue−uw1 )∗ ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Using the Baker-Campbell-Hausdorff formula and the fact that w1 > 0, we see that eue−uw1 = ev, where the lowest weight of v is strictly greater than w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence the result follows by induction on w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let γ ∈ H1(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Claim: The inclusion (H●(U0),adγ) → (U ● 0,δS +adγ) is a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This follows by the homological perturbation lemma [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let a ∶ H●(U0) → U ● 0 be the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4, this is a quasi-isomorphism with respect to the 0 differential on the domain and δS on the codomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We have the decomposition U 1 0 = H1(U0) ⊕ Im(δS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let C be a complement to H0(U0), so that U 0 0 = H0(U0) ⊕ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is possible to choose this complement compatible with the weight decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Using the decomposition we define the projection b ∶ U ● 0 → H●(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The restriction δS∣C ∶ C → Im(δS) is an isomorphism, and the inverse defines a map h ∶ U 1 0 → U 0 0 which has weight −w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' These maps satisfy b ○ a = id, id − a ○ b = [δS,h], as well as the side conditions h ○ a = 0, b ○ h = 0 and h ○ h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now consider the map adγ ∶ U 0 0 → U 1 0 which will serve as a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that it restricts to a map H0(U0) → H1(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Expanding in the weights, γ = ∑w>w0 γw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since the weight of h is −w0, it follows that adγ ○h is an endomorphism of U 1 0 which raises the weight of an element by at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It follows that adγ ○h is nilpotent, allowing us to apply the perturbation lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Using the formulas appearing above Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4, we see that a remains unperturbed, δS is deformed to δS + adγ and the zero differential is deformed to adγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The moduli stack [W(A)/Aut(S)] Now we choose an element A = S + N0 ∈ g, which we write using the Jordan decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This determines the derived stack [W(A)/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The base of the derived manifold is W(A) = {Cβ + Nα0 ∈ U 1 0 ⊕ U 0 0 α0 ∣ N(0) ∈ GS ∗ N0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The bundle of curved dgla’s is the trivial bundle W(A) × U 1 0 α0 with trivial dgla structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The curvature section is given by F(Cβ + Nα0) = (V (N) + [C,N])β ∧ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='6 we can construct a smaller model for this derived stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5, the vector space H1(U0) is a linear representation of PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The Lie algebra Lie(PS) is likewise a representation, and the subspace dχ−1(GS ∗ N0) is preserved by this action (recall that χ ∶ PS → GS is the projection to the reductive quotient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let Q(A) = H1(U0) × dχ−1(GS ∗ N0), equipped with the action of PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The infinitesimal action of PS on H1(U0) defines a PS-equivariant map FS ∶ Q(A) → H1(U0), (C,N) ↦ [C,N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Viewing this as a section of the bundle Q(A) × H1(U0) we get a derived manifold Q(A), which represents the derived vanishing locus of FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This defines the derived stack [Q(A)/PS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5 this maps into [W(A)/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' There is a map of derived stacks i ∶ [Q(A)/PS] → [W(A)/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For points (0,N) ∈ Q(A) the derivative di(0,N) is a quasi-isomorphism of tangent complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Fur- thermore, if S is large enough, then i is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' To begin, assume that S is large enough, so that PS ⋉ H1(U0) ⊆ Aut(S) ⋉ U 1 0 is a Morita equivalence by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now let Cβ + Nα0 ∈ MC(W(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Claim: If C ∈ H1(U0), then (C,N) ∈ MC(Q(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Indeed, observe that F(Cβ + Nα0) = (δS(N) + adCβ(N)) ∧ α0, and that (H●(U0),adCβ) → (U ● 0,δS + adCβ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1) is a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It follows that N ∈ H0(U0), and therefore that the claim is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is straightforward to deduce from this claim that the induced morphism PS ⋉ MC(Q(A)) → Aut(A) ⋉ MC(W(A)) is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now given a point (C,N) ∈ MC(Q(A)), we need to show that the morphism of tangent complexes di(C,N) ∶ T(C,N)[Q(A)/PS] → T(C,N)[W(A)/Aut(S)] is a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The differentials on these tangent complexes have the form d + adNα0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By an argument involving the perturbation lemma, it suffices to prove that di(C,N) is a quasi-isomorphism with respect to the differentials d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' But for these differentials, di(C,N) is a direct sum of Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1 and a shift of a subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, it is a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4, Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='1 is quasi-isomorphism when C = 0 even if S is not large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Connections of the form (0,N) ∈ Q(A) are pullbacks by f of connections on C with a logarithmic pole at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The condition in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='7 that S is large enough is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In the following Example we see that the moduli space can have extra components when the condition is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let f = x2 − y5, let g = gl3, and let A = S be a diagonal matrix with entries 0,1 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that S is not large enough since 1 is a positive eigenvalue of adS which is smaller than w0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The subalgebra gS consists of the diagonal matrices and U 0 0 = gS ⊕ spanC(x2E23,y5E23,xy3E13), U 1 0 = spanC(yE21,y2E12,xy4E23,x2y2E13,y7E13), where Eij are the elementary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' On the other hand, the δS cohomology is given by H0(U0) = gS ⊕ spanC(fE23), H1(U0) = spanC(yE21,y2E12,y2fE13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' An arbitrary element of W(A) has the form Cβ + Nα0, where C = C21yE21 + C12y2E12 + C23xy4E23 + (C(1) 13 x2y2 + C(2) 13 y7)E13, N = N13xy3E13 + (N (1) 23 f + N (2) 23 (x2 + y5))E23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The Maurer-Cartan equation consists of the following coupled system of equations: 20N (2) 23 = −C21N13 5N13 = −C12(N (2) 23 − N (1) 23 ) 6N13 = −C12(N (2) 23 + N (1) 23 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Adding the last two equations and substituting the result into the first yields (110−C21C12)N (2) 23 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Assume first that C21C12 ≠ 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then the equations simplify to N (2) 23 = N13 = C12N (1) 23 = 0, and N ∈ H0(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The result is a 5-dimensional variety Ma with 2 irreducible components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Next, assume that C21C12 = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then we can solve the equations to obtain N13 = −2C12N (1) 23 and N (2) 23 = 11N (1) 23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, the result is a smooth irreducible 5-dimensional variety Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 17 An element of Aut(S) has the form g = ⎛ ⎜ ⎝ u 0 λxy3 0 v ax2 + by5 0 0 w ⎞ ⎟ ⎠ , and it acts on C = (C21,C12,C23,C(1) 13 ,C(2) 13 ) and N = (N13,N (1) 23 ,N (2) 23 ) in the following way: g ∗ C = (v uC21, u v C12, v wC23 − λv uwC21 − 10(a + b) w , u wC(1) 13 − au vwC12 − 6λ w , u wC(2) 13 − bu vwC12 − 5λ w ) g ∗ N = ( u wN13, v wN (1) 23 , v wN (2) 23 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' There are two things we can immediately note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' First, by looking at the action on N, we can see that there are orbits in Mb which do not intersect Q(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, [Mb/Aut(S)] is an extra component of the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Second, the product C21C12 is invariant under the action and if we assume that C21C12 ≠ 110, then it is always possible to send C into H1(U0) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' C23 = 0 and C(1) 13 + C(2) 13 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The subgroup of Aut(S) which stabilizes this locus is PS, and hence [Ma/Aut(S)] is contained in [MC(Q(A))/PS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' ∎ Relation to the Grothendieck-Springer resolution Recall that PS ⊆ CS is a parabolic subgroup with Levi factor GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Denote their Lie algebras pS, cS and gS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let P denote the partial flag variety, defined as the set of parabolic subalgebras of cS which are conjugate to pS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It is isomorphic to CS/PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Consider the incidence variety ˜c = {(x,p) ∈ cS × P ∣ x ∈ p}, which is isomorphic to (CS × pS)/PS via the map which sends (g,x) ∈ CS × pS to the pair (Adg(x),Adg(pS)) ∈ ˜c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' When PS is a Borel subgroup, ˜c is the Grothendieck-Springer resolution (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' [11] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The element N0 ∈ gS defines an adjoint orbit O in the Levi quotient of every parabolic p ∈ P [5, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This lets us define the following subspace of ˜c: ˜cN0 = {(x,p) ∈ ˜c ∣ dχ(x) ∈ O}, where dχ denotes the projection to the Levi quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The space Q(A) is a PS-equivariant vector bundle over dχ−1(GS ∗ N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence π ∶ EA = (CS × Q(A))/PS → ˜cN0 is a CS-equivariant vector bundle and the map FS gives rise to an equivariant section σS of π∗(EA) → EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this way, we obtain a derived stack [EA/CS] which is equivalent to [Q(A)/PS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let G = GLn and let A = S = pqD, where D is a diagonal matrix with distinct integer eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this case, S is large enough, CS = G and PS = B is a Borel subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence P is the flag variety Fln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since N0 = 0, ˜g0 is the Springer resolution, which is isomorphic to T ∗Fln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We use Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4 to compute the cohomology space H1(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The weights w showing up in the decomposition have the form w = (p − 1)q + (q − 1)p + cqp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' But (C[f] ⊗ C)w = 0 in this case, so that H1(U0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, the moduli space [W(A)/Aut(S)] is equivalent to the quotient stack [T ∗Fln/GLn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that the connections corresponding to the points of T ∗Fln are all pulled-back from logarithmic connections on C, where a similar classification is given by [5, Theorem A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' ∎ 18 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let G = GLn and A = S = qp r D, where D is a diagonal integer matrix with distinct eigenvalues and r is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this case, the eigenvalues of e −2πi qp S are rth roots of unity, implying that CS = ∏r−1 i=0 GLmi, with the factors indexed by the roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Since the eigenvalues of S are assumed to be distinct, PS = ∏r−1 i=0 Bmi is a product of Borels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, P ≅ ∏r−1 i=0 Flmi is a product of flag varieties and ˜c0 ≅ ∏r−1 i=0 T ∗Flmi is the product of their cotangent bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Writing the eigenvalues of D as rk+u, with 0 ≤ u < r, the eigenvalues of adS have the form qp(k1 − k2) + qp r (u1 − u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We can ensure that S is large enough by restricting the possible values of ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' For example, this is guaranteed if k1 − k2 ≠ 0,±1 for all pairs of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='4, the weights w showing up in the decomposition of H1(U0) have the form w = qp(1 + k2 − k1) + qp r (u2 − u1) − p − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' As in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='10, we must have u2 ≠ u1 in order to get a non-zero contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now we specialise to GL4, with r = pq and S = D a diagonal matrix with entries pqk1, pqk2, pqk3 + p + q, pqk4 + p + q, such that k1 ≪ k2 ≪ k3 ≪ k4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then CS = GL2 × GL2 and P ≅ P1 × P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let T1 and T2 be the tautological rank 2 vector bundles over P1 × P1 (corresponding respectively to the first and sec- ond factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' They are both trivial, but are equipped with tautological line subbundle bundles Li with respective degrees (−1,0) and (0,−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The cotangent bundle ˜c0 can be identified with Hom(T1/L1,L1) ⊕ Hom(T2/L2,L2), or alternatively, the nilpotent filtration-preserving endomor- phisms of T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The weights in the decomposition of H1(U0) have the form qp(1 + kj − ki), for j = 3,4 and i = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, H1(U0) can be identified with the subspace of gl4 consisting of the upper right 2×2 block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, EA = Hom(T1/L1,L1) ⊕ Hom(T2/L2,L2) ⊕ Hom(T2,T1), with section σS(a,b,c) = c ○ b − a ○ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' ∎ Tangent Lie bialgebra and shifted Poisson geometry Let A = S ∈ g be a semisimple element and consider the moduli space [W(S)/Aut(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' There is a distinguished point 0 ∈ MC(W(S)) corresponding to the connection 1-form α0S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this section we focus our attention on a formal neighbourhood of this point in the moduli space and sketch the construction of a −2-shifted Poisson structure on this neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' According to the fundamental principal of derived deformation theory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' [17, 20, 27, 18]) this formal neighbourhood is encoded by the shifted tangent complex T0[W(S)/Aut(S)][−1], equipped with its structure as a differential graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='7 this dgla is quasi-isomorphic to T0[Q(A)/PS][−1] = H0(U0) → H1(U0) ⊕ H0(U0)+ → H1(U0), where the differential is 0 and H0(U0)+ = ⊕c>0 f cg−cpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By [26, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='5], a −3-shifted Lie bialgebra structure on T0[Q(A)/PS][−1] gives rise to a −2-shifted Poisson structure on the formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence, our strategy is to construct a Lie bialgebra structure on the tangent complex by embedding it into H●(U0) ⋉ H●(U0)[−1], and then to realize this larger Lie algebra as a Lagrangian inside a −3-shifted Manin triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We construct the Manin triple in stages, starting with the following input data: Choose an invariant inner product k on the reductive Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This induces a perfect pairing between the eigenspaces gλ and g−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 19 The C-vector space C = C[x,y]/(xp−1,yq−1) is canonically isomorphic to the degree 1 coho- mology of f −1(1), which is a curve of genus g = 1 2(p − 1)(q − 1) with a single puncture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The isomorphism is given as follows: C → H1(f −1(1)), xayb ↦ (xaybβ)∣f −1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By pulling back the intersection pairing on the curve, we obtain a symplectic form I on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Up to a scaling constant, it is given by the following formula I(xayb,xa′yb′) = δ(a + a′ + 2 − p)δ(b + b′ + 2 − q) aq + bp − w0 , where δ is the delta function which evaluates to 1 at 0 and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We can extend this to a C[f]-linear pairing on C[f] ⊗C C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now consider the Lie algebra c = ⊕c∈Z f cg−cpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This has two distinguished subalgebras: first b = H0(U0), where c ≥ 0, and second b−, where c ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that c can be viewed as a subalgebra of g and so inherits the pairing k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This defines a perfect pairing between b and b−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Next, define the following vector space K = ⊕ c∈Z ⊕ 0≤a≤p−2 0≤b≤q−2 f cxaybgw0−cpq−aq−bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This decomposes as K = n+⊕h⊕n− according to whether c is positive, zero, or negative, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that H1(U0) = n+ ⊕ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Viewing both c and K as subspaces of the Lie algebra C[x,y] ⊗ g, we can show that K is a representation of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By combining the symplectic form I on C with the Lie bracket on g, we define a symmetric c-equivariant map ω ∶ S2(K) → c as follows ω(f c1xa1yb1X1,f c2xa2yb2X2) = f c1+c2I(xa1yb1,xa2yb2)[X1,X2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Post-composing this with the natural projection to b− defines a map µ ∶ S2(K) → b−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' This is b- equivariant, where b− is a b-representation by using the inner product to identify it with b∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' With respect to the b action on K, both n+ and n+⊕h are sub-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Hence the quotient h = n+⊕h n+ is naturally a b-representation, and µ descends to define a b-equivariant map ν ∶ S2(h) → b−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We now define a graded Lie algebra L = b ⊕ (K ⊕ h)[−1] ⊕ b−[−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The bracket is constructed from the bracket on b, the b-action on the other summands, the sym- metric pairing µ on K and the symmetric pairing −ν on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We set the bracket between K and h to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The vector space L with the bracket described above defines a graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let p+ = b ⊕ (n+ ⊕ h)[−1] and let p− = (n− ⊕ h)[−1] ⊕ b−[−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' There are morphisms p+ → L, (b,n,h) ↦ (b,n + h,h,0) p− → L, (n,h,u) ↦ (0,n + h,−h,u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' These embed p± as complementary subalgebras of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, p+ is isomorphic to H●(U0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Next we construct an inner product on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' By combining the inner product k on g with the symplectic pairing I on C, we define a skew symmetric pairing Ω ∶ ∧2K → C as follows Ω(f c1xa1yb1X1,f c2xa2yb2X2) = f c1+c2I(xa1yb1,xa2yb2)k(X1,X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 20 This pairing is non-degenerate, restricts to a non-degenerate pairing on h and defines a perfect pairing between n±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' We can now define a non-degenerate graded symmetric bilinear pairing B ∶ S2(L) → C[−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' More precisely, given y = (b1,k1,h1,u1),z = (b2,k2,h2,u2) ∈ L, we set B(y,z) = k(b1,u2) + k(u1,b2) + Ω(k1,k2) − Ω(h1,h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The pairing B is a non-degenerate invariant inner product on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Furthermore, p± are complementary Lagrangian subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In other words, (L,p+,p−) is a −2-shifted Manin triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Now let C[ϵ] be the cdga generated by a degree +1 variable ϵ and let tr ∶ C[ϵ] → C[−1] be the trace map defined by sending the element a + bϵ to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Tensoring with C[ϵ] defines a new triple of graded Lie algebras (C[ϵ]⊗L,C[ϵ]⊗p+,C[ϵ]⊗p−) and the bilinear form B extends in the following way B ∶ S2(C[ϵ] ⊗ L) → C[−3], (fy,gz) ↦ (−1)∣y∣∣g∣tr(fg)B(y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' In this way, we obtain a −3-shifted Manin triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Note that C[ϵ]⊗p+ ≅ H●(U0)⋉H●(U0)[−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The tangent complex T0[Q(A)/PS][−1] is isomorphic to the subalgebra b ⊕ (n+ ⊕ h) ⊕ b>0ϵ ⊕ (n+ ⊕ h)ϵ, where b>0 consists of the elements f cX with c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Let M be the direct sum of this subalgebra with C[ϵ] ⊗ p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Then M is a coisotropic subalgebra of L and M ⊥ = g0 ⊂ b− ⊂ M is an isotropic ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' It follows that (M/M −1,T0[Q(A)/PS][−1],C[ϵ] ⊗ p−/M ⊥) is a −3-shifted Manin triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, by [26, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='3], the tangent complex obtains a −3-shifted Lie bialgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' The tangent complex T0[Q(A)/PS][−1] admits the structure of a −3-shifted Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Therefore, the formal neighbourhood of 0 ∈ [W(S)/Aut(S)] admits a −2-shifted Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} 