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+DUALITY IN MONOIDAL CATEGORIES
+SEBASTIAN HALBIG AND TONY ZORMAN
+Abstract. We compare closed and rigid monoidal categories.
+Closedness is
+defined by the tensor product having a right adjoint: the internal-hom functor.
+Rigidity on the other hand generalises the concept of duals in the sense of finite-
+dimensional vector spaces. A consequence of these axioms is that the internal-hom
+functor is implemented by tensoring with the respective duals. This raises the
+question: can one decide whether a closed monoidal category is rigid, simply by
+verifying that the internal-hom is tensor-representable? At the Research School on
+Bicategories, Categorification and Quantum Theory, Heunen suggested that this
+is not the case. In this note, we will prove his claim by constructing an explicit
+counterexample.
+1. Introduction: Closed and Rigid Monoidal Categories
+Monoidal categories are a ubiquitous tool in mathematics, physics, and computer
+science [BS11].
+Often, they come equipped with additional structures, such as
+braidings or twists, see the previously cited article. In the following, we will compare
+two notions of duality for monoidal categories: closedness and rigidity.
+We assume the reader’s familiarity with standard concepts of category theory; in
+particular, adjunctions and monoidal categories as discussed for example in [ML98]
+and [EGNO15]. As rigidity and closedness are preserved, as well as reflected, by
+monoidal equivalences, see [Lin78], we restrict ourselves to the strict setting. As
+such, let C be a strict monoidal category with − ⊗ −: C × C −→ C as its tensor
+product and 1 ∈ C as its unit.
+The category C is called (right) closed if it admits a functor [−, −]: Cop × C −→ C,
+the (right) internal-hom, such that for all objects x ∈ C there exists an adjunction
+(1.1)
+− ⊗ x: C ⇄ C :[x, −].
+On the other hand, C is said to be (right) rigid if every object x ∈ C has a (right)
+dual x∗ equipped with an evaluation and coevaluation morphism
+evx : x∗ ⊗ x −→ 1
+and
+coevx : 1 −→ x ⊗ x∗,
+subject to the snake identities
+(1.2)
+idx = (idx ⊗ evx)(coevx ⊗idx)
+and
+idx∗ = (evx ⊗idx∗)(idx∗ ⊗ coevx).
+Rigid monoidal categories are closed, see for example Section 2.10 of [EGNO15].
+Date: January 10, 2023.
+2020 Mathematics Subject Classification. 18D15(primary), 18M10(secondary).
+Key words and phrases. closed monoidal categories, rigid monoidal categories, autonomous
+categories, Grothendieck–Verdier categories.
+We would like to thank Robert Allen for fruitful discussions in the early stages of this project,
+as well as Chris Heunen and Jean-Simon Lemay for their comments on a draft of this note. T.Z. is
+supported by the DFG grant KR 5036/2-1.
+1
+arXiv:2301.03545v1  [math.CT]  9 Jan 2023
+
+DUALITY IN MONOIDAL CATEGORIES
+2
+Lemma 1.1. If C is rigid, the internal-hom is implemented by the adjunction
+(1.3)
+− ⊗ x: C ⇄ C :− ⊗ x∗
+for all x ∈ C.
+The main concern of this note is to show that the converse of the above result
+does not hold. That is, we will prove that the internal-hom being given by tensoring
+with the dual of an object does not imply rigidity.
+In order to elucidate the underlying problem, let us assume that we are given
+objects x, y ∈ C such that − ⊗ x: C ⇄ C :− ⊗ y. The unit and counit of the
+adjunction provide us with natural candidates for the coevaluation and evaluation
+morphisms:
+coevx := η1 : 1 −→ x ⊗ y
+and
+evx := ε1 : y ⊗ x −→ 1.
+The triangle identities of this adjunction evaluated at the monoidal unit state that
+idx = εx ◦ (η1 ⊗ x) and idy = (ε1 ⊗ x) ◦ ηy. However, since we a priori do not know
+whether εx ∼= idx ⊗ ε1 and ηy ∼= idy ⊗ η1, the snake identities do not necessarily
+follow.
+2. A counterexample
+First, we define a strict monoidal category (D, ⊕, 0) in terms of generators and
+relations. For details of this type of construction we refer the reader to [Kas98,
+Chapter XII]. The objects of D are the natural numbers N0 with addition as the
+tensor product and 0 ∈ N0 as monoidal unit.1 Its arrows are tensor products and
+compositions of identities, and the generating morphisms
+(2.1)
+ηm,n : m −→ m ⊕ n ⊕ n,
+εm,n : m ⊕ n ⊕ n −→ m,
+n, m ∈ N0, n ≥ 1.
+These are for all i, j, k, l, n ∈ N0 with n, k ≥ 1 subject to the relations
+ηi+j+2k+l,n(idi ⊕ ηj,k ⊕ idl) = ((idi ⊕ ηj,k ⊕ idl) ⊕ id2n)ηi+j+l,n,
+(2.2)
+ηi+j+l,n(idi ⊕ εj,k ⊕ idl) = ((idi ⊕ εj,k ⊕ idl) ⊕ id2n)ηi+j+2k+l,n,
+(2.3)
+εi+j+2k+l,n((idi ⊕ ηj,k ⊕ idl) ⊕ id2n) = (idi ⊕ ηj,k ⊕ idl)εi+j+l,n,
+(2.4)
+εi+j+l,n((idi ⊕ εj,k ⊕ idl) ⊕ id2n) = (idi ⊕ εj,k ⊕ idl)εi+j+2k+l,n.
+(2.5)
+These relations are tailored to implement for any n ∈ N natural transformations
+ηx,n : x −→ x ⊕ (n ⊕ n),
+εx,n : x ⊕ (n ⊕ n) −→ x,
+for all x ∈ D.
+For example, let i, j, k, l, n be as above. Further, define x := i⊕j⊕l, y := i⊕j⊕2k⊕j,
+and f := idi ⊕ ηj,k ⊕ idj : x −→ y. In this setting, Equation (2.2) translates to the
+usual naturality condition, expressed by the commutativity of the following diagram:
+x
+y
+x ⊕ (n ⊕ n)
+y ⊕ (n ⊕ n)
+f
+ηy,n
+ηx,n
+f⊕(idn⊕idn)
+By quotienting out the triangle identities, we obtain a category C in which tensoring
+with any fixed object gives rise to a self-adjoint functor. Explained in more detail,
+1A strict monoidal category whose monoid of objects is (isomorphic to) the natural numbers is
+also called a PRO.
+
+DUALITY IN MONOIDAL CATEGORIES
+3
+the monoidal category (C, ⊕, 0) has the same objects and generating morphisms as
+D and the same identities hold. In addition, for any i, n ∈ N0 with n ≥ 1 we require
+(2.6)
+εi+n,n(ηi,n ⊕ idn) = idi+n,
+and
+(εi,n ⊕ idn)(ηi+n,n) = idi+n.
+The next result succinctly summarises the observations made so far concerning the
+internal-hom of C.
+Lemma 2.1. The category C is closed monoidal; its internal-hom functor is given by
+(2.7)
+− ⊗ n: C ⇄ C :− ⊗ n,
+for all n ∈ C.
+In order to analyse the morphisms in C and show that it is not rigid monoidal, we
+will rely on two tools. The first is the length of an arrow f ∈ C(n, m). It is defined as
+the minimal number of generating morphisms needed to present f. The second tool
+will be given by invariants for morphisms in C arising from functors into the category
+vectk of finite-dimensional vector spaces over a field k. Note that for any such vector
+space V there exists an isomorphism φ: V −→ V ∗ to its dual V ∗. The morphisms
+coevV := (idV ⊗ φ−1) coevV : k −→ V ⊗ V,
+evV := (φ ⊗ idV ) evV : V ⊗ V −→ k
+satisfy the snake identities, turning V into its own dual. The next theorem is an
+application of [Kas98, Proposition XII.1.4].
+Theorem 2.2. For any V ∈ vectk and isomorphism φ: V −→ V ∗ there exists a
+strong monoidal functor F(V,φ) : C −→ vectk such that for all n, m ∈ N0 with n ≥ 1
+F(V,φ)(ηm,n) = idm ⊗ coevV ⊗n
+and
+F(V,φ)(εm,n) = idm ⊗ evV ⊗n.
+To prove the statement, one has to show that relations in C are mapped to relations
+in vectk. This amounts to verifying that V is its own right dual, in the rigid sense.
+Corollary 2.3. The category C is skeletal. Furthermore, for any g ∈ C(m, n) the
+following arrows cannot be isomorphisms:
+(2.8)
+(idj1 ⊗ ηl,m ⊗ idj2)g,
+g(idi1 ⊗ εj,k, idi2).
+Proof. Let V ∈ vectk of dimension at least 2 and fix an isomorphism φ: V −→ V ∗.
+For any n, m ∈ C we have F(V,φ)(n) = V ⊗n = V ⊗m = F(V,φ)(m) if and only if n = m.
+Thus, C must be skeletal.
+Now suppose that g ∈ C(m, n) and consider the morphism f := g(idi1 ⊗ εj,k, idi2).
+Applying F(V,φ) to f, we get F(V,φ)(f) = F(V,φ)(g)F(V,φ)(idi1 ⊗ εj,k, idi2). However, due
+to the difference in the dimensions of its source and target, F(V,φ)(idi1 ⊗ εj,k, idi2)
+must have a non-trivial kernel and thus f cannot be an isomorphism.
+A similar argument involving the cokernel proves that (idj1 ⊗ ηl,m ⊗ idj2)g is not
+invertible.
+□
+We can now state and prove our main theorem.
+Theorem 2.4. The category C is not rigid.
+Proof. We assume that 1 ∈ C admits a right dual. Due to the uniqueness of adjoints,
+there exist isomorphisms ϑ: 2n −→ 2n and θ: n −→ n such that the evaluation and
+coevaluation morphisms are given by
+coev1 := ϑη0,1 : 0 −→ 2,
+ev1 := ε0,1(θ ⊗ idn): 2 −→ 0.
+We now want to consider the following subset of homomorphisms of D:
+S :=
+�
+(id1 ⊗ ε0,1) φ (η0,1 ⊗ id1) ∈ D(1, 1)
+��� φ ∈ D(3, 3) such that π(φ) is invertible
+�
+,
+
+DUALITY IN MONOIDAL CATEGORIES
+4
+where π: D −→ C is the ‘projection’ functor.
+By construction, the morphism
+s = (id1 ⊗ ev1)(coev1 ⊗id1) corresponding to one of the two snake-identities is an
+element of S. Furthermore, every element of S has length at least two.2 Thus, by
+proving that S is closed under the relations arising from Equation (2.6), it follows
+that π(s) ̸= id1, which concludes the proof.
+To that end, let us consider an element x = (id1 ⊗ ε0,1) φ (η0,1 ⊗ id1) ∈ S. There
+are two types of ‘moves’ we have to study. First, suppose we expand an identity into
+one of the triangle-morphisms. This equates to either pre- or postcomposing φ with
+an arrow ψ ∈ D(3, 3) which projects onto an isomorphism in C, leading to another
+element in S. Second, a triangle-morphism might be contracted to an identity. A
+priori, there are three ways in which this might occur
+x = (id1 ⊗ ε0,1)ε1,1(η0,1 ⊗ id1),
+where φ = φ′ ε1,1, or
+(2.9)
+x = (id1 ⊗ ε0,1)η1,1φ′′(η0,1 ⊗ id1),
+with φ = η1,1 φ′′, or
+(2.10)
+x = (id1 ⊗ ε0,1)φ2tφ1(η0,1 ⊗ id1)
+with φ = φ2tφ1 and π(t) = id.
+(2.11)
+Due to Corollary 2.3, neither π(φ′)π(ε1,1) nor π(η1,1)π(φ′′) are isomorphisms, contra-
+dicting Cases (2.9) and (2.10). Now assume x = (id1 ⊗ ε0,1) φ2tφ1 (η0,1 ⊗ id1) and
+φ = φ2tφ1. Using the functoriality of π: D −→ C, we get
+π(φ) = π(φ2tφ1) = π(φ2)π(t)π(φ1) = π(φ2)π(φ1) = π(φ2φ1).
+Since π(φ2φ1) is an isomorphism, (id1 ⊗ ε0,1)φ2φ1(η0,1 ⊗ id1) is an element of S.
+□
+3. Tensor-Representability and Grothendieck–Verdier Categories
+Although the internal-hom of a closed monoidal category C being tensor-represent-
+able does not imply rigidity, C often admits additional structure.
+Definition 3.1 ([BD13, Section 1.1]). A Grothendieck–Verdier category is a pair
+(C, d) of a monoidal category C and an object d ∈ C, such that there exists an
+antiequivalence D: C −→ Cop and for all x ∈ C the functor C(−⊗x, d) is representable
+by D(x).
+If d = 1 is the monoidal unit, one speaks of an r-category.
+Symmetric Grothendieck–Verdier categories are also called ⋆-autonomous cate-
+gories, see [Bar95]. Any rigid monoidal category is an instance of an r-category. The
+converse does not hold, as shown by the counterexamples [BD13, Example 1.9] and
+[BD13, Example 3.3].
+We conclude this note by showing that any monoidal category where tensoring
+with an object has tensor-reprensentable left and right adjoints is an r-category. To
+this end, we fix a monoidal category C such that for any x ∈ C there exist objects
+L(x) and R(x) such that
+− ⊗ L(x) ⊣ − ⊗ x ⊣ − ⊗ R(x).
+Theorem 3.2. If C is as described above, it is an r-category.
+Proof. By the parameter theorem, see for example [ML98, Theorem IV.7.3], the
+object maps L, R: Ob(C) −→ Ob(C) can be promoted to functors
+R: C −→ Cop
+and
+L: Cop −→ C.
+2Note that the relations of D leave the number of generating morphisms in any presentation of a
+given arrow invariant.
+
+DUALITY IN MONOIDAL CATEGORIES
+5
+We verify that L and R are quasi-inverses of each other. By assumption, for all
+y, z ∈ C we have
+C(y ⊗ LR(x), z) ∼= C(y, z ⊗ R(x)) ∼= C(y ⊗ x, z).
+Setting y = 1, the Yoneda embedding gives rise to a natural isomorphism LR −→ IdC.
+A similar argument gives RL ∼= IdCop.
+In order to show that C(− ⊗ x, 1) is representable by R(x), we have to prove that
+for all y ∈ C there exists a natural isomorphism
+C(y ⊗ x, 1) ∼= C(y, R(x)).
+By assumption, we have C(y ⊗ x, z) ∼= C(y, z ⊗ Rx). The claim follows by setting
+z = 1.
+□
+References
+[Bar95] Michael Barr. Nonsymmetric ∗-autonomous categories. Theor. Comput. Sci., 139(1-
+2):115–130, 1995.
+[BD13] Mitya Boyarchenko and Vladimir Drinfeld.
+A duality formalism in the spirit of
+Grothendieck and Verdier. Quantum Topol., 4(4):447–489, 2013.
+[BS11] John Baez and Mike Stay. Physics, topology, logic and computation: a Rosetta Stone.
+In New structures for physics, volume 813 of Lecture Notes in Phys., pages 95–172.
+Springer, Heidelberg, 2011.
+[EGNO15] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor categories,
+volume 205 of Mathematical Surveys and Monographs. American Mathematical Society,
+Providence, RI, 2015.
+[Kas98] Christian Kassel. Quantum groups. In Algebra and operator theory (Tashkent, 1997),
+pages 213–236. Kluwer Acad. Publ., Dordrecht, 1998.
+[Lin78] Harald Lindner. Adjunctions in monoidal categories. Manuscr. Math., 26:113–139, 1978.
+[ML98] Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate
+Texts in Mathematics. Springer-Verlag, New York, second edition, 1998.