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+page_content=' Homotopical algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Geometric stacks and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=', 193(902):x+224, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfCfrw/content/2301.00962v1.pdf'} diff --git a/x9FJT4oBgHgl3EQfhSy6/content/tmp_files/2301.11565v1.pdf.txt b/x9FJT4oBgHgl3EQfhSy6/content/tmp_files/2301.11565v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..29ffc22242e8360694fbdbc1ed5043fdccf7da0f --- /dev/null +++ b/x9FJT4oBgHgl3EQfhSy6/content/tmp_files/2301.11565v1.pdf.txt @@ -0,0 +1,643 @@ + +1 +Three decades’ trends and extremes of building cooling demand in +Hong Kong, Sydney, Montreal, Zurich and London +Haiwei Li a, b, Yongling Zhao a, *, Ronita Bardhan b, *, Pak-Wai Chan c, +Dominique Derome d, Zhiwen Luo e, Diana Ürge-Vorsatz f, Jan Carmeliet a + +a Department of Mechanical and Process Engineering, ETH Zürich, Zürich, Switzerland +b Department of Architecture, University of Cambridge, Cambridge, United Kingdom +c Hong Kong Observatory, Kowloon, Hong Kong, China +d Department of Civil and Building Engineering, Université de Sherbrooke, Sherbrooke, Canada +e Welsh School of Architecture, Cardiff University, Cardiff, Wales, UK +f Department of Environmental Sciences and Policy, Central European University, Austria +*Corresponding authors: Yongling Zhao (yozhao@ethz.ch), Ronita Bardhan (rb867@cam.ac.uk) +Abstract +This brief communication interprets three decades’ evolution of building cooling demand of urban +and rural areas through the lens of five representative cities, i.e., Hong Kong, Sydney, Montreal, +Zurich, and London. The upward trend and extremes in building cooling demand, estimated from +cooling degree hours (CDH) using meteorological data from 1990 to 2021, largely explained by +global warming, urban heat islands, and extreme heat events. The quantification of the impact of +base temperatures further reveals that 20% energy saving could be achieved by increasing one +degree setpoint temperatures, which could be potentially reached by regulatory intervention on +building system operation. +Main +The building sector is responsible for 30% of global energy consumption, and 27% of total energy +emissions1. At present, 56% of the world’s population lives in urban areas, which is expected to +continually increase to 68% by 20502, escalating the energy burden in cities. The energy demand +for space heating or cooling is determined by building design and operation, building physical +properties and occupancy activities, socio-economic development, and most importantly, climatic +conditions3–5. Climate change encompasses global warming. The earth’s average surface +temperature has risen by approximately 1 °C since the late 19th century6. Measures to mitigate and +adapt the warming climate are in urgent need7. Furthermore, the frequency, duration, and severity + + +2 +of extreme temperature events such as heatwaves and record-breaking high temperatures are +aggravated8. Moreover, urban areas experience higher air or surface temperatures than their +surrounding sub-urban or rural areas, which is called the urban heat island (UHI)9,10. This +phenomenon appears frequently alongside urbanization, with enlarged urban heat storage capacity, +long-wave radiation trapping, reduced evapotranspiration and convection efficiency, and increased +anthropogenic heat. UHI exacerbates the warming effects in cities, leading to high heat-related +illness and mortality rates, increasing energy demand, air pollution associated with hot temperature +and reduced air circulation, and a potential energy crisis among a large portion of the global +population. Understanding the long-term evolution of cooling energy demands and the causal +factors of the trends and extremes is crucial for predicting future energy emissions, formulating +mitigation strategies, and facilitating the implementation of the Paris Agreement11 and Sustainable +Development Goals (SDGs)12. +We analyze three decades of building cooling demand and the principal drivers, such as warming +background climate by global warming, urban heat island (UHI), heatwaves, cities’ population, and +some potential mitigation measures for five cities residing in different climates: Hong Kong, +Sydney, Zurich, Montreal, and London. The cooling demand for urban and rural (or suburban) +areas is quantified by yearly cooling degree hours (CDH) calculated from climatological standard +30-year (from 1990 to 2021) observation data13. The calculation of CDH is summarized in the +Methods section. We remark that compared to the commonly used average temperature, CDH, a +cumulated calculation measure over time, is a more solid metric to assess heat-related cooling +demand. +Fig. 1 shows the yearly CDH in 5 cities from 1990 to 2021, demonstrating yearly cooling energy +demand in all five cities during the last three decades. The cooling season refers to the period that +requires cooling inducing an air conditioning load. The main climate driver of energy demand is +the background ambient temperature during the cooling season14,15. Other climate factors, such as +humidity levels and wind speeds, accompanied by the air temperature, reflect human heat sensation +and influence the climate nexus of cooling demand16. The background climatology determines the +distinct difference in the cooling demand magnitudes in five cities. Hong Kong, showing CDH’s +in the range of 25,000 to 33,000 C∙h is in the humid summer subtropical zone (Köppen climate +classification: Cwa), and Sydney, showing a CDH in the range of 2,000 to 10,000 C∙h, is in the +all-year humid subtropical zone (Cfa). Both Hong Kong and Sydney have a cooling season from +spring to autumn, while the cooling season for Montreal, Zurich, and London is mostly summer, +from June to August. Montreal (1,000-6,000 C∙h) is in the humid continental zone (Dfb); Zurich + + +3 +(1,000-7,000 C∙h)and London (0-3,000 C∙h) belong to the marine west coast (oceanic) climate +zone (Cfb). The results show that the warmer subtropical cities, Hong Kong and Sydney, have +distinctly higher cooling demand and longer cooling seasons than continental and oceanic climate +cities, Montreal, Zurich, and London. Also, the influence of latitude is clearly observed for the last +three cities, showing lower cooling demand for the higher latitude city London compared to +Montreal and Zurich. + +Fig. 1: (a) Yearly cooling degree hours (CDH) of urban and rural areas (or suburban) in Hong Kong, +Sydney, Montreal, Zurich, and London from 1990 to 2021. The base temperature for CDH +calculation is 22 °C. The graphs in the center of each circular plot display the land types about 10 +km around the weather stations. Remark the different maximum scale for Hong Kong (40,000) +compared to the other cities maximum of 8,000. (b) Population growth (bars) and annual mean +urban air temperature rise (lines) from 1990 to 2021. We remark that the sub-figures have different +scales of the y-axis for different cities. +Statistically, average CDH increase for each decade is 14,619 C∙h in urban Hong Kong, 16,997 +C∙h in rural Hong Kong; 15,907 C∙h in urban Sydney, 13,491 C∙h in rural Sydney; 5,783 C∙h +in urban Montreal, 3,285 C∙h in rural Montreal; 7184 C∙h in urban Zurich, 4,365 C∙h in rural + +HongKong +Sydney +Montreal +Zurich +London +(Cwa) +(Cfa) +(Dfb) +(Cfb) +(Cfb) +40000 +30000 +97 +532 +8000 +8000 +8000 +6000 +0008 +3062 +20000 +6000 +4347 +6000 +6000 +Urban Area CDH +10000 +4000 +4000 +4000 +4000 +2000 +527 +2000 +2000 +82000 +0 +1 +29 +4386 +3175 +A0 +266717558 +3984 +3264 +2025 +524g +2951 +659 +975 +247126935 +4986 +2604 +1988 +1947 +2189 +912 +114 +298907949 +86792766 +3490 +1301 +27912 +4368 +25 +2841 +6305 +870 +A +n +21 +:8 +2552 +7854 +679 +(a) +0 +R. +4000 +30000 +8000 +8000 +6000- +8000 +8000 +6000 +6000 +20000- +4000 +4000 +A +4000 +4000 +10000- +Rural Area CDH +2000 +2000 +273 +722 +4546 +287 +3826 +2 +2125 +510 +27001 +21873 +6081 +4125 +1143 +99 +234 +19024 +4963 +6060 +1787 +1347 +1530 +270 +25392 +1893 +2565 +2138 +613 +6008 +043 +3955 +2954 +99 +54 +2 +A +density (people/km) +400 +20 +6000 +15 +6000 +24 +19 +800 +800 +5000 +14 .5000 +(b) +OOE +4000 +23 +18 +600 +8 +600 +13 +4000 +3000 +200 +3000 +22 +17 +400 +7 +400 +12 +12 +population +2000 +2000 +1000 +21 +100 +16 +200 +6 +200 +1990 +2000 +2010 +2020 +20 +0- +1990 +2000 +2010 +2020 +15 +0 +1990 +2000 +2010 +2020 +5 +1990 +2000 +2010 +2020 +10 +0 +1990 +2000 +2010 +2020 +10 +year +year +year +year +year +(a) +Sp +(b) +al +meantemperature +eget +den +4 +Zurich; and 2,604 C∙h in urban London, 793 C∙h in rural London. All five cities experienced +pronounced growing trends in cooling energy demand in the last three decades. The increasing +trend in CDH values has a robust association with temperature change, including increasing time- +averaged temperature, increasing peak temperatures, and heat events during the cooling season. +The increase in cooling demand occurs mainly during the cooling season, particularly in the +summer. The percentage of CDH increase in the summer period is 47.6% (Hong Kong), 56.5% +(Sydney), 72.5% (Montreal), 86.4% (Zurich), and 78.7% (London) of the total CDH increase in all +four seasons. Cities that have hot summer climates, Hong Kong and Sydney are more sensitive to +temperature change, presenting more evident growth in cooling demand. In other words, the +increased cooling load driven by climate change is placed on top of the high-demand cooling +seasons and is more evident in the high-demand cities. +Urban vs. rural areas +The level of cooling demand in urban areas is higher than that in rural areas in terms of both the +magnitudes and growth rate. The reduction of evapotranspiration and convection efficiency and the +increase of anthropogenic heat in urban areas are considered the main contributors to urban +warming and UHI17. As presented in Fig.1, the land types of urban areas of the cities have distinct +differences compared to the land types of rural areas. Urban areas have more impervious heat- +storing built-ups and less vegetation or water bodies than rural areas, meaning low water +availability and evapotranspiration in urban environments, leading to high urban-rural temperature +differences and higher cooling demand in urban areas. The convection efficiency, which is +associated with aerodynamic resistance changes, represents the heat dissipation or heat transfer +from building surfaces to the atmosphere9. High aerodynamic resistance of urban areas results in +low efficient convection in comparison with rural areas, which reduces the convection efficiency +and increases the UHI intensity and cooling demand18. +Meanwhile, the growing population density (Fig.1b) and some other socio-economic factors, such +as increased annual income and energy prices, also contribute to anthropogenic heat generation and +increasing trends of cooling demand19. Some literature has incorporated population weighting and +other weightings to analyze the socio-economic sensitivity20,21. Although it is not available to +clearly distinguish the urban and rural socio-economic factors in all five cities, the general growth +of population and GDP, and the relative higher growth in urban areas indicate potential higher +cooling demand in urban areas than the actual CDH values of Fig. 1. Re-introducing green spaces +and water surfaces into the urban area could increase both evapotranspiration and convection + + +5 +efficiency, reduce the energy demand for cooling to the electricity grid and hence reduce +anthropogenic heat emission. + +Fig. 2: (a) Urban CDH of Hong Kong, Sydney, Montreal, Zurich, and London for different base +temperatures from 22°C to 27°C. The increase in base temperature can be interpreted as potential +impact of building retrofitting (e.g., envelope insulation). (b) The number of days with maximum +air temperature ≥25, 30, 35°C in the urban areas of Hong Kong, Sydney, Montreal, Zurich, and +London, based on the calculation of climate norms stated in WMO 2017 Guidelines13. +The base temperature is a fundamental consideration in CDH analysis, which is chosen based on +the relationship between local climate, occupancy activities, building properties, and the cooling +applications in a building22. Prior research uses base temperatures ranging from 18 °C to 28 °C for +CDH calculations23–25. Upgrading the building cooling system, using higher cooling setpoints, and +improving the thermal insulation of the building could increase the base temperature of the building. +Fig. 2 (a) reports that, as the base temperature is increased, the value of CDH decreases significantly. +The cooling demand is reduced the most, by about 20%, as the base temperature is increased by +the first degree, from 22 °C to 23 °C. The results imply that a relatively small improvement by +carrying out building envelope or energy system retrofitting could achieve huge energy saving +potential. With the implementation of building passive designs and retrofitting, higher setpoint +temperatures of 27-28 °C could be adapted in future scenarios26. Renovating the existing building +stocks in terms of improving energy performance is crucial worldwide, as pointed out by the + +HongKong +Sydney +Montreal +Zurich +London +Urban Area CDH +10000 +7000 +5000 +6000 +3000 +25000 +8000 +4000 +5000 +2500 +20000 +6000 +4000 +2000 +15000 +3000 +4000 +3000 +1500 +10000 +2000 +2000 +1000 +2000 +1000 +1000 +500 +(a) +199520002005201020152020 +199520002005201020152020 +199520002005201020152020 +199520002005201020152020 +19952000 2005201020152020 +yeat +year +year +year +year +Rural Area CDH +10000 +7000 +5000 +6000 +3000 +25000 +8000 +4000 +5000 +2500 +20000 +6000 +2000 +15000 +3000 +4000 +1500 +10000 +2000 +2000 +1000 +5000 +2000 +1000 +1000 +500 +199520002005201020152020 +199520002005201020152020 +199520002005201020152020 +199520002005201020152020 +199520002005201020152020 +year +year +year +year +year +150days +(b) +100 +20 +50 +20 +10 +1990 +1990 +1990 +1990 +1990 +2000 +2000 +2000 +$S.SEE +2000 +2000 +2010 +2010 +2010 +2010 +2020 +2020 +2020 +2020 +2020 +(a) +Base +(b) +Numberofdays +temperatures: +withmaximumtemperature: +6 +Commercial Building Disclosure (CBD) program in Australia27 and the Annex projects launched +by International Energy Agency (IEA)28. Effective regulatory intervention on the building energy +retrofitting and operation codes should be a preferred instrument for policymakers aiming for +reduction of building cooling energy demand and related emissions in the building sector29. +Spikes of CDH and high-temperature events +High-temperature events can be distinguished by the number of days with maximum temperature +exceeding 25 °C, 30 °C, and 35 °C, shown in Fig. 2 (b), as proposed in the WMO Guidelines. These +events can be also seen in the yearly CDH, where spikes in CDH can be interpreted as indicators +of the occurrence of extreme heat events, for instance, heatwaves. The cooling load is more than +doubled in the years with exceptionally high frequency and high duration of summer heat events. +The increasing trend of CDH is relatively smooth in Hong Kong, Sydney, and Montreal; while the +number of spikes in CDH is more frequently seen in western Europe, e.g., London and Zurich. +Scientists have identified Europe as a ‘heatwave spot’ since its increase in heat extreme occurrences +has been much faster than for other regions in the world over the past decades30, which is due not +only to natural climate drivers, such as atmospheric circulation and jet stream states, oceanic +circulation and change of sea-surface temperatures, but also anthropogenic drivers, such as the +increasing greenhouse gas emissions. Zurich and London suffered from the record-breaking +heatwave that prevailed in Europe in the year 2003. That severe heatwave was considered to be the +warmest period of the last 500 years, which not only caused energy consumption to increase , but +also burdened health and emergency services in Europe, leading to over tens of thousands of +deaths31,32. Recently, an exceptional heatwave event affected the U.K. in July 2022, reaching 40 °C +for the first time and causing over 2,800 excess deaths in the elder population33. Urgent and +effective climate-sensitive urban planning with sustainable and resilient mitigation measures is +critical to tackling future energy demand spikes. +Methods +Cooling degree hour (CDH) calculation utilizes the outdoor air temperature by quantifying to what +degree and for how long the outdoor air temperature is higher than a base temperature with a +resolution of one hour. The mathematical expression is explained in the Equation. 1, as defined by +ASHRAE34. +CDH = (1hour) ∑ +(𝑡𝑜𝑎 − 𝑡𝑏)+ +hours + +(1) + + +7 +where 𝑡𝑏 is the base temperature and 𝑡𝑜𝑎 is outdoor ambient temperature for every hour. The +positive sign (+) above the parenthesis means that only positive values are counted. + +Data availability +The ambient temperatures and CDH dataset are accessible on the website of the Chair of Building +Physics, ETH Zurich. The dataset includes 30 years of hourly data, from 1990 to 2021, in urban +and rural areas of Hong Kong, Sydney, Zurich, Montreal, and London. 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Book (2005). + + diff --git a/x9FJT4oBgHgl3EQfhSy6/content/tmp_files/load_file.txt b/x9FJT4oBgHgl3EQfhSy6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..235f4dc5013de5f38e561e6a7c66c851673461a5 --- /dev/null +++ b/x9FJT4oBgHgl3EQfhSy6/content/tmp_files/load_file.txt @@ -0,0 +1,596 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf,len=595 +page_content='1 Three decades’ trends and extremes of building cooling demand in Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Sydney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Montreal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Zurich and London Haiwei Li a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Yongling Zhao a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' *,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Ronita Bardhan b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' *,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Pak-Wai Chan c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Dominique Derome d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Zhiwen Luo e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Diana Ürge-Vorsatz f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Jan Carmeliet a a Department of Mechanical and Process Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' ETH Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Switzerland b Department of Architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' United Kingdom c Hong Kong Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Kowloon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' China d Department of Civil and Building Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Université de Sherbrooke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Sherbrooke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Canada e Welsh School of Architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Cardiff University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Cardiff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Wales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' UK f Department of Environmental Sciences and Policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Central European University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Austria Corresponding authors: Yongling Zhao (yozhao@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='ch), Ronita Bardhan (rb867@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='uk) Abstract This brief communication interprets three decades’ evolution of building cooling demand of urban and rural areas through the lens of five representative cities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=', Hong Kong, Sydney, Montreal, Zurich, and London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The upward trend and extremes in building cooling demand, estimated from cooling degree hours (CDH) using meteorological data from 1990 to 2021, largely explained by global warming, urban heat islands, and extreme heat events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The quantification of the impact of base temperatures further reveals that 20% energy saving could be achieved by increasing one degree setpoint temperatures, which could be potentially reached by regulatory intervention on building system operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Main The building sector is responsible for 30% of global energy consumption, and 27% of total energy emissions1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' At present, 56% of the world’s population lives in urban areas, which is expected to continually increase to 68% by 20502, escalating the energy burden in cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The energy demand for space heating or cooling is determined by building design and operation, building physical properties and occupancy activities, socio-economic development, and most importantly, climatic conditions3–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Climate change encompasses global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The earth’s average surface temperature has risen by approximately 1 °C since the late 19th century6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Measures to mitigate and adapt the warming climate are in urgent need7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Furthermore, the frequency, duration, and severity 2 of extreme temperature events such as heatwaves and record-breaking high temperatures are aggravated8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Moreover, urban areas experience higher air or surface temperatures than their surrounding sub-urban or rural areas, which is called the urban heat island (UHI)9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' This phenomenon appears frequently alongside urbanization, with enlarged urban heat storage capacity, long-wave radiation trapping, reduced evapotranspiration and convection efficiency, and increased anthropogenic heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' UHI exacerbates the warming effects in cities, leading to high heat-related illness and mortality rates, increasing energy demand, air pollution associated with hot temperature and reduced air circulation, and a potential energy crisis among a large portion of the global population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Understanding the long-term evolution of cooling energy demands and the causal factors of the trends and extremes is crucial for predicting future energy emissions, formulating mitigation strategies, and facilitating the implementation of the Paris Agreement11 and Sustainable Development Goals (SDGs)12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' We analyze three decades of building cooling demand and the principal drivers, such as warming background climate by global warming, urban heat island (UHI), heatwaves, cities’ population, and some potential mitigation measures for five cities residing in different climates: Hong Kong, Sydney, Zurich, Montreal, and London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The cooling demand for urban and rural (or suburban) areas is quantified by yearly cooling degree hours (CDH) calculated from climatological standard 30-year (from 1990 to 2021) observation data13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The calculation of CDH is summarized in the Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' We remark that compared to the commonly used average temperature, CDH, a cumulated calculation measure over time, is a more solid metric to assess heat-related cooling demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 1 shows the yearly CDH in 5 cities from 1990 to 2021, demonstrating yearly cooling energy demand in all five cities during the last three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The cooling season refers to the period that requires cooling inducing an air conditioning load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The main climate driver of energy demand is the background ambient temperature during the cooling season14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Other climate factors, such as humidity levels and wind speeds, accompanied by the air temperature, reflect human heat sensation and influence the climate nexus of cooling demand16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The background climatology determines the distinct difference in the cooling demand magnitudes in five cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Hong Kong, showing CDH’s in the range of 25,000 to 33,000 \uf0b0C∙h is in the humid summer subtropical zone (Köppen climate classification: Cwa), and Sydney, showing a CDH in the range of 2,000 to 10,000 \uf0b0C∙h, is in the all-year humid subtropical zone (Cfa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Both Hong Kong and Sydney have a cooling season from spring to autumn, while the cooling season for Montreal, Zurich, and London is mostly summer, from June to August.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Montreal (1,000-6,000 \uf0b0C∙h) is in the humid continental zone (Dfb);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Zurich 3 (1,000-7,000 \uf0b0C∙h)and London (0-3,000 \uf0b0C∙h) belong to the marine west coast (oceanic) climate zone (Cfb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The results show that the warmer subtropical cities, Hong Kong and Sydney, have distinctly higher cooling demand and longer cooling seasons than continental and oceanic climate cities, Montreal, Zurich, and London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Also, the influence of latitude is clearly observed for the last three cities, showing lower cooling demand for the higher latitude city London compared to Montreal and Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 1: (a) Yearly cooling degree hours (CDH) of urban and rural areas (or suburban) in Hong Kong, Sydney, Montreal, Zurich, and London from 1990 to 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The base temperature for CDH calculation is 22 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The graphs in the center of each circular plot display the land types about 10 km around the weather stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Remark the different maximum scale for Hong Kong (40,000) compared to the other cities maximum of 8,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' (b) Population growth (bars) and annual mean urban air temperature rise (lines) from 1990 to 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' We remark that the sub-figures have different scales of the y-axis for different cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Statistically, average CDH increase for each decade is 14,619 \uf0b0C∙h in urban Hong Kong, 16,997 \uf0b0C∙h in rural Hong Kong;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 15,907 \uf0b0C∙h in urban Sydney, 13,491 \uf0b0C∙h in rural Sydney;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 5,783 \uf0b0C∙h in urban Montreal, 3,285 \uf0b0C∙h in rural Montreal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 7184 \uf0b0C∙h in urban Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='365 \uf0b0C∙h in rural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='HongKong ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='Sydney ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='Montreal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='Zurich ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='London ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='(Cwa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='(Cfa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='(Dfb) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='(Cfb) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='(Cfb) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='97 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='532 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='0008 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='3062 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='6000 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 4000 30000 8000 8000 6000- 8000 8000 6000 6000 20000- 4000 4000 A 4000 4000 10000- Rural Area CDH 2000 2000 273 722 4546 287 3826 2 2125 510 27001 21873 6081 4125 1143 99 234 19024 4963 6060 1787 1347 1530 270 25392 1893 2565 2138 613 6008 043 3955 2954 99 54 2 A density (people/km) 400 20 6000 15 6000 24 19 800 800 5000 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='5000 (b) OOE 4000 23 18 600 8 600 13 4000 3000 200 3000 22 17 400 7 400 12 12 population 2000 2000 1000 21 100 16 200 6 200 1990 2000 2010 2020 20 0- 1990 2000 2010 2020 15 0 1990 2000 2010 2020 5 1990 2000 2010 2020 10 0 1990 2000 2010 2020 10 year year year year year (a) Sp (b) al meantemperature eget den 4 Zurich;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' and 2,604 \uf0b0C∙h in urban London, 793 \uf0b0C∙h in rural London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' All five cities experienced pronounced growing trends in cooling energy demand in the last three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The increasing trend in CDH values has a robust association with temperature change, including increasing time- averaged temperature, increasing peak temperatures, and heat events during the cooling season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The increase in cooling demand occurs mainly during the cooling season, particularly in the summer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The percentage of CDH increase in the summer period is 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='6% (Hong Kong), 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='5% (Sydney), 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='5% (Montreal), 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='4% (Zurich), and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='7% (London) of the total CDH increase in all four seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Cities that have hot summer climates, Hong Kong and Sydney are more sensitive to temperature change, presenting more evident growth in cooling demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' In other words, the increased cooling load driven by climate change is placed on top of the high-demand cooling seasons and is more evident in the high-demand cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Urban vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' rural areas The level of cooling demand in urban areas is higher than that in rural areas in terms of both the magnitudes and growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The reduction of evapotranspiration and convection efficiency and the increase of anthropogenic heat in urban areas are considered the main contributors to urban warming and UHI17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' As presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='1, the land types of urban areas of the cities have distinct differences compared to the land types of rural areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Urban areas have more impervious heat- storing built-ups and less vegetation or water bodies than rural areas, meaning low water availability and evapotranspiration in urban environments, leading to high urban-rural temperature differences and higher cooling demand in urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The convection efficiency, which is associated with aerodynamic resistance changes, represents the heat dissipation or heat transfer from building surfaces to the atmosphere9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' High aerodynamic resistance of urban areas results in low efficient convection in comparison with rural areas, which reduces the convection efficiency and increases the UHI intensity and cooling demand18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Meanwhile, the growing population density (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='1b) and some other socio-economic factors, such as increased annual income and energy prices, also contribute to anthropogenic heat generation and increasing trends of cooling demand19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Some literature has incorporated population weighting and other weightings to analyze the socio-economic sensitivity20,21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Although it is not available to clearly distinguish the urban and rural socio-economic factors in all five cities, the general growth of population and GDP, and the relative higher growth in urban areas indicate potential higher cooling demand in urban areas than the actual CDH values of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Re-introducing green spaces and water surfaces into the urban area could increase both evapotranspiration and convection 5 efficiency, reduce the energy demand for cooling to the electricity grid and hence reduce anthropogenic heat emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 2: (a) Urban CDH of Hong Kong, Sydney, Montreal, Zurich, and London for different base temperatures from 22°C to 27°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The increase in base temperature can be interpreted as potential impact of building retrofitting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=', envelope insulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' (b) The number of days with maximum air temperature ≥25, 30, 35°C in the urban areas of Hong Kong, Sydney, Montreal, Zurich, and London, based on the calculation of climate norms stated in WMO 2017 Guidelines13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The base temperature is a fundamental consideration in CDH analysis, which is chosen based on the relationship between local climate, occupancy activities, building properties, and the cooling applications in a building22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Prior research uses base temperatures ranging from 18 °C to 28 °C for CDH calculations23–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Upgrading the building cooling system, using higher cooling setpoints, and improving the thermal insulation of the building could increase the base temperature of the building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 2 (a) reports that, as the base temperature is increased, the value of CDH decreases significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The cooling demand is reduced the most, by about 20%, as the base temperature is increased by the first degree, from 22 °C to 23 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The results imply that a relatively small improvement by carrying out building envelope or energy system retrofitting could achieve huge energy saving potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' With the implementation of building passive designs and retrofitting, higher setpoint temperatures of 27-28 °C could be adapted in future scenarios26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Renovating the existing building stocks in terms of improving energy performance is crucial worldwide,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' as pointed out by the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='HongKong ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='Sydney ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='Montreal ' metadata={'source': 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+page_content='199520002005201020152020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='19952000 2005201020152020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='yeat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='year ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='$S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='SEE 2000 2000 2010 2010 2010 2010 2020 2020 2020 2020 2020 (a) Base (b) Numberofdays temperatures: withmaximumtemperature: 6 Commercial Building Disclosure (CBD) program in Australia27 and the Annex projects launched by International Energy Agency (IEA)28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Effective regulatory intervention on the building energy retrofitting and operation codes should be a preferred instrument for policymakers aiming for reduction of building cooling energy demand and related emissions in the building sector29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Spikes of CDH and high-temperature events High-temperature events can be distinguished by the number of days with maximum temperature exceeding 25 °C, 30 °C, and 35 °C, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 2 (b), as proposed in the WMO Guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' These events can be also seen in the yearly CDH, where spikes in CDH can be interpreted as indicators of the occurrence of extreme heat events, for instance, heatwaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The cooling load is more than doubled in the years with exceptionally high frequency and high duration of summer heat events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The increasing trend of CDH is relatively smooth in Hong Kong, Sydney, and Montreal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' while the number of spikes in CDH is more frequently seen in western Europe, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=', London and Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Scientists have identified Europe as a ‘heatwave spot’ since its increase in heat extreme occurrences has been much faster than for other regions in the world over the past decades30, which is due not only to natural climate drivers, such as atmospheric circulation and jet stream states, oceanic circulation and change of sea-surface temperatures, but also anthropogenic drivers, such as the increasing greenhouse gas emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Zurich and London suffered from the record-breaking heatwave that prevailed in Europe in the year 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' That severe heatwave was considered to be the warmest period of the last 500 years, which not only caused energy consumption to increase , but also burdened health and emergency services in Europe, leading to over tens of thousands of deaths31,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Recently, an exceptional heatwave event affected the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' in July 2022, reaching 40 °C for the first time and causing over 2,800 excess deaths in the elder population33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Urgent and effective climate-sensitive urban planning with sustainable and resilient mitigation measures is critical to tackling future energy demand spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Methods Cooling degree hour (CDH) calculation utilizes the outdoor air temperature by quantifying to what degree and for how long the outdoor air temperature is higher than a base temperature with a resolution of one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The mathematical expression is explained in the Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' 1, as defined by ASHRAE34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' CDH = (1hour) ∑ (𝑡𝑜𝑎 − 𝑡𝑏)+ hours (1) 7 where 𝑡𝑏 is the base temperature and 𝑡𝑜𝑎 is outdoor ambient temperature for every hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The positive sign (+) above the parenthesis means that only positive values are counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Data availability The ambient temperatures and CDH dataset are accessible on the website of the Chair of Building Physics, ETH Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The dataset includes 30 years of hourly data, from 1990 to 2021, in urban and rural areas of Hong Kong, Sydney, Zurich, Montreal, and London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' The weather stations are Hong Kong Observatory Headquarters (Hong Kong urban, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='30° N, 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='17° E), Ta Kwu Ling (Hong Kong rural, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='53° N, 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='16° E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Bankstown (Sydney urban, -33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='92° S, 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='98° E), Observatory Hill (Sydney rural, -33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='86° S, 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='20° E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Trudeau (Montreal urban, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='47° N, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='74° W), Mirabel (Montreal rural, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='68° N, 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='04° W);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Zurich Kaserne (Zurich urban, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='38° N, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='53° E), Kloten (Zurich rural, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='29° N, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='32° E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' St James’ Park (London urban, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='50° N, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='23° W), Kenley (London rural, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='303° N, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='09° W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' IEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' Tracking buildings 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf'} +page_content='iea.' 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0000000000000000000000000000000000000000..503b54f4d671839e129aeedc1b19b02a6d686a47 --- /dev/null +++ b/xdAyT4oBgHgl3EQfa_cC/content/tmp_files/2301.00251v1.pdf.txt @@ -0,0 +1,1613 @@ +Feature Selection for Personalized Policy Analysis +Maria Nareklishvili +Booth School of Business +University of Chicago +Nicholas Polson* +Booth School of Business +University of Chicago +Vadim Sokolov +Department of Systems Engineering +and Operations Research +George Mason University +First Draft: July 5, 2022 +This Draft: December 31, 2022 +Abstract +In this paper, we propose Forest-PLS, a feature selection method for analyzing pol- +icy effect heterogeneity in a more flexible and comprehensive manner than is typically +available with conventional methods. In particular, our method is able to capture pol- +icy effect heterogeneity both within and across subgroups of the population defined +by observable characteristics. To achieve this, we employ partial least squares to iden- +tify target components of the population and causal forests to estimate personalized +policy effects across these components. We show that the method is consistent and +leads to asymptotically normally distributed policy effects. To demonstrate the effi- +cacy of our approach, we apply it to the data from the Pennsylvania Reemployment +Bonus Experiments, which were conducted in 1988-1989. The analysis reveals that fi- +nancial incentives can motivate some young non-white individuals to enter the labor +market. However, these incentives may also provide a temporary financial cushion +for others, dissuading them from actively seeking employment. Our findings high- +light the need for targeted, personalized measures for young non-white male partici- +pants. +1 +Introduction +Randomized control trials play an important role for treatment, policy, or a program ef- +fect analysis in economics, statistics, medicine, and other fields (Banerjee and Duflo, 2009; +*Email: ngp@chicagobooth.edu +1 +arXiv:2301.00251v1 [econ.EM] 31 Dec 2022 + +Bertrand and Duflo, 2017; Chernozhukov et al., 2018a). To design and implement effective +interventions, policymakers and researchers are often interested in partitions of the pop- +ulation that are particularly susceptible to a new program or a policy. Identifying such +subgroups can be challenging, especially when there are a large number of observable +characteristics that can influence the outcome. In such cases, conventional estimation +methods, such as ordinary least squares, may produce inefficient estimates. This is be- +cause these methods often struggle to identify relevant variables in a sparse feature space +(Johnstone and Titterington, 2009; Belloni and Chernozhukov, 2013). +We propose Forest-PLS, a data-driven approach for selecting the target components of +the population for personalized policy analysis. In our approach, the target components +represent linear combinations of the explanatory variables. These components are char- +acterized by large weights on the variables that are most strongly associated with the out- +come, and carry a significant amount of information. As a second step, we identify and +estimate personalized policy effects across the chosen components. By focusing on these +key components, rather than considering the full set of characteristics, policymakers can +design targeted interventions tailored to the most diversified segments of the population. +The procedure is a combination of two distinct methods, the partial least squares +(Geladi and Kowalski, 1986; Vinzi et al., 2010) and causal forest (Athey and Imbens, 2016; +Wager and Athey, 2018) algorithms. The partial least squares method is used to detect the +policy-relevant components in the first step. These components represent a reduced ex- +planatory variable space. The reduced space reflects the highest variation in explanatory +features and the most relevant information for predicting the outcome. These compo- +nents are continuous rather than discrete clusters of the data space, allowing us to ana- +lyze policy effects across a full spectrum of the population segments. In the second step, +we use the causal forest algorithm to identify different quantiles of policy effects within +each component value. This allows us to capture heterogeneity in the policy effects at a +finer granularity. +The primary contribution of this paper is to advance our understanding of the distri- +bution of policy effects from a theoretical and empirical perspective. The theoretical com- +ponent of the article demonstrates that our approach is consistent and leads to asymp- +totically normally distributed policy effects. Our framework and findings extend beyond +a single coefficient of interest (Wager and Athey, 2018) to multiple (plausibly) correlated +policy effects. The empirical contribution of the paper is to identify and analyze two +types of heterogeneity of the policy effects: within-group heterogeneity and between- +group heterogeneity. The proposed method