+S.H., Philipps-Universit¨at Marburg, Arbeitsgruppe Algebraische Lie-Theorie,
+Hans-Meerwein-Straße 6, 35043 Marburg
+Email address: sebastian.halbig@uni-marburg.de
+T.Z., Technische Universit¨at Dresden, Institut f¨ur Geometrie, Zellescher Weg
+12–14, 01062 Dresden
+Email address: tony.zorman@tu-dresden.de
+

-tE1T4oBgHgl3EQf8gUY/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf,len=194
+page_content='DUALITY IN MONOIDAL CATEGORIES  SEBASTIAN HALBIG AND TONY ZORMAN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' We compare closed and rigid monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Closedness is  defined by the tensor product having a right adjoint: the internal-hom functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Rigidity on the other hand generalises the concept of duals in the sense of finite-  dimensional vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' A consequence of these axioms is that the internal-hom  functor is implemented by tensoring with the respective duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' This raises the  question: can one decide whether a closed monoidal category is rigid, simply by  verifying that the internal-hom is tensor-representable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' At the Research School on  Bicategories, Categorification and Quantum Theory, Heunen suggested that this  is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' In this note, we will prove his claim by constructing an explicit  counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Introduction: Closed and Rigid Monoidal Categories  Monoidal categories are a ubiquitous tool in mathematics, physics, and computer  science [BS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Often, they come equipped with additional structures, such as  braidings or twists, see the previously cited article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' In the following, we will compare  two notions of duality for monoidal categories: closedness and rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  We assume the reader’s familiarity with standard concepts of category theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' in  particular, adjunctions and monoidal categories as discussed for example in [ML98]  and [EGNO15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' As rigidity and closedness are preserved, as well as reflected, by  monoidal equivalences, see [Lin78], we restrict ourselves to the strict setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' As  such, let C be a strict monoidal category with − ⊗ −: C × C −→ C as its tensor  product and 1 ∈ C as its unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  The category C is called (right) closed if it admits a functor [−, −]: Cop × C −→ C,  the (right) internal-hom, such that for all objects x ∈ C there exists an adjunction  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='1)  − ⊗ x: C ⇄ C :[x, −].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  On the other hand, C is said to be (right) rigid if every object x ∈ C has a (right)  dual x∗ equipped with an evaluation and coevaluation morphism  evx : x∗ ⊗ x −→ 1  and  coevx : 1 −→ x ⊗ x∗,  subject to the snake identities  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='2)  idx = (idx ⊗ evx)(coevx ⊗idx)  and  idx∗ = (evx ⊗idx∗)(idx∗ ⊗ coevx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Rigid monoidal categories are closed, see for example Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='10 of [EGNO15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' 18D15(primary), 18M10(secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' closed monoidal categories, rigid monoidal categories, autonomous  categories, Grothendieck–Verdier categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  We would like to thank Robert Allen for fruitful discussions in the early stages of this project,  as well as Chris Heunen and Jean-Simon Lemay for their comments on a draft of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' is  supported by the DFG grant KR 5036/2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='03545v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='CT]  9 Jan 2023  DUALITY IN MONOIDAL CATEGORIES  2  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' If C is rigid, the internal-hom is implemented by the adjunction  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='3)  − ⊗ x: C ⇄ C :− ⊗ x∗  for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  The main concern of this note is to show that the converse of the above result  does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' That is, we will prove that the internal-hom being given by tensoring  with the dual of an object does not imply rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  In order to elucidate the underlying problem, let us assume that we are given  objects x, y ∈ C such that − ⊗ x: C ⇄ C :− ⊗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The unit and counit of the  adjunction provide us with natural candidates for the coevaluation and evaluation  morphisms:  coevx := η1 : 1 −→ x ⊗ y  and  evx := ε1 : y ⊗ x −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  The triangle identities of this adjunction evaluated at the monoidal unit state that  idx = εx ◦ (η1 ⊗ x) and idy = (ε1 ⊗ x) ◦ ηy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' However, since we a priori do not know  whether εx ∼= idx ⊗ ε1 and ηy ∼= idy ⊗ η1, the snake identities do not necessarily  follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' A counterexample  First, we define a strict monoidal category (D, ⊕, 0) in terms of generators and  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' For details of this type of construction we refer the reader to [Kas98,  Chapter XII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The objects of D are the natural numbers N0 with addition as the  tensor product and 0 ∈ N0 as monoidal unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='1 Its arrows are tensor products and  compositions of identities, and the generating morphisms  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='1)  ηm,n : m −→ m ⊕ n ⊕ n,  εm,n : m ⊕ n ⊕ n −→ m,  n, m ∈ N0, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  These are for all i, j, k, l, n ∈ N0 with n, k ≥ 1 subject to the relations  ηi+j+2k+l,n(idi ⊕ ηj,k ⊕ idl) = ((idi ⊕ ηj,k ⊕ idl) ⊕ id2n)ηi+j+l,n,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='2)  ηi+j+l,n(idi ⊕ εj,k ⊕ idl) = ((idi ⊕ εj,k ⊕ idl) ⊕ id2n)ηi+j+2k+l,n,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='3)  εi+j+2k+l,n((idi ⊕ ηj,k ⊕ idl) ⊕ id2n) = (idi ⊕ ηj,k ⊕ idl)εi+j+l,n,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='4)  εi+j+l,n((idi ⊕ εj,k ⊕ idl) ⊕ id2n) = (idi ⊕ εj,k ⊕ idl)εi+j+2k+l,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='5)  These relations are tailored to implement for any n ∈ N natural transformations  ηx,n : x −→ x ⊕ (n ⊕ n),  εx,n : x ⊕ (n ⊕ n) −→ x,  for all x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  For example, let i, j, k, l, n be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Further, define x := i⊕j⊕l, y := i⊕j⊕2k⊕j,  and f := idi ⊕ ηj,k ⊕ idj : x −→ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' In this setting, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='2) translates to the  usual naturality condition, expressed by the commutativity of the following diagram:  x  y  x ⊕ (n ⊕ n)  y ⊕ (n ⊕ n)  f  ηy,n  ηx,n  f⊕(idn⊕idn)  By quotienting out the triangle identities, we obtain a category C in which tensoring  with any fixed object gives rise to a self-adjoint functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Explained in more detail,  1A strict monoidal category whose monoid of objects is (isomorphic to) the natural numbers is  also called a PRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  DUALITY IN MONOIDAL CATEGORIES  3  the monoidal category (C, ⊕, 0) has the same objects and generating morphisms as  D and the same identities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' In addition, for any i, n ∈ N0 with n ≥ 1 we require  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='6)  εi+n,n(ηi,n ⊕ idn) = idi+n,  and  (εi,n ⊕ idn)(ηi+n,n) = idi+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  The next result succinctly summarises the observations made so far concerning the  internal-hom of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The category C is closed monoidal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' its internal-hom functor is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='7)  − ⊗ n: C ⇄ C :− ⊗ n,  for all n ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  In order to analyse the morphisms in C and show that it is not rigid monoidal, we  will rely on two tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The first is the length of an arrow f ∈ C(n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' It is defined as  the minimal number of generating morphisms needed to present f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The second tool  will be given by invariants for morphisms in C arising from functors into the category  vectk of finite-dimensional vector spaces over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Note that for any such vector  space V there exists an isomorphism φ: V −→ V ∗ to its dual V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The morphisms  coevV := (idV ⊗ φ−1) coevV : k −→ V ⊗ V,  evV := (φ ⊗ idV ) evV : V ⊗ V −→ k  satisfy the snake identities, turning V into its own dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The next theorem is an  application of [Kas98, Proposition XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' For any V ∈ vectk and isomorphism φ: V −→ V ∗ there exists a  strong monoidal functor F(V,φ) : C −→ vectk such that for all n, m ∈ N0 with n ≥ 1  F(V,φ)(ηm,n) = idm ⊗ coevV ⊗n  and  F(V,φ)(εm,n) = idm ⊗ evV ⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  To prove the statement, one has to show that relations in C are mapped to relations  in vectk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' This amounts to verifying that V is its own right dual, in the rigid sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The category C is skeletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Furthermore, for any g ∈ C(m, n) the  following arrows cannot be isomorphisms:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='8)  (idj1 ⊗ ηl,m ⊗ idj2)g,  g(idi1 ⊗ εj,k, idi2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Let V ∈ vectk of dimension at least 2 and fix an isomorphism φ: V −→ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  For any n, m ∈ C we have F(V,φ)(n) = V ⊗n = V ⊗m = F(V,φ)(m) if and only if n = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Thus, C must be skeletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Now suppose that g ∈ C(m, n) and consider the morphism f := g(idi1 ⊗ εj,k, idi2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Applying F(V,φ) to f, we get F(V,φ)(f) = F(V,φ)(g)F(V,φ)(idi1 ⊗ εj,k, idi2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' However, due  to the difference in the dimensions of its source and target, F(V,φ)(idi1 ⊗ εj,k, idi2)  must have a non-trivial kernel and thus f cannot be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  A similar argument involving the cokernel proves that (idj1 ⊗ ηl,m ⊗ idj2)g is not  invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  □  We can now state and prove our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The category C is not rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' We assume that 1 ∈ C admits a right dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Due to the uniqueness of adjoints,  there exist isomorphisms ϑ: 2n −→ 2n and θ: n −→ n such that the evaluation and  coevaluation morphisms are given by  coev1 := ϑη0,1 : 0 −→ 2,  ev1 := ε0,1(θ ⊗ idn): 2 −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  We now want to consider the following subset of homomorphisms of D:  S :=  �  (id1 ⊗ ε0,1) φ (η0,1 ⊗ id1) ∈ D(1, 1)  ��� φ ∈ D(3, 3) such that π(φ) is invertible  �  ,  DUALITY IN MONOIDAL CATEGORIES  4  where π: D −→ C is the ‘projection’ functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  By construction, the morphism  s = (id1 ⊗ ev1)(coev1 ⊗id1) corresponding to one of the two snake-identities is an  element of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Furthermore, every element of S has length at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='2 Thus, by  proving that S is closed under the relations arising from Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='6), it follows  that π(s) ̸= id1, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  To that end, let us consider an element x = (id1 ⊗ ε0,1) φ (η0,1 ⊗ id1) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' There  are two types of ‘moves’ we have to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' First, suppose we expand an identity into  one of the triangle-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' This equates to either pre- or postcomposing φ with  an arrow ψ ∈ D(3, 3) which projects onto an isomorphism in C, leading to another  element in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Second, a triangle-morphism might be contracted to an identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' A  priori, there are three ways in which this might occur  x = (id1 ⊗ ε0,1)ε1,1(η0,1 ⊗ id1),  where φ = φ′ ε1,1, or  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='9)  x = (id1 ⊗ ε0,1)η1,1φ′′(η0,1 ⊗ id1),  with φ = η1,1 φ′′, or  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='10)  x = (id1 ⊗ ε0,1)φ2tφ1(η0,1 ⊗ id1)  with φ = φ2tφ1 and π(t) = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='11)  Due to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='3, neither π(φ′)π(ε1,1) nor π(η1,1)π(φ′′) are isomorphisms, contra-  dicting Cases (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Now assume x = (id1 ⊗ ε0,1) φ2tφ1 (η0,1 ⊗ id1) and  φ = φ2tφ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Using the functoriality of π: D −→ C, we get  π(φ) = π(φ2tφ1) = π(φ2)π(t)π(φ1) = π(φ2)π(φ1) = π(φ2φ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Since π(φ2φ1) is an isomorphism, (id1 ⊗ ε0,1)φ2φ1(η0,1 ⊗ id1) is an element of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Tensor-Representability and Grothendieck–Verdier Categories  Although the internal-hom of a closed monoidal category C being tensor-represent-  able does not imply rigidity, C often admits additional structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='1 ([BD13, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' A Grothendieck–Verdier category is a pair  (C, d) of a monoidal category C and an object d ∈ C, such that there exists an  antiequivalence D: C −→ Cop and for all x ∈ C the functor C(−⊗x, d) is representable  by D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  If d = 1 is the monoidal unit, one speaks of an r-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Symmetric Grothendieck–Verdier categories are also called ⋆-autonomous cate-  gories, see [Bar95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Any rigid monoidal category is an instance of an r-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The  converse does not hold, as shown by the counterexamples [BD13, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='9] and  [BD13, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  We conclude this note by showing that any monoidal category where tensoring  with an object has tensor-reprensentable left and right adjoints is an r-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' To  this end, we fix a monoidal category C such that for any x ∈ C there exist objects  L(x) and R(x) such that  − ⊗ L(x) ⊣ − ⊗ x ⊣ − ⊗ R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' If C is as described above, it is an r-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' By the parameter theorem, see for example [ML98, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='3], the  object maps L, R: Ob(C) −→ Ob(C) can be promoted to functors  R: C −→ Cop  and  L: Cop −→ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  2Note that the relations of D leave the number of generating morphisms in any presentation of a  given arrow invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  DUALITY IN MONOIDAL CATEGORIES  5  We verify that L and R are quasi-inverses of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' By assumption, for all  y, z ∈ C we have  C(y ⊗ LR(x), z) ∼= C(y, z ⊗ R(x)) ∼= C(y ⊗ x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Setting y = 1, the Yoneda embedding gives rise to a natural isomorphism LR −→ IdC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  A similar argument gives RL ∼= IdCop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  In order to show that C(− ⊗ x, 1) is representable by R(x), we have to prove that  for all y ∈ C there exists a natural isomorphism  C(y ⊗ x, 1) ∼= C(y, R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  By assumption, we have C(y ⊗ x, z) ∼= C(y, z ⊗ Rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' The claim follows by setting  z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  □  References  [Bar95] Michael Barr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Nonsymmetric ∗-autonomous categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=', 139(1-  2):115–130, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  [BD13] Mitya Boyarchenko and Vladimir Drinfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  A duality formalism in the spirit of  Grothendieck and Verdier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Quantum Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=', 4(4):447–489, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  [BS11] John Baez and Mike Stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Physics, topology, logic and computation: a Rosetta Stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  In New structures for physics, volume 813 of Lecture Notes in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=', pages 95–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  Springer, Heidelberg, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  [EGNO15] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Tensor categories,  volume 205 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' American Mathematical Society,  Providence, RI, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  [Kas98] Christian Kassel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' In Algebra and operator theory (Tashkent, 1997),  pages 213–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Kluwer Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=', Dordrecht, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  [Lin78] Harald Lindner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Adjunctions in monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Manuscr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=', 26:113–139, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  [ML98] Saunders Mac Lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Categories for the working mathematician, volume 5 of Graduate  Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=' Springer-Verlag, New York, second edition, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=', Philipps-Universit¨at Marburg, Arbeitsgruppe Algebraische Lie-Theorie,  Hans-Meerwein-Straße 6, 35043 Marburg  Email address: sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='halbig@uni-marburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='de  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content=', Technische Universit¨at Dresden, Institut f¨ur Geometrie, Zellescher Weg  12–14, 01062 Dresden  Email address: tony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='zorman@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}
+page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQf8gUY/content/2301.03545v1.pdf'}

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+Recurrent Structure Attention Guidance for Depth Super-Resolution
+Jiayi Yuan*, Haobo Jiang*, Xiang Li, Jianjun Qian†, Jun Li†, Jian Yang
+PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education
+Jiangsu Key Lab of Image and Video Understanding for Social Security
+School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China
+{jiayiyuan, jiang.hao.bo, xiang.li.implus, csjqian, junli, csjyang}@njust.edu.cn
+Abstract
+Image guidance is an effective strategy for depth super-
+resolution. Generally, most existing methods employ hand-
+crafted operators to decompose the high-frequency (HF) and
+low-frequency (LF) ingredients from low-resolution depth
+maps and guide the HF ingredients by directly concatenat-
+ing them with image features. However, the hand-designed
+operators usually cause inferior HF maps (e.g., distorted or
+structurally missing) due to the diverse appearance of com-
+plex depth maps. Moreover, the direct concatenation often re-
+sults in weak guidance because not all image features have
+a positive effect on the HF maps. In this paper, we de-
+velop a recurrent structure attention guided (RSAG) frame-
+work, consisting of two important parts. First, we introduce a
+deep contrastive network with multi-scale filters for adaptive
+frequency-domain separation, which adopts contrastive net-
+works from large filters to small ones to calculate the pixel
+contrasts for adaptive high-quality HF predictions. Second,
+instead of the coarse concatenation guidance, we propose a
+recurrent structure attention block, which iteratively utilizes
+the latest depth estimation and the image features to jointly
+select clear patterns and boundaries, aiming at providing re-
+fined guidance for accurate depth recovery. In addition, we
+fuse the features of HF maps to enhance the edge structures in
+the decomposed LF maps. Extensive experiments show that
+our approach obtains superior performance compared with
+state-of-the-art depth super-resolution methods.
+Introduction
+Depth super-resolution (DSR) is a fundamental low-level vi-
+sion topic in computer vision as it plays an important role in
+a variety of applications, such as 3D reconstruction (Hou,
+Dai, and Nießner 2019), autonomous driving (Caesar et al.
+2020), and virtual reality (Meuleman et al. 2020). Gener-
+ally, DSR is to recover a high-resolution depth map precisely
+from a given low-resolution depth map. Recently, an image
+guidance DSR framework becomes more and more popu-
+lar since it has demonstrated remarkable progress by bor-
+rowing the structures and boundaries in high-resolution im-
+age to improve the depth map (Hui, Loy, and Tang 2016).
+As shown in Fig. 1 (a), most efforts (Hui, Loy, and Tang
+*These authors contributed equally.
+†corresponding authors
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Hand-crafted HF
+& LF Separation
+RGB & HF
+Feature Fusion
+������������������������������������
+Adaptive HF & LF 
+Separation
+Structure  
+Attention
+(a) Image Guided Residual Framework
+(b) Recurrent Structure Attention Guided Framework
+HF & LF
+Feature Fusion
+������������������������������������
+������������������������������������
+������������������������������������
+������������������������������������
+������������������������������������
+Bicubic
+Interpolation
+������������������������������������
+������������������������������������
+������������������������������������
+������������������������������������
+Recurrent
+Figure 1: Image guided DSR framework. (a) Popular image
+guided residual DSR framework; (b) Our recurrent structure
+attention guided DSR framework.
+2016; Guo et al. 2019; Zuo et al. 2019b; Li et al. 2019) usu-
+ally 1) use hand-crafted operators (e.g., hand-designed fil-
+ters) to perform an early spectral decomposition of the low-
+resolution depth map (i.e., high-frequency (HF) and low-
+frequency (LF)), 2) coarsely implement the image guidance
+by directly concatenating the image features into the HF
+maps, and 3) run a simple up-sampling like bicubic interpo-
+lation on the decomposed low-resolution LF map to a high-
+resolution one. However, this framework still suffers from
+three challenging problems as follows.
+Firstly, the hand-designed operators often cause a weak
+spectral decomposition as they are difficult to handle the di-
+verse structures in the complex depth map, resulting in lost
+object structures in the HF map of Fig. 2 (b). Secondly, the
+direct feature concatenation results in weak image guidance
+since the complex textures of the image usually produce in-
+ferior features based on our observation. For example, Fig. 2
+(f-h) show clear features of the ceramic bottle (white box)
+and poor features of the poster (yellow box), corresponding
+to the complex and simple textures of the image in Fig. 2 (j),
+respectively. Thirdly, the bicubic interpolation also results
+in blurred edges in the LF map of Fig. 2 (d), because it is
+unsuitable for up-sampling of all kinds of structures.
+arXiv:2301.13419v1  [cs.CV]  31 Jan 2023
+
+(c) HF (Ours)
+(b) HF (hand-crafted)
+(d) LF (hand-crafted)
+(e) LF (Ours)
+(f) Fea. map (DMSG)
+(i) Fea. map (Ours)
+(j) Color image
+(a) LR depth
+(h) Fea. map (Ours-direct)
+(g) Fea. map (DJFR)
+Figure 2: Visualizations of the decomposed HF&LF and guidance feature maps. (b) and (d) show a weak frequency-domain
+separation using the hand-designed operators(Hui, Loy, and Tang 2016). (f-h) show image guidance features with redundant
+textures and noise in DMSG (Hui, Loy, and Tang 2016), DJFR (Li et al. 2019) and our network using direct concatenation.
+Compared with them, our method produces better HF structure in (c), sharper LF boundaries in (e), and clearer guidance
+structure in (i). LR depth map and color image are plotted in (a) and (g).
+To address these problems, we develop a novel recur-
+rent structure attention guided (RSAG) framework for high-
+quality DSR in Fig. 1 (b) through three aspects. First of all,
+we introduce a deep contrastive network with multi-scale fil-
+ters (DCN) to effectively decompose the HF and LF compo-
+nents of the input depth, instead of the hand-designed opera-
+tors. DCN is to subtly stack simple contrastive networks (Xu
+et al. 2020) three times from large filters to small ones for a
+coarse-to-fine HF prediction with contextual structures, and
+to calculate the LF component by subtracting the HF predic-
+tion from the input depth map. To better guide the depth fea-
+tures, in addition, we propose a recurrent structure attention
+(SA) block to select the useful image features, instead of the
+direct concatenation guidance. The key step of SA is to add
+absolute values of contrastive attention features of the image
+and the latest depth prediction, and then calculate an atten-
+tion map by employing channel and spatial attention oper-
+ators. Finally, we present an HF&LF feature fusion (HLF)
+block to improve the blurred edges in the LF component
+by concatenating the HF feature produced by our SA block,
+as its contextual structure can enhance the edges. Overall,
+our RSAG framework has a significant improvement on the
+hand-crafted spectral decomposition and image guidance. In
+summary, our contributions are as follows:
+• We introduce a deep contrastive network with multi-scale
+filters (DCN) for the robust HF and LF reconstruction,
+where the HF structure is implemented by stacking the
+pixel-wise contrast from large to small kernels.
+• We propose a novel recurrent structure attention (SA)
+block by combining the latest depth prediction with the
+image feature to select useful image guidance features.
+• Extensive experiments on the benchmark datasets verify
+the superior effectiveness of the proposed framework and
+achieve state-of-the-art restoration performance.
+Related work
+In this section, we mainly review the previous spectral de-
+composition and cross-modality fusion mechanisms used in
+depth map super-resolution (DSR).
+Spectral Decomposition in DSR
+Since the HF component of the depth map can provide suf-
+ficient structure information which coincides well with the
+image boundaries, most methods adopt early spectral de-
+composition for efficient DSR. A line of methods (Makarov,
+Aliev, and Gerasimova 2017; Xiao et al. 2018; Li et al. 2019;
+Zuo et al. 2019b; Guo et al. 2019) regard the interpolated
+depth input as the LF component and add a jump connection
+to transfer it to the end of the network. This global residual
+learning forces the network to focus on recovering the HF
+details. Another line of methods adopt the hand-designed
+filters (Hui, Loy, and Tang 2016; Yang et al. 2017) or edge-
+attention (Ye, Duan, and Li 2018; Chen and Jung 2018)
+blocks to extract HF information. However, these methods
+require additional completion operation, since the HF out-
+puts always include broken edges and holes. Recently, oc-
+tave convolution (Chen et al. 2019) is utilized for frequency
+division operation in DSR network (He et al. 2021), which is
+a plug-and-play convolutional unit. However, it separates the
+frequency domain in embedding space, which does not guar-
+antee that HF information is completely extracted. Instead,
+we propose a simple, fast, and adaptive separation method
+at the pixel level to provide reliable HF and LF maps.
+Cross-modality Fusion Mechanism
+Multi-path/scale Learning. Previous methods (Li et al.
+2016; Lutio et al. 2019; Zhu et al. 2018; Chen and Jung
+2018; Hao et al. 2019; Su et al. 2019) extract features in
+color space and depth space through two independent paths
+
+Copy
+Add
+SA
+Bicubic
+������������������������������������
+������������������������������������������������
+������������������������
+������������������������
+DCN
+������������������������������������
+������������������������������������
+SA
+������������������������������������
+������������������������������������
+������������������������−������������
+������������������������
+������������������������������������
+Recurrent
+������������������������������������
+Figure 3: The pipeline of our RSAG framework. It consists of a green DCN module for the adaptive frequency-domain separa-
+tion, an orange recurrent SA module for the HF component recovery, and a blue module for the LF component recovery.
+respectively, and transfer common structures through a joint
+branch. However, the multi-path methods may cause details
+missing since the cross-modality features are only fused in
+one specific layer. To handle the abovementioned problem,
+recent methods (Hui, Loy, and Tang 2016; Guo et al. 2019;
+He et al. 2021; Zuo et al. 2019b,a; Yan et al. 2022) adopt a
+multi-scale fusion strategy to merge the cross-modality fea-
+tures at different levels. Although the multi-scale methods
+have achieved considerable performance, the coarse aggre-
+gation may cause texture copying and depth bleeding.
+Recursive Learning. In order to generate higher-level
+details without introducing excessive parameters, recursive
+learning repeatedly applies similar modules for progres-
+sive image reconstruction. Existing recursive DSR meth-
+ods (Wen et al. 2019; Yang et al. 2019; Song et al. 2020)
+construct the depth map in a coarse-to-fine manner by re-
+garding the previous crude depth output as the input of the
+DSR network. Even though the multi-supervision and resid-
+ual learning avoid vanishing or exploding gradient problems
+to a certain extent, there still exists the risk of falling into a
+local optimum. However, we propose a recurrent guidance
+for DSR, which considers the previous depth prediction as
+the guidance information for the next recursion. As the re-
+cursion progresses, the continuously refined guidance is a
+strong constraint for better choosing the image features.
+Attention Mechanism. In recent years, the attention
+mechanism (Zhang et al. 2019; Guo et al. 2020; Wang et al.
+2021) has achieved significant improvements in the low-
+level vision field. In DSR task, Song et al. (Song et al. 2020)
+utilize the channel attention to focus on HF depth. Mean-
+while, Tang et al. (Tang et al. 2021) also design an HF
+attention bridge to extract the useful HF information dur-
+ing the depth estimation process and input it into the re-
+construction network. Although these attention operations
+selectively highlight the HF information, they do not es-
+sentially solve the problem of texture copying and incon-
+sistent boundaries in guidance images. The most related
+to our method is (Zhong et al. 2021), which also aims to
+find the consistent structure with an attention mechanism.
+However, there are big differences between them. 1) The
+proposed method uses contrastive networks to explore the
+cross-modality correlation in the HF layer since the HF
+modalities of the depth map and image are closer. 2) Com-
+pared with single image guidance, we complement the guid-
+ance with progressively refined depth prediction in a recur-
+sive fashion to accurately mine the consistent structure.
+Approach
+In this section, we introduce our recurrent structure attention
+guided (RSAG) framework for DSR. As shown in Fig. 3,
+RSAG contains three modules, including a deep contrastive
+network with multi-scale filters (DCN), a recurrent structure
+attention module (SA), and an HF&LF feature fusion (HLF)
+module. DCN adaptively learns the HF and LF decomposi-
+tion by cascading contrastive networks from large filters to
+small ones. Then, by introducing the last depth prediction
+to complement the image guidance, recurrent SA jointly se-
+lects the useful and clear structure features of the image for
+accurate HF depth reconstruction. Furthermore, during the
+reconstruction process, HF features guided by recurrent SA
+are integrated with LF features to refine the LF edges.
+Before presenting our method, we denote a high-
+resolution (HR) image by Y hr ∈ RH×W , where H and
+W are the sizes of the image, an HR depth map by
+Dhr ∈ RH×W , a low-resolution (LR) depth map by Dlr ∈
+RpH×pW , where 0 < p ≤ 1 is the downscaling factor
+(e.g., 1/4, 1/8, and 1/16). For Dhr, Dlf ∈ RH×W and
+Dhf ∈ RH×W are denoted as its LF and HF components.
+Deep Contrastive Network with Multi-scale Filters
+As shown in Fig. 4 (a), we aim to explore a DCN network
+for high-quality frequency components, instead of the hand-
+designed operator for the frequency-domain decomposition.