allows us to estimate quantiles of treatment +effects within and across the values of target components. Analyzing individual explana- +tory variables separately (Meinshausen and Ridgeway, 2006) can be challenging in high- +dimensional data. Our approach allows us to focus on aggregate aspects of these charac- +teristics without losing the economic interpretation of the resulting subgroups. +Our framework is closely related to the papers dedicated to the estimation of per- +sonalized treatment effects (Athey and Imbens, 2015, 2016; Wager and Athey, 2018; Cher- +nozhukov et al., 2018a,b; K¨unzel et al., 2019; Hahn et al., 2020; Nie and Wager, 2021; Xiong +et al., 2021), and feature selection for the inference on treatment effects (Belloni et al., 2012, +2014; Chernozhukov et al., 2015b,a; Urminsky et al., 2016; Banerjee et al., 2021). Previous +work for personalized treatment effect analysis considers a single source of heterogeneity, +2 + +such as quantile treatment effects, or treatment effects across the original set of covariates +(features). We unify the feature selection methods with personalised policy analysis. This +allows us to investigate a full density of policy effects within and across a pooled variable +space (target components). +Other related methods are proposed by Hahn et al. (2002); Chun and Keles¸ (2010); +Mehmood et al. (2012, 2020); Polson et al. (2021); Nareklishvili et al. (2022); Dixon et al. +(2022) that use the partial least squares algorithm for feature selection. These methods +suggest that partial least squares as a precursor to a more general framework of deep +learning and instrumental variables can increase efficiency. By comparison, our study +shows that the method can pool statistically and economically significant variables for +policy effect heterogeneity. Additionally, Nekipelov et al. (2018); Li (2020); Nareklishvili +(2022) investigate large sample properties for random forests under multiple outcomes, +coefficients or network effects. We show that the theoretical properties hold even after the +feature selection procedure by partial least squares. It is important to note that Bayesian +approach is an alternative to policy effect heterogeneity. Ansari et al. (2000); Taddy et al. +(2016); Santos and Lopes (2018); Hahn et al. (2020); Woody et al. (2020); Starling et al. +(2021); Krantsevich et al. (2022) formulate Bayesian Additive Regression Trees (BART) for +heterogeneous treatment or policy effect analysis. The advantage of the approach lies in +the regularization effect through predetermined priors of the tree parameters. Our work +can be extended to accommodate Bayesian priors. +This article examines the impact of financial incentives on unemployment duration +based on data from the Pennsylvania ”Reemployment Bonus” Demonstration, a random- +ized control trial conducted in 1988-1989. The analysis reveals significant variation in the +policy effects both within and across different subgroups of the population. Specifically, +the results show that the effects of the policy are more significantly dispersed for young, +non-white male claimants who joined the experiment early on, compared to middle-age +and older female participants with a high number of dependents. The difference between +the 97.5th and 2.5th percentiles of policy effects is 92.8% for the first vigintile of the target +component, and decreases to 22.1% for the final vigintile of the component. These find- +ings highlight the need for targeted, personalized measures for specific subgroups, such +as young non-white male participants. +2 +The Forest-PLS Framework +Consider the outcome yi ∈ R (e.g., unemployment duration) for a subject i = 1, . . . , N. +Each subject is characterized with an observable vector of features Xi ∈ Rp (e.g., age, +gender, occupation). We assume that p = Nψ, with ψ < 1. A policy is denoted by +Pi ∈ {0, 1}, and we let yi(1) and yi(0) denote potential outcomes with and without the +policy, respectively. We assume, the unconfoundedness holds: +Assumption 2.1 (Unconfoundedness). The policy is independent of the potential outcomes: +yi(1), yi(0) ⊥⊥ Pi|Xi. +A policymaker wishes to identify the dimensions of the feature space that contain the +most relevant information about the policy effects. To this end, we consider a mapping +3 + +f : Xi �→ Ci ∈ Rq that maps the original features to a set of target components Ci. In +other words, Ci is a collection of q-dimensional linear combinations of the features xj for +j = 1, . . . , p (with q ≤ p). This transformation allows us to focus on a smaller, more +interpretable set of features while preserving the information about the policy effects. +The coefficient of interest is the effect of Pi on the outcome: +θi = yi(1) − yi(0), +Personalized policy effects are not directly observable, as an individual is only ex- +posed to one policy state (either with or without the policy). Therefore, we typically +consider the expectations of the potential outcomes: +θ(Ci) = E(θi|Ci) + εi, +E(θi|Ci) = E +� +yi(1) − yi(0) +��Ci]. +Due to Assumption 2.1, the policy effect of interest is given as +τ(Ci) = E +� +yi(1) − yi(0)|Ci +� = E +� +yi|Pi = 1, Ci +� − E +� +yi|Pi = 0, Ci +� +, +(1) +where E +� +yi|Pi = d, Ci +� +for d ∈ {1, 0} denotes the observed expected outcomes with and +without the policy, respectively. +To estimate the average policy effect in (1), we need to determine the optimal target +components Ci and use a method that estimates group-level policy effects, conditional on +the chosen components. +2.1 +Identification of Target Components +We seek to identify the linear combinations of features, also known as target compo- +nents/scores/factors, C = [c1, c2, . . . , cq] (i.e., c1 = Xw1) that explain the highest varia- +tion in covariates X, as well as the outcome y (Tobias et al., 1995; Abdi, 2003). X and y can +be decomposed as: +X +(N×p) = +C +(N×k) VT +(k×p) + +E +(N×p), +(2) +y +(N×1) += +C +(N×k) +b +(k×1) + +e +(N×1), +(3) +where X = [x1, x2, . . . , xp] is the matrix of covariates, VT is the matrix of loadings (weights), +and E is the matrix of errors for the covariates. y denotes the outcome as before, b is the +vector of coefficients (the influence of components on the outcome), and e is the vector of +errors for the response. +We use the iterative procedure to obtain the target components (aka partial least squares). +Consider, the weight ˜w1 = +� +cov(x1, y), cov(x2, y), . . . , cov(xp, y) +� = +� ˜w11, ˜w21, . . . , ˜wp1 +� +. +We normalize it to get a unit vector: +w1 = +˜w1 +|| ˜w1||, +4 + +where || ˜w1|| denotes the Euclidean norm. We use these weights to compute the first +principal component: +c1 = w11x1 + w21x2 + · · · + wp1xp = Xw1 = Xw1 +wT +1 w1 +. +(4) +The last equality in (4) follows by the fact that the weights are unit vectors. A linear +regression of a j−th covariate on the first component yields a loading. The vector of +loadings associated with the first component is given by regressing the covariates on it: +v1 = XTc1 +cT +1 c1 +. +(5) +Similarly, the first coefficient b1 is obtained by regressing the outcome on the first +component: +b1 = yTc1 +cT +1 c1 +. +(6) +The next step is to obtain the approximation of the covariate matrix and the outcome, +and predict residuals: +X1 = X − ˆX, y1 = y − ˆy, +(7) +where ˆX = c1vT +1 and ˆy = b1c1. We obtain the subsequent components by repeating +the described procedure for the first, second, and higher order residuals of the covariate +matrix X1, X2 . . . , and the outcome y1, y2 . . . , respectively. +A desirable property of the procedure is that the coefficients have a closed-form so- +lution. The estimator of these coefficients is given as (Helland, 1990; Stone and Brooks, +1990): +�b = ˆR( ˆRTSxx ˆR)−1 ˆRTsxy, +(8) +where ˆR = (sxy, Sxxsxy, . . . , Sq−1 +xx sxy) is the p × q matrix of the Krylov sequence with a +p × p matrix Sxx and a p × 1 vector sxy defined as follows: +Sxx = XT(I − 11T/N)X +N − 1 +, +sxy = (X − E(X))T(y − E(y)) +N − 1 +, +where I is an identity matrix and 1 is a matrix of ones. Intuitively, the algorithm searches +for factors that capture the highest variability in X, and at the same time maximizes the +covariance between X and y. If the number of components equals the dimension of the +covariates, q = p, the method is equivalent to the ordinary least squares (Helland, 1990). +5 + +2.2 +Estimation of Personalized Policy Effects +To identify smaller subgroups within the population, we use a causal forest algorithm. A +tree in causal forests recursively partitions the feature space, in this setting, the space of +identified target components C, and makes axis-aligned splits to estimate the conditional +mean of the outcome µ(c) = E(yi|Pi, Ci = c) at a point c for Pi ∈ {0, 1}. +An axis-aligned split is a pair s = (j, c), where j = 1, . . . , q is a specific component +(the splitting coordinate) and c ∈ R is the corresponding value (the splitting index). The +recursive partitioning procedure begins by considering the set P(0) = C ∈ Rq (the parent +node of the tree). For this set, we select the splitting coordinate j : 1 ≤ j ≤ q and the +splitting index c that divide P(0) into two non-overlapping rectangles (child nodes): +P(1,1) = P(0) ∩ {�c ∈ P(0) : �cj ≤ c} and P(1,2) = P(0) ∩ {�c ∈ P(0) : �cj > c}, +(9) +After the first split, the process is repeated for P(1,1) and P(1,2)separately until the desired +level of partitioning is achieved. +The sequence of k splits defines a partition of the component space C, which we de- +note by Π. This partition (or equivalently, a tree) consists of non-overlapping rectangular +regions ℓn called the leaves or terminal nodes of the tree. These leaves represent the final +subgroups or subpopulations identified by the algorithm. The union of all these partitions +is the entire component space: +Π = {ℓ1, ℓ2, . . . , ℓ|Π|} and ∪|Π| +n=1 ℓn = C. +Athey and Imbens (2016) propose a method for estimating heterogeneous policy ef- +fects under the assumption of unconfoundedness. To implement this method, we split +the data into two different samples: a training sample Str used to build and find the split- +ting variables and values, and an estimation sample Sest used to estimate policy effects +across different subgroups of the population. The unbiased sample analogue of E(θ(Ci)) +is denoted as follows: +�θ(Ci, Sest, Π) = +|Π| +∑ +n=1 +� +1(c ∈ ℓn, Pi = 1) +1 +|i : Ci ∈ ℓn, Pi = 1| ∑ +i:Ci∈ℓn +Yi(1)− +1(c ∈ ℓn, Pi = 0) +1 +|i : Ci ∈ ℓn, Pi = 0| ∑ +i:Ci∈ℓn +Yi(0) +� +, +(10) +where |Π| is the total number of the terminal nodes. 1(c ∈ ℓn, Pi = d) is a binary variable +and equals one when, for a given d ∈ {0, 1}, a generic test data point c belongs to a +terminal leaf ℓn, and zero otherwise. Additionally, let Σ be the variance of �θ(Xi, Sest, Π). 1 +1While our analysis is based on a single policy variable and a single outcome, the proposed framework +can handle multiple policy variables and outcomes with correlated coefficients. +6 + +To estimate policy effects from the available data, we aim to maximize the variance of +the policy effect estimator: +ˆθ(Ci, Sest, Π) = arg max +�θ +1 +Ntr ∑ +ℓ +Ntr +ℓ �θ(Ci, Π)T �Σ−1�θ(Ci, Π). +(11) +The proof of (11) is provided in Appendix A.1. Intuitively, the objective function in (11) +encourages the causal forest algorithm to search for subsets of target components with the +highest variation in policy effects. To further increase the robustness of the estimates, we +build multiple trees on bootstrapped data and average the resulting coefficients. This +approach, known as the causal forest algorithm, has been described in detail by Athey +and Imbens (2016) and Wager and Athey (2018). +3 +Large Sample Properties +To show the asymptotic normality of the estimated policy effects, we need to make certain +assumptions about the underlying data-generating process. +Assumption 3.1 (Data Generating Process). Let y = g(b0 + Xb) + ε where b is a p × 1 +vector of coefficients, b0 is a constant and g is a non-linear mapping. Assume, X have a joint +Elliptical distribution with the mean µX and a variance ΣXX. Assume X is independent of ε. +Moreover, let Sxx and sxy converge in probability to ΣXX (the population variance of X) and +σXy (the population covariance of X and y) when N → ∞. Moreover, let there exist a pair +of eigenvectors and eigenvalues (vj, λj) for which σXy = ∑M +j=1 γjvj (with γj non-zero for each +j = 1, . . . , M). Assume also E(|g(U)|) < ∞ and E(U|g(U)|) < ∞ with U = b0 + Xb and q = +M. +Under Assumption 3.1, the relation between the response and the independent charac- +teristics follows a predetermined functional form. Additionally, the subject characteristics +are assumed to have an elliptical distribution, meaning they are shaped like an ellipse in +a multi-dimensional coordinate system. While this assumption is not always satisfied in +practice, it has been shown that the results obtained under this assumption do not signifi- +cantly differ from those obtained when the features have other types of distributions (see +Brillinger, 2012). +Lemma 3.1. Let Assumption 3.5 hold. Then ˆb in (8) is consistent up to a proportionality constant. +The proof of Lemma 3.1 is provided in Appendix A.2. Lemma 3.1 shows that the +identified target components are consistent. The causal forest method described in this +article relies on the same assumptions as those introduced by Wager and Athey (2018). +One of them is the ”honesty” of the tree. +Assumption 3.2 (Honesty). The outcome yi and the splitting parameters (the splitting coor- +dinates and indices, s = (j, c)) are independent of each other, conditional on the observed com- +ponents Ci. This independence holds for each subject i whose outcome yi is used in the final +7 + +prediction: +F(yi|Ci, s) = F(yi|Ci). +F denotes the density of the outcome variable.2 +There are various ways to satisfy Assumption 3.2. In this article, we use a two-sample +approach, where we split the data into a training sample Str and an estimation sample +Sest. The splitting coordinates and indices (s) of the trees are determined based on the ob- +servations in Str, while the predicted outcomes are based on the observations in Sest. This +separation of the data into two different samples ensures that the splitting parameters +and the outcomes are independent of each other. +Assumption 3.3 (Random Split Trees). At each recursive step, the probability of choosing the +j-th component as the splitting coordinate is lower bounded by π/d for π ∈ (0, 1] and for all +j = 1, . . . , q. +In order to guarantee the consistency of the causal forest method, it is necessary for +the leaves of the trees to become small in all dimensions of the component space as the +sample size N increases. To ensure this, we adopt Assumption 3.3, which is based on the +assumptions of Meinshausen and Ridgeway (2006) and Wager and Athey (2018). This +assumption states that for all splitting steps, each component has a probability of at least +π/d of being selected as the splitting coordinate, for some 0 < π ≤ 1. +Assumption 3.4 (The Splitting Algorithm is (α, k)-regular). There exists a positive constant +α such that at each split, at least a fraction α of the available training examples are left on each side +of the split. Additionally, we require that the splitting process ceases at a node when it contains +less than k observations for some k ∈ N. +Assumption 3.4 ensures that each half-space produced by a split in the tree construc- +tion process contains a sufficient number of observations. As shown by Wager and Walther +(2015), this assumption also implies that the half-spaces are large in Euclidean volume. +Assumption 3.4 places an upper bound on the number of observations that can be con- +tained in a terminal node of the tree. Specifically, when a tree is fully grown to depth k, we +have that each terminal node contains between [k, 2k − 1] observations. One important +consequence of this assumption is that it places an upper bound on the variance of the +tree estimator at any test point c. +Assumption 3.5 (Distributional Assumptions on the Data Generating Process). The target +components Ci are supported on the unit cube Ci ∈ [0, 1]p, and the density of these components is +bounded away from zero and infinity. The first and second moments of the outcome, E(yi|Ci = c) +and E +� +y2 +i |Ci = c +� +, are Lipschitz-continuous functions of the target components. The variance of +the outcome, Var(yi|Ci = c), is bounded away from zero for all values of the target components. +Specifically, we have in fc∈CVar(yi|Ci = c) > 0. +2If we have access to multiple outcomes, this assumption holds for each one individually. +8 + +Lipschitz continuity and bounded variances are widely used assumptions in the field +of statistics and machine learning (Wager and Athey, 2018; Biau, 2012). In the context +of this paper, the results do not depend explicitly on the distributional assumptions of +Ci, however, they affect the constants that we carry throughout this paper (constants +borrowed from Lemma 2 and Theorem 3 in Section 3.2 in Wager and Athey, 2018). +Assumption 3.6 (Overlap). Let 0 < ϵ < 1, and consider any element c ∈ [0, 1]q. Then the +following holds: +ϵ < P(Pi = 1 | Ci = c) < 1 − ϵ. +Assumption 3.6 ensures that, as the number of observations N increases, there will be +a sufficient number of subjects with and without a policy at any given test point c. +In their work, Wager and Athey (2018) derive the lower bound of the variance of +the Hajek projection of a conventional random forest, ˙F(c, A1, . . . , AN), and demonstrate +that it converges to zero. We follow a similar approach in this article, with the objective +of showing that Σ−1/2 �F(c, A1, . . . , AN) − ˙F(c, A1, . . . , AN) +� +converges in squared mean +(Σ denotes the variance of the random forest projection; see Appendix A.3 for definitions +and further details). For simplicity, we denote F(c, A1, . . . , AN) and ˙F(c, A1, . . . , AN) as +F and ˙F, respectively. It is important to mention that the theoretical properties are shown +for the conventional random forest estimator as by Wager and Athey (2018). The proofs, +however, generalize well to the coefficients, such as policy effects. Wager and Athey +(2018) discuss additional details. +Lemma 3.2 (Hajek Projection). The Hajek projection of a random forest estimator and the co- +variance of this projection are given as: +˙F(c, A1, . . . , AN) − µ = s +N +N +∑ +i=1 +�T1(Ai) − µ +� +, +Σ = s +NV ˙ +(T ) ∈ RM×M, +where AN +i=1 = {yi, Ci}N +i=1 defines data, s = Nβ with β sufficiently close to one, and M is +the dimension of the outcome variables in each terminal node (M = 1 in this article). V de- +notes the covariance matrix of the projected elements of the tree. Lastly, ˙T = ∑s +i=1 T1(Ai) with +T1(a) = Eξ,A2,...,ANT (c, ξ, a, A2, . . . , AN) is the Hajek projection of a tree T (c, A1, . . . , AN) = +EξT (c, ξ, A1, . . . , AN) ∈ RM. +Lemma 3.3 (Error Bound). The mean squared difference between F and ˙F has the following +upper bound: +E +�F − ˙F +�TΣ−1�F − ˙F +� ≤ s +N tr +�� +V( ˙T ) +�−1V(T ) +� +, +where tr is a trace operator, and V(T ) and V( ˙T ) denote the variance of a vector-valued tree and +its’ Hajek projection, respectively. +9 + +Lemma 3.2 derives the explicit expression for the projection of a forest and its’ vari- +ance, respectively. Lemma 3.3 illustrates the upper bound of the squared deviation be- +tween the random forest (for multiple policy effects) and its’ projection. The proofs are +provided in Subsections A.3.2 and A.3.1 of Appendix A.3, respectvely. The Hajek projec- +tion of a random forest meets Lindeberg central limit theorem conditions, and is asymp- +totically normally distributed. As a result, by showing that this bound converges to zero +in the limit, we establish the asymptotic normality of the random forest estimator. +Proposition 3.1. The entries of V(T ) are bounded, and its diagonal elements are bounded away +from zero. Additionally, the lower bound of the off-diagonal terms of V( ˙T ) is on the order of +o +� +1 +logq(s) +� +. As a result, the upper bound in Lemma 3.3 approaches zero as N tends towards infinity: +s +N tr +�� +V( ˙T ) +�−1V(T ) +� +−→ 0 when N → ∞. +Proof. The boundedness of the elements of V(T ) is a direct consequence of the proposed +assumptions. Specifically, according to Assumption 3.4, the number of observations in +each terminal node is upper bounded, which implies that the variance of the tree is upper +bounded by a constant times V(yi|Ci = c). Furthermore, Assumption 3.5 ensures that +V(yi|Ci = c) is lower bounded away from zero. +In this proof, we utilize the result from Wager and Athey (2018) concerning the asymp- +totic behavior of the variance terms. In our framework with q−dimensional target com- +ponents: +V( ˙T )ii = +K +logq(s), for some constant K. +(12) +V( ˙T )ii denotes the diagonal terms of the variance of the projection of a tree estimator. We +show that the off-diagonal terms V( ˙T )ij = o +� +1 +logq(s) +� +for all i ̸= j. +We begin by defining the Hajek projection of a tree: +˙T − µ = +s +∑ +i=1 +E(T |Ai) +(13) +Since the observations are i.i.d., then: +V +� ˙T +� = sV +� +E(T |A1 +� +. +(14) +It is clear to see that: +V +� +E(T |A1) +� = V +� +E(T |A1) − E(T |C1) +� + V +� +E(T |C1 +� +. +(15) +Let us consider the m-th outcome variable, where m = 1, . . . , M. As the tree is hon- +est, the diagonal terms in equation (15) simplify to the following form (see the Proof of +Theorem 5 in Wager and Athey, 2018): +V +� +E(T |A1) − E(T |C1) +� +mm = V +� +E(Sℓn|C1)(y1m − E(y1m|C1) +� +mm ≈ +E +�� +E(Sℓn|C1) +�2� +E +�� +y1m − E(y1m|C1) +�2� = +(16) +E +�� +E(Sℓn|C1) +�2� +Var(ym|C1 = c), +10 + +and +V +� +E(T |C1) +� +mm = E +�� +E(Sℓn|C1) +�2� +Var +� +E(ym|C1 = c) +� +. +(17) +where Tm is the estimator of a tree at a test point c. Sℓn is the indicator function and equals +one if C1 ∈ ℓn(c, Π), and zero otherwise. +The off-diagonal terms equal to: +V +� +E(T |A1) − E(T |C1) +� +mm′ = +E +�� +E(Sℓn|C1) +�2� +E +�(y1m − E(y1m|C1))(y1m′ − E(C1)) +� +. (18) +Under Assumption 3.5, the variance of each outcome variable is lower bounded away +from zero. By applying the Cauchy-Schwarz inequality, we see that the absolute value of +the covariance between yim and yim′ given C1 is also lower bounded away from zero: 3 +|Cov(y1m, y1m′|C1)| ≤ +� +Var(y1m|C1 = c)Var(y1m′|C1 = c). +In our setting, E +�� +E(Sℓn|C1) +�2� +, is lower bounded as (Wager and Athey, 2018): +E +�� +E(Sℓn|C1) +�2� ≥ +(q − 1)! +2q+1 logq(s) · 1 +ks, +(19) +where k is the minimum number of observations in a given terminal node. Combining +(14) and (19) yields the order of diagonal and off-diagonal terms: +V +� ˙�T +� +mm = o +� +1 +logq(s) +� +, and V +� ˙�T +� +mm′ = o +� +1 +logq(s) +� +. +(20) +In the following, we prove that s +N tr +�� +V( ˙�T) +�−1 +V(�T) +� +−→ 0 in a more general frame- +work. Let B and D be two square matrices with diagonal elements bii and dii, and non- +diagonal elements bij and dij, respectively. We assume that these matrices possess the +following properties: +1. dii ≥ η for some η ∈ R+ and for all i = 1, . . . M, +(21) +2. bii ≥ +dii +log(N), +(22) +3. bij = o +� +1 +log(N) +� +. +(23) +3An alternative argument is to notice that the term in the integrand consists of multiples of the first +and second moments of the outcome variables y1m and y1m′. Since these moments are continuous, they are +bounded. Thus, their expectation is also bounded. +11 + +Then we show that s +N tr(B−1D) −→ 0. Recall that the Leibniz formula for the determi- +nant is given as follows: +det(B) = ∑ +π +� +sgn(π) +M +∏ +i=1 +bi,πi +� +, +(24) +where π is a permutation function that reorders the set {1, . . . , M}. Diagonal and off- +diagonal terms are on the same order. Therefore, det(B) is