+Inspired by the contrast learning operator (Xu et al. 2020),
+which is designed for RGB image decomposition, we stack
+
+Conv7
+������������������������������������������������
+������������������������������������
+������������������������������������
+(a) Deep Contrastive Network with Multi-scale Filters
+Conv5
+Conv3
+Conv5
+Conv1
+Conv3
+LFE
+Channel Attention
+Space Attention
+γ
+c
+������������������������������������
+������������������������−������������
+������������������������
+(b) Structure Attention
+Conv1
+*
+Conv1
+Conv3
+Conv1
+Conv3
+������������������������
+������������
+LFE
+Figure 4: The architectures of (a) DCN and (b) SA. DCN
+aims to decompose HF and LF maps of a depth map by
+stacking three contrastive networks from large to small fil-
+ters. SA tends to adaptively filter out unwanted textures and
+highlight the useful HF regions of the image.
+it three times to a DCN with multi-scale filters for extracting
+high-quality HF components of the depth map.
+Specifically, given an LR depth map Dlr ∈ RpH×pW as
+input, we first upscale it to the desired resolution map Dbic ∈
+RH×W by bicubic interpolation. We denote the number of
+layers of our DCN network as I, and the HF map Dhf is
+defined as a recursive formulation:
+Dhf = HI
+I ,
+(1)
+HI
+i = Sigmoid
+�
+Convk(HI
+i−1) − Convk−2(HI
+i−1)
+�
+,
+(2)
+where HI
+0 = Dbic; HI
+i is the HF feature of the i-th layer
+in the DCN network with I layers (1 ≤ i ≤ I); Convk(·)
+represents a k × k convolutional operation followed by
+PReLU (He et al. 2015) activation, k = 2(I −i)+3, and we
+set I = 3 in this paper. Then, the LF map Dlf is calculated
+by subtracting the HF map Dhf from Dbic:
+Dlf = Dbic − Dhf.
+(3)
+To better understand the DCN network with different layers
+(I = 1, 2, 3), Fig. 5 shows their HF features. Fig. 5 (a-c) plot
+the HF features H1
+1, H2
+1 and H2
+2 of shallow DCN networks.
+Compared to H1
+1 and H2
+2, the HF feature H3
+3 of deeper DCN
+network are shown in Fig. 5 (f), which has the clearest and
+most complete edges. According to Fig. 5 (d-f), it is worth
+noticing that deeper DCN is prone to weaken depth informa-
+tion and enhance structural information (e.g., edge of plaster
+statue behind the teapot).
+Recurrent Structure Attention
+Removing textures while making full use of consistent
+boundaries in the image is a key challenge for guided DSR.
+Instead of trivial cross-modality feature concatenation, we
+propose a novel recurrent structure attention (SA) mecha-
+nism to bridge the modality gap between depth input and
+image guidance. As shown in Fig. 4 (b), we put our efforts
+into the following two aspects: (1) A cross-modality struc-
+ture feature attention is designed, where the consistent struc-
+tures are highlighted by contrast operators and the redundant
+features (e.g., textures and inconsistent boundaries) are sup-
+pressed in channel and space levels. (2) For better guiding
+depth details restoration, useful image features are selected
+with the progressively refined depth prediction recursively.
+Structure Attention. Given the image Y hr ∈ RH×W
+and the same size depth map Dhr ∈ RH×W as input, we
+first use the learnable feature extractor to produce a set of
+hierarchical image and depth features, which match with the
+corresponding HF features in decoder path. Then, sharing
+the same spirit as contrastive networks used in DCN, we ex-
+ploit the contextual information under multiple-level recep-
+tive fields and calculate the high contrastive features as HF
+components. We further sum the depth and image contrast
+maps and use absolute operations to enforce their consistent
+structures and prevent HF smoothing caused by positive and
+negative cancellations. This process can be formulated as:
+J = |Fy1 − Fy2| + |Fd1 − Fd2|,
+(4)
+Fyi = Conv2i−1(LFE(Y hr)),
+(5)
+Fdi = Conv2i−1(LFE(Dhr)), i ∈ {1, 2} ,
+(6)
+where LFE(·) denotes the learnable feature extractor for ini-
+tial hierarchical features learning. Conv2i−1(·) are the con-
+volutions with kernel size 2i − 1 followed by PReLU (He
+et al. 2015) activation. Fyi and Fdi are extracted image fea-
+tures and depth features under different receptive fields, re-
+spectively. J denotes the joint HF features, which are further
+fed into the channel and spatial attention blocks (Woo et al.
+2018). Such a design encourages learning the interaction be-
+tween different channels and focusing on the important spa-
+tial locations. The features after the attention block denoted
+as structure-aware features Sa, can be formulated as:
+Sa = SpatA(CA(J)),
+(7)
+where SpatA(·) and CA(·) represent the spatial attention
+and the channel attention blocks, respectively. At last, Sa
+is added to the image features and combined with the depth
+features. The SA process is formulated as follows:
+G =SA(Dhr, Y hr)
+=Cat(LFE(Dhr), Sa + γ ∗ LFE(Conv1(Y hr))),
+(8)
+where γ denotes a learnable parameter for controlling the
+degree of highlighting and Cat(·) means concatenation of
+features. Conv1(·) is a 1 × 1 convolutional kernel followed
+by PReLU (He et al. 2015) activation. G represents the fused
+guidance features for feeding into the decoder path.
+Recurrent Mechanism with Refined Depth Guidance.
+As mentioned above, compared to the single image guid-
+ance, the HR depth guidance owns the same modality as the
+LR depth input, which facilitates our attention module to lo-
+cate and select consistent edge structures in image guidance.
+More clear depth structures can achieve more accurate guid-
+ance information for better details restoration, thence we re-
+fine the depth guidance in a recursive manner.
+
+������������1
+1
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+������������1
+2
+������������2
+2
+������������1
+3
+������������2
+3
+������������3
+3
+Figure 5: Visual HF features of our DCN network with dif-
+ferent layers (I = 1, 2, 3).
+Specifically, for the first recursion, the input depth guid-
+ance is the up-sampled version of LR depth map Dbic ∈
+RH×W by bicubic interpolation, i.e., G0 = SA(Dbic, Y hr).
+For the k-th recursion, the latest output of our DSR network
+is taken as the input of the attention module next time. The
+recurrent SA can be formulated as follows:
+Gk =SA(HLF(Gk−1, Dlf, Dhf), Y hr),
+(9)
+where HLF(·) is the HF&LF feature fusion operation. As
+shown in Fig. 6, the image feature map before being inputted
+into the SA module contains complex patterns and unclear
+boundaries. As the recursion progresses, complex textures
+are removed (e.g., background pattern and cylinder label).
+HF&LF feature fusion module
+Different from previous methods directly up-sampling LF
+component by bicubic interpolation, we propose an HF&LF
+feature fusion (HLF) module to reconstruct the HF compo-
+nent and improve the blurred LF edges. The HF reconstruc-
+tion module is built upon the U-Net architecture, including
+an encoder path, an attention-based guidance branch, and
+a decoder branch (See orange blocks of Fig. 3). Rich hi-
+erarchical features extracted from the guidance branch and
+encoder-decoder structure are fused by using repeated resid-
+ual convolutional block attention modules (Woo et al. 2018;
+Guo et al. 2020). Then, the achieved contextual features in
+the decoder branch are concatenated with the LF features at
+multiple levels for the edges refining during the LF recon-
+struction (See blue blocks of Fig. 3).
+Loss Function
+We train our model by minimizing the smooth-L1 loss be-
+tween the network output Dhr of each recursion and the
+ground-truth depth map Dgt. For the k-th recursion, the loss
+function Lk(·) is defined as below:
+Lk(Dhr
+k , Dgt) =
+N
+�
+i=1
+smoothL1(Dhr
+k,i, Dgt
+i ),
+(10)
+where smoothL1(x) =
+�
+0.5x2,
+if |x| < 1
+|x| − 0.5,
+otherwise. Dhr
+k
+de-
+notes the network output of the k-th recursion. N and i in-
+������������ = 0
+������������ = 1
+������������ = 2
+Figure 6: Visual image features calculated by the Eq. (8)
+when the recursive step is varied from k = 0 to 2.
+dicate the pixel number and the pixel index in the map, re-
+spectively. We can obtain K depth outputs and the overall
+loss is expressed as:
+Ls =
+K
+�
+k=1
+λkLk,
+(11)
+where λk is the weight coefficient of the k-th loss.
+Experiment
+Experimental Setting
+To evaluate the performance of our framework, we conduct
+sufficient experiments on five datasets:
+• Middlebury (Hirschmuller and Scharstein 2007) & MPI
+Sintel (Butler et al. 2012): Training dataset consists of 34
+RGB/D pairs from Middlebury dataset and 58 RGB/D
+pairs from MPI Sintel dataset. Testing dataset includes
+6 RGB/D pairs (Art, Books, Dolls, Laundry, Mobeius,
+Reindeer) from Middlebury 2005.
+• NYU-v2 (Silberman et al. 2012): Following the widely
+used data splitting manner, we sample 1000 pairs for
+training and the rest 449 pairs for testing.
+• Lu (Lu, Ren, and Liu 2014): We test 6 RGB/D pairs from
+this dataset with the training model on NYU-v2.
+• RGB-D-D (He et al. 2021): Following FDSR (He et al.
+2021), we use 405 RGB/D pairs for evaluation with the
+training model on NYU-v2.
+We compare our method with 3 traditional methods: TGV
+(Ferstl et al. 2013), FBS (Barron and Poole 2016), SDF
+(Ham, Cho, and Ponce 2017), 3 classical methods: DJF (Li
+et al. 2016), DMSG (Hui, Loy, and Tang 2016), DGDIE
+(Gu et al. 2017) and 12 state-of-the-art (SOTA) methods:
+SVLRM (Pan et al. 2019), GSPRT (Lutio et al. 2019), DJFR
+(Li et al. 2019), PacNet (Su et al. 2019), GbFT (AlBahar
+and Huang 2019), CUNet (Deng and Dragotti 2020), PM-
+BAN (Ye et al. 2020), DKN (Kim, Ponce, and Ham 2021),
+FDKN (Kim, Ponce, and Ham 2021), FDSR (He et al. 2021),
+AHMF (Zhong et al. 2021) and CTKT (Sun et al. 2021).
+Mean Absolute Error (MAD) and Root Mean Squared Error
+(RMSE) are used to evaluate the performance.
+During training, we randomly extract patches with stride
+= {96, 96, 128} for the scale = {4, 8, 16} respectively as
+ground truth and use bicubic interpolation to get LR in-
+puts. The training and testing data are normalized to the
+range [0, 1]. To balance the training time and network perfor-
+mance, we set the recurrent steps of the SA blocks as k = 2
+in this paper. The loss weights are set as λk = 0.5. The
+
+Model
+Art
+Books
+Dolls
+Laundry
+Mobeius
+Reindeer
+×4
+×8
+×16
+×4
+×8
+×16
+×4
+×8
+×16
+×4
+×8
+×16
+×4
+×8
+×16
+×4
+×8
+×16
+Bicbuic
+1.15
+2.15
+4.04
+0.41
+0.72
+1.32
+0.44
+0.76
+1.31
+0.65
+1.17
+2.17
+0.41
+0.76
+1.37
+0.66
+1.16
+2.26
+DJF
+0.40
+1.07
+2.78
+0.16
+0.45
+1.00
+0.20
+0.49
+0.99
+0.28
+0.71
+1.67
+0.18
+0.46
+1.02
+0.23
+0.60
+1.36
+DMSG
+0.46
+0.76
+1.53
+0.15
+0.41
+0.76
+0.25
+0.51
+0.87
+0.30
+0.46
+1.12
+0.21
+0.43
+0.76
+0.31
+0.52
+0.99
+DGDIE
+0.48
+1.20
+2.44
+0.30
+0.58
+1.02
+0.34
+0.63
+0.93
+0.35
+0.86
+1.56
+0.28
+0.58
+0.98
+0.35
+0.73
+1.29
+GSPRT
+0.48
+0.74
+1.48
+0.21
+0.38
+0.76
+0.28
+0.48
+0.79
+0.33
+0.56
+1.24
+0.24
+0.49
+0.80
+0.31
+0.61
+1.07
+DJFR
+0.33
+0.71
+1.72
+0.19
+0.38
+0.78
+0.25
+0.44
+0.79
+0.22
+0.50
+1.12
+0.20
+0.38
+0.76
+0.24
+0.45
+0.96
+PacNet
+0.40
+0.82
+1.59
+0.22
+0.49
+0.84
+0.28
+0.53
+0.85
+0.28
+0.56
+1.08
+0.23
+0.44
+0.79
+0.29
+0.53
+1.00
+CUNet
+0.47
+1.06
+2.34
+0.33
+0.63
+1.41
+0.40
+0.67
+1.27
+0.41
+0.80
+1.88
+0.29
+0.65
+1.12
+0.35
+0.69
+1.14
+PMBAN
+0.28
+0.55
+1.11
+0.19
+0.30
+0.53
+0.23
+0.37
+0.64
+0.21
+0.36
+0.74
+0.18
+0.31
+0.57
+0.22
+0.39
+0.75
+DKN
+0.25
+0.51
+1.22
+0.16
+0.30
+0.52
+0.21
+0.35
+0.61
+0.17
+0.34
+0.81
+0.16
+0.28
+0.54
+0.20
+0.38
+0.70
+AHMF
+0.22
+0.50
+1.04
+0.14
+0.30
+0.50
+0.18
+0.35
+0.62
+0.15
+0.34
+0.73
+0.14
+0.28
+0.53
+0.18
+0.37
+0.64
+CTKT
+0.25
+0.53
+1.44
+0.11
+0.26
+0.67
+0.16
+0.36
+0.65
+0.16
+0.36
+0.76
+0.13
+0.27
+0.69
+0.17
+0.35
+0.77
+RSAG
+0.13
+0.23
+0.88
+0.09
+0.14
+0.50
+0.15
+0.20
+0.57
+0.10
+0.19
+0.58
+0.12
+0.17
+0.42
+0.13
+0.18
+0.52
+Table 1: Quantitative comparisons (in MAD) on Middlebury dataset.
+Bicubic
+TGV
+DJF
+FBS
+DMSG
+DJFR
+GbFT
+PacNet
+FDKN
+DKN
+FDSR
+CTKT
+DCTNet
+RSAG
+×4
+8.16
+4.98
+3.54
+4.29
+3.02
+2.38
+3.35
+2.39
+1.86
+1.62
+1.61
+1.49
+1.59
+1.23
+×8
+14.22
+11.23
+6.20
+8.94
+2.99
+4.94
+5.73
+4.59
+3.58
+3.26
+3.18
+2.73
+3.16
+2.51
+×16
+22.32
+28.13
+10.21
+14.59
+9.17
+9.18
+9.01
+8.09
+6.96
+6.51
+5.86
+5.11
+5.84
+5.27
+Table 2: Quantitative comparisons (in RMSE (cm)) on NYU-v2 dataset.
+Model
+Lu
+RGB-D-D
+×4
+×8
+×16
+×4
+×8
+×16
+DJF
+1.65
+3.96
+6.75
+3.41
+5.57
+8.15
+DJFR
+1.15
+3.57
+6.77
+3.35
+5.57
+7.99
+FDKN
+0.82
+2.10
+5.05
+1.18
+1.91
+3.41
+DKN
+0.96
+2.16
+5.11
+1.30
+1.96
+3.42
+FDSR
+0.81
+1.91
+4.64
+1.16
+1.82
+3.06
+RSAG
+0.79
+1.67
+4.30
+1.14
+1.75
+2.96
+Table 3: Quantitative comparisons (in RMSE) on Lu dataset
+and RGB-D-D dataset.
+proposed method is implemented using PyTorch with one
+RTX 2080Ti GPU. For simplicity, we name our Recurrent
+Structure Attention Guided framework as RSAG.
+Comparing to State-of-the-Arts
+Quantitative Comparisons.
+We first show the quantita-
+tive evaluation results with SOTA methods under the same
+conditions. Table 1 shows the results on Middlebury dataset
+under three up-scaling factors. It can be observed that the
+proposed RSAG outperforms the SOTA methods by signifi-
+cant margins for all up-scaling factors. For example, RSAG
+decreases the average MAD by 25%(×4), 48%(×8), and
+30%(×16) compared to CTKT (Sun et al. 2021). We fur-
+ther evaluate the proposed method on NYU-v2 dataset in
+Table 2. The proposed method yields the best performance
+for ×4 and ×8 DSR and comparable performance for ×16
+DSR. Compared with the second-best method, RSAG de-
+creases the average RMSE by 17% for ×4 DSR.
+To verify the generalization ability of our method on Lu
+dataset and RGB-D-D dataset, we test RSAG for ×4, ×8,
+and ×16 DSR, which is trained on NYU dataset. As shown
+Model
+Middlebury
+NYU-v2
+baseline
+0.26
+3.60
+baseline + DCN
+0.24
+3.10
+baseline + DCN + HLF
+0.23
+3.02
+baseline + DCN + HLF + SA
+0.19
+2.51
+Table 4: Ablation studies of RSAG (in MAD) on Middlebury
+dataset and (in RMSE) on NYU-v2 dataset for ×8 DSR.
+in Table 3, we can see that RSAG performs the competi-
+tive generalization results for all up-sampling cases, which
+demonstrates the accuracy and effectiveness of our method.
+Visual Comparisons.
+We provide the visual comparisons
+of the ×8 upsampled results on Middlebury dataset in Fig. 7.
+It is worth noted that edges and luxuriant details are hard to
+be reconstructed by interpolation or simple feature concate-
+nation. Even though CUNet (Deng and Dragotti 2020) and
+DKN (Kim, Ponce, and Ham 2021) can recover most bound-
+aries, they fail to reconstruct some complex structures, such
+as texture beside pencils in Art and boundaries of antlers in
+Reindeer. In contrast, our results show sharper edges and
+smaller errors with the ground truth. Fig. 8 shows ×8 re-
+sults on NYU-v2 dataset. Boundaries and details generated
+by RSAG are more accurate without introducing the texture
+copying artifacts, which demonstrates that RSAG can well
+recover both HF structures and LF content.
+Furthermore, Fig. 9 demonstrates the good generalization
+ability of the proposed method on Lu dataset for ×16 DSR.
+Most methods generally tend to over-smooth the results and
+fail to recover the depth details with low-light guidance im-
+ages, while our method produces more convincing results.
+
+(g) GT
+(f) Ours
+(e) DKN
+(d) CUNet
+(c) DJF
+(b) Bicubic
+(a) GT and image
+Figure 7: Visual comparisons of Art and Laundry on Middlebury dataset (×8 case).
+(a) Image
+(b) DKN
+(c) FDSR
+(d) Ours            (e) GT
+Figure 8: Visual comparisons on NYU-v2 dataset (×8 case).
+(d) Ours
+(b) DKN 
+ (c) FDSR
+(e) GT 
+(b) DKN 
+ (c) FDSR
+(d) Ours 
+  (e) GT
+(a) Images
+(a) Images
+Figure 9: Visual comparisons on Lu dataset (×16 case).
+Ablation Study
+Effect of DCN and HLF modules. Table 4 reports the abla-
+tion studies on the DCN and HLF modules in our frame-
+work. As shown in the first row of Table 4, the baseline
+model uses a hand-designed operator for frequency-domain
+decomposition and direct concatenation for cross-modality
+feature fusion. The second row demonstrates that the pro-
+posed DCN module, which selects HF component adap-
+tively in a coarse-to-fine manner, can significantly improve
+the performance over the baseline. When the HLF module is
+added, the average RMSE of the NYU-v2 dataset shown in
+the third row can be reduced from 3.60 to 3.02, which further
+verifies the effectiveness of high-quality frequency-domain
+separation and HF&LF feature fusion modules.
+Effect of SA module. The last row in Table 4 demon-
+strates the effectiveness of the SA module, which iteratively
+utilizes the latest depth estimation to choose clear and con-
+sistent image features. We can see that the SA module can
+outperform them by a large margin. From the results of Ta-
+2.50
+MAD
+NYU-v2
+0(w/o RMA)   1 
+2 
+  3
+4
+RMSE
+0(w/o RMA)   1  
+ 2 
+  3  
+  4
+Middlebury
+Recurrent Steps
+3.00
+3.50
+0.25
+0.20
+0.15
+Figure 10: Ablation studies of SA with different recursive
+steps on Middlebury and NYU-v2 datasets (×8 case).
+ble 4, it is observed that all the modules proposed in the
+RSAG framework have made a positive contribution to the
+ultimate success of our method. To further study the impact
+of the recurrent steps of SA, we conduct experiments on
+Middlebury and NYU-v2 datasets by varying the step from
+0 (w/o SA) to 4, as illustrated in Fig. 10. It can be found that
+the method achieves better performance when the recursion
+steps increase, where 2 recurrent steps obtain the best trade-
+off between speed and accuracy. It also proves that higher-
+quality depth information can help obtain a more reliable
+guidance structure for subsequent depth reconstruction.
+Conclusion
+In this paper, we proposed a novel recurrent structure atten-
+tion guided (RSAG) framework for depth super-resolution.
+In our framework, a deep contrastive network with multi-
+scale filters (DCN) block was designed to adaptively de-
+compose the high-quality HF and LF components by us-
+ing contrastive networks from large kernels to small ones.
+In addition, by leveraging the latest depth output and high-
+resolution image as guidance, we introduced recurrent struc-
+ture attention (SA) block, instead of the trivial feature con-
+catenation, to select consistent and clear image features
+for subsequent cross-modality fusion. Furthermore, we pre-
+sented the HF&LF feature fusion block to refine the blurred
+edges of the LF component. Extensive experiments on var-
+ious benchmark datasets demonstrated the superiority and
+effectiveness of the proposed framework.
+
+Acknowledgement
+This work was supported by the National Science Fund of
+China under Grant Nos. U1713208 and 62072242.
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+3D GENOME RECONSTRUCTION FROM PARTIALLY
+PHASED HI-C DATA
+DIEGO CIFUENTES, JAN DRAISMA, OSKAR HENRIKSSON,
+ANNACHIARA KORCHMAROS, AND KAIE KUBJAS
+Abstract. The 3-dimensional (3D) structure of the genome is of significant importance for
+many cellular processes. In this paper, we study the problem of reconstructing the 3D struc-
+ture of chromosomes from Hi-C data of diploid organisms, which poses additional challenges
+compared to the better-studied haploid setting. With the help of techniques from algebraic
+geometry, we prove that a small amount of phased data is sufficient to ensure finite identifi-
+ability, both for noiseless and noisy data. In the light of these results, we propose a new 3D
+reconstruction method based on semidefinite programming, paired with numerical algebraic ge-
+ometry and local optimization. The performance of this method is tested on several simulated
+datasets under different noise levels and with different amounts of phased data. We also apply
+it to a real dataset from mouse X chromosomes, and we are then able to recover previously
+known structural features.
+1. Introduction
+The eukaryotic chromatin has a three-dimensional (3D) structure in the cell nucleus which
+has been shown to be important in regulating basic cellular functions, including gene regulation,
+transcription, replication, recombination, and DNA repair [41, 43]. The 3D DNA organization is
+also associated to brain development and function; in particular, it is shown to be misregulated
+in schizophrenia [32, 34] and Alzheimer’s disease [28].
+All genetic material is stored in chromosomes which interact in the cell nucleus, and the 3D
+chromatin structure influences the frequencies of such interactions. A benchmark tool to measure
+such frequencies is high-throughput chromosome conformation capture (Hi-C) [16]. Hi-C first
+crosslinks cell genomes, which “freezes” contacts between DNA segments. Then the genome is
+cut in fragments, the fragments are ligated together and then are associated to equally-sized
+segments of the genome using high-throughput sequencing [33]. These segments of the genome
+are called loci and their size is known as resolution (e.g., bins of size 1Mb or 50Kb). The result of
+Hi-C is stored in a matrix called contact matrix whose elements are the contact counts between
+pairs of loci.
+According to the structure they generate, computational methods for inferring the 3D chro-
+matin structure from a contact matrix fall into two classes: ensemble and consensus methods.
+In a haploid setting (organisms having a single set of chromosomes), ensemble models such as
+MCMC5C [35], BACH-MIX [11] and Chrom3D [30], try to account for structure variations on
+the genome across cells by inferring a population of 3D structures. On the other hand, consensus
+methods aim at reconstructing one single 3D structure which may be used as a model for fur-
+ther analysis. In this category, probability-based methods such as PASTIS [42, 4] model contact
+counts as Poisson random variables of the Euclidean distances between loci, and distance-based
+methods such as ChromSDE [46] and ShRec3D [17] model contact counts as functions of the
+Euclidean distances. An extensive overview of different 3D genome reconstruction techniques is
+given in [29].
+Date: January 30, 2023.
+1
+arXiv:2301.11764v1  [q-bio.GN]  27 Jan 2023
+
+2
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+Most of the methods for 3D genome reconstructions from Hi-C data are for haploid organisms.
+However, humans like most mammals are diploid organisms, in which the genetic information is
+stored in pairs of chromosomes called homologs. Homologous chromosomes are almost identical
+besides some single nucleotide polymorphisms (SNPs) [18]. In the case of diploid organisms,
+the Hi-C data does not generally differentiate between homologous chromosomes. If we model
+each chromosome as a string of beads, then we associate two beads to each locus i ∈ {1, . . . , n},
+one bead for each homolog. Therefore, each observed contact count ci,j between loci i and j
+represents aggregated contacts of four different types of interactions, more precisely one of the
+two homologous beads associated to locus i gets in contact with one of the two homologous
+beads associated to locus j, see Figure 1. This means that the Hi-C data is unphased. Phased
+Hi-C data that distinguishes contacts for homologs is rare. In our setting, we assume that the
+data is partially phased, i.e., some of the contact counts can be associated with a homolog. For
+example, in the (mouse) Patski (BL6xSpretus) [6, 45] cell line, 35.6% of the contact counts are
+phased; while this value is as low as 0.14% in the human GM12878 cell line [33, 45]. Therefore,
+methods for inferring diploid 3D chromatin structure need to take into account the ambiguity
+of diploid Hi-C data to avoid inaccurate reconstructions.
+Figure 1. Ambiguity of phased data.
+Each entry ci,j of the Hi-C matrix corresponds to four
+different contacts between the two pairs (xi, yi) for locus i and (xj, yj) for locus j.
+Methods for 3D genome reconstruction in diploid organisms have been studied in [40, 4, 23, 2,
+22, 37]. One approach is to phase Hi-C data [40, 23, 22], for example by assigning haplotypes to
+contacts based on assignments at neighboring contacts [40, 22]. Cauer et al. [4] models contact
+counts as Poisson random variables. To find the optimal 3D chromatin structure, the associated
+likelihood function combined with two structural constraints is maximized. The first constraint
+imposes that the distances between neighboring beads are similar and the second one requires
+that homologous chromosomes are located in different regions of the cell nucleus. Belyaeva et
+al. [2] shows identifiability of the 3D structure when the Euclidean distances between neighboring
+beads and higher-order contact counts between three or more loci simultaneously are given.
+Under these assumptions, the 3D reconstruction is obtained by combining distance geometry
+with semidefinite programming. Segal [37] applies recently developed imaging technology, in
+situ genome sequencing (IGS) [31], to point out issues in the assumptions made in [40, 4, 2], and
+suggests as alternative assumptions that intra-homolog distances are smaller than corresponding
+inter-homolog distances and intra-homolog distances are similar for homologous chromosomes.
+
+ci
+Reference Genome
+Homologous Chromosomes
+HiC-matrix3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+3
+IGS [31] provides yet another method for inferring the 3D structure of the genome, however, at
+present the resolution and availability of IGS data is limited.
+Contributions. In this work, we focus on a distance-based approach for partially phased Hi-C
+data. In particular, we assume that contacts only for some loci are phased. In the string of beads
+model, the locations of the pair of beads associated to i-th loci are denoted by xi, yi ∈ R3. Then
+homologs are represented by two sequences x1, x2, . . . , xn and y1, x2, . . . , yn in R3; see Figure 1.
+Inferring the 3D chromatin structure corresponds to estimating the bead coordinates. Based
+on Lieberman-Aiden et al. [21], we assume the power law dependency ci,j = γdα
+i,j, where α is
+a negative conversion factor, between the distance di,j and contact count ci,j of loci i and j.
+Following Cauer et al. [4], we assume that a contact count between loci is given by the sum of
+all possible contact counts between the corresponding beads. We call a bead unambiguous if
+the contacts for the corresponding locus are phased; otherwise we call a bead ambiguous.
+Our first main contribution is to show that for negative rational conversion factors α, knowing
+the locations of six unambiguous beads ensures that there are generically finitely many possible
+locations for the other beads, both in the noiseless (Theorem 3.1) and noisy (Corollary 3.5)
+setting. Moreover, we prove finite identifiability also in the fully ambiguous setting when α = −2
+and the number of loci is at least 13 (Theorem 3.6). Note that the identifiability does not hold
+for α = 2 as shown in [2].
+Our second main contribution is to provide a reconstruction method when α = −2, based
+on semidefinite programming combined with numerical algebraic geometry and local optimiza-
+tion (section 4). The general idea is the following: We first estimate the coordinates of the
+unambiguous beads using only the unambiguous contact counts (which precisely corresponds
+to the haploid setting) using the SDP-based solver implemented in ChromSDE [46]. We then
+exploit our theoretical result on finite identifiability to estimate the coordinates of the ambigu-
+ous beads, one by one, by solving several polynomial systems numerically. These estimates are
+then improved by a local estimation step that take into account all contact counts. Finally, a
+clustering algorithm is used to overcome the symmetry (xi, yi) �→ (yi, xi) in the estimation for
+the ambiguous beads.
+The paper is organized as follows. In section 2, we introduce our mathematical model for
+the 3D genome reconstruction problem. In section 3, we recall identifiability results in the un-
+ambigous setting (section 3.1), and then prove identifiability results in the partially ambiguous
+setting (section 3.2) and in the fully ambiguous setting (section 3.3). We describe our recon-
+struction method in section 4. We test the performance of our method on synthetic datasets
+and on a real dataset from the mouse X chromosomes in section 5. We conclude with a dis-
+cussion about future research directions in section 6. The code for computations and exper-
+iments is available at https://github.com/kaiekubjas/3D-genome-reconstruction-from-
+partially-phased-HiC-data.
+2. Mathematical model for 3D genome reconstruction
+In this section we introduce the distance-based model under which we study 3D genome re-
+construction. In section 2.1 we give the background on contact count matrices. In section 2.2 we
+describe a power-law between contacts and distances, which allows to translate the information
+about contacts into distances.
+2.1. Contact count matrices. We model the genome as a string of 2n beads, corresponding
+to n pairs of homologous beads. The positions of the beads are recorded by a matrix
+Z = [x1, . . . , xn, y1, . . . , yn]T ∈ R2n×3.
+
+4
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+The positions xi and yi correspond to homologous beads. When convenient, we use the notation
+z1 := x1, . . . , zn := xn, zn+1 := y1, . . . , z2n := yn. In this notation,
+Z = [z1, . . . , zn, zn+1, . . . , z2n]T ∈ R2n×3.
+Let U be the subset of pairs that are unambiguous, i.e., beads in the pair can be distinguished,
+and let A be the subset of pairs that are ambiguous, i.e., beads in the pair cannot be distin-
+guished. The sets U and A form a partition of [n].
+A Hi-C matrix C is a matrix with each row and column corresponding to a genomic locus.
+Following Cauer et al. [4], we call these contact counts ambiguous and denote the corresponding
+contact count matrix by CA. If parental genotypes are available, then one can use SNPs to
+map some reads to each haplotype [6, 24, 33].
+If both ends of a read contains SNPs that
+can be associated to a single parent, then the contact count is called unambiguous and the
+corresponding contact count matrix is denoted by CU. Finally, if only one of the genomic loci
+present in an interaction can be mapped to one of the homologous chromosomes, then the count
+is called partially ambiguous and the contact count matrix is denoted by CP .
+The unambiguous count matrix CU is a 2n×2n matrix with the first n indices corresponding
+to x1, . . . , xn and the last n indices corresponding to y1, . . . , yn. The ambiguous count matrix
+CA is an n×n matrix and we assume that each ambiguous count is the sum of four unambiguous
+counts:
+cA
+i,j = cU
+i,j + cU
+i,j+n + cU
+i+n,j + cU
+i+n,j+n.
+The partially ambiguous count matrix CP is a 2n×n matrix and each partially ambiguous count
+is the sum of two unambiguous counts:
+cP
+i,j = cU
+i,j + cU
+i,j+n.
+xi
+xj
+yi
+yj
+(a) cA
+i,j for i, j ∈ A
+xi
+xj
+yi
+yj
+(b) cP
+i,j for i ∈ U, j ∈ A
+xi
+xj
+yi
+yj
+(c) cP
+i+n,j for i ∈ U, j ∈ A
+xi
+xj
+yi
+yj
+(d) cU
+i,j for i, j ∈ U
+xi
+xj
+yi
+yj
+(e) cU
+i,j+n for i, j ∈ U
+xi
+xj
+yi
+yj
+(f) cU
+i+n,j for i, j ∈ U
+xi
+xj
+yi
+yj
+(g) cU
+i+n,j+n for i, j ∈ U
+Figure 2. Seven different types of contacts between the ith and jth locus.
+2.2. Contacts and distances. Denoting the distance ∥zi − zj∥ between zi and zj by di,j, the
+power law dependency observed by Lieberman-Aiden et al. [21] can be written as
+cU
+i,j = γdα
+i,j,
+(2.1)
+where α < 0 is a conversion factor and γ > 0 is a scaling factor. This relationship between
+contact counts and distances is assumed in [2, 46], while in [4, 42] the contact counts ci,j are
+modeled as Poisson random variables with the Poisson parameter being βdα
+i,j.
+
+3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+5
+In our paper, we assume that contact counts are related to distances by (2.1). Similarly to [2],
+we set γ = 1 and in parts of the article α = −2. In general, the conversion factor α depends on
+a dataset and its estimation can be part of the reconstruction problem [42, 46]. Setting γ = 1
+means that we recover the configuration up to a scaling factor. In practice, the configuration
+can be rescaled using biological knowledge, e.g., the radius of the nucleus.
+Our approach to 3D genome reconstruction builds on the power law dependency between
+contacts and distances between unambiguous beads. We convert the empirical contact counts
+to Euclidean distances and then aim to reconstruct the positions of beads from the distances.
+This leads us to the following system of equations:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+cA
+i,j = ∥xi − xj∥α + ∥xi − yj∥α + ∥yi − xj∥α + ∥yi − yj∥α
+∀i, j ∈ A
+cP
+i,j = ∥xi − xj∥α + ∥xi − yj∥α,
+cP
+i+n,j = ∥yi − xj∥α + ∥yi − yj∥α
+∀i ∈ U, j ∈ A
+cU
+i,j = ∥xi − xj∥α,
+cU
+i,j+n = ∥xi − yj∥α,
+cU
+i+n,j = ∥yi − xj∥α,
+cU
+i+n,j+n = ∥yi − yj∥α
+∀i, j ∈ U
+(2.2)
+If α is an even integer, then (2.2) is a system of rational equations.
+Determining the points xi, yi, where i ∈ U, is the classical Euclidean distance problem: We
+know the (noisy) pairwise distances between points and would like to construct the locations of
+points, see section 3.1 for details. Hence after section 3.1 we assume that we have estimated the
+locations of points xi, yi, where i ∈ U, and we would like to determine the points xi, yi, where
+i ∈ A.
+3. Identifiability
+In this section, we study the uniqueness of the solutions of the system (2.2) up to rigid
+transformations (translations, rotations and reflections), or in other words, the identifiability of
+the locations of beads. We study the unambiguous, partially ambiguous and ambiguous settings
+in sections 3.1, 3.2 and 3.3, respectively.
+3.1. Unambiguous setting and Euclidean distance geometry. If all pairs are unambigu-
+ous, i.e., U = [n], then constructing the original points translates to a classical problem in
+Euclidean distance geometry. The principal task in Euclidean distance geometry is to construct
+original points from pairwise distances between them. In the rest of the subsection, we will recall
+how to solve this problem. Since pairwise distances are invariant under translations, rotations
+and reflections (rigid transformations), then the original points can be reconstructed up to rigid
+transformations. For an overview of distance geometry and Euclidean distance matrices, we
+refer the reader to [7, 15, 20, 26].
+The Gram matrix of the points z1, . . . , z2n is defined as
+G = ZZT = [z1, . . . , z2n]T · [z1, . . . , z2n] ∈ R2n×2n.
+Let z =
+1
+2n
+�2n
+i=1 zi and ˜zi = zi − z for i = 1, . . . , 2n. The matrix ˜Z = [˜z1, . . . , ˜z2n]T gives the
+locations of points after centering them around the origin. Let ˜G denote the Gram matrix of
+the centered point configuration ˜z1, . . . , ˜z2n.
+Let Di,j = ∥zi−zj∥2 denote the squared Euclidean distance between the points zi and zj. The
+Euclidean distance matrix of the points z1, . . . , z2n is defined as D = (Di,j)1≤i,j≤2n ∈ R2n×2n.
+To express the centered Gram matrix in terms of the Euclidean distance matrix, we define the
+geometric centering matrix
+J = I2n − 1
+2n11T ,
+
+6
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+where I2n is the 2n × 2n identity matrix and 1 is the vector of ones. The linear relationship
+between ˜G and D is given by
+˜G = −1
+2JDJ.
+Therefore, given the Euclidean distance matrix, we can construct the centered Gram matrix for
+the points z1, . . . , z2n.
+The centered points up to rigid transformations are extracted from the centered Gram matrix
+˜G using the eigendecomposition ˜G = QΛQ−1, where Q is orthonormal and Λ is a diagonal
+matrix with entries ordered in decreasing order λ1 ≥ λ2 ≥ . . . ≥ λ2n ≥ 0. We define Λ1/2
+3
+:=
+[diag(√λ1, √λ2, √λ3), 03×(2n−3)]T and set ˆZ = QΛ1/2
+3
+. In the case of noiseless distance matrix
+D, the Gram matrix ˜G has rank three and the diagonal matrix Λ has precisely three non-zero
+entries. Hence we could obtain ˆZ also from QΛ1/2 by truncating zero columns. Using Λ1/2
+3
+has
+the advantage that it gives an approximation for the points also for a noisy distance matrix D.
+The uniqueness of ˆZ up to rotations and reflections follows from [14, Proposition 3.2], which
+states that AAT = BBT if and only if A = BQ for some orthogonal matrix Q.
+The procedure that transforms the distance matrix to origin centered Gram matrix and then
+uses eigendecomposition for constructing original points is called classical multidimensional scal-
+ing (cMDS) [5]. Although cMDS is widely used in practice, it does not always find the distance
+matrix that minimizes the Frobenius norm to the empirical noisy distance matrix [39]. Other
+approaches to solving the Euclidean distance and Euclidean completion problems include non-
+convex [9, 25] as well semidefinite formulations [1, 10, 27, 44, 46, 47].
+3.2. Partially ambiguous setting. The next theorem establishes the uniqueness of the solu-
+tions of the system (2.2) in the presence of ambiguous pairs. In particular, it states that there
+are finitely many possible locations for beads in one ambiguous pair given the locations of six
+unambiguous beads. The identifiability results in this subsection hold for all negative rational
+numbers α. In the rest of the paper, we denote the true but unknown coordinates by x∗ and the
+symbol x stands for a variable that we want to solve for. We write ∥ · ∥ for the standard inner
+product on R3.
+Theorem 3.1. Let α be a negative rational number. Then for a∗, b∗, . . . , f∗, x∗, y∗ ∈ R3 suffi-
+ciently general, the system of six equations
+∥x − t∗∥α + ∥y − t∗∥α = ∥x∗ − t∗∥α + ∥y∗ − t∗∥α for t∗ = a∗, b∗, . . . , f∗
+(3.1)
+in the six unknowns x1, x2, x3, y1, y2, y3 ∈ R has only finitely many solutions.
+Remark 3.2. The proof will show that this system has only finitely many solutions over the
+complex numbers.
+We believe that the theorem holds for general nonzero rational α.
+Indeed, our argument
+works, with a minor modification, also for α > 2, but for α in the range (0, 2] a refinement of
+the argument is needed.
+Proof. First write Q(x) := x2
+1 + x2
+2 + x2
+3, so that ∥x∥ =
+�
+Q(x) for x ∈ R3. The advantage of Q
+over ∥x∥ is that it is well-defined on C3.
+Write α
+2 =
+m
+n with m, n integers, m ̸= 0, and n > 0.
+Consider the affine variety X ⊆
+(C3)8 × (C2)6 consisting of all tuples
+((a∗, . . . , f∗, x∗, y∗), (rt∗, st∗)t∗=a∗,...,f∗)
+such that
+Q(x∗ − t∗)m = rn
+t∗ ̸= 0 and Q(y∗ − t∗)m = sn
+t∗ ̸= 0 for t∗ = a∗, . . . , f∗.
+
+3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+7
+Note that, if x∗, t∗ are real, then it follows that
+Q(x∗ − t∗)m = (��x∗ − t∗∥α)n,
+and similarly for Q(y∗ − t∗). Hence if a∗, . . . , y∗ are all real, then the point
+((a∗, . . . , f∗, x∗, y∗), (∥x∗ − t∗∥α, ∥y∗ − t∗∥α)t∗)
+(3.2)
+is a point in X with real-valued coordinates.
+The projection π from X to the open affine subset U ⊆ (C3)8 where all Q(x∗−t∗) and Q(y∗−t∗)
+are nonzero is a finite morphism with fibres of cardinality n12; to see this cardinality note that
+there are n possible choices for each of the numbers rt∗, st∗. Each irreducible component of X
+is a smooth variety of dimension 24.
+Consider the map ψ : X → (C3 × C1)6 defined by
+((a∗, . . . , f∗, x∗, y∗), (rt∗, st∗)t∗) �→ ((t∗, rt∗ + st∗))t∗
+We claim that for q in some open dense subset of X, the derivative dqψ has full rank 24. For
+this, it suffices to find one point p ∈ U such that dqψ has rank 24 at each of the n12 points
+q ∈ π−1(p). We take a real-valued point p := (a∗, b∗, . . . , f∗, x∗, y∗) ∈ (R3)8 to be specified later
+on. Let q ∈ π−1(p). Then, near q, the map ψ factorises via π and the unique algebraic map
+ψ′ : U → (C3 × C1)6 (defined near p) which on a neighbourhood of p in U ∩ (R3)8 equals
+ψ′(a, . . . , f, x, y) = ((t, ξt∗ · Q(x − t)α/2 + ηt∗ · Q(y − t)α/2))t=a,...,f ∈ (C3 × C1)6
+where ξt∗ and ζt∗ are n-th roots of unity in C depending on which q is chosen among the n12
+points in π−1(p). The situation is summarised in the following diagram:
+(X, q)
+π
+�
+ψ
+�
+(U, p)
+ψ′
+� ((C3 × C1)6, ψ(q)).
+Now, dqψ = dpψ′ ◦ dqπ, and since dqπ is a linear isomorphism, it suffices to prove that dpψ′
+is a linear isomorphism. Suppose that (a′, . . . , f′, x′, y′) ∈ ker dpψ′. Then, since the map ψ′
+remembers a, . . . , f, it follows immediately that a′ = . . . = f′ = 0. On the other hand, by
+differentiating we find that, for each t∗ ∈ {a∗, . . . , f∗},
+ξt∗ · (α/2) · Q(x∗ − t∗)α/2−1 · 2 · ⟨x′, x∗ − t∗⟩
++ηt∗ · (α/2) · Q(y∗ − t∗)α/2−1 · 2 · ⟨y′, y∗ − t∗⟩ = 0,
+where ⟨·, ·⟩ stands for the standard bilinear form on C3. In other words, the vector (x′, y′) ∈ C6
+is in the kernel of the 6 × 6-matrix
+M :=
+�
+��
+∥x∗ − a∗∥α−2 · ξa∗ · (x∗ − a∗)
+∥y∗ − a∗∥α−2 · ηa∗ · (y∗ − a∗)
+...