asymptotically equivalent to +either ∏M +i=1 bii or ∏M +i=1 bij where i ̸= j. For simplicity, we keep the notation that det(B) ∼a +∏M +i=1 bii, where ” ∼a ” denotes asymptotic equivalence. Based on Cramer’s rule, we can +write i-th diagonal term of the inverse of B: +(B−1)ii = det(B−i) +det(B) . +B−i is the matrix where we remove the i-th row and the i-th column. By the same argu- +ment, det(B−i) ∼a ∏M−1 +j=1 bjj. Then we end up with: +(B−1)ii ∼a ∏M−1 +i=1 bjj +∏M +j=1 bii += 1 +bii +. +The i-th diagonal entry of the matrix +(B−1D)ii = (b−1)iidii + ∑ +j̸=i +(b−1)ijdji ∼a dii +bii +≤ log(N). +The last equality follows from Property 2 in (22). Therefore, the trace of (B−1D) is also +on the order of log(N). We take the limit of +s +N tr(B−1D), where s = Nβ and β < 1. +L’Hˆopital’s rule yields: +lim +N−→∞ +s +N log(N) = lim +N−→∞ +1 +(1 − β)N1−β −→ 0. +(25) +The proof is complete by letting B = +� +V( ˙T ) +�−1 and D = V(T ). +Theorem 3.1 (Asymptotic Normality). Let Assumptions 3.1- 3.6 hold. Then the causal forest +estimator in Subsection 2.2 is asymptotically normally distributed. This result also holds for +multiple correlated policy effects. +See Appendix A.3 for the proof. We quantify the uncertainty of the policy effects +based on the jackknife variance estimator (Wager and Athey, 2018). Subsection A.3.3 in +Appendix A.3 provides further details regarding the uncertainty. +12 + +4 +Reemployment Experiment in Pennsylvania +The Pennsylvania ”Reemployment Bonus” Demonstration was a randomized controlled +trial in 1988-89 that aimed to investigate the impact of financial incentives on the reem- +ployment outcomes of unemployed individuals. The study population was divided into +a control group, which received the usual benefits provided by the Unemployment In- +surance System, and six treatment groups. Treated individuals were offered a cash bonus +for fulfilling certain criteria related to finding and retaining employment. Specifically, +participants in the treatment groups were required to accept a bonus that would be paid +to them if they were able to secure a full-time job of at least 32 hours per week within a +specified period (the qualification period) and maintain that employment for at least 16 +weeks. +Two bonus levels were tested. These two levels were a low bonus and a high bonus, +which were respectively three and six times the weekly benefit amount (WBA) received +by the participants. The low bonus was on average $500, while the high bonus was $997. +In addition to these two levels of bonus, the study also considered two different qualifi- +cation periods, starting from the date on which the bonus offer was made. These periods +were a short one of 6 weeks and a long one of 12 weeks. +In addition to testing the impact of financial incentives on reemployment outcomes, +the Pennsylvania ”Reemployment Bonus” Demonstration also aimed to investigate the +effectiveness of providing job-search assistance to unemployed individuals. To this end, +participants in the treatment groups were offered a workshop and an individualized as- +sessment session as part of the treatment design. However, attendance at the workshop +and completion of the assessment session were not mandatory for claimants. +In this article, we focus on treatment Group 4 which received a high bonus amount +and a long qualification period, as well as an offer of a workshop. The primary out- +come of interest is the logarithm of unemployment duration in weeks. The data include +twenty different characteristics of the claimants, such as age, gender, the quarter of the +experiment in which they enrolled, and unemployment rates in the local area. 4 Further +information about the experiment and data can be found in the article by Bilias (2000). +4.1 +Target Components +In this section, we identify and characterize policy-relevant target components. Accord- +ing to Figure 2 in Appendix A.4, the optimal number of components equals two. To +characterize and interpret the chosen scores, Table 1 in Appendix A.4 illustrates the effect +of the claimant characteristics on each target component. The negative value of a coeffi- +cient indicates that there is an inverse relationship between the characteristic in question +and the outcome being measured. For instance, a black claimant is associated with a 1.3% +lower score on average relative to a white claimant. Based on the sign of the coefficients +in Table 1, we can interpret the scores. +The lowest values of the target components identified in this article correspond to +4The variables are described in detail at the following url: +http://qed.econ.queensu.ca/jae/ +2000-v15.6/bilias/readme.b.txt. +13 + +a subgroup of young, non-white male claimants in the non-durable manufacturing sec- +tor who enrolled in the experiment early on. On the other hand, the highest values of +the components reflect a subgroup of middle-aged and older female individuals in the +durable manufacturing sector. These claimants enrolled in the experiment late in the final +quarter and tend to have a high number of dependents (as indicated by the positive co- +efficient for the ”dep” variable in Table 1 of Appendix A.4). (26) and (27) summarize the +characteristics of subgroups. 5 Components are continuous, therefore, they characterize +a full spectrum of individuals from one group to another. +Low score values ∼ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Young (age <= 35) +Male +Non-white +Joined the experiment early +In the sector of non-durable manufacturing +Few or no dependents +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(26) +High score values ∼ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Middle-age and older +Female +White +Joined the experiment late +In the sector of durable manufacturing +Many dependents +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(27) +4.2 +Effect Heterogeneity +In this section, we investigate the heterogeneity, or variability, in the effect of financial +incentives on unemployment duration within and across different values of the compo- +nents. This is done by examining various percentiles of the reemployment bonus effect +on unemployment duration across the corresponding component vigintiles, as shown in +Figure 1. +The results depicted in Figure 1 reveal considerable variation in the policy effects both +within and between groups. In particular, financial incentives have been found to poten- +tially motivate some young non-white individuals (the group represented by low score +values in (26)) to enter the labor market. However, these incentives may also provide a +temporary financial cushion for others, potentially dissuading them from actively seeking +employment. In contrast, this variation is less pronounced among older white claimants +(upper vigintiles of each component, the group corresponding to high score values in +(27)). +5According to Table 1 in Appendix A.4, these two components are almost identical, therefore, (26) and +(27) hold for each. +14 + +-0.5 +0.0 +0.5 +0 +5 +10 +15 +20 +policy effect +Percentiles +.025 +.5 +.975 +(a) Component 1 +-0.4 +0.0 +0.4 +0 +5 +10 +15 +20 +policy effect +Percentiles +.025 +.5 +.975 +(b) Component 2 +Figure 1: x-axis represents vigintiles of the corresponding component, while y-axis mea- +sures the ”Reemployment Bonus” effect on the logarithm of unemployment duration in +weeks. The plot shows the error bars with the color corresponding to a given percentile +of the policy effect. The number of trees equals 1000. +15 + +Our analysis shows that the ”Reemployment Bonus” policy has, on average, a neg- +ative effect on unemployment duration. However, significant variation is captured by +different percentiles of policy effects. The difference between the 97.5th and 2.5th per- +centiles of policy effects is 92.8% in the first vigintile of Component 2, and this difference +decreases to 22.1% in the 20th vigintile of the same component. These findings suggest +the need for targeted, personalized measures for younger non-white male claimants. +5 +Conclusion +Policymakers frequently seek to understand the impact of interventions on specific sub- +groups or segments of the population, defined by certain characteristics or attributes +known as covariates. In this article, we present a method for analyzing the density of +policy effects within these target segments, which are defined as linear combinations of +the explanatory variables. To achieve this, we combine two existing techniques, partial +least squares and causal forests, that allow us to identify and analyze policy effects for +the full range of the target segments. This approach enables policymakers to understand +how treatment effects vary within and across these segments, providing valuable insights +for personalized policy analysis. +We show that the method is consistent and leads to asymptotically normally dis- +tributed policy effects. Additionally, our approach generalizes beyond a single policy +effect to multiple (plausibly) correlated policy effects. Our analysis based on data from +Pennsylvania ”Reemployment Bonus” Demonstration reveals a significant variation in +the effect of financial incentives on the logarithm of unemployment duration. The find- +ings highlight the need for targeted measures for young non-white male participants. +One potential extension of our method is to incorporate quantile regression forests +(Meinshausen and Ridgeway, 2006), conditional on the target components. 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Federated causal inference in heterogeneous observational data. arXiv +preprint arXiv:2107.11732, 2021. +20 + +A.1 +The Loss Function +Minimize the difference between the personalized and group-level parameters, +arg +min +�θ(Ci,Sest) +EStr,Sest +�� +θ(Ci) − �θ(Ci, Sest, Π) +�TΣ−1� +θ(Ci) − �θ(Ci, Sest, Π) +� − θ(Ci)TΣ−1θ(Ci) +� += +(28) +EStr,Sest +� � +θ(Ci) − θ(Ci, Π) +� +�� +� +A ++ θ(Ci, Π) − θ(Ci, Sest, Π) +� +�� +� +B +�TΣ−1� +θ(Ci) − θ(Ci, Π) +� +�� +� +A ++ +θ(Ci, Π) − θ(Ci, Sest, Π) +� +�� +� +B +� − θ(Ci)TΣ−1θ(Ci) +� += +(29) +EStr +� +θ(Ci)TΣ−1θ(Ci) − 2θ(Ci)TΣ−1θ(Ci, Π) + +θ(Ci, Π)TΣ−1θ(Ci, Π) − θ(Ci)TΣ−1θ(Ci) +� ++ +(30) +ECi,Sest +�� +θ(Ci, Π) − �θ(Ci, Sest, Π) +�TΣ−1� +θ(Ci, Π) − �θ(Ci, Sest, Π) +�� += +(31) +− ECi +� +θ(Ci, Π)TΣ−1θ(Ci, Π) +� + E(tr(I))2×2. +(32) +The second equality follows after taking into account the independence of the train +and estimation data, cov(A, B) = 0. The final equality is based on the fact that θ(Ci, Π) = +E(θ(Ci)|Ci ∈ ℓ(c, Π)), E +��θ(Ci, Sest, Π) +� = θ(Ci, Π), and: +ECi,Sest +�� +θ(Ci, Π) − �θ(Ci, Sest, Π) +�TΣ−1� +θ(Ci, Π) − �θ(Ci, Sest, Π) +�� += +tr +� +Σ−1E +� +θ(Ci, Π) − �θ(Ci, Sest, Π) +�T� +θ(Ci, Π) − �θ(Ci, Sest, Π) +�� += +tr(Σ−1Σ) = tr(I)2×2, +where tr(I)2×2 is the trace of a 2 × 2 identity matrix. Since E(tr(I))2×2 does not depend on +the parameter of interest, we can disregard it. Hence, the optimal parameter maximizes +the unbiased estimator of the negative mean squared error: +ˆθ(Ci, Sest, Π) = arg max +�θ +1 +Ntr ∑ +ℓ +Ntr +ℓ �θ(Ci, Π)T �Σ−1�θ(Ci, Π), +(33) +where the covariance matrix can be estimated as ˆΣ = ˆΣ +� ˜θ(Ci, Str, Π)|Nest� +. In this study, +training and estimation samples have an equal number of observations, Ntr = Nest. +21 + +A.2 +Consistency of Component Weights +Proof. We adopt the approach of Naik and Tsai (2000) in which +b⋆ = (ΣXX)−1σXy, +where ΣXX is the covariance matrix of the predictors and σXy is the covariance vector +between the predictors and the response. We define the matrix R as follows: +R = (σXy, ΣXXσXy, . . . , Σq−1 +XX σXy), +where q ≤ p is a positive integer. +Under the assumption that Sxx approaches ΣXX and sxy approaches σXy as the sample +size N approaches infinity, we have: +ˆb → R(RTΣXXR)−1ΣXXα⋆ in probability when N → ∞. +The assumptions q = M and σXy = ∑M +j=1 γjvj imply that b⋆ is contained in the space +spanned by R. Consequently, Σ1/2 +XX b⋆ is contained in the space spanned by R⋆ = Σ1/2 +XX R. +Therefore, +R⋆(R⋆TR⋆)−1R⋆TΣ1/2 +XX α⋆ = Σ1/2 +XX α⋆, +and +R(R⋆TR⋆)−1RTΣXXα⋆ = α⋆. +Hence, ˆb → b⋆. Brillinger (2012) in Sections 3 and 4 shows that b = ξΣXXb, where ξ = +cov +� +g(b0 + Xb), b0 + Xb) +� +/var +� +b0 + Xb +� +. Therefore, the proof is complete by noting +that ˆb → Σ−1 +XXσXy = ξb. +A.3 +Consistency and Asymptotic Normality of Policy Ef- +fects +The large sample theory of causal forests is largely based on the works of Wager and +Athey (2018) and Wager and Walther (2015), which present the H¨oeffding decomposition +of conventional random forests in a univariate setup. In this article, we demonstrate that +the deviation between the random forest and its H¨oeffding decomposition approaches +zero in the limit, and prove the asymptotic normality of the random forest estimator. It is +important to note that the outcome variable yi in this context may be a vector-valued vari- +able, in which case most of the operations and results apply coordinate-wise. The lemmas +and proofs discussed in this article have also been extensively analyzed by Nareklishvili +(2022) for partially identified policy effects, and by Li (2020) under network effects. +For notational simplicity, we will define Xi = Xi and Ci = Ci throughout the remain- +der of the section. +22 + +A.3.1 +H¨oeffding decomposition: Lemma 3.2 +Proof. Given a collection of ℓ|Π| +n=1 terminal nodes that form a partition of the component +space C, we define the prediction of a tree at a generic test point c as: +T = T (c, ξ, Ai, . . . , AN) = +|Π| +∑ +n=1 +1(c ∈ ℓn) 1 +Nℓn ∑ +i:Ci∈ℓn +yi. +(34) +ξ is an external source of randomization, to allow for the randomized split selection pro- +cedures. 1(c ∈ ℓn) is an indicator function and equals one if a point c ∈ ℓn, and zero +otherwise. Nℓn denotes the number of observations in a terminal node ℓn. +A tree T (c, ξ, A1, . . . , AN) represents a prediction at a point c based on data AN +i=1 = +{yi, Ci}N +i=1 and a randomization parameter ξ. As described in Lewis (2000) and Kingsford +and Salzberg (2008), trees are a popular choice for classification and regression tasks due +to their interpretability, ease of implementation, and robustness to outliers and missing +data. However, trees also have a high variance and are prone to overfitting, which makes +it difficult to determine the optimal tree structure. To address these issues, Breiman (2001) +introduced the random forest algorithm. +Let s < N be a subset of size s from a population i = 1, . . . , N, where s = Nβ and β +is sufficiently close to 1 (Wager and Athey, 2018). Following the work of Breiman (2001) +and Wager and Athey (2018), we define the random forest estimator as the average of +the tree estimators aggregated over all possible size-s subsamples of the training data, +marginalized over the auxiliary noise ξ. Specifically, the prediction of the random forest +estimator at a particular test data point c is defined as: +F(c, A1, . . . , AN) = +1 +� Ns +� +∑ +1≤i1≤···≤is≤N +EξT (c, ξ, Ai1, . . . , Ais), +(35) +where i1, . . . , is are the size-s subsamples of the population {i = 1, . . . , N}. In practice, we +estimate such a random forest by Monte Carlo averaging: +F(c, A1, . . . , AN) ≈ 1 +B +B +∑ +b=1 +T (c, ξ∗, A∗ +1, . . . A∗ +N) +(36) +where {A∗ +1, . . . A∗ +N} is drawn without replacement from {A1, . . . AN}. ξ∗ is an auxiliary +noise in a given sample and B is the number of sub-samples. F(c, A1, . . . , AN) is a 1 × M +vector. Therefore, most of the arithmetic operations in this section are defined coordinate- +wise in RM. +A random forest estimator can be represented as a U-statistic (Hoeffding, 1961; Ko- +rolyuk and Borovskich, 2013). A common approach to studying the large sample prop- +erties of random forests is to derive the lower bound of its H¨oeffding decomposition. +H¨oeffding decomposition (also known as the Hajek projection) in a univariate setting is +described by H´ajek (1968). Specifically, consider a vector-valued function T ∈ RM which +23 + +is measurable and permutation symmetric, that is, T(πc) = T(c) for all π ∈ Π (a tree in +this setting). Then the Hajek projection of this function is defined as: +˙T = E(T) + +N +∑ +i=1 +� +E(T|Ci) − E(T) +� = +N +∑ +i=1 +E(T|Ci) − (N − 1)E(T). +(37) +Intuitively, the Hajek projection in (37) represents a projection of T onto the linear sub- +space of all random variables of the form ∑N +i=1 gi(Ci), where gi : Rd → R are arbitrary +measurable functions such that E(g2 +i (Ci)) < ∞ for i = 1, . . . , N. It is clear that the con- +ditional expectation of the centered and symmetric component ˙T in (37) is equal to the +conditional expectation of T: +E( ˙T|Ci) = E(T|Ci), and +(38) +E( ˙T) = E(T). +Now consider the random forest estimator, F(c, A1, . . . , AN) ∈ RM, and let the corre- +sponding vector of means be µ. Moreover, let ˙F(c, A1, . . . , AN) and Σ denote the Hajek +projection of the random forest estimator, and the covariance matrix of the Hajek projec- +tion, respectively. Assume also that the trees in ˙F(c, A1, . . . , AN) are symmetric and the +observations are i.i.d. Then Lemma 3.2 holds: +We define the Hajek projection of the random forest estimator as +˙F(c, A1, . . . , AN) − µ = +N +∑ +i=1 +E +�F(c, A1, . . . , AN) − µ|Ai +� = +(39) +1 +� Ns +� +N +∑ +i=1 +E +� +∑ +1≤i1≤···≤is≤N +EξT (c, ξ, Ai1, . . . , Ais) − µ|Ai +� +, +where (N +s ) is the number of size-s subsets i1 ≤ · · · ≤ is that can be selected from the +N observations. Then Lemma 6.1 of Nareklishvili (2022) applies. Moreover, this is also +shown by Li (2020) for in a setup with network effects. +Since the required conditions for the Lindeberg central limit theorem to hold are satis- +fied (Billingsley, 2008), the Hajek projection of the random forest estimator is asymptoti- +cally normally distributed: +Σ−1/2� ˙˙F(c, A1, . . . , AN) − µ +� +d−→ N (0, IM), +where 0 is a RM vector of zeros and IM is an identity matrix. Our goal is to prove +that the random forest estimator is asymptotically normal. By adding and subtracting +Σ−1/2 ˙F(c, A1, . . . , AN) to Σ−1/2�F(c, A1, . . . , AN) − µ +� +, we can see that the random forest +estimator is related to its projection in the following way: +24 + +Σ−1/2�F(c, A1, . . . , AN) − µ +� = Σ−1/2�F(c, A1, . . . , AN) − ˙F(c, A1, . . . , AN) +� + +Σ−1/2� ˙F(c, A1, . . . , AN) − µ +� +. +The main goal of this article is to show that: +Σ−1/2�F(c, A1, . . . , AN) − ˙F(c, A1, . . . , AN) +� +p−→ 0. +Then, by Slutsky’s theorem, it follows that the random forest estimator is asymptotically +normally distributed. +A.3.2 +Asymptotic Normality: Proposition 3.3 and Theorem 3.1 +Proof. We define the mean squared deviation of the random forest estimator and its pro- +jection as: +E(F − ˙F)TΣ−1(F − ˙F) = E +� +trΣ−1(F − ˙F)(F − ˙F)T� = +(40) +trΣ−1E(F − ˙F)(F − ˙F)T = tr Σ−1/2V +�F − ˙F +� +Σ−1/2. +In order for the weak independence condition to hold for the exchangeable sequence +of Xi, the following equation must be satisfied: +E +�Ti(Ci ∈ B)|Ci /∈ B) +� = 0. +(41) +Assuming that Ti(Ci ∈ B) are symmetric, square-integrable, vector-valued functions, +then each Ti and Ti′ are pairwise independent. As i = 1, . . . , N is an exchangeable (i.i.d.) +sequence, Theorem 6 of Peccati (2004) and Proposition 1 of Li (2020) both apply. Further- +more, this lemma is shown by Nareklishvili (2022) (Lemma 6.2). +Under Assumptions 3.2-3.6, Proposition 3.1 shows that the upper bound of the devi- +ation between the random forest and its’ own projection converges to zero when N → 0. +According to Slutsky’s theorem, Proposition 3.1 implies that the random forest estimator +is asymptotically normally distributed. This result holds for the case of policy effects as +well, where the difference appears in the constant term K that we have carried throughout +the proof (see Wager and Athey (2018) for more details). +A.3.3 +Inference +We quantify the uncertainty of policy effects by using the jackknife variance estimator (as +described in Wager and Athey, 2018). Let g = 1, . . . , G be the g-th bootstrapped sample. +We use a tree Πg and the corresponding estimation sample Sest +g to obtain ˆθg(c, Sest +g , Πg) at +a generic test point c. Next, the average of the individual tree estimates is give as: +25 + +ˆθ +� +c, {Sest +g }B +g=1}, {Πg}G +g=1 +� = 1 +G +G +∑ +g=1 +ˆθg(c, Sest +g , Πg). +We define Nig as the number of times an observation i appears in the g-th bootstrapped +sample, either in the training sample Str or the estimation sample Sest. The following +variance estimator can be used to construct valid confidence intervals:: +Var +� ˆθ +� +c, {Sest +g }G +g=1}, {Πg}G +g=1 +�� = +N +∑ +i=1 +∆g − N +G2 +G +∑ +g=1 +� +θg(c, Sest +g , Πg) − ˆθ +� +c, {Sest +g }G +g=1} +��2, +(42) +where ∆g = +�∑G +g=1(Nig − 1) +� ˆθg(c, Sest +G , Πg) − ˆθ +� +c, {Sest +g }G +g=1}, {Πg}G +g=1 +�� +G +� +. +A.4 +Interpreting Target Components +0 +5 +10 +15 +20 +1.190 +1.205 +1.220 +number of components +RMSEP +CV +adjCV +Figure 2: Root-mean-squared error (RMSE) based on five-fold cross-validation. +26 + +Table 1: Linear regression of each target component on independent characteristics. +Dependent variable: +(Component 1) +(Component 2) +abdt +−0.001∗∗∗ +0.004∗∗∗ +(0.000) +(0.000) +female +0.362∗∗∗ +0.365∗∗∗ +(0.000) +(0.000) +black +−1.270∗∗∗ +−0.955∗∗∗ +(0.000) +(0.000) +hispanic +−0.901∗∗∗ +−0.631∗∗∗ +(0.000) +(0.000) +othrace +−0.827∗∗∗ +−0.699∗∗∗ +(0.000) +(0.000) +dep +0.166∗∗∗ +0.082∗∗∗ +(0.000) +(0.000) +q1 +−0.467∗∗∗ +−2.588∗∗∗ +(0.000) +(0.000) +q2 +−0.066∗∗∗ +−1.716∗∗∗ +(0.000) +(0.000) +q3 +−0.516∗∗∗ +−1.334∗∗∗ +(0.000) +(0.000) +q4 +−0.586∗∗∗ +−1.344∗∗∗ +(0.000) +(0.000) +q5 +−0.847∗∗∗ +−0.602∗∗∗ +(0.000) +(0.000) +recall +1.473∗∗∗ +0.724∗∗∗ +(0.000) +(0.000) +agelt35 +−0.938∗∗∗ +−0.086∗∗∗ +(0.000) +(0.000) +agegt54 +1.238∗∗∗ +−0.361∗∗∗ +(0.000) +(0.000) +durable +0.190∗∗∗ +0.018∗∗∗ +(0.000) +(0.000) +nondurable +−0.724∗∗∗ +−0.989∗∗∗ +(0.000) +(0.000) +lusd +−0.396∗∗∗ +−0.356∗∗∗ +(0.000) +(0.000) +husd +0.221∗∗∗ +−0.812∗∗∗ +(0.000) +(0.000) +muld +0.123∗∗∗ +0.938∗∗∗ +(0.000) +(0.000) +Constant +10.317∗∗∗ +−39.629∗∗∗ +(0.000) +(0.000) +Observations +13,913 +13,913 +R2 +1.000 +1.000 +Adjusted R2 +1.000 +1.000 +Residual Std. Error (df = 13893) +0.000 +0.000 +Note: +∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 +27 + diff --git a/xdAyT4oBgHgl3EQfa_cC/content/tmp_files/load_file.txt b/xdAyT4oBgHgl3EQfa_cC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae396982db44de1fa0575998c8494a60e0321536 --- /dev/null +++ b/xdAyT4oBgHgl3EQfa_cC/content/tmp_files/load_file.txt @@ -0,0 +1,1012 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf,len=1011 +page_content='Feature Selection for Personalized Policy Analysis Maria Nareklishvili Booth School of Business University of Chicago Nicholas Polson* Booth School of Business University of Chicago Vadim Sokolov Department of Systems Engineering and Operations Research George Mason University First Draft: July 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 2022 This Draft: December 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 2022 Abstract In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' we propose Forest-PLS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' a feature selection method for analyzing pol- icy effect heterogeneity in a more flexible and comprehensive manner than is typically available with conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In particular, our method is able to capture pol- icy effect heterogeneity both within and across subgroups of the population defined by observable characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To achieve this, we employ partial least squares to iden- tify target components of the population and causal forests to estimate personalized policy effects across these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We