+...
+∥x∗ − f∗∥α−2 · ξf∗ · (x∗ − f∗)
+∥y∗ − f∗∥α−2 · ηf∗ · (y∗ − f∗)
+�
+��
+where we have interpreted a∗, . . . , f∗, x∗, y∗ as row vectors. It suffices to show that, for some
+specific choice of p = (a∗, . . . , f∗, x∗, y∗) ∈ (R3)8, this matrix is nonsingular for all n12 choices
+of ((ξt∗, ηt∗))t∗.
+We choose a∗, . . . , f∗, x∗, y∗ as the vertices of the unit cube, as follows:
+a∗ = (1, 0, 0)
+b∗ = (0, 1, 0)
+c∗ = (0, 0, 1)
+c∗ = (0, 1, 1)
+d∗ = (1, 0, 1)
+f∗ = (1, 1, 0)
+x∗ = (0, 0, 0)
+y∗ = (1, 1, 1).
+
+8
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+Then the matrix M becomes, with β = α − 2:
+�
+���������
+−ξa∗
+0
+0
+0
+2
+β
+2 · ηa∗
+2
+β
+2 · ηa∗
+0
+−ξb∗
+0
+2
+β
+2 · ηb∗
+0
+2
+β
+2 · ηb∗
+0
+0
+−ξc∗
+2
+β
+2 · ηc∗
+2
+β
+2 · ηc∗
+0
+0
+−(2
+β
+2 · ξd∗)
+−(2
+β
+2 · ξd∗)
+ηd∗
+0
+0
+−(2
+β
+2 · ξe∗)
+0
+−(2
+β
+2 · ξe∗)
+0
+ηe∗
+0
+−(2
+β
+2 · ξf∗)
+−(2
+β
+2 · ξf∗)
+0
+0
+0
+ηf∗
+�
+���������
+.
+Now, det(M) equals
+− ξa∗ · ξb∗ · ξc∗ · ηd∗ · ηe∗ · ηf∗ + 22+3β · ηa∗ · ηb∗ · ηc∗ · ξd∗ · ξe∗ · ξf∗ + 22β · R
+(3.3)
+where R is a sum of (products of) roots of unity. Now α < 0 implies that β < −2, so that
+2 + 3β < 2β < 0. Since roots of unity have 2-adic valuation 0, the second term in the expression
+above is the unique term with minimal 2-adic valuation. Hence det(M) ̸= 0, as desired.
+It follows that ψ is a dominant morphism from each irreducible component of X into (C3 ×
+C1)6, and hence for all q in an open dense subset of X, the fibre ψ−1(ψ(q)) is finite. This then
+holds, in particular, for q in an open dense subset of the real points as in (3.2). This proves the
+theorem.
+□
+Remark 3.3. If α > 2, then β > 0, and hence the unique term with minimal 2-adic valuation in
+(3.3) is the first term. This can be used to show that the theorem holds then, as well. The only
+subtlety is that for positive α, solutions where x or y equal one of the points a∗, . . . , f∗ are not
+automatically excluded, and these are not seen by the variety X. But a straightforward argument
+shows that such solutions do not exist for sufficiently general choices of a∗, . . . , f∗, x∗, y∗.
+We now consider the setting when we know locations of seven unambiguous beads. In the
+special case when α = −2, we construct the ideal generated by the polynomials obtained from
+rational equations (3.1) for seven unambiguous beads after moving all terms to one side and
+clearing the denominators. Based on symbolic computations in Macaulay2 for the degree of
+this ideal, we conjecture that the location of a seventh unambiguous bead guarantees unique
+identifiability of an ambiguous pair of beads:
+Conjecture 3.4. Let a∗, b∗, c∗, d∗, e∗, f∗, g∗, x∗, y∗ ∈ R3 be sufficiently general. The system of
+rational equations
+1
+∥t∗ − x∗∥2 +
+1
+∥t∗ − y∗∥2 =
+1
+∥t∗ − x∥2 +
+1
+∥t∗ − y∥2 for t∗ = a∗, b∗, c∗, d∗, e∗, f∗, g∗
+(3.4)
+has precisely two solutions (x∗, y∗) and (y∗, x∗).
+In practice, we only have noisy estimates a, b, . . . , f ∈ R3 of the true positions of unambiguous
+beads a∗, b∗, . . . , f∗ ∈ R3, and we have noisy observations ct of the true contact counts c∗
+t :=
+∥x∗ − t∗∥α + ∥y∗ − t∗∥α. We aim to find x, y ∈ R3 such that
+∥x − t∥α + ∥y − t∥α = ct for t = a, b, . . . , f.
+We may write ct = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for some ϵt that depends on the noise level. Hence,
+the above system of equations can be rephrased as
+∥x − t∥α + ∥y − t∥α = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for t = a, b, . . . , f.
+(3.5)
+In the following corollary we show that this system has generically finitely many solutions.
+Corollary 3.5. Let α be a negative rational number.
+Then for a, b, . . . , f, x∗, y∗ ∈ R3 and
+ϵa, ϵb, . . . , ϵf ∈ R sufficiently general, the system of six equations
+∥x − t∥α + ∥y − t∥α = ∥x∗ − t∥α + ∥y∗ − t∥α + ϵt for t = a, b, . . . , f
+(3.6)
+
+3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+9
+in the six unknowns x1, x2, x3, y1, y2, y3 ∈ R has only finitely many solutions.
+Proof. Recall the map ψ : X → (C3 × C1)6 from the proof of Theorem 3.1 defined by
+((a, . . . , f, x∗, y∗), (rx∗,t, sy∗,t)t) �→ ((t, rx∗,t + sy∗,t))t.
+We showed that ψ is a dominant morphism from each irreducible component of X into (C3×C1)6,
+and that each irreducible component of X is 24-dimensional. Every solution to (3.6) is the (x, y)-
+component of a point in the fibre
+ψ−1((t, ||x∗ − t||α + ||y∗ − t||α + ϵt))t.
+Since this is a fibre over a sufficiently general point, the fibre is finite.
+□
+Corollary 3.5 will be the basis of a numerical algebraic geometric based reconstruction method
+in section 4.
+3.3. Ambiguous setting. Finally we consider the ambiguous setting, where one would like to
+reconstruct the locations of beads only from ambiguous contact counts. It is shown in [2] that
+for α = 2, one does not have finite identifiability no matter how many pairs of ambiguous beads
+one considers. We show finite identifiability for the locations of beads given contact counts for
+13 pairs of ambiguous beads for α = −2. We believe that the result might be true for further
+conversion factors α’s, however our proof technique does not directly generalize.
+Theorem 3.6. Let α = −2. Then for x∗
+1, y∗
+1, . . . , x∗
+12, y∗
+12 ∈ R3 sufficiently general, the system
+of 66 equations
+∥xi − xj∥α + ∥xi − yj∥α + ∥yi − xj∥α + ∥yi − yj∥α =
+∥x∗
+i − x∗
+j∥α + ∥x∗
+i − y∗
+j ∥α + ∥y∗
+i − x∗
+j∥α + ∥y∗
+i − y∗
+j ∥α for 1 ≤ i < j ≤ 12
+(3.7)
+in the 72 unknowns x1,1, x1,2, x1,3, y1,1, y1,2, y1,3, . . . , x12,1, x12,2, x12,3, y12,1, y12,2, y12,3 ∈ R has
+only finitely many solutions up to rigid transformations.
+Proof. As before, we write Q(x) := x2
+1 + x2
+2 + x2
+3, so that ∥x∥ =
+�
+Q(x) for x ∈ R3. Consider
+the affine open subset X ⊆ (C3)24 consisting of all tuples (x∗
+1, y∗
+1, . . . , x∗
+12, y∗
+12) such that
+Q(x∗
+i − x∗
+j) ̸= 0, Q(x∗
+i − y∗
+j ) ̸= 0, Q(y∗
+i − x∗
+j) ̸= 0 and Q(y∗
+i − y∗
+j ) ̸= 0 for 1 ≤ i < j ≤ 12.
+Consider also the map ψ : X → C66 defined by
+(x∗
+1, y∗
+1, . . . , x∗
+12, y∗
+12) �→ (Q(x∗
+i − x∗
+j)−1 + Q(x∗
+i − y∗
+j )−1 + Q(y∗
+i − x∗
+j)−1 + Q(y∗
+i − y∗
+j )−1)i<j
+By a computer calculation (with exact arithmetic) we found that at a randomly chosen q ∈ X
+with rational coordinates, the derivative dqψ had full rank 66. It then follows that for q in
+some open dense subset of X, dqψ has rank 66. Hence ψ is dominant, and for any sufficiently
+general q ∈ X, all irreducible components of the fibre ψ−1(ψ(q)) through q have dimension 6.
+Moreover, each such component C is preserved by the 6-dimensional group G = SO(3, C) ⋉ C3.
+If the stabilizer in G of a sufficiently general point in C has dimension 0, then it follows that
+C is a 6-dimensional G-orbit. That this stabilizer is indeed zero-dimensional follows from a
+Lie algebra argument: if a point (x∗
+1, . . . , y∗
+12) in X has a positive-dimensional stabiliser in G,
+then there exists a nonzero element A in the Lie algebra of SO(3) that maps all di��erences
+x∗
+i − x∗
+j, x∗
+i − y∗
+j , y∗
+i − y∗
+j to zero. But A is a skew-symmetric matrix of rank 2, and it follows
+that all 24 points lie on a line parallel to the kernel of A. Such 24-tuples in X, consisting of
+collinear points, cannot map dominantly into C66 for dimension reasons, hence we may assume
+that the fibre through q does not containing any such tuple. Thus we have shown that, for q ∈ X
+sufficiently general, ψ−1(ψ(q)) is a finite union of G-orbits C. If, furthermore, q has real-valued
+coordinates, then a finite number of these G-orbits C contain a real-valued point q′. It then
+readily follows that C ∩ (R3)24 = (SO(3, R) ⋉ R3) · q′, as desired.
+□
+
+10
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+Remark 3.7. When α = 2, which corresponds to the setting studied in [2], then computationally
+we found that for some special choices of x∗
+1, y∗
+1, . . . , x∗
+12, y∗
+12 ∈ R3 the rank of the Jacobian matrix
+in Theorem 3.6 is 42. This is consistent with the fact that Theorem 3.6 fails for α = 2 [2].
+4. A new reconstruction method
+In this section, we outline a new approach to diploid 3D genome reconstruction for partially
+phased data, based on the theoretical results discussed in subsection 3.2. The method consists
+of the following main steps:
+(1) Estimation of the unambiguous beads {xi, yi}i∈U through semidefinite programming (dis-
+cussed in subsection 4.1).
+(2) A preliminary estimation of the ambiguous beads using numerical algebraic geometry,
+based on Corollary 3.5 (discussed in subsection 4.2).
+(3) A refinement of this estimation using local optimization (discussed in subsection 4.3).
+(4) A final clustering step, where we disambiguate between the estimations (xi, yi) and
+(yi, xi) for each i ∈ A, based on the assumption that homolog chromosomes are separated
+in space (discussed in subsection 4.4).
+In what follows, we will refer to this method by the acronym SNLC (formed from the initial letters
+in semidefinite programming, numerical algebraic geometry, local optimization and clustering).
+4.1. Estimation of the positions of unambiguous beads. As discussed in section 3.1, the
+unambiguous bead coordinates {xi, yi}i∈U = {zi}i∈U∪(n+U) can be estimated with semidefinite
+programming. More specifically, we use ChromSDE [46, Section 2.1] for this part of our re-
+construction, which relies on a specialized solver from [13], to solve an SDP relaxation of the
+optimization problem
+min
+{zi}i∈U∪(n+U)
+�
+i,j∈U∪(n+U)
+cU
+ij̸=0
+�
+cU
+ij
+�
+1
+cU
+ij
+− ∥zi − zj∥2
+�2
++ λ
+�
+i,j∈U∪(n+U)
+cU
+ij=0
+∥zi − zj∥2
+(4.1)
+with λ = 0.01 (cf. [46, Equation 4]). The terms in the first sum are weighted by the square root
+for the corresponding contact counts, in order to account for the fact that higher counts can be
+assumed to be less susceptible to noise.
+4.2. Preliminary estimation using numerical algebraic geometry. To estimate the co-
+ordinates of the ambiguous beads {xi, yi}i∈A, we will use a method based on numerical equation
+solving, where we estimate the ambiguous bead pairs one by one.
+Let x, y be the unknown coordinates in R3 of a pair of ambiguous beads. We pick six unam-
+biguous beads with already estimated coordinates a, b, c, d, e, f ∈ R3. For each t ∈ {a, . . . , f},
+let ct ∈ R be the corresponding partially ambiguous counts between f and the ambiguous bead
+pair (x, y). Clearing the denominators in the system (3.6), we obtain a system of polynomial
+equations
+∥x − t∥2 + ∥y − t∥2 = ct∥x − t∥2∥y − t∥2 for t = a, b, c, d, e, f.
+(4.2)
+By Corollary 3.5, this system has finitely many complex solutions both in the noiseless and noisy
+setting, which can be found using homotopy continuation.
+We observe that the system (4.2) generally has 80 complex solutions, and we only expect one
+pair of solutions (x, y), (y, x) to correspond to an accurate estimation. Naively adding another
+polynomial arising from a seventh unambiguous bead (as in Conjecture 3.4) does not work; in
+the noisy setting this over-determined system typically lacks solutions. Instead, we compute an
+estimation based on the following two heuristic assumptions:
+
+3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+11
+(1) The most accurate estimation should be approximately real, in the sense that the norm
+of the imaginary part is below a certain tolerance (in this work, we used 0.2 for the
+experiments in subsection 5.1, and 0.15 for the experiments in subsection 5.2).
+(2) The most accurate estimation should be consistent when we change the choice of six
+unambiguous beads.
+Based on these assumptions, we apply the following strategy. We make a number N ≥ 2, choices
+of sets of six unambiguous beads, and solve the corresponding N square systems of the form (4.2).
+Since larger contact counts can be expected to have smaller relative noise, we make the choices
+of beads among the 20 unambiguous beads t that have highest contact count ct to the ambiguous
+locus at hand. For each system, we pick out the approximately real solutions, and obtain N
+sets S1, . . . , SN ⊆ R6 consisting of the real parts of the approximately real solutions. Up to
+the symmetry (x, y) �→ (y, x), we expect these sets to have a unique “approximately common”
+element. We therefore compute, by an exhaustive search, the tuple (w1, . . . , wN) ∈ S1 ×· · ·×SN
+that minimizes the sum
+����w1 − w1 + · · · + wN
+N
+���� + · · · +
+����wN − w1 + · · · + wN
+N
+���� ,
+and use w1+···+wN
+N
+as our estimation of (x, y). For the computations presented in section 5, we
+use N = 5.
+To solve the systems, we use the Julia package HomotopyContinuation.jl [3], and follow the
+two-phase procedure described in [38, Section 7.2]. For the first phase, we solve (4.2) with ran-
+domly chosen parameters a∗, . . . , f∗ ∈ C3 and ca∗, . . . , cf∗ ∈ C, using a polyhedral start system
+[12]. We trace 1280 paths in this first phase, since the Newton polytopes of the polynomials
+appearing in the system (4.2) all contain the origin, and have a mixed volume of 1280, which
+makes 1280 an upper bound on the number of complex solutions by [19, Theorem 2.4]. For the
+second phase, we use a straight-line homotopy in parameter space from the randomly chosen
+parameters a∗, . . . , f∗ ∈ C3 and ca∗, . . . , cf∗ ∈ C, to the values a, . . . , f and ca, . . . , cf ∈ C at
+hand. We observe that we generally find 80 complex solutions in the first phase, which means
+40 orbits with respect to the symmetry (x, y) �→ (y, x). By the discussion in [38, Section 7.6], it
+is enough to only trace one path per orbit, so in the end, we only trace 40 paths in the second
+phase.
+Remark 4.1. If the noise levels are sufficiently high, there could be choices of six unambiguous
+beads for which the system lacks approximately-real solutions. If this situation is encountered,
+we try to redraw the six unambiguous beads until we find an approximately-real solution. If this
+does not succeed within a certain number of attempts (100 in the experiments conducted for this
+paper), we use the average of the closest neighboring unambiguous beads instead.
+4.3. Local optimization. A disadvantage of the numerical algebraic geometry based estima-
+tion discussed in the previous subsection is that it only takes into account “local” information
+about the interactions for one ambiguous locus at a time, which might make it more sensitive to
+noise. In our proposed method, we therefore refine this preliminary estimation of {xi, yi}i∈A fur-
+ther in a local optimization step that takes into account the “global” information of all available
+data.
+The idea is to estimate {xi, yi}i∈A by solving the optimization problem
+min
+{xi,yi}i∈A
+�
+i∈U,j∈A
+��
+cP
+i,j −
+1
+∥xi−xj∥2 −
+1
+∥xi−yj∥2
+�2
++
+�
+cP
+i+n,j −
+1
+∥yi−xj∥2 −
+1
+∥yi−yj∥2
+�2�
+(4.3)
+
+12
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+while keeping the estimates of {xi, yi}i∈U from the ChromSDE step fixed. We use the quasi-
+Newton method for unconstrained optimization implemented in the Matlab Optimization Tool-
+box for this step. The already estimated coordinates of {xi, yi}i∈A from the numerical algebraic
+geometry step are used for the initialization.
+4.4. Clustering to break symmetry. Our objective function remains invariant if we exchange
+xi and yi for any i ∈ A. We can break symmetry by relying on the empirical observation that
+homologous chromosomes typically are spatially separated in different so-called compartments
+of the nucleus [8].
+Let (¯xi, ¯yi)n
+i=1 denote the estimates from the previous steps.
+Our final
+estimations will be obtained by solving the minimization problem
+min
+{xi,yi}i∈A
+n−1
+�
+i=1
+gi,i+1(x, y),
+with
+gi,i+1(x, y) :=
+�
+∥xi − xi+1∥2 + ∥yi − yi+1∥2�
+,
+(4.4)
+where (xi, yi) = (¯xi, ¯yi) for i ∈ U are fixed, and (xi, yi) ∈ {(¯xi, ¯yi), (¯yi, ¯xi)} for i ∈ A are the
+optimization variables. The optimal solution can be computed efficiently, as explained next.
+We first decompose the problem into contiguous chunks of ambiguous beads. Let (i1, . . . , iL) :=
+U be the indices of the unambiguous beads and let i0 := 1, iL+1 := n. The optimization problem
+can be phrased as
+min
+{xi,yi}i∈A
+L
+�
+ℓ=0
+Gℓ(x, y),
+with
+Gℓ(x, y) :=
+iℓ+1−1
+�
+i=iℓ
+gi,i+1(x, y)
+(4.5)
+where there is one summand Gℓ(x, y) for each contiguous chunk of ambiguous beads. Since the
+summands Gℓ(x, y) do not share any ambiguous bead, we can minimize them independently.
+We proceed to describe the optimal solution of the problem. Let
+si =
+�
+1,
+if (xi, yi) = (¯xi, ¯yi)
+−1,
+if (xi, yi) = (¯yi, ¯xi) ,
+wi,i+1 = (¯xi − ¯yi)T (¯xi+1 − ¯yi+1).
+The variable si indicates whether we keep using (¯xi, ¯yi) or we reverse it.
+Note that si = 1
+for i ∈ U. The next lemma gives the optimal assignment of si for i ∈ A. This assignment is
+constructed by using inner products wi,i+1.
+Lemma 4.2. The optimal solution of (4.4) can be constructed as follows:
+(1) For the last chunk (ℓ = L) we have
+s∗
+iℓ = 1,
+s∗
+i+1 = sgn(wi,i+1)s∗
+i
+for i = iℓ, iℓ+1, . . . , iℓ+1−1
+where sgn(·) is the sign function and sgn(0) can be either 1 or −1.
+(2) For the first chunk (ℓ = 0) we have
+s∗
+iℓ+1 = 1,
+s∗
+i = sgn(wi,i+1)s∗
+i+1
+for i = iℓ+1−1, iℓ+1−2, . . . , iℓ
+(3) For any other chunk, let k be the index of the smallest absolute value |wk,k+1|, among
+iℓ ≤ k ≤ iℓ+1 − 1. The solution is
+s∗
+iℓ = 1,
+s∗
+i+1 = sgn(wi,i+1)s∗
+i
+for i = iℓ, iℓ+1, . . . , k−1
+s∗
+iℓ+1 = 1,
+s∗
+i = sgn(wi,i+1)s∗
+i+1
+for i = iℓ+1−1, iℓ+1−2, . . . , k+1
+Proof. Denoting ¯ui := 1
+2(¯xi + ¯yi), ¯vi := 1
+2(¯xi − ¯yi), then xi = ui + sivi, yi = ui − sivi. Note that
+∥¯xi∥2 + ∥¯yi∥2 + ∥¯xi+1∥2 + ∥¯yi+1∥2 − gi,i+1(x, y) = 2(xT
+i xi+1 + yT
+i yi+1)
+= 2(¯ui + si¯vi)T (¯ui+1 + si+1¯vi+1) + 2(¯ui − si¯vi)T (¯ui+1 − si+1¯vi+1)
+= 4(¯uT
+i ¯ui+1) + 4(¯vT
+i ¯vi+1)sisi+1
+
+3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+13
+= 4(¯uT
+i ¯ui+1) + wi,i+1sisi+1
+Since ¯xi, ¯yi, ¯ui, ¯vi are constants, minimizing gi,i+1(x, y) is equivalent to maximizing wi,i+1sisi+1.
+Then for each chunk we have to solve the optimization problem
+max
+si∈{1,−1}
+iℓ+1−1
+�
+i=iℓ
+wi,i+1sisi+1 ,
+(4.6)
+The formulas from the first and last chunk are such that wi,i+1s∗
+i s∗
+i+1 ≥ 0 for all i. This is
+possible because in these cases only one of the endpoints has a fixed value, and the remaining
+values are computed recursively starting from such a fixed point.
+Since all summands are
+nonnegative, the sum in (4.6) is maximized.
+For the inner chunks, the two endpoints are fixed, so it may not be possible to have that
+wi,i+1s∗
+i s∗
+i+1 ≥ 0 for all indices. In an optimal assignment we should pick at most one term to
+be negative, and such a term (if it exists) should be the one with the smallest absolute value
+|wi,i+1|. This leads to the formula from the lemma.
+□
+5. Experiments
+In this section, we apply the SNLC scheme described in section 4 to synthetic and real datasets,
+and compare its performance with the preexisting software package PASTIS.
+All experiments are done using Julia 1.6.1, with ChromSDE being run in Matlab 2021a
+and PASTIS in Python 3.8.10, and the Julia package MATLAB.jl (v0.8.3) acting as interface
+to Matlab. The numerical algebraic geometry part of the estimation procedure is done with
+HomotopyContinuation.jl (v2.5.5) [3].
+For the PASTIS computations, we fix α = −2 to ensure compatibility with the modelling
+assumptions made in this paper. We run PASTIS without filtering, in order to make it possible
+to compare RMSD values. Since PASTIS only takes integer inputs, we multiply the theoretical
+contact counts calculated by (2.2) by a factor 105 and round them to the nearest integer.
+Following the approach taken in [4], we use a coarse grid search to find the optimal coefficients
+for the homolog separating constraint and bead connectivity constraints. Specifically, we fix a
+structure simulated with the same method as used in the experiments, and compute the RMSD
+values for all λ1, λ2 ∈ {1, 101, 102, . . . , 1012}. In this way, we find that λ1 = 1011 and λ2 = 1012
+give optimal results.
+5.1. Synthetic data. We conduct a number of experiments where we simulate a single chromo-
+some pair (referred to as X and Y in figures) through Brownian motion with fixed step length,
+compute unambiguous, partially ambiguous and ambiguous contact counts according to (2.2),
+add noise, and then try to recover the structure of the chromosomes through the SNLC scheme
+described in section 4. Following [2], we model noise by multiplying each entry of CU, CP and
+CA by a factor 1 + δ, where δ is sampled uniformly from the interval (−ε, ε) for some chosen
+noise level ε ∈ [0, 1].
+As a measure of the quality of the reconstruction, we use the minimal root-mean square
+distance (RMSD) between, on the one hand, the true coordinates (x∗
+i , y∗
+i )n
+i=1, and, on the other
+hand, the estimated coordinates (xi, yi)n
+i=1 after rigid transformations and scaling, i.e., we find
+the minimum
+min
+R∈O(3)
+s>0, b∈R3
+�
+�
+�
+� 1
+2n
+n
+�
+i=1
+�
+∥(sRxi + b) − x∗
+i ∥2 + ∥(sRyi + b) − y∗
+i ∥2
+�
+.
+
+14
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+This can be seen as a version of the classical Procrustes problem solved in [36], which is imple-
+mented in Matlab as the function procrustes.
+Specific examples of reconstructions of the Brownian motion and helix-shaped chromosomes
+obtained with SNLC at varying noise levels and 50% of ambiguous beads are shown in Figure 3.
+For low noise levels the reconstructions by SNLC and the original structure highly overlap. For
+higher noise levels the general region occupied by the reconstructions overlaps with the original
+structure, while the local features become less aligned. Analogous reconstructions obtained with
+SNLC without the local optimization step are shown in Figure S1.
+A comparison of how the quality of the reconstruction depends on the noise level and pro-
+portion of ambiguous beads for SNLC and PASTIS is done in Figure 4. We measure the RMSD
+value between the reconstructed and original 3D structure for different noise levels over 20 runs.
+The RMSD values obtained by SNLC are consistently lower than the ones obtained by PASTIS.
+The difference is specially large for low to medium noise levels. While our method outperforms
+PASTIS in the setting considered in this paper, it is worth mentioning that PASTIS works also
+in a more general setting, where there might be contacts of all three types (ambiguous, partially
+ambiguous and unambiguous) between every pair of loci.
+-5
+0
+5
+10
+-4
+-2
+0
+2
+4
+6
+8
+RMSD = 0.17757
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(a) ε = 0.10
+-5
+0
+5
+10
+-4
+-2
+0
+2
+4
+6
+8
+RMSD = 0.5478
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(b) ε = 0.50
+-5
+0
+5
+10
+-4
+-2
+0
+2
+4
+6
+8
+RMSD = 0.9856
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(c) ε = 0.90
+0.5
+RMSD = 0.052324
+0
+-1
+-0.5
+-0.5
+0
+0.5
+-2
+1
+1.5
+0
+2
+2.5
+2
+3
+-1
+4
+6
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(d) ε = 0.10
+1
+RMSD = 0.19914
+0
+-1
+-0.5
+0
+0.5
+-2
+1
+1.5
+0
+2
+2.5
+2
+-1
+3
+4
+6
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(e) ε = 0.50
+1
+RMSD = 0.54979
+0
+-1
+-1
+-0.5
+0
+0.5
+-2
+1
+1.5
+0
+2
+2.5
+2
+3
+-2
+4
+6
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(f) ε = 0.90
+Figure 3. Examples of reconstructions for varying noise levels, for a chromosome pair with 60 loci, out
+of which 50% are ambiguous. Subfigures (a)–(c) show chromosomes simulated with Brownian motion
+(projected onto the xy-plane), whereas figure (d)–(e) show helix-shaped chromosomes.
+5.2. Experimentally obtained data. We compute SNLC reconstructions based on the real
+dataset explored in [4], which is obtained from Hi-C experiments on the X chromosomes in the
+
+3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+15
+0
+0.2
+0.4
+0.6
+0.8
+1
+epsilon
+-0.5
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+RMSD
+PASTIS
+SNLC
+(a) 25% ambiguous loci
+0
+0.2
+0.4
+0.6
+0.8
+1
+epsilon
+-0.5
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+RMSD
+PASTIS
+SNLC
+(b) 50% ambiguous loci
+Figure 4. Comparison between our reconstruction method and PASTIS. The values are the average
+over 20 runs, with the error bars showing the standard deviation.
+All experiments took place with
+60 loci, with varying levels of noise, as well as varying number of ambiguous loci, uniformly randomly
+distributed over the chromosomes.
+Patski (BL6xSpretus) cell line. The data has been recorded at a resolution of 500 kb, which
+corresponds to 343 bead pairs in our model.
+For some of these pairs, no or only very low contact counts have been recorded. Since such low
+contact counts are susceptible to high uncertainty and can be assumed to be a consequence of
+experimental errors, we exclude the 47 loci with the lowest total contact counts from the analysis.
+To select the cutoff, the loci are sorted according to the total contact counts (see Figure S2 (a)),
+and the ratios between the total contact counts for consecutive loci are computed. A peak for
+these ratios is observed at the 47th contact count, as shown in Figure S2 (b). After applying
+this filter, we obtain a dataset with 296 loci. Out of these, we consider as ambiguous all loci i
+for which less than 40% of the total contact count comes from contacts where xi and yi were
+not distinguishable. These proportions for all loci are shown in Figure S2 (c). For the Patski
+dataset, we obtain 46 ambiguous loci and 250 unambiguous loci in this way.
+In the PASTIS dataset, a locus can simultaneously participate in unambiguous, partially am-
+biguous and ambiguous contacts. To obtain the setting of our paper where loci are partitioned
+into unambiguous or ambiguous, we reassign the contacts according to whether a locus is unam-
+biguous or ambiguous. Our reassignment method is motivated by the assignment of haplotype
+to unphased Hi-C reads in [22]. The exact formulas are given in Supplementary Material.
+The reconstruction obtained via SNLC can be found in Figure 5 (a).
+The logarithmic
+heatmaps for contact count matrices for original data and the SNLC reconstruction are shown
+in Figure S3.
+It was discovered in [6] that the inactive homolog in the Patski X chromosome pair has a
+bipartite structure, consisting of two superdomains with frequent intra-chromosome contacts
+within the superdomains and a boundary region between the two superdomains. The active
+homolog does not exhibit the same behaviour. The boundary region on the inactive X chromo-
+some is centered at 72.8-72.9 MB [6] which at the 500 kB resolution corresponds to the bead
+146 [4]. We show in Figure 5 (b) that the two chromosomes reconstructed using SNLC exhibit
+this structure by computing the bipartite index for the respective homologs as in [4, 6]. We
+recall that, in the setting of a single chromosome with beads z1, . . . , zn ∈ R3, the bipartite index
+is defined as the ratio of intra-superdomain to inter-superdomain contacts in the reconstruction:
+BI(h) =
+1
+h2
+�h
+i=1
+�h
+j=1
+1
+∥zi−zj∥2 +
+1
+(n−h)2
+�n
+i=h+1
+�n
+j=h+1
+1
+∥zi−zj∥2
+2
+h(n−h)
+�h
+i=1
+�n
+j=h+1
+1
+∥zi−zj∥2
+.
+
+16
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+-0.5
+0
+0.5
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+Inactive homolog
+Active homolog
+Ambiguous
+(a)
+0
+50
+100
+150
+200
+250
+300
+350
+h
+0
+5
+10
+15
+20
+25
+Bipartite index
+Inactive homolog
+Active homolog
+(b)
+Figure 5. (a) Reconstruction from a real dataset using our reconstruction method.
+A dashed line
+between two beads is used to indicate that there is one or more beads between them, for which we
+have not given an estimation (due to low contact counts). (b) Bipartite index for the reconstructed
+chromosomes. The dashed vertical line indicates the known hinge point at locus 146.
+6. Discussion
+In this article we study the finite identifiability of 3D genome reconstruction from contact
+counts under the model where the distances di,j and contact counts ci,j between two beads i
+and j follow the power law dependency ci,j = dα
+i,j for a conversion factor α < 0. We show that
+if at least six beads are unambiguous, then the locations of the rest of the beads can be finitely
+identified from partially ambiguous contact counts for rational α satisfying α < 0 or α > 2.
+In the fully ambiguous setting, we prove finite identifiability for α = −2, given ambiguous
+contact counts for at least 12 pairs of beads. From [2] it is known that finite identifiability
+does not hold in the fully ambiguous setting for α = 2. It is an open question whether finite
+identifiability of 3D genome reconstruction holds for other α ∈ R\{−2, 2} in the fully ambiguous
+setting and for rational α ∈ (0, 2] in the partially ambiguous setting. We conjecture that in the
+partially ambiguous setting seven unambiguous loci guarantee unique identifiability of the 3D
+reconstruction for rational α < 0 or α > 2. When α = −2, then one approach to studying the
+unique identifiability might be via the degree of a parametrized family of algebraic varieties.
+After establishing the identifiability, we suggest a reconstruction method for the partially am-
+biguous setting with α = −2 that combines semidefinite programming, homotopy continuation
+in numerical algebraic geometry, local optimization and clustering. To speed up the homotopy
+continuation based part, we observe that the parametrized system of polynomial equations cor-
+responding to six unambiguous beads has 40 pairs of complex solutions and we trace one path
+for each orbit. It is an open question to prove that for sufficiently general parameters the sys-
+tem has 40 pairs of complex solution. This question again reduces to studying the degree of a
+family of algebraic varieties. While our goal is to highlight the potential of our method, one
+could further regularize its output and use interpolation for the beads that are far away from
+the neighboring beads. A future research direction is to explore whether numerical algebraic
+geometry or semidefinite programming based methods can be proposed also for other conversion
+factors α < 0.
+Acknowledgements
+Oskar Henriksson and Kaie Kubjas were partially supported by the Academy of Finland Grant
+No. 323416. We thank Anastasiya Belyaeva, Gesine Cauer, AmirHossein Sadegemanesh, Luca
+Sodomaco, and Caroline Uhler for very helpful discussions and answers to our questions.
+
+3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+17
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+Authors’ addresses:
+Diego Cifuentes, Georgia Institute of Technology
+diego.cifuentes@isye.gatech.edu
+Jan Draisma, University of Bern
+jan.draisma@math.unibe.ch
+Oskar Henriksson, University of Copenhagen
+oskar.henriksson@math.ku.dk
+Annachiara Korchmaros, University of Leipzig
+annachiara@bioinf.uni-leipzig.de
+Kaie Kubjas, Aalto University
+kaie.kubjas@aalto.fi
+Supplementary Material
+In this part of the paper, we include additional details and figures for the experiments in
+section 5.
+Figure S1 shows reconstructions of the same chromosomes as displayed in Figure 3 but with-
+out the local optimization step, indicating that semidefinite programming, numerical algebraic
+geometry and clustering alone can recover the main features of the 3D structure.
+Figure S2 illustrates the preprocessing steps of the real dataset where loci with low contact
+counts are removed and the rest of the loci are partitioned into unambiguous and ambiguous.
+The total contact count for the ith locus is defined as the sum of all contacts where it participates:
+T(i) =
+�
+j∈[n]
+�
+cA(i, j) + cP (i, j) + cP (i + n, j)
+�
++
+�
+j∈[2n]
+�
+cP (j, i) + cU(i, j) + cU(i + n, j)
+�
+.
+Similarly, we define the unambiguity quotient as the proportion of T(i) that consists of contacts
+where xi and yi could be distinguished:
+UQ(i) =
+1
+T(i)
+�
+� �
+j∈[n]
+�
+cP (i, j) + cP (i + n, j)
+�
++
+�
+j∈[2n]
+�
+cU(i, j) + cU(i + n, j)
+�
+�
+� .
+To obtain the setting of our paper where loci are partitioned into unambiguous or ambiguous,
+we reassign the contact counts of ˜CU ˜CP and ˜CA of the Patski dataset according to whether a
+
+20
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+-5
+0
+5
+10
+-4
+-2
+0
+2
+4
+6
+8
+RMSD = 0.58575
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(a) ε = 0.10
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+-4
+-2
+0
+2
+4
+6
+8
+RMSD = 0.86406
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(b) ε = 0.50
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+-4
+-2
+0
+2
+4
+6
+8
+RMSD = 1.2564
+X true
+X estimated
+Y true
+Y estimated
+Start
+Unambiguous
+(c) ε = 0.90
+Figure S1. SNLC reconstructions, without the local optimization step.
+0
+50
+100
+150
+200
+250
+300
+350
+i
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+104
+The i-th smallest total contact count
+(a)
+0
+50
+100
+150
+200
+250
+300
+350
+i
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+2.8
+3 Ratio between the i-th and (i+1)-th smallest total contact count
+(b)
+0
+50
+100
+150
+200
+250
+300
+i
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+0.55
+0.6
+The i-th smallest unambiguity quotient
+(c)
+Figure S2. (a) Total contact counts sorted in increasing order. (b) Ratios between total contact counts.
+The peak corresponding to the ratio between the 48th and the 47th smallest count is used as a motivation
+for excluding the 47 loci with smallest total contact from the analysis. (c) Unambiguity quotients for
+each of the remaining 296 loci, sorted in increasing order. We consider a locus as ambiguous if this ratio
+is less than 0.4; otherwise, we consider it as unambiguous.
+locus is unambiguous or ambiguous. For i, j ∈ U, we define
+cU
+i,j = ˜cU
+i,j + ˜cP
+i,j
+˜cU
+i,j
+˜cU
+i,j + ˜cU
+i,j+n
++ ˜cP
+j,i
+˜cU
+i,j
+˜cU
+i,j + ˜cU
+i+n,j
++ ˜cA
+i,j
+˜cU
+i,j
+˜cU
+i,j + ˜cU
+i,j+n + ˜cU
+i+n,j + ˜cU
+i+n,j+n
+,
+cU
+i,j+n = ˜cU
+i,j+n + ˜cP
+i,j
+˜cU
+i,j+n
+˜cU
+i,j + ˜cU
+i,j+n
++ ˜cP
+j+n,i
+˜cU
+i,j+n
+˜cU
+i,j+n + ˜cU
+i+n,j+n
++ ˜cA
+i,j
+˜cU
+i,j+n
+˜cU
+i,j + ˜cU
+i,j+n + ˜cU
+i+n,j + ˜cU
+i+n,j+n
+,
+cU
+i+n,j = ˜cU
+i+n,j + ˜cP
+i+n,j
+˜cU
+i+n,j
+˜cU
+i+n,j + ˜cU
+i+n,j+n
++ ˜cP
+j,i
+˜cU
+i+n,j
+˜cU
+i,j + ˜cU
+i+n,j
++ ˜cA
+i,j
+˜cU
+i+n,j
+˜cU
+i,j + ˜cU
+i,j+n + ˜cU
+i+n,j + ˜cU
+i+n,j+n
+,
+cU
+i+n,j+n = ˜cU
+i+n,j+n + ˜cP
+i+n,j
+˜cU
+i+n,j+n
+˜cU
+i+n,j + ˜cU
+i+n,j+n
++ ˜cP
+j+n,i
+˜cU
+i+n,j+n
+˜cU
+i,j+n + ˜cU
+i+n,j+n
++ ˜cA
+i,j
+˜cU
+i+n,j+n
+˜cU
+i,j + ˜cU
+i,j+n + ˜cU
+i+n,j + ˜cU
+i+n,j+n
+.
+For i ∈ U, j ∈ A, we define
+cP
+i,j = ˜cU
+i,j + ˜cU
+i,j+n + ˜cP
+i,j + ˜cP
+j,i
+˜cU
+i,j
+˜cU
+i,j + ˜cU
+i+n,j
++ ˜cP
+j+n,i
+˜cU
+i,j+n
+˜cU
+i,j+n + ˜cU
+i+n,j+n
++ ˜cA
+i,j
+˜cP
+i,j
+˜cP
+i,j + ˜cP
+i+n,j
+,
+
+3D GENOME RECONSTRUCTION FROM PARTIALLY PHASED HI-C DATA
+21
+cP
+i+n,j = ˜cU
+i+n,j + ˜cU
+i+n,j+n + ˜cP
+i+n,j + ˜cP
+j,i
+˜cU
+i+n,j
+˜cU
+i,j + ˜cU
+i+n,j
++ ˜cP
+j+n,i
+˜cU
+i+n,j+n
+˜cU
+i,j+n + ˜cU
+i+n,j+n
++ ˜cA
+i,j
+˜cP
+i+n,j
+˜cP
+i,j + ˜cP
+i+n,j
+.
+Finally, for i, j ∈ A, we define
+cA
+i,j = ˜cU
+i,j + ˜cU
+i,j+n + ˜cU
+i+n,j + ˜cU
+i+n,j+n + ˜cP
+i,j + ˜cP
+i+n,j + ˜cP
+j,i + ˜cP
+j+n,i + ˜cA
+i,j.
+In Figure S3, the experimental contact counts from the Patski dataset are compared with the
+contact counts from the SNLC reconstruction.
+
+22
+D. CIFUENTES, J. DRAISMA, O. HENRIKSSON, A. KORCHMAROS, AND K. KUBJAS
+Patski data
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+Figure S3. Logarithmic heat maps for the reassigned contact count matrices obtained from the original
+Patski dataset and from the SNLC reconstruction: (a) and (b) CU; (c) and (d) CP ; (e) and (f) CA.
+The axis labels correspond to the 500 unambiguous beads, and the 46 ambiguous loci.
+

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+ Method to deterministically generate large‐amplitude Optical Schrödinger‐cat states  
+Zheng-Hong Li,1,2,* Zhen-Ya Li,1 Fei Yu,1 M. Al-Amri,3,4,5 and M. Suhail Zubairy3 
+1 Department of Physics, Shanghai University, Shanghai 200444, China 
+2 Shanghai Key Laboratory of High Temperature Superconductors, Shanghai University, Shanghai 200444, 
+China 
+3 Institute for Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy, Texas 
+A&M University, College Station, Texas 77843‐4242, USA 
+4 NCQOQI, KACST, P.O.Box 6086, Riyadh 11442, Saudi Arabia 
+5 The National Center for Quantum Optics and Quantum Informatics, KACST, Riyadh 11442, Saudi Arabi 
+ 
+A deterministic preparation method for large‐amplitude optical Schrödinger‐cat state is proposed. The 
+key ingredient is to entangle an atom buried in a single‐side cavity with a large‐amplitude coherent light 
+pulse. To achieve this purpose, a multiple reflection Michelson interferometer is used. The light pulse can 
+go back and forth inside the interferometer and interact with the atom many times. However, in every 
+interaction, the average photon number of the light field that manipulated by the atom is much less than 
+1, which ensures that the atom‐cavity system can properly control the phase of the reflected field, and thus 
+achieve the entanglement. Not only that, but we also further demonstrate that due to quantum Zeno effect, 
+our scheme is insensitive to both atomic spontaneous emission and detuning between the atom and the 
+cavity. Therefore, the fidelity of the cat state can be increased by improving the linear optical system.  
+ 
+Introduction 
+Schrödinger’s gedanken experiment involving a cat in a superposition of dead and alive states 
+played a crucial role in elucidating certain counterintuitive aspects of quantum mechanics [1]. In 
+modern physics, this Schrödinger cat state (CS) is usually represented by the superposition of two 
+distinct  coherent  states  |�𝛼⟩ .  With  the  increase  of  amplitude  |𝛼| ,  CS  gets  closer  to  the 
+macroscopic superposition. It is not only attractive from a fundamental point of view [2,3], but 
+also valuable for applications including quantum teleportation [4‐7], quantum computing [8‐13], 
+quantum error correction [14‐17] and quantum metrology [18‐24].  
+It is well‐known that a quasi‐ideal CS requires |𝛼| to be large. In response to the above demands, 
+after decades of efforts, CS has been generated on various platforms [3,25‐28]. However, in the 
+optical domain, in the best experimental results so far, |𝛼| remains less than 2 [29‐37]. Needless 
+to say, optical field is an excellent medium for information transmission [37]. It is necessary and 
+valuable to create optical CS of large amplitudes with propagation properties that are on demand.  