show that the method is consistent and leads to asymptotically normally distributed policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To demonstrate the effi- cacy of our approach, we apply it to the data from the Pennsylvania Reemployment Bonus Experiments, which were conducted in 1988-1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The analysis reveals that fi- nancial incentives can motivate some young non-white individuals to enter the labor market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' However, these incentives may also provide a temporary financial cushion for others, dissuading them from actively seeking employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Our findings high- light the need for targeted, personalized measures for young non-white male partici- pants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 1 Introduction Randomized control trials play an important role for treatment, policy, or a program ef- fect analysis in economics, statistics, medicine, and other fields (Banerjee and Duflo, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Email: ngp@chicagobooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='00251v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='EM] 31 Dec 2022 Bertrand and Duflo, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To design and implement effective interventions, policymakers and researchers are often interested in partitions of the pop- ulation that are particularly susceptible to a new program or a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Identifying such subgroups can be challenging, especially when there are a large number of observable characteristics that can influence the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In such cases, conventional estimation methods, such as ordinary least squares, may produce inefficient estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This is be- cause these methods often struggle to identify relevant variables in a sparse feature space (Johnstone and Titterington, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Belloni and Chernozhukov, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We propose Forest-PLS, a data-driven approach for selecting the target components of the population for personalized policy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In our approach, the target components represent linear combinations of the explanatory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These components are char- acterized by large weights on the variables that are most strongly associated with the out- come, and carry a significant amount of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' As a second step, we identify and estimate personalized policy effects across the chosen components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' By focusing on these key components, rather than considering the full set of characteristics, policymakers can design targeted interventions tailored to the most diversified segments of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The procedure is a combination of two distinct methods, the partial least squares (Geladi and Kowalski, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Vinzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2010) and causal forest (Athey and Imbens, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Wager and Athey, 2018) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The partial least squares method is used to detect the policy-relevant components in the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These components represent a reduced ex- planatory variable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The reduced space reflects the highest variation in explanatory features and the most relevant information for predicting the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These compo- nents are continuous rather than discrete clusters of the data space, allowing us to ana- lyze policy effects across a full spectrum of the population segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In the second step, we use the causal forest algorithm to identify different quantiles of policy effects within each component value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This allows us to capture heterogeneity in the policy effects at a finer granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The primary contribution of this paper is to advance our understanding of the distri- bution of policy effects from a theoretical and empirical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The theoretical com- ponent of the article demonstrates that our approach is consistent and leads to asymp- totically normally distributed policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Our framework and findings extend beyond a single coefficient of interest (Wager and Athey, 2018) to multiple (plausibly) correlated policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The empirical contribution of the paper is to identify and analyze two types of heterogeneity of the policy effects: within-group heterogeneity and between- group heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The proposed method allows us to estimate quantiles of treatment effects within and across the values of target components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Analyzing individual explana- tory variables separately (Meinshausen and Ridgeway, 2006) can be challenging in high- dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Our approach allows us to focus on aggregate aspects of these charac- teristics without losing the economic interpretation of the resulting subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Our framework is closely related to the papers dedicated to the estimation of per- sonalized treatment effects (Athey and Imbens, 2015, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Wager and Athey, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Cher- nozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2018a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' K¨unzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Nie and Wager, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2021), and feature selection for the inference on treatment effects (Belloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2012, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2015b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Urminsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Previous work for personalized treatment effect analysis considers a single source of heterogeneity, 2 such as quantile treatment effects, or treatment effects across the original set of covariates (features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We unify the feature selection methods with personalised policy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This allows us to investigate a full density of policy effects within and across a pooled variable space (target components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Other related methods are proposed by Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Chun and Keles¸ (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Mehmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2012, 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Polson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Nareklishvili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Dixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2022) that use the partial least squares algorithm for feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These methods suggest that partial least squares as a precursor to a more general framework of deep learning and instrumental variables can increase efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' By comparison, our study shows that the method can pool statistically and economically significant variables for policy effect heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Additionally, Nekipelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Li (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Nareklishvili (2022) investigate large sample properties for random forests under multiple outcomes, coefficients or network effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We show that the theoretical properties hold even after the feature selection procedure by partial least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' It is important to note that Bayesian approach is an alternative to policy effect heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Ansari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Taddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Santos and Lopes (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Woody et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Starling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Krantsevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (2022) formulate Bayesian Additive Regression Trees (BART) for heterogeneous treatment or policy effect analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The advantage of the approach lies in the regularization effect through predetermined priors of the tree parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Our work can be extended to accommodate Bayesian priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This article examines the impact of financial incentives on unemployment duration based on data from the Pennsylvania ”Reemployment Bonus” Demonstration, a random- ized control trial conducted in 1988-1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The analysis reveals significant variation in the policy effects both within and across different subgroups of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Specifically, the results show that the effects of the policy are more significantly dispersed for young, non-white male claimants who joined the experiment early on, compared to middle-age and older female participants with a high number of dependents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The difference between the 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5th and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5th percentiles of policy effects is 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='8% for the first vigintile of the target component, and decreases to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1% for the final vigintile of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These find- ings highlight the need for targeted, personalized measures for specific subgroups, such as young non-white male participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 2 The Forest-PLS Framework Consider the outcome yi ∈ R (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', unemployment duration) for a subject i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Each subject is characterized with an observable vector of features Xi ∈ Rp (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', age, gender, occupation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We assume that p = Nψ, with ψ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A policy is denoted by Pi ∈ {0, 1}, and we let yi(1) and yi(0) denote potential outcomes with and without the policy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We assume, the unconfoundedness holds: Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 (Unconfoundedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The policy is independent of the potential outcomes: yi(1), yi(0) ⊥⊥ Pi|Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A policymaker wishes to identify the dimensions of the feature space that contain the most relevant information about the policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To this end, we consider a mapping 3 f : Xi �→ Ci ∈ Rq that maps the original features to a set of target components Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In other words, Ci is a collection of q-dimensional linear combinations of the features xj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , p (with q ≤ p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This transformation allows us to focus on a smaller, more interpretable set of features while preserving the information about the policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The coefficient of interest is the effect of Pi on the outcome: θi = yi(1) − yi(0), Personalized policy effects are not directly observable, as an individual is only ex- posed to one policy state (either with or without the policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Therefore, we typically consider the expectations of the potential outcomes: θ(Ci) = E(θi|Ci) + εi, E(θi|Ci) = E � yi(1) − yi(0) ��Ci].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Due to Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1, the policy effect of interest is given as τ(Ci) = E � yi(1) − yi(0)|Ci � = E � yi|Pi = 1, Ci � − E � yi|Pi = 0, Ci � , (1) where E � yi|Pi = d, Ci � for d ∈ {1, 0} denotes the observed expected outcomes with and without the policy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To estimate the average policy effect in (1), we need to determine the optimal target components Ci and use a method that estimates group-level policy effects, conditional on the chosen components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 Identification of Target Components We seek to identify the linear combinations of features, also known as target compo- nents/scores/factors, C = [c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , cq] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', c1 = Xw1) that explain the highest varia- tion in covariates X, as well as the outcome y (Tobias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Abdi, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' X and y can be decomposed as: X (N×p) = C (N×k) VT (k×p) + E (N×p), (2) y (N×1) = C (N×k) b (k×1) + e (N×1), (3) where X = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , xp] is the matrix of covariates, VT is the matrix of loadings (weights), and E is the matrix of errors for the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' y denotes the outcome as before, b is the vector of coefficients (the influence of components on the outcome), and e is the vector of errors for the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We use the iterative procedure to obtain the target components (aka partial least squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Consider, the weight ˜w1 = � cov(x1, y), cov(x2, y), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , cov(xp, y) � = � ˜w11, ˜w21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , ˜wp1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We normalize it to get a unit vector: w1 = ˜w1 || ˜w1||, 4 where || ˜w1|| denotes the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We use these weights to compute the first principal component: c1 = w11x1 + w21x2 + · · · + wp1xp = Xw1 = Xw1 wT 1 w1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (4) The last equality in (4) follows by the fact that the weights are unit vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A linear regression of a j−th covariate on the first component yields a loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The vector of loadings associated with the first component is given by regressing the covariates on it: v1 = XTc1 cT 1 c1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (5) Similarly, the first coefficient b1 is obtained by regressing the outcome on the first component: b1 = yTc1 cT 1 c1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (6) The next step is to obtain the approximation of the covariate matrix and the outcome, and predict residuals: X1 = X − ˆX, y1 = y − ˆy, (7) where ˆX = c1vT 1 and ˆy = b1c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We obtain the subsequent components by repeating the described procedure for the first, second, and higher order residuals of the covariate matrix X1, X2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , and the outcome y1, y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A desirable property of the procedure is that the coefficients have a closed-form so- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The estimator of these coefficients is given as (Helland, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Stone and Brooks, 1990): �b = ˆR( ˆRTSxx ˆR)−1 ˆRTsxy, (8) where ˆR = (sxy, Sxxsxy, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , Sq−1 xx sxy) is the p × q matrix of the Krylov sequence with a p × p matrix Sxx and a p × 1 vector sxy defined as follows: Sxx = XT(I − 11T/N)X N − 1 , sxy = (X − E(X))T(y − E(y)) N − 1 , where I is an identity matrix and 1 is a matrix of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Intuitively, the algorithm searches for factors that capture the highest variability in X, and at the same time maximizes the covariance between X and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' If the number of components equals the dimension of the covariates, q = p, the method is equivalent to the ordinary least squares (Helland, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 Estimation of Personalized Policy Effects To identify smaller subgroups within the population, we use a causal forest algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A tree in causal forests recursively partitions the feature space, in this setting, the space of identified target components C, and makes axis-aligned splits to estimate the conditional mean of the outcome µ(c) = E(yi|Pi, Ci = c) at a point c for Pi ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' An axis-aligned split is a pair s = (j, c), where j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , q is a specific component (the splitting coordinate) and c ∈ R is the corresponding value (the splitting index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The recursive partitioning procedure begins by considering the set P(0) = C ∈ Rq (the parent node of the tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' For this set, we select the splitting coordinate j : 1 ≤ j ≤ q and the splitting index c that divide P(0) into two non-overlapping rectangles (child nodes): P(1,1) = P(0) ∩ {�c ∈ P(0) : �cj ≤ c} and P(1,2) = P(0) ∩ {�c ∈ P(0) : �cj > c}, (9) After the first split, the process is repeated for P(1,1) and P(1,2)separately until the desired level of partitioning is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The sequence of k splits defines a partition of the component space C, which we de- note by Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This partition (or equivalently, a tree) consists of non-overlapping rectangular regions ℓn called the leaves or terminal nodes of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These leaves represent the final subgroups or subpopulations identified by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The union of all these partitions is the entire component space: Π = {ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , ℓ|Π|} and ∪|Π| n=1 ℓn = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Athey and Imbens (2016) propose a method for estimating heterogeneous policy ef- fects under the assumption of unconfoundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To implement this method, we split the data into two different samples: a training sample Str used to build and find the split- ting variables and values, and an estimation sample Sest used to estimate policy effects across different subgroups of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The unbiased sample analogue of E(θ(Ci)) is denoted as follows: �θ(Ci, Sest, Π) = |Π| ∑ n=1 � 1(c ∈ ℓn, Pi = 1) 1 |i : Ci ∈ ℓn, Pi = 1| ∑ i:Ci∈ℓn Yi(1)− 1(c ∈ ℓn, Pi = 0) 1 |i : Ci ∈ ℓn, Pi = 0| ∑ i:Ci∈ℓn Yi(0) � , (10) where |Π| is the total number of the terminal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 1(c ∈ ℓn, Pi = d) is a binary variable and equals one when, for a given d ∈ {0, 1}, a generic test data point c belongs to a terminal leaf ℓn, and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Additionally, let Σ be the variance of �θ(Xi, Sest, Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 1 1While our analysis is based on a single policy variable and a single outcome, the proposed framework can handle multiple policy variables and outcomes with correlated coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 6 To estimate policy effects from the available data, we aim to maximize the variance of the policy effect estimator: ˆθ(Ci, Sest, Π) = arg max �θ 1 Ntr ∑ ℓ Ntr ℓ �θ(Ci, Π)T �Σ−1�θ(Ci, Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (11) The proof of (11) is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Intuitively, the objective function in (11) encourages the causal forest algorithm to search for subsets of target components with the highest variation in policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To further increase the robustness of the estimates, we build multiple trees on bootstrapped data and average the resulting coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This approach, known as the causal forest algorithm, has been described in detail by Athey and Imbens (2016) and Wager and Athey (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 3 Large Sample Properties To show the asymptotic normality of the estimated policy effects, we need to make certain assumptions about the underlying data-generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 (Data Generating Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Let y = g(b0 + Xb) + ε where b is a p × 1 vector of coefficients, b0 is a constant and g is a non-linear mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assume, X have a joint Elliptical distribution with the mean µX and a variance ΣXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assume X is independent of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Moreover, let Sxx and sxy converge in probability to ΣXX (the population variance of X) and σXy (the population covariance of X and y) when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Moreover, let there exist a pair of eigenvectors and eigenvalues (vj, λj) for which σXy = ∑M j=1 γjvj (with γj non-zero for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assume also E(|g(U)|) < ∞ and E(U|g(U)|) < ∞ with U = b0 + Xb and q = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1, the relation between the response and the independent charac- teristics follows a predetermined functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Additionally, the subject characteristics are assumed to have an elliptical distribution, meaning they are shaped like an ellipse in a multi-dimensional coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' While this assumption is not always satisfied in practice, it has been shown that the results obtained under this assumption do not signifi- cantly differ from those obtained when the features have other types of distributions (see Brillinger, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Then ˆb in (8) is consistent up to a proportionality constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 shows that the identified target components are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The causal forest method described in this article relies on the same assumptions as those introduced by Wager and Athey (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' One of them is the ”honesty” of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 (Honesty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The outcome yi and the splitting parameters (the splitting coor- dinates and indices, s = (j, c)) are independent of each other, conditional on the observed com- ponents Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This independence holds for each subject i whose outcome yi is used in the final 7 prediction: F(yi|Ci, s) = F(yi|Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' F denotes the density of the outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 There are various ways to satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In this article, we use a two-sample approach, where we split the data into a training sample Str and an estimation sample Sest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The splitting coordinates and indices (s) of the trees are determined based on the ob- servations in Str, while the predicted outcomes are based on the observations in Sest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This separation of the data into two different samples ensures that the splitting parameters and the outcomes are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 (Random Split Trees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' At each recursive step, the probability of choosing the j-th component as the splitting coordinate is lower bounded by π/d for π ∈ (0, 1] and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In order to guarantee the consistency of the causal forest method, it is necessary for the leaves of the trees to become small in all dimensions of the component space as the sample size N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To ensure this, we adopt Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3, which is based on the assumptions of Meinshausen and Ridgeway (2006) and Wager and Athey (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This assumption states that for all splitting steps, each component has a probability of at least π/d of being selected as the splitting coordinate, for some 0 < π ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4 (The Splitting Algorithm is (α, k)-regular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' There exists a positive constant α such that at each split, at least a fraction α of the available training examples are left on each side of the split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Additionally, we require that the splitting process ceases at a node when it contains less than k observations for some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4 ensures that each half-space produced by a split in the tree construc- tion process contains a sufficient number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' As shown by Wager and Walther (2015), this assumption also implies that the half-spaces are large in Euclidean volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4 places an upper bound on the number of observations that can be con- tained in a terminal node of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Specifically, when a tree is fully grown to depth k, we have that each terminal node contains between [k, 2k − 1] observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' One important consequence of this assumption is that it places an upper bound on the variance of the tree estimator at any test point c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5 (Distributional Assumptions on the Data Generating Process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The target components Ci are supported on the unit cube Ci ∈ [0, 1]p, and the density of these components is bounded away from zero and infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The first and second moments of the outcome, E(yi|Ci = c) and E � y2 i |Ci = c � , are Lipschitz-continuous functions of the target components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The variance of the outcome, Var(yi|Ci = c), is bounded away from zero for all values of the target components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Specifically, we have in fc∈CVar(yi|Ci = c) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 2If we have access to multiple outcomes, this assumption holds for each one individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 8 Lipschitz continuity and bounded variances are widely used assumptions in the field of statistics and machine learning (Wager and Athey, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Biau, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In the context of this paper, the results do not depend explicitly on the distributional assumptions of Ci, however, they affect the constants that we carry throughout this paper (constants borrowed from Lemma 2 and Theorem 3 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 in Wager and Athey, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='6 (Overlap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Let 0 < ϵ < 1, and consider any element c ∈ [0, 1]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Then the following holds: ϵ < P(Pi = 1 | Ci = c) < 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='6 ensures that, as the number of observations N increases, there will be a sufficient number of subjects with and without a policy at any given test point c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In their work, Wager and Athey (2018) derive the lower bound of the variance of the Hajek projection of a conventional random forest, ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN), and demonstrate that it converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We follow a similar approach in this article, with the objective of showing that Σ−1/2 �F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) � converges in squared mean (Σ denotes the variance of the random forest projection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 for definitions and further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' For simplicity, we denote F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) and ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) as F and ˙F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' It is important to mention that the theoretical properties are shown for the conventional random forest estimator as by Wager and Athey (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The proofs, however, generalize well to the coefficients, such as policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Wager and Athey (2018) discuss additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 (Hajek Projection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The Hajek projection of a random forest estimator and the co- variance of this projection are given as: ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − µ = s N N ∑ i=1 �T1(Ai) − µ � , Σ = s NV ˙ (T ) ∈ RM×M, where AN i=1 = {yi, Ci}N i=1 defines data, s = Nβ with β sufficiently close to one, and M is the dimension of the outcome variables in each terminal node (M = 1 in this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' V de- notes the covariance matrix of the projected elements of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Lastly, ˙T = ∑s i=1 T1(Ai) with T1(a) = Eξ,A2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=',ANT (c, ξ, a, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) is the Hajek projection of a tree T (c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) = EξT (c, ξ, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 (Error Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The mean squared difference between F and ˙F has the following upper bound: E �F − ˙F �TΣ−1�F − ˙F � ≤ s N tr �� V( ˙T ) �−1V(T ) � , where tr is a trace operator, and V(T ) and V( ˙T ) denote the variance of a vector-valued tree and its’ Hajek projection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 derives the explicit expression for the projection of a forest and its’ vari- ance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 illustrates the upper bound of the squared deviation be- tween the random forest (for multiple policy effects) and its’ projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The proofs are provided in Subsections A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3, respectvely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The Hajek projec- tion of a random forest meets Lindeberg central limit theorem conditions, and is asymp- totically normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' As a result, by showing that this bound converges to zero in the limit, we establish the asymptotic normality of the random forest estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The entries of V(T ) are bounded, and its diagonal elements are bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Additionally, the lower bound of the off-diagonal terms of V( ˙T ) is on the order of o � 1 logq(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' As a result, the upper bound in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 approaches zero as N tends towards infinity: s N tr �� V( ˙T ) �−1V(T ) � −→ 0 when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The boundedness of the elements of V(T ) is a direct consequence of the proposed assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Specifically, according to Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4, the number of observations in each terminal node is upper bounded, which implies that the variance of the tree is upper bounded by a constant times V(yi|Ci = c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Furthermore, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5 ensures that V(yi|Ci = c) is lower bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In this proof, we utilize the result from Wager and Athey (2018) concerning the asymp- totic behavior of the variance terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In our framework with q−dimensional target com- ponents: V( ˙T )ii = K logq(s), for some constant K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (12) V( ˙T )ii denotes the diagonal terms of the variance of the projection of a tree estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We show that the off-diagonal terms V( ˙T )ij = o � 1 logq(s) � for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We begin by defining the Hajek projection of a tree: ˙T − µ = s ∑ i=1 E(T |Ai) (13) Since the observations are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=', then: V � ˙T � = sV � E(T |A1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (14) It is clear to see that: V � E(T |A1) � = V � E(T |A1) − E(T |C1) � + V � E(T |C1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (15) Let us consider the m-th outcome variable, where m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' As the tree is hon- est, the diagonal terms in equation (15) simplify to the following form (see the Proof of Theorem 5 in Wager and Athey, 2018): V � E(T |A1) − E(T |C1) � mm = V � E(Sℓn|C1)(y1m − E(y1m|C1) � mm ≈ E �� E(Sℓn|C1) �2� E �� y1m − E(y1m|C1) �2� = (16) E �� E(Sℓn|C1) �2� Var(ym|C1 = c), 10 and V � E(T |C1) � mm = E �� E(Sℓn|C1) �2� Var � E(ym|C1 = c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (17) where Tm is the estimator of a tree at a test point c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Sℓn is the indicator function and equals one if C1 ∈ ℓn(c, Π), and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The off-diagonal terms equal to: V � E(T |A1) − E(T |C1) � mm′ = E �� E(Sℓn|C1) �2� E �(y1m − E(y1m|C1))(y1m′ − E(C1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (18) Under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5, the variance of each outcome variable is lower bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' By applying the Cauchy-Schwarz inequality, we see that the absolute value of the covariance between yim and yim′ given C1 is also lower bounded away from zero: 3 |Cov(y1m, y1m′|C1)| ≤ � Var(y1m|C1 = c)Var(y1m′|C1 = c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In our setting, E �� E(Sℓn|C1) �2� , is lower bounded as (Wager and Athey, 2018): E �� E(Sℓn|C1) �2� ≥ (q − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 2q+1 logq(s) · 1 ks, (19) where k is the minimum number of observations in a given terminal node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Combining (14) and (19) yields the order of diagonal and off-diagonal terms: V � ˙�T � mm = o � 1 logq(s) � , and V � ˙�T � mm′ = o � 1 logq(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (20) In the following, we prove that s N tr �� V( ˙�T) �−1 V(�T) � −→ 0 in a more general frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Let B and D be two square matrices with diagonal elements bii and dii, and non- diagonal elements bij and dij, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We assume that these matrices possess the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' dii ≥ η for some η ∈ R+ and for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' M, (21) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' bii ≥ dii log(N), (22) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' bij = o � 1 log(N) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (23) 3An alternative argument is to notice that the term in the integrand consists of multiples of the first and second moments of the outcome variables y1m and y1m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Since these moments are continuous, they are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Thus, their expectation is also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 11 Then we show that s N tr(B−1D) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Recall that the Leibniz formula for the determi- nant is given as follows: det(B) = ∑ π � sgn(π) M ∏ i=1 bi,πi � , (24) where π is a permutation function that reorders the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Diagonal and off- diagonal terms are on the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Therefore, det(B) is asymptotically equivalent to either ∏M i=1 bii or ∏M i=1 bij where i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' For simplicity, we keep the notation that det(B) ∼a ∏M i=1 bii, where ” ∼a ” denotes asymptotic equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Based on Cramer’s rule, we can write i-th diagonal term of the inverse of B: (B−1)ii = det(B−i) det(B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' B−i is the matrix where we remove the i-th row and the i-th column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' By the same argu- ment, det(B−i) ∼a ∏M−1 j=1 bjj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Then we end up with: (B−1)ii ∼a ∏M−1 i=1 bjj ∏M j=1 bii = 1 bii .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The i-th diagonal entry of the matrix (B−1D)ii = (b−1)iidii + ∑ j̸=i (b−1)ijdji ∼a dii bii ≤ log(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The last equality follows from Property 2 in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Therefore, the trace of (B−1D) is also on the order of log(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We take the limit of s N tr(B−1D), where s = Nβ and β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' L’Hˆopital’s rule yields: lim N−→∞ s N log(N) = lim N−→∞ 1 (1 − β)N1−β −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (25) The proof is complete by letting B = � V( ˙T ) �−1 and D = V(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 (Asymptotic Normality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Let Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='6 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Then the causal forest estimator in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 is asymptotically normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This result also holds for multiple correlated policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We quantify the uncertainty of the policy effects based on the jackknife variance estimator (Wager and Athey, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 provides further details regarding the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 12 4 Reemployment Experiment in Pennsylvania The Pennsylvania ”Reemployment Bonus” Demonstration was a randomized controlled trial in 1988-89 that aimed to investigate the impact of financial incentives on the reem- ployment outcomes of unemployed individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The study population was divided into a control group, which received the usual benefits provided by the Unemployment In- surance System, and six treatment groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Treated individuals were offered a cash bonus for fulfilling certain criteria related to finding and retaining employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Specifically, participants in the treatment groups were required to accept a bonus that would be paid to them if they were able to secure a full-time job of at least 32 hours per week within a specified period (the qualification period) and maintain that employment for at least 16 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Two bonus levels were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These two levels were a low bonus and a high bonus, which were respectively three and six times the weekly benefit amount (WBA) received by the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The low bonus was on average $500, while the high bonus was $997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In addition to these two levels of bonus, the study also considered two different qualifi- cation periods, starting from the date on which the bonus offer was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These periods were a short one of 6 weeks and a long one of 12 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In addition to testing the impact of financial incentives on reemployment outcomes, the Pennsylvania ”Reemployment Bonus” Demonstration also aimed to investigate the effectiveness of providing job-search assistance to unemployed individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To this end, participants in the treatment groups were offered a workshop and an individualized as- sessment session as part of the treatment design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' However, attendance at the workshop and completion of the assessment session were not mandatory for claimants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In this article, we focus on treatment Group 4 which received a high bonus amount and a long qualification period, as well as an offer of a workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The primary out- come of interest is the logarithm of unemployment duration in weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The data include twenty different characteristics of the claimants, such as age, gender, the quarter of the experiment in which they enrolled, and unemployment rates in the local area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 4 Further information about the experiment and data can be found in the article by Bilias (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 Target Components In this section, we identify and characterize policy-relevant target components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Accord- ing to Figure 2 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4, the optimal number of components equals two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To characterize and interpret the chosen scores, Table 1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4 illustrates the effect of the claimant characteristics on each target component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The negative value of a coeffi- cient indicates that there is an inverse relationship between the characteristic in question and the outcome being measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' For instance, a black claimant is associated with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3% lower score on average relative to a white claimant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Based on the sign of the coefficients in Table 1, we can interpret the scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The lowest values of the target components identified in this article correspond to 4The variables are described in detail at the following url: http://qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='queensu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='ca/jae/ 2000-v15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='6/bilias/readme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='txt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 13 a subgroup of young, non-white male claimants in the non-durable manufacturing sec- tor who enrolled in the experiment early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' On the other hand, the highest values of the components reflect a subgroup of middle-aged and older female individuals in the durable manufacturing sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These claimants enrolled in the experiment late in the final quarter and tend to have a high number of dependents (as indicated by the positive co- efficient for the ”dep” variable in Table 1 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (26) and (27) summarize the characteristics of subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 5 Components are continuous, therefore, they characterize a full spectrum of individuals from one group to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Low score values ∼ � � � � � � � � � � � � � � � Young (age <= 35) Male Non-white Joined the experiment early In the sector of non-durable manufacturing Few or no dependents � � � � � � � � � � � � � � � (26) High score values ∼ � � � � � � � � � � � � � � � Middle-age and older Female White Joined the experiment late In the sector of durable manufacturing Many dependents � � � � � � � � � � � � � � � (27) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 Effect Heterogeneity In this section, we investigate the heterogeneity, or variability, in the effect of financial incentives on unemployment duration within and across different values of the compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This is done by examining various percentiles of the reemployment bonus effect on unemployment duration across the corresponding component vigintiles, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The results depicted in Figure 1 reveal considerable variation in the policy effects both within and between groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In particular, financial incentives have been found to poten- tially motivate some young non-white individuals (the group represented by low score values in (26)) to enter the labor market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' However, these incentives may also provide a temporary financial cushion for others, potentially dissuading them from actively seeking employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In contrast, this variation is less pronounced among older white claimants (upper vigintiles of each component, the group corresponding to high score values in (27)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 5According to Table 1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4, these two components are almost identical, therefore, (26) and (27) hold for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5 0 5 10 15 20 policy effect Percentiles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='025 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='975 (a) Component 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4 0 5 10 15 20 policy effect Percentiles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='025 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='975 (b) Component 2 Figure 1: x-axis represents vigintiles of the corresponding component, while y-axis mea- sures the ”Reemployment Bonus” effect on the logarithm of unemployment duration in weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The plot shows the error bars with the color corresponding to a given percentile of the policy effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The number of trees equals 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 15 Our analysis shows that the ”Reemployment Bonus” policy has, on average, a neg- ative effect on unemployment duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' However, significant variation is captured by different percentiles of policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The difference between the 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5th and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='5th per- centiles of policy effects is 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='8% in the first vigintile of Component 2, and this difference decreases to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1% in the 20th vigintile of the same component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' These findings suggest the need for targeted, personalized measures for younger non-white male claimants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 5 Conclusion Policymakers frequently seek to understand the impact of interventions on specific sub- groups or segments of the population, defined by certain characteristics or attributes known as covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In this article, we present a method for analyzing the density of policy effects within these target segments, which are defined as linear combinations of the explanatory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To achieve this, we combine two existing techniques, partial least squares and causal forests, that allow us to identify and analyze policy effects for the full range of the target segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This approach enables policymakers to understand how treatment effects vary within and across these segments, providing valuable insights for personalized policy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We show that the method is consistent and leads to asymptotically normally dis- tributed policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Additionally, our approach generalizes beyond a single policy effect to multiple (plausibly) correlated policy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Our analysis based on data from Pennsylvania ”Reemployment Bonus” Demonstration reveals a significant variation in the effect of financial incentives on the logarithm of unemployment duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The find- ings highlight the need for targeted measures for young non-white male participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' One potential extension of our method is to incorporate quantile regression forests (Meinshausen and Ridgeway, 2006), conditional on the target components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In addition to using randomized control trials, we also plan to explore the application of our method to observational data with an endogenous policy (Vella and Verbeek, 1999;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='06388, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Spencer Woody, Carlos M Carvalho, P Richard Hahn, and Jared S Murray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Estimating het- erogeneous effects of continuous exposures using bayesian tree ensembles: revisiting the impact of abortion rates on crime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='09845, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Ruoxuan Xiong, Allison Koenecke, Michael Powell, Zhu Shen, Joshua T Vogelstein, and Susan Athey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Federated causal inference in heterogeneous observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='11732, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 The Loss Function Minimize the difference between the personalized and group-level parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' arg min �θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='Sest) EStr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='Sest �� θ(Ci) − �θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Sest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) �TΣ−1� θ(Ci) − �θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Sest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) � − θ(Ci)TΣ−1θ(Ci) � = (28) EStr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='Sest � � θ(Ci) − θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) � �� � A + θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) − θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Sest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) � �� � B �TΣ−1� θ(Ci) − θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) � �� � A + θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) − θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Sest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) � �� � B � − θ(Ci)TΣ−1θ(Ci) � = (29) EStr � θ(Ci)TΣ−1θ(Ci) − 2θ(Ci)TΣ−1θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) + θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π)TΣ−1θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) − θ(Ci)TΣ−1θ(Ci) � + (30) ECi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='Sest �� θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) − �θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Sest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) �TΣ−1� θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) − �θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Sest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) �� = (31) − ECi � θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π)TΣ−1θ(Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Π) � + E(tr(I))2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (32) The second equality follows after taking into account the independence of the train and estimation data, cov(A, B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The final equality is based on the fact that θ(Ci, Π) = E(θ(Ci)|Ci ∈ ℓ(c, Π)), E ��θ(Ci, Sest, Π) � = θ(Ci, Π), and: ECi,Sest �� θ(Ci, Π) − �θ(Ci, Sest, Π) �TΣ−1� θ(Ci, Π) − �θ(Ci, Sest, Π) �� = tr � Σ−1E � θ(Ci, Π) − �θ(Ci, Sest, Π) �T� θ(Ci, Π) − �θ(Ci, Sest, Π) �� = tr(Σ−1Σ) = tr(I)2×2, where tr(I)2×2 is the trace of a 2 × 2 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Since E(tr(I))2×2 does not depend on the parameter of interest, we can disregard it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Hence, the optimal parameter maximizes the unbiased estimator of the negative mean squared error: ˆθ(Ci, Sest, Π) = arg max �θ 1 Ntr ∑ ℓ Ntr ℓ �θ(Ci, Π)T �Σ−1�θ(Ci, Π), (33) where the covariance matrix can be estimated as ˆΣ = ˆΣ � ˜θ(Ci, Str, Π)|Nest� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In this study, training and estimation samples have an equal number of observations, Ntr = Nest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 Consistency of Component Weights Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We adopt the approach of Naik and Tsai (2000) in which b⋆ = (ΣXX)−1σXy, where ΣXX is the covariance matrix of the predictors and σXy is the covariance vector between the predictors and the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We define the matrix R as follows: R = (σXy, ΣXXσXy, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , Σq−1 XX σXy), where q ≤ p is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Under the assumption that Sxx approaches ΣXX and sxy approaches σXy as the sample size N approaches infinity, we have: ˆb → R(RTΣXXR)−1ΣXXα⋆ in probability when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The assumptions q = M and σXy = ∑M j=1 γjvj imply that b⋆ is contained in the space spanned by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Consequently, Σ1/2 XX b⋆ is contained in the space spanned by R⋆ = Σ1/2 XX R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Therefore, R⋆(R⋆TR⋆)−1R⋆TΣ1/2 XX α⋆ = Σ1/2 XX α⋆, and R(R⋆TR⋆)−1RTΣXXα⋆ = α⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Hence, ˆb → b⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Brillinger (2012) in Sections 3 and 4 shows that b = ξΣXXb, where ξ = cov � g(b0 + Xb), b0 + Xb) � /var � b0 + Xb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Therefore, the proof is complete by noting that ˆb → Σ−1 XXσXy = ξb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 Consistency and Asymptotic Normality of Policy Ef- fects The large sample theory of causal forests is largely based on the works of Wager and Athey (2018) and Wager and Walther (2015), which present the H¨oeffding decomposition of conventional random forests in a univariate setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In this article, we demonstrate that the deviation between the random forest and its H¨oeffding decomposition approaches zero in the limit, and prove the asymptotic normality of the random forest estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' It is important to note that the outcome variable yi in this context may be a vector-valued vari- able, in which case most of the operations and results apply coordinate-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The lemmas and proofs discussed in this article have also been extensively analyzed by Nareklishvili (2022) for partially identified policy effects, and by Li (2020) under network effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' For notational simplicity, we will define Xi = Xi and Ci = Ci throughout the remain- der of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 H¨oeffding decomposition: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Given a collection of ℓ|Π| n=1 terminal nodes that form a partition of the component space C, we define the prediction of a tree at a generic test point c as: T = T (c, ξ, Ai, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) = |Π| ∑ n=1 1(c ∈ ℓn) 1 Nℓn ∑ i:Ci∈ℓn yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (34) ξ is an external source of randomization, to allow for the randomized split selection pro- cedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 1(c ∈ ℓn) is an indicator function and equals one if a point c ∈ ℓn, and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Nℓn denotes the number of observations in a terminal node ℓn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A tree T (c, ξ, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) represents a prediction at a point c based on data AN i=1 = {yi, Ci}N i=1 and a randomization parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' As described in Lewis (2000) and Kingsford and Salzberg (2008), trees are a popular choice for classification and regression tasks due to their interpretability, ease of implementation, and robustness to outliers and missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' However, trees also have a high variance and are prone to overfitting, which makes it difficult to determine the optimal tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' To address these issues, Breiman (2001) introduced the random forest algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Let s < N be a subset of size s from a population i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , N, where s = Nβ and β is sufficiently close to 1 (Wager and Athey, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Following the work of Breiman (2001) and Wager and Athey (2018), we define the random forest estimator as the average of the tree estimators aggregated over all possible size-s subsamples of the training data, marginalized over the auxiliary noise ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Specifically, the prediction of the random forest estimator at a particular test data point c is defined as: F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) = 1 � Ns � ∑ 1≤i1≤···≤is≤N EξT (c, ξ, Ai1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , Ais), (35) where i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , is are the size-s subsamples of the population {i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In practice, we estimate such a random forest by Monte Carlo averaging: F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) ≈ 1 B B ∑ b=1 T (c, ξ∗, A∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A∗ N) (36) where {A∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A∗ N} is drawn without replacement from {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' AN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' ξ∗ is an auxiliary noise in a given sample and B is the number of sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) is a 1 × M vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Therefore, most of the arithmetic operations in this section are defined coordinate- wise in RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A random forest estimator can be represented as a U-statistic (Hoeffding, 1961;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Ko- rolyuk and Borovskich, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A common approach to studying the large sample prop- erties of random forests is to derive the lower bound of its H¨oeffding decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' H¨oeffding decomposition (also known as the Hajek projection) in a univariate setting is described by H´ajek (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Specifically, consider a vector-valued function T ∈ RM which 23 is measurable and permutation symmetric, that is, T(πc) = T(c) for all π ∈ Π (a tree in this setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Then the Hajek projection of this function is defined as: ˙T = E(T) + N ∑ i=1 � E(T|Ci) − E(T) � = N ∑ i=1 E(T|Ci) − (N − 1)E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (37) Intuitively, the Hajek projection in (37) represents a projection of T onto the linear sub- space of all random variables of the form ∑N i=1 gi(Ci), where gi : Rd → R are arbitrary measurable functions such that E(g2 i (Ci)) < ∞ for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' It is clear that the con- ditional expectation of the centered and symmetric component ˙T in (37) is equal to the conditional expectation of T: E( ˙T|Ci) = E(T|Ci), and (38) E( ˙T) = E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Now consider the random forest estimator, F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) ∈ RM, and let the corre- sponding vector of means be µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Moreover, let ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) and Σ denote the Hajek projection of the random forest estimator, and the covariance matrix of the Hajek projec- tion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Assume also that the trees in ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) are symmetric and the observations are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 holds: We define the Hajek projection of the random forest estimator as ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − µ = N ∑ i=1 E �F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − µ|Ai � = (39) 1 � Ns � N ∑ i=1 E � ∑ 1≤i1≤···≤is≤N EξT (c, ξ, Ai1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , Ais) − µ|Ai � , where (N s ) is the number of size-s subsets i1 ≤ · · · ≤ is that can be selected from the N observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 of Nareklishvili (2022) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Moreover, this is also shown by Li (2020) for in a setup with network effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Since the required conditions for the Lindeberg central limit theorem to hold are satis- fied (Billingsley, 2008), the Hajek projection of the random forest estimator is asymptoti- cally normally distributed: Σ−1/2� ˙˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − µ � d−→ N (0, IM), where 0 is a RM vector of zeros and IM is an identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Our goal is to prove that the random forest estimator is asymptotically normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' By adding and subtracting Σ−1/2 ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) to Σ−1/2�F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − µ � , we can see that the random forest estimator is related to its projection in the following way: 24 Σ−1/2�F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − µ � = Σ−1/2�F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) � + Σ−1/2� ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The main goal of this article is to show that: Σ−1/2�F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) − ˙F(c, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , AN) � p−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Then, by Slutsky’s theorem, it follows that the random forest estimator is asymptotically normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2 Asymptotic Normality: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We define the mean squared deviation of the random forest estimator and its pro- jection as: E(F − ˙F)TΣ−1(F − ˙F) = E � trΣ−1(F − ˙F)(F − ˙F)T� = (40) trΣ−1E(F − ˙F)(F − ˙F)T = tr Σ−1/2V �F − ˙F � Σ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' In order for the weak independence condition to hold for the exchangeable sequence of Xi, the following equation must be satisfied: E �Ti(Ci ∈ B)|Ci /∈ B) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' (41) Assuming that Ti(Ci ∈ B) are symmetric, square-integrable, vector-valued functions, then each Ti and Ti′ are pairwise independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' As i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , N is an exchangeable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=') sequence, Theorem 6 of Peccati (2004) and Proposition 1 of Li (2020) both apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Further- more, this lemma is shown by Nareklishvili (2022) (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Under Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 shows that the upper bound of the devi- ation between the random forest and its’ own projection converges to zero when N → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' According to Slutsky’s theorem, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1 implies that the random forest estimator is asymptotically normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' This result holds for the case of policy effects as well, where the difference appears in the constant term K that we have carried throughout the proof (see Wager and Athey (2018) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='3 Inference We quantify the uncertainty of policy effects by using the jackknife variance estimator (as described in Wager and Athey, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Let g = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' , G be the g-th bootstrapped sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We use a tree Πg and the corresponding estimation sample Sest g to obtain ˆθg(c, Sest g , Πg) at a generic test point c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Next, the average of the individual tree estimates is give as: 25 ˆθ � c, {Sest g }B g=1}, {Πg}G g=1 � = 1 G G ∑ g=1 ˆθg(c, Sest g , Πg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' We define Nig as the number of times an observation i appears in the g-th bootstrapped sample, either in the training sample Str or the estimation sample Sest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' The following variance estimator can be used to construct valid confidence intervals:: Var � ˆθ � c, {Sest g }G g=1}, {Πg}G g=1 �� = N ∑ i=1 ∆g − N G2 G ∑ g=1 � θg(c, Sest g , Πg) − ˆθ � c, {Sest g }G g=1} ��2, (42) where ∆g = �∑G g=1(Nig − 1) � ˆθg(c, Sest G , Πg) − ˆθ � c, {Sest g }G g=1}, {Πg}G g=1 �� G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='4 Interpreting Target Components 0 5 10 15 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='190 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='205 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='220 number of components RMSEP CV adjCV Figure 2: Root-mean-squared error (RMSE) based on five-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' 26 Table 1: Linear regression of each target component on independent characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Dependent variable: (Component 1) (Component 2) abdt −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='001∗∗∗ 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='270∗∗∗ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='955∗∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) hispanic −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='901∗∗∗ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='631∗∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) durable 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='190∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='018∗∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) nondurable −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='724∗∗∗ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='989∗∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) lusd −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='396∗∗∗ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='356∗∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) husd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='221∗∗∗ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='812∗∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) muld 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='123∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='938∗∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) Constant 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='317∗∗∗ −39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='629∗∗∗ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000) Observations 13,913 13,913 R2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000 Adjusted R2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000 Residual Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' Error (df = 13893) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='000 Note: ∗p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' ∗∗p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content=' ∗∗∗p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'} +page_content='01 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAyT4oBgHgl3EQfa_cC/content/2301.00251v1.pdf'}