+Although there have been some probabilistic methods, for example, the photon subtraction 
+method [29‐33,38,39], they have low probability of success for generating large‐amplitude CS. As 
+for  synthesis  method  proposed  in  Refs.[36,40,41],  it  is  limited  by  the  amplitude  of  the  pre‐
+prepared CS. In addition, the light‐matter interaction to generate CS has become an important 
+research direction recently [37,42].  
+In the experiment of Ref.[37], CS is deterministically generated by the interaction of an incident 
+coherent pulse with a single‐side cavity containing a single atom.  According to different atomic 
+states,  the  reflected  light  field  evolves  in  different  ways,  and  eventually  produces  π  phase 
+difference leading to entanglement between the atom and the field. Applying the measurement 
+
+on the atom collapses the wave function into the optical CS. It is worth noting that in such scheme 
+[37,42], only one reflection happens between the light and the atom‐cavity system (Hereinafter 
+we call it the single reflection scheme). This means preparing a large‐amplitude CS requires the 
+atom to control a strong coherent light field through a single interaction. It is obviously unrealistic, 
+and in the experiment [37], the amplitude of the output CS is only |𝛼| � 1.4.  
+In light of this discussion, it is clear that a deterministic generation of large‐amplitude CS in the 
+optical regime remains elusive. In this article, we propose a deterministic method to generate 
+flying optical CS whose amplitude��can be arbitrarily large.  Our  starting point  is  Refs. [37,42]. 
+However, our approach differs distinctly by employing a multiple reflection model to achieve 
+multiple phase operations. This allows, on one hand, for more interactions between the light field 
+and the atom, but, on the other hand, only a small fraction of light is reflected by the atom‐cavity 
+system during each interaction. Through repeated interactions, we demonstrate that just one 
+atom is possible to control a macroscopic light field and become entangled with it. In addition, it 
+is worth emphasizing that the atom‐cavity system presented in Refs. [37,42] is not the keystone 
+for our multiple reflection scheme. It can be replaced by any other quantum systems say Rydberg 
+blockade [43‐45], photon blockade [46], nondemolition measurement of an optical photon [47,48] 
+and so on. The physics behind our scheme is similar to the interaction free measurement along 
+with quantum Zeno effect [49‐51], which explain another important result of this work. When the 
+atom‐cavity system is used, the simulation shows that our scheme becomes insensitive to both 
+atomic spontaneous emission and detuning between the atom and the cavity. The insensitivity 
+increases as the number of interactions increases. Consequently, our scheme can achieve better 
+performance by just enhancing the quality of the linear optical system.  
+ 
+RESULTS 
+Multiple reflection scheme 
+As shown in Fig.1, the scheme consists of a Michelson interferometer and a single‐side cavity‐
+atom system [37,42,52‐54]. The interference occurs between the light fields in Zones 0 and 1, 
+which are located on the left and right sides of the beam splitter �𝐵𝑆�, respectively, separated by 
+a dotted line. Assume that 𝑎�
+� (𝑧 � 0,1) represents the creation operator of the light field in Zone 
+𝑧.  The  function  of 𝐵𝑆 can  be  described  by 𝑎�
+� → 𝑎�
+� cos 𝜃� � 𝑎�
+� sin 𝜃� and 𝑎�
+� → 𝑎�
+� cos 𝜃� �
+𝑎�
+� sin 𝜃�  [51],  where  cos� 𝜃�  represents  the  reflectivity  of 𝐵𝑆  and    𝜃� � 𝜋 2𝑀
+⁄
+ ( 𝑀  is  an 
+integer).  In  addition, 𝑆 stands  for  light  source, 𝐶 stands  for  optical  circulator, 𝑆𝑀 stands  for 
+switchable mirror (In the experiment, it can be realized by fiber switch and mirrors [55]), which is 
+transparent when it is turned off, and 𝑃𝑆 stands for phase shifter, which adds a π phase shift to 
+the light field only as it propagates from 𝐵𝑆 to single‐side cavity 𝑆𝑆𝐶. When 𝑃𝑆 works, its function 
+can be described as 𝑎�
+� → �𝑎�
+�. As for 𝑆𝑆𝐶����, it is constituted by two facing mirrors 𝐶𝑀����� and 
+𝐶𝑀�����.  Ideally, 𝐶𝑀� is  assumed  to  have  perfect  reflection,  but 𝐶𝑀� is  allowed  for  in‐  and 
+outcoupling of light. 𝑆𝑆𝐶� is an empty cavity, while 𝑆𝑆𝐶� traps a three‐level atom whose level 
+configuration is shown in Fig.1. Only the transition between levels |↑⟩ and |𝑒⟩ is strongly coupled 
+by the cavity mode. According to Refs. [37,42], when the atom is in |↑⟩, due to normal‐mode 
+splitting [53], an incident weak coherent light pulse |𝛼⟩, which is resonant with the empty cavity, 
+does not  enter the  cavity, but  is  reflected  directly  with  no phase change. The corresponding 
+description of the reflection due to 𝑆𝑆𝐶� is 𝑎�
+� → 𝑎�
+�. As for the transition between |↓⟩ and |↑⟩, it 
+
+is decoupled from the cavity mode due to large detuning. Therefore, when the atom is in |↓⟩, the 
+cavity can be treated as empty. The incident pulse enters the cavity and is reflected back but with 
+a 𝜋 phase [37,42], i.e., 𝑎�
+� → �𝑎�
+�. Last but not least, the feature of our scheme is that 𝑆𝑀 can be 
+turned on so that a coherent light pulse travels back and forth inside the interferometer and hence 
+interacts with the atom 𝑀 cycles. One cycle is defined as a wave packet starting at 𝑆𝑀, going 
+through 𝐵𝑆 twice, and returning to 𝑆𝑀. 
+ 
+ 
+FIG. 1 Multiple reflection scheme based on a Michelson interferometer. When the switchable mirrors (𝑆𝑀) 
+are turned on, the coherent light pulse is bounced inside the interferometer and interact with the single‐
+side cavity (𝑆𝑆𝐶) for 𝑀 times. Inside 𝑆𝑆𝐶� there is an atom whose level structure is shown on the up‐left 
+side.  
+At the beginning of the preparation, 𝑆𝑀 is transparent. The light source emits a coherent pulse 
+into the interferometer, while the light field in Zone 1 is in a vacuum state. The corresponding 
+initial state of the light field is |𝛼, 0⟩ � exp�𝛼𝑎�
+� � 𝛼∗𝑎��|0,0⟩. When the pulse passes, 𝑆𝑀 turns 
+on to start 𝑀 cycles. Supposing that the atom is prepared in a superposition state �|↑⟩ � |↓⟩�/√2 
+initially, after 𝑚 cycles, the wave‐function of the whole system becomes [56] 
+�𝜓����� � 1
+√2
+�|𝛼, 0⟩|↑⟩ � |𝛼 cos 2 𝑚𝜃�, 𝛼 sin 2 𝑚𝜃�⟩|↓⟩�.
+�1� 
+When  𝑚 � 𝑀, we have the light‐atom entangled state �|𝛼, 0⟩|↑⟩ � |�𝛼, 0⟩|↓⟩� √2
+⁄
+. Apparently, 
+no  photons  appear  at  𝑆𝑀�  side.  After  measuring  the  atom  with  basis  �|↑⟩ � |↓⟩� √2
+⁄
+,  the 
+corresponding even/odd optical CS, i.e., �|𝛼⟩ � |�𝛼⟩� √2
+⁄
+, is output from 𝑆𝑀� side.  
+So far, we have only focused on the ideal case. In the following, nonetheless, we analyze the 
+performance of the multiple reflection scheme for non‐ideal situation. We show that our scheme 
+highly durable when it comes to parameter variations such as atomic spontaneous emission decay 
+and atom‐cavity detuning. 
+ 
+Practical parameter analysis 
+Regarding the practical atom‐cavity system (𝑆𝑆𝐶�), the incident light field is not only reflected, 
+but also transmitted and scattered [37]. To evaluate these effects, we set that 2𝛾 and 𝜔� as the 
+
+Atomiclevelstructure
+CMTO
+ISSCO
+e)
+Cavity
+Empty
+CMROD
+11>
+PS
+(/+<)PS
+Input
+■
+SM
+BS
+c
+Atom
+S
+CMR1
+CMT1
+Output
+SM,
+Zone 0
+Zone 1spontaneous emission decay rate and transition frequency of the atomic transition between |𝑒⟩ 
+and |↑⟩, respectively. The coupling constant between the cavity mode with frequency 𝜔� and the 
+atomic transition is 𝑔. The atom‐cavity detuning is Δ � 𝜔� � 𝜔�. Moreover, we set 𝜅���� as the 
+cavity field decay rate into the external light field on the 𝐶𝑀���� side. Considering that the atom 
+is hardly excited  in  our scheme, as long as the  condition of slowly varying light intensities is 
+satisfied [37,54,57], 𝑆𝑆𝐶� can be well described by the input‐output theory [58,59]. Suppose that 
+|𝛼�,�↑⟩ is the incident coherent light field from 𝐶𝑀�� side when the atom is in |↑⟩. The cavity 
+reflection |𝛼�,�↑⟩ satisfies [56] 
+𝛼�,�↑ � �1 �
+2𝜅��𝑖𝛥 � 𝛾�
+𝜅�𝑖𝛥 � 𝛾� � 𝑔�� 𝛼�,�↑ � �𝜂�,�↑�𝑒���,�↑𝛼�,�↑,
+�2� 
+where 𝜅 � 𝜅� � 𝜅�, �𝜂�,�↑�
+� is the reflectivity and 𝛽�,�↑ describes the  phase  of the reflection. 
+Similarly, for the transmission of the cavity �𝛼�,�↑�, we have  
+𝛼�,�↑ � � 2�𝑖𝛥 � 𝛾�√𝜅�𝜅�
+𝜅�𝑖𝛥 � 𝛾� � 𝑔� 𝛼�,�↑.
+�3� 
+Regarding the scattering field �𝛼�,�↑� due to the atomic spontaneous emission, we have 
+𝛼�,�↑ �
+2𝑔√𝜅�𝛾
+𝜅�𝑖𝛥 � 𝛾� � 𝑔� 𝑎�,�↑.
+�4� 
+As for the situation that the atom is in |↓⟩, we still assume that the atom is completely unaffected 
+by the cavity mode due to the large detuning. Therefore, 𝑆𝑆𝐶� in such case can be treated the 
+same as the empty cavity 𝑆𝑆𝐶�. By setting 𝑔 � 0 in Eqs. (2)‐(4), we can immediately obtain the 
+corresponding reflection and transmission. As for the scattering light field, it is obviously 0. 
+Based on the above mathematical description of 𝑆𝑆𝐶� and 𝑆𝑆𝐶�, we can numerically simulate 
+the dynamic evolution process of the input coherent pulse |𝛼⟩ and the fidelity of the output. We 
+suppose that the target state is |𝜓�⟩ � �|𝛼⟩|↑⟩ � |�𝛼⟩|↓⟩� √2
+⁄
+, and the final state of the whole 
+system  after  𝑀  cycles  is  �𝜓�� � �|𝐶�↑⟩|𝑙𝑜𝑠𝑠↑⟩|↑⟩ � |𝐶�↓⟩|𝑙𝑜𝑠𝑠↓⟩|↓⟩� √2
+⁄
+with  �𝑙𝑜𝑠𝑠↑�↓�� �
+�𝐶�↑�↓�� ⊗ ∏
+�𝛼�,�↑�↓�
+���
+� ������,�↑�↓�
+���
+� �𝛼�,�↑�↓�
+���
+� �𝛼�,�↑�↓�
+���
+�
+�
+���
+. Here state |𝐶�↑�↓�⟩ (𝑧 � 0,1) denotes the 
+outputs  appearing  at 𝑆𝑀� side  when  the  atom  is  in  state |↑ �↓�⟩,  and   �𝛼�,�↑�↓�
+���
+�  ��𝛼�,�↑�↓�
+���
+�� 
+denotes the transmission (scattering) field generated by 𝑆𝑆𝐶�  in 𝑚‐th cycle. Therefore, �𝑙𝑜𝑠𝑠↑�↓�� 
+includes  all  optical  losses,  while  the  fidelity  is  obtained  by  tracing  �𝑙𝑜𝑠𝑠↑�↓�� ,  i.e.,  𝐹 �
+𝑇𝑟�����⟨𝜓�|𝜓�〉�𝜓��𝜓�〉� � �|⟨𝛼|𝐶�↑⟩|� � |⟨�𝛼|𝐶�↓⟩|� � 2Re�⟨𝛼|𝐶�↑〉⟨𝐶�↓|�𝛼⟩⟨𝑙𝑜𝑠𝑠↓|𝑙𝑜𝑠𝑠↑⟩��/4. 
+As a comparison, we also consider the single reflection model in Ref.[37]. More specifically, the 
+input  |𝛼⟩  is  directly  reflected  by  𝑆𝑆𝐶� ,  and  the  corresponding  output  state  is 
+��𝛼�,�↑�|𝑙𝑜𝑠𝑠↑⟩|↑⟩ � �𝛼�,�↓�|𝑙𝑜𝑠𝑠↓⟩|↓⟩� √2
+⁄
+ with   �𝑙𝑜𝑠𝑠↑�↓�� � �𝛼�,�↑�↓���𝛼�,�↑�↓��.  In  this  model, 
+the constraints on the atomic parameters 𝛾 and Δ can be directly obtained from Eq. (2). For the 
+empty  cavity  case  (atom  is  in |↓⟩),  as  long  as 𝜅� � 0,  the  ideal  reflection 𝛼� � �𝛼�  can  be 
+obtained. As for the case where the atom is in |↑⟩, the condition for ideal reflection 𝛼� � 𝛼� is 
+Δ � 𝛾 � 𝜅� � 0. If only 𝛾 is non‐zero, we can see that the ideal reflection can be approximately 
+achieved  when  𝛾 ≪ 𝑔�/𝜅� .  As  𝛾  increases,  the  cavity  reflectivity  �𝜂�,�↑�
+� decreases 
+monotonically until it drops to 0 when 𝛾 � 𝑔�/𝜅�. If we focus on Δ, however, it only affects 𝛽�,�↑ 
+
+when 𝛾 � 𝜅� � 0, since �𝜂�,�↑�
+� � 1. As Δ varies from �∞ to ∞, 𝛽�,↑ decreases monotonically 
+from 𝜋 to �𝜋.  In order to ensure that 𝛽�,�↑ is close to 0, the constraint Δ ≪ 𝑔�/𝜅� is required.  
+ 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+=2x3.0MHz
+max
+Color: 
+ ||2=4  
+ ||2=10  
+ ||2=16
+Style:  
+ M=5  
+ M=20  
+ M=100  
+ max
+Multiple reflection model
+Fidelity
+
+ ||2=4
+Single reflection model
+ 
+Fig.2.  Fidelity 𝐹 versus  dimensionless 𝛾� � 𝜅�𝛾/𝑔�with 𝑔 � 2𝜋 � 7.8𝑀𝐻𝑧, 𝜅� � 𝜅 � 2𝜋 � 2.3𝑀𝐻𝑧 and 
+𝛥 � 0. The dashed double doted pink curve is for the single reflection case. Other curves are for the multiple 
+reflection case. Different colors represent different |𝛼|�. Different styles represent different 𝑀, except that 
+the dotted curves are plotted for 𝑣��� with 𝑀 � 20, which is the maximum value of the average photon 
+number reaching 𝑆𝑆𝐶� in each cycle when the atom is in |↑⟩. 
+ 
+In our multiple reflection scheme, however, the above constraints are relaxed. In the following, 
+we show that our scheme can be insensitive to atomic parameters 𝛾 and Δ, thus the fidelity of 
+the CS depends only on the quality of the linear optical system. 
+In  order  to  analyze  the  effect  of  𝛾 ,  we  plot  the  fidelity  against  𝛾� � 𝜅�𝛾/𝑔�  with  𝑔 �
+2𝜋 � 7.8𝑀𝐻𝑧 , 𝜅� � 𝜅 � 2𝜋 � 2.3𝑀𝐻𝑧  and  Δ � 0  in  Fig.  2.  The  pink  dot‐dot‐dash  curve  is 
+plotted for the single reflection model with |𝛼|� � 4, which has almost reached the upper limit 
+of  such  model  [37].  Other  curves  are  plotted  for  the  multiple  reflection  model.  The  color 
+black/red/blue represents |𝛼|� � 4 10 16
+⁄
+⁄
+. The curve style dash/solid/dot‐dash denotes 𝑀 �
+5 20 100
+⁄
+⁄
+, while the dotted curves are drawn for 𝑣��� with 𝑀 � 20 instead of fidelity, where 
+𝑣��� � max ��𝛼�,�↑
+��� �
+�
+, �𝛼�,�↑
+��� �
+�
+, … , �𝛼�,�↑
+����
+�
+… �  is  the  maximum  value  of  the  average  photon 
+number reaching 𝑆𝑆𝐶� in each cycle when the atom is in |↑⟩. As shown in the figure,  𝑣��� is 
+always less than 1 (For other 𝑀, the situation is similar), which validates the low atomic excitation 
+probability condition, hence Eqs. (2)‐(4) are valid for simulations.  
+By  comparison,  we  can  see  that  the  multiple  reflection  scheme  outperforms  the  single 
+reflection scheme. In our scheme, it is evident that fidelity increases as 𝑀 increases. Whereas for 
+larger |𝛼|�, larger 𝑀 is required to achieve the same fidelity. More importantly, for 𝛾 much larger 
+than 2𝜋 � 3.0𝑀𝐻𝑧 (This value is taken from the experiment in Ref. [37]. It corresponds to 𝛾� �
+0.11 and has been marked in the figure), our scheme can still provide large 𝐹. To better explain 
+
+the result, we consider the extreme case when 𝛾� � 1, which means all photons reaching 𝑆𝑆𝐶� in 
+a single cycle are lost when the atom is in |↑⟩. Under such conditions, the interference between 
+Zone 0 and Zone 1 is continuously interrupted, resulting in the output light field state in Zone 0 
+becomes |𝛼 cos� 𝜃� cos��� 2 𝜃�⟩ [50,51]. Since cos� 𝜃� cos��� 2 𝜃� � 1 � 𝜋� 2𝑀
+⁄
+ tends to 1 
+as 𝑀 tends to infinity, this implies that the light field is frozen in its initial state. Such result is 
+exactly what we look for, and the mechanism is called the quantum Zeno effect [49,50]. In practice, 
+𝑀 is finite, hence, the quantum Zeno effect is inevitably accompanied by photon loss, which is 
+proportional to |𝛼|�, but tends to 0 as 𝑀 increases. Subsequently, this can explain that in Fig.2, 
+the larger 𝑀 and the smaller |𝛼|�, the better the fidelity. So far, our discussion is about 𝛾� � 1. As 
+for the case of 0 � 𝛾� � 1, the situation is similar. There is a mixture of two physical mechanisms. 
+The first is to maintain the initial state by phase modulation, which does not bring any photon 
+loss. The second is the quantum Zeno effect. It is worth  mentioning that the fidelity in Fig.2 
+decreases monotonically as 𝛾� increases, which implies that the upper limit of the total photon 
+loss of our scheme is determined by the quantum Zeno effect, i.e., 𝑀 and |𝛼| only. Together, the 
+two mechanisms ensure that our scheme has higher fidelity and higher tolerance to 𝛾 than the 
+single reflection scheme as 𝑀 increases. In addition, since the condition 𝛾 ≪ 𝑔�/𝜅� is relaxed, it 
+implies that our scheme does not require strong coupling between atom and cavity. 
+Following the analysis of 𝛾, we discuss the impact of Δ. We have shown that by interrupting the 
+interference, the transmission of the light field from Zone 0 to Zone 1 can be suppressed. Note 
+that the phase mismatch between the two Zones also interrupts the interference, we expect that 
+our scheme can have high tolerance of Δ as well. In Fig. 3, we plot the fidelity against Δ� � 𝜅�Δ/𝑔�. 
+Solid curves are for 𝛾 � 0. Dotted dashed curves are for 𝛾 � 2𝜋 � 3.0𝑀𝐻𝑧. The values of 𝑔, 𝜅� 
+and 𝜅� are the same as in Fig. 2. In addition, the pink curves are plotted for the single reflection 
+model with |𝛼|� � 4. As for the multiple reflection model, the black curves are for |𝛼|� � 4, 𝑀 �
+5, the red curves are for |𝛼|� � 10, 𝑀 � 20 and the blue curves are for |𝛼|� � 16, 𝑀 � 100, 
+respectively.  As  shown  in  Fig.3,  even  for  large |𝛼|,  as  long  as 𝑀 is  large,  our  scheme  can  be 
+insensitive to Δ. 
+ 
+ 
+
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Style:   
+ =0     
+ =2x3.0MHz
+Color:  
+ Single reflection model  
+ ||2=4, M=5
+             
+ ||2=16, M=100               
+ ||2=10, M=20
+Fidelity
+
+ 
+Fig.3. Fidelity 𝐹 versus dimensionless Δ� � 𝜅�Δ/𝑔� with 𝑔 � 2𝜋 � 7.8𝑀𝐻𝑧 and 𝜅� � 𝜅 � 2𝜋 � 2.3𝑀𝐻𝑧. 
+The solid curves are for 𝛾 � 0, and the dotted dashed curves are for 𝛾 � 2𝜋 � 3.0𝑀𝐻𝑧. The pink curves 
+are for the single reflection model with |𝛼|� � 4, and other curves are for the multiple reflection cases.  
+ 
+In  the  above  analyses,  we  ignore  the  influence  of  the  cavity  parameter 𝜅�,  which  will  be 
+discussed  below.  According  to  Eq.  (2),  the  reflectivity  of  an  empty  cavity  is  
+|�𝜅� � 𝜅�� �𝜅� � 𝜅�
+⁄
+�|� .  In  Ref.  [37],  𝜅� � 2𝜋 � 0.2 𝑀𝐻𝑧  and  𝜅� � 2𝜋 � 2.3 𝑀𝐻𝑧 ,  which 
+results in a reflectivity of only about 0.7 for single reflection, while after a few reflections, almost 
+all photons are lost. Therefore, the cavity employed in Ref. [37] is unfortunately not suitable for 
+our scheme. To increase reflectivity, one needs either decrease 𝜅�, or increase 𝜅�. The latter is 
+simpler  in  practice.  However,  although  increasing 𝜅� can  reduce  the  photon  loss  during  the 
+interference of two empty cavities (The atom is in state |↓⟩), it also increases the photon loss in 
+the presence of atom‐cavity coupling (The atom is in state |↑⟩). To verify this, we plot effective 
+fidelity 𝐹�� � 𝑇𝑟������𝜓���𝜓�⟩ � against 𝜅� with |𝛼|� � 8, 𝑀 � 10, 𝛾 � 2𝜋 � 3.0𝑀𝐻𝑧 and Δ � 0 
+in Fig.4. Here, the target state is set as �𝜓��� � ��𝛼���|↑⟩ � ��𝛼���|↓⟩� √2
+⁄
+ with 𝛼�� � �𝐶�↓ �
+𝛼 ��𝜅� � 𝜅�� �𝜅� � 𝜅��
+⁄
+�� .  Note  that  |𝐶�↓⟩  is  the  output  when  the  atom  is  in  |↓⟩ ,  where 
+interference  occurs  between  the  two  empty  cavities.  If  the  optical  parameters  of  these  two 
+cavities are the same, only intensity of the output is affected and reduced from |𝛼|� to �𝛼���
+�. In 
+the  figure,  the  solid  (dashed)  curves  are  plotted  for  𝜅� � 2𝜋 � 0.02 �0.002�𝑀𝐻𝑧 .  The 
+black/red/blue curves are plotted for 𝑔 � 2𝜋 � 7.8 15 30
+⁄
+⁄
+𝑀𝐻𝑧. In addition, the pink curves are 
+plotted for �𝛼���
+�. We can see that 𝐹�� can be significantly improved as 𝜅� decreases. As for 𝜅�, 
+when it increases at the beginning, �𝛼���
+� rapidly rises to its maximum value 8, which causes 𝐹�� 
+to increase. Subsequently, photon loss due to atom‐cavity coupling plays a major role, resulting 
+in the decrease of 𝐹��. Particularly, we note that for the black curve, when 𝐹�� starts to decrease, 
+its corresponding �𝛼���
+� is not close to 8. The reason is that 𝜅� is approaching to the limit 𝑔� 𝛾
+⁄ . 
+Under such limit, the photon loss of a single reflection on 𝑆𝑆𝐶� when the atom is in |↑⟩ is almost 
+
+100%. Therefore, we plot for larger 𝑔 in order to increase the limit so that �𝛼���
+� can get closer 
+to the maximum value 8. We can see that the maximum value of 𝐹�� increases as 𝑔 increases.  
+Moreover, 𝜅� maintains wide range of high fidelity (see blue curve). This is because the constraint 
+𝑔� ≫ 𝜅�𝛾 in  our  scheme  is  relaxed.  However,  we  must  emphasize  that  the  larger 𝑔 is  not 
+necessary  for  high  fidelity.  By  decreasing 𝜅�,  we  can  achieve  the  same  purpose.  In  fact,  the 
+motivation of this work is to reduce the influence of the atom, and to show that the performance 
+of  our  protocol  can  be  improved  by  just  upgrading  the  linear  optical  system,  such  as  the 
+parameters 𝑀 and 𝜅�. 
+Besides the atomic parameters (𝛾, Δ) and linear optical system parameters (𝜅�, 𝑀), next we 
+provide a discussion about the influence of the decoherence between the atomic states |↓⟩ and 
+|↑⟩. Obviously, our scheme requires the atom to remain in superposition at least until the end of 
+𝑀 cycles. Nevertheless, we need to mention that the multiple reflection processes hardly affect 
+the atomic decoherence. When the atom is in |↑⟩, the low atomic excitation probability can be 
+satisfied. As for the atom in |↓⟩, it is not coupled to the light field. While the atomic superposition 
+state has been reported to last about 400𝜇𝑠 [60,61]. The full‐width at half‐maximum of the light 
+pulse that is employed in the experiment of single reflection model is 2.3𝜇𝑠 [37]. Therefore, it is 
+possible for our scheme to be completed before the decoherence.  
+10
+20
+30
+40
+50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+10
+20
+30
+40
+50
+0
+2
+4
+6
+8
+Effective Fidelity, Fef
+Color : 
+ Fef , g=2x7.8MHz    
+ Fef , g=2x15MHz         
+             
+ Fef , g=2x30MHz
+Style :  
+  =2x0.02MHz       
+ =2x0.002MHz   
+Multiple reflection model  ||2=8  M=10
+|ef|2
+R(x2MHz)
+ |ef|2
+0.5
+ 
+Fig.4.  Effective  fidelity 𝐹��  and  the  output  intensity���𝛼���
+�  versus 𝜅�  with  different 𝑔  and 𝜅�  for  the 
+multiple reflection model. In addition, |𝛼|� � 8, 𝑀 � 10, 𝛾 � 2𝜋 � 3.0𝑀𝐻𝑧 and 𝛥 � 0. The initial state of 
+the system is  �|𝛼⟩|↑⟩ � |�𝛼⟩| ↓⟩� √2
+⁄
+ and the target state is ��𝛼���|↑⟩ � ��𝛼���� ↓�� √2
+⁄
+. 
+ 
+DISCUSSION 
+Advantages of multiple reflection scheme 
+Compared  with  the  single  reflection  model,  our  multiple  reflection  scheme  has  two  main 
+advantages.  
+
+First, our scheme provides the single atom with the means to manipulate a strong coherent 
+light field. When the atom is in |↓ �↑�⟩, the light field |𝛼⟩ evolves to | � 𝛼⟩�|𝛼⟩�. We emphasizes 
+that in the single reflection model [37], the above phase manipulation can only be realized when 
+|𝛼|� is small. As |𝛼|� increases, the single atom can no longer prevent the light field from entering 
+the cavity (In this case, regardless of the state of the atom, the reflected light field carries a 𝜋 
+phase shift just like the empty cavity case), causing the atom to be excited from state |↑⟩ to |𝑒⟩. 
+As a result, Eq. (2) is no longer valid. In our scheme, however, when the atom is in |↑⟩, only a small 
+fraction  of  light  touches  𝑆𝑆𝐶�  in  each  cycle.  Its  average  photon  number  is  |𝛼 sin 𝜃�|� .  By 
+adjusting the transmittance of 𝐵𝑆, this value can be far less than 1, thus preventing the atom from 
+being excited. Consequently, even after a large number of cycles, the phase of a strong coherent 
+light field still can be manipulated by the single atom. This result illustrates that our multiple 
+reflection scheme provides a single qubit with the ability to control large amplitude light field, 
+even at macroscopic level. 
+Second, our scheme does not require a high‐quality atom‐cavity coupling system, and it has a 
+high  tolerance  for  atomic  parameters  (𝛾 and Δ ).  In  the  single  reflection  model,  the  phase 
+manipulation depends on the interaction between the atom and the cavity. Hence, the constraint 
+𝑔� ≫ 𝜅�𝛾  is  necessary.  In  our  multiple  reflection  model,  however,  the  phase  manipulation 
+depends  on  the  interference  of  light  between  Zones  0  and  1.  If  the  interference  continues 
+uninterrupted,  the  light  field  eventually  carries  a  𝜋  phase  shift  ( |𝛼⟩ → | � 𝛼⟩ ),  whereas  if 
+interrupted, the phase remains unchanged (|𝛼⟩ → |𝛼⟩). It is worth noting that in the process of 
+generating 𝜋 phase, the interference occurs only between two empty cavities and the atom is not 
+involved. Unlike the interference case, the interruption of the interference is more likely to occur, 
+bearing  in  mind  that  atom‐cavity  system  from  Ref.  [37]  is  not  the  only  way  to  realize  the 
+interruption. For example, if we replace 𝑆𝑆𝐶�(𝑆𝑆𝐶�) by a photon‐absorbing object (mirror), the 
+scheme in Fig.1 becomes a typical interaction‐free measurement scheme based on quantum Zeno 
+effect [49] (the difference from Ref. [49] is that here we use a Michelson interferometer instead 
+of  a  chain  of  Mach‐Zehnder  interferometers,  and  a  coherent  light  source  instead  of  a  single 
+photon source). Since the photons entering Zone 1 are absorbed in each cycle, the light field is 
+suppressed in Zone 0, maintaining its initial state |𝛼⟩. In fact, some studies have further shown 
+that even if the object causes only a partial loss of light, it still can interrupt the interference 
+process and prevent the evolution of the light field [62], which is consistent with our numerical 
+analysis results. Note that in our CS preparation scheme, the main role of atom‐cavity system is 
+just to interrupt the interference. Therefore, our scheme does not require a high‐quality atom‐
+cavity coupling. Even if the atom‐cavity system has imperfections such as photon scattering by 
+the atom, CS can be still prepared.  
+ 
+Scalability of multiple reflection scheme 
+From the above analysis, we can see that the atom‐cavity system from Ref. [37] is not necessary 
+to accomplish our CS preparation. It can certainly be replaced by any quantum object that is in a 
+superposition  of  passing/absorbing  photons  such  as  Rydberg  blockade  [43‐45]  and  photon 
+blockade  [46].  Moreover,  Fig.  3  implies  that  if  the  object  adds  an  additional  phase  to  those 
+photons passing through it instead of absorbing them, it also leads to freezing the evolution of 
+the  initial  state |𝛼⟩.  This  suggests  that  the  three‐level  atomic  model,  used  in  nondemolition 
+measurement [47,48], can also be used to replace the atom‐cavity system. Therefore, in our CS 
+
+preparation method, the multiple reflection model is more indispensable. In addition, our scheme 
+can be used beyond the preparation of CS, and realize the entangled coherent state required in 
+Ref. [20,21]. To do so, we just need to turn off 𝑆𝑀 when 𝑚 � 𝑀/2 instead of 𝑚 � 𝑀,  so that Eq. 
+(2) becomes  �|𝛼, 0⟩|↑⟩ � |0, 𝛼⟩|↓⟩� √2
+⁄
+. Last but not least, we focus on the optical platform so far, 
+nevertheless, our method also works for other platforms such as superconducting microwave 
+resonator [63,64]. 
+ 
+In  summary,  we  have  proposed  a  deterministic  method  to  entangle  an  atom  to  a  large‐
+amplitude  coherent  pulse,  thus  realizing  the  preparation  of  a  large‐amplitude  optical  CS.  A 
+multiple reflection scheme is used, which brings two advantages. First, in each reflection, the 
+actual number of photons manipulated by the atom is very small, which ensures that the single 
+atom can properly control the phase of the reflected field. Second, due to quantum Zeno effect, 
+our  scheme  becomes  insensitive  to  atomic  parameters 𝛾 and Δ.  The  sensitivity  continues  to 
+decrease as the number of reflections 𝑀 increases. This allows our scheme to improve the fidelity 
+of the output CS only by improving the linear optical system. 
+ 
+Data availability: 
+Data sharing not applicable to this article as no datasets were generated or analyzed during the 
+current study. 
+ 
+Code availability: 
+The code generated to analyze the protocol is available from the corresponding author upon 
+reasonable request. 
+ 
+Acknowledgements: 
+This work is supported by a grant from the King Abdulaziz City for Science and Technology (KACST), 
+and Project No. NPRP 13S‐0205‐200258 of the Qatar National Research Fund (QNRF). 
+ 
+Author contributions:  
+The theory was conceived by Z‐H.L. Numerical calculations were performed by Z‐Y.L. and F.Y. 
+under the supervision of Z‐H.L. The project was supervised by M.A. and M.S.Z. All the authors 
+participated in the manuscript preparation, discussions, and checks of the results. 
+ 
+Competing interests: 
+The authors declare no competing interests. 
+ 
+Additional information: 
+Correspondence and requests for materials should be addressed to Z‐H.L. 
+ 
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+ 
+
+Supplementary material 
+A. Calculations for Equation 1 
+In the main text, we have mentioned that 𝑎� and 𝑎� represent the annihilation operators of 
+the  light  field  in  Zone  0  and  Zone  1,  respectively.  Based  on  this,  the  function  of 𝐵𝑆 can  be 
+described as  𝑎�
+� → 𝑎�
+� cos 𝜃� � 𝑎�
+� sin 𝜃� and 𝑎�
+� → 𝑎�
+� cos 𝜃� � 𝑎�
+� sin 𝜃� where 𝜃� � 𝜋/2𝑀. 
+Now, we consider an arbitrary initial photon state  
+ 
+|𝐼𝑛𝑖𝑡𝑖𝑎𝑙⟩ � |𝑢, 𝑣⟩ � exp�𝑢𝑎�
+� � 𝑢∗𝑎��exp�𝑣𝑎�
+� � 𝑣∗𝑎��|0,0⟩, 
+(A1) 
+which represents that a coherent state |𝑢⟩ is in Zone 0 and a coherent state |𝑣⟩ is in Zone 1. After 
+passing through the 𝐵𝑆, we have the final state 
+ 
+|𝐹𝑖𝑛𝑎𝑙⟩ � exp�𝑢�𝑎�
+� cos 𝜃� � 𝑎�
+� sin 𝜃�� � 𝑢∗�𝑎� cos 𝜃� � 𝑎� sin 𝜃���,  
+ 
+                             � exp�𝑣�𝑎�
+� cos 𝜃� � 𝑎�
+� sin 𝜃�� � 𝑣∗�𝑎� cos 𝜃� � 𝑎� sin 𝜃��� |0,0⟩ 
+� |𝑢 cos 𝜃� � 𝑣 sin 𝜃� , 𝑢 sin 𝜃� � 𝑣 cos 𝜃�⟩.  
+(A2) 
+Similarly, consider an arbitrary phase operation 𝑎� → 𝑒��𝑎�. For the initial state |𝐼𝑛𝑖𝑡𝑖𝑎𝑙⟩ �
+|𝑢⟩, after the operation, the final state is 
+|𝐹𝑖𝑛𝑎𝑙⟩ � exp �� 1
+2 |𝑢|�� � 𝑢�
+√𝑛!
+1
+√𝑛!
+�𝑒��𝑎��
+�|0⟩
+∞
+���
+ 
+� exp ��
+�
+� |𝑢|�� ∑
+�
+√�! �𝑢𝑒���
+�|𝑛⟩
+∞
+���
+� |𝑢𝑒��⟩                          (A3) 
+Based on Eqs. (A2) and (A3), we provide the calculation of Eq. (1) in the main text.  
+At the beginning of the preparation, the wave‐function of the whole system is 
+ 
+�𝜓���� � √�
+� |𝛼, 0⟩�|↑⟩ � |↓⟩�  
+(A4) 
+In the first cycle, after the photons pass through  𝐵𝑆 for the first time, the system state is 
+ 
+�𝜓���� � √�
+� |𝛼 cos 𝜃� , 𝛼 sin 𝜃�⟩�|↑⟩ � |↓⟩�  
+(A5) 
+Before 
+the 
+photons 
+are 
+reflected 
+by 
+𝑆𝑆𝐶
+, 
+the 
+system 
+state 
+becomes 
+√�
+� |�𝛼 cos 𝜃� , �𝛼 sin 𝜃�⟩�|↑⟩ � |↓⟩� due  to 𝑃𝑆.  Regarding  the  reflection,  we  emphasize  that 
+only when the atom is in |↑⟩, 𝑆𝑆𝐶� does not change the phase of the reflected field. As a result, 
+the 
+wave‐function 
+of 
+the 
+whole 
+system 
+becomes 
+√�
+� |𝛼 𝑐𝑜𝑠 𝜃� , ���� 𝑠𝑖𝑛 𝜃�⟩|↑⟩ �
+√�
+� |𝛼 𝑐𝑜𝑠 𝜃� , 𝛼 𝑠𝑖𝑛 𝜃�⟩|↓⟩. Subsequently, after the second time that the photons pass through 
+𝐵𝑆, we have 
+ 
+�𝜓���� � √�
+� |𝛼, 0⟩|↑⟩ � √�
+� |cos 2 𝜃�𝛼, sin 2 𝜃�𝛼⟩|↓⟩  
+(A6) 
+
+This state becomes the initial state of the second cycle, and the process is repeated. It is not 
+difficult to obtain that after 𝑚 cycles, the wave‐function of the whole system is 
+ 
+�𝜓����� � √�
+� |𝛼, 0⟩|↑⟩ � √�
+� |𝛼 cos 2 𝑚𝜃�, 𝛼 sin 2 𝑚𝜃�⟩|↓⟩ 
+(A7) 
+Here the superscript of �𝜓����� represents the photons pass through 𝐵𝑆 2𝑚 times. 
+ 
+B. Calculations for Equations 2‐4 
+The Hamiltonian of cavity‐atom system (𝑆𝑆𝐶�) can be described as 
+𝐻 � ℏ𝜔�𝜎�� � ℏ𝜔↑𝜎↑↑ � ℏ𝜔�𝑎�𝑎 � ℏ �
+�
+𝜔�𝑏�
+��𝜔��𝑏��𝜔��𝑑𝜔�
+�
+－�
+���,�,�
+ 
+�ℏ𝑔�𝜎↑�𝑎� � 𝜎�↑𝑎� � ℏ�
+�
+� �
+�𝜎↑�𝑏�
+��𝜔�� � 𝜎�↑𝑏��𝜔���𝑑𝜔�
+�
+－�
+  
+     �𝑖ℏ�
+��
+� �
+�𝑎𝑏�
+��𝜔�� � 𝑎�𝑏��𝜔���𝑑𝜔�
+�
+－�
+� 𝑖ℏ�
+��
+� �
+�𝑎𝑏�
+��𝜔�� � 𝑎�𝑏��𝜔���𝑑𝜔�
+�
+－�
+       (B1) 
+where ℏ𝜔� is the energy of excited atomic state |𝑒⟩, ℏ𝜔↑ is the energy of the atomic state |↑⟩, 𝜔𝑐 
+is the frequency of the cavity mode described by annihilation operator 𝑎, 𝜔𝐽 is the frequency of 
+external field described by annihilation operator 𝑏�𝜔�� with �𝑏𝐽�𝜔𝐽�, 𝑏𝐽
+†�𝜔𝐽
+′�� � 𝛿�𝜔𝐽 � 𝜔𝐽
+′�, and 
+the subscript 𝑅 represents the external multi‐mode field on 𝐶𝑀� side, 𝑇 represents the external 
+field on 𝐶𝑀� side, 𝑆 represents the scattering field due to the atomic spontaneous emission. In 
+addition, 𝑔 is coupling constant between the cavity and the atomic transition between |𝑒⟩ and |↑⟩, 
+2𝛾 is the spontaneous atomic decay rate on the same transition, 𝜅𝑅 and 𝜅𝑇 are cavity field decay 
+rates. We also set that 𝜎↑𝑒 � |↑⟩⟨𝑒|, 𝜎𝑒𝑒 � |𝑒⟩⟨𝑒| and 𝜎↑↑ � |↑⟩⟨↑|.       
+Based on the above Hamiltonian, it is not difficult to obtain the following Heisenberg equations 
+ 
+𝑑𝑎�𝑡�
+𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � ∑
+�
+𝜅𝐽
+𝜋
+𝐽�𝑅,𝑇
+�
+𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽
+∞
+－∞
+,  
+(B2) 
+𝑑
+𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔�𝜎𝑒𝑒�𝑡� � 𝜎↑↑�𝑡��𝑎�𝑡�  
+�𝑖�
+�
+� �
+�𝜎���𝑡� � 𝜎↑↑�𝑡��
+�
+－�
+𝑏��𝜔�, 𝑡�𝑑𝜔�  
+ 
+� �𝑖�𝜔� � 𝜔↑�𝜎↑��𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖�
+�
+� �
+𝑏��𝜔�, 𝑡�𝑑𝜔�
+�
+－�
+.  
+(B3) 
+In the approximation, we have assumed that [1‐3] 
+ 
+⟨�𝜎𝑒𝑒 � 𝜎↑↑�𝑎⟩ � �⟨𝑎⟩,  
+(B4) 
+which indicates that the atom stays in the state |↑⟩ most of the time. This can be satisfied when 
+the input is weak. 
+
+In addition, we can also obtain Heisenberg equations for 𝑏�𝜔�. They are 
+ 
+𝑑𝑏𝐽�𝜔𝐽,𝑡�
+𝑑𝑡
+� �𝑖𝜔𝐽𝑏𝐽�𝜔𝐽, 𝑡� � �
+𝜅𝑅
+𝜋 𝑎�𝑡�,     𝐽 � 𝑅, 𝑇, 
+(B5) 
+ 
+𝑑𝑏𝑆�𝜔𝑆,𝑡�
+𝑑𝑡
+� �𝑖𝜔𝑆𝑏𝑆�𝜔𝑆, 𝑡� � 𝑖�
+𝛾
+𝜋 𝜎↑𝑒�𝑡�. 
+(B6) 
+Eqs. (B5) and (B6) can be rewritten in integral form. If we assume that the atom‐light interaction 
+begins at time 𝑇𝑖𝑛 � 𝑡, we have 
+ 
+𝑏𝐽�𝜔𝐽, 𝑡� � 𝑏𝐽�𝜔𝐽, 𝑇𝑖𝑛�𝑒𝑖𝜔𝐽�𝑇𝑖𝑛�𝑡� � �
+𝜅𝐽
+𝜋 �
+𝑎�𝑡′�𝑒𝑖𝜔𝐽�𝑡′�𝑡�𝑑𝑡′
+𝑡
+𝑇𝑖𝑛
+,  
+(B7) 
+ 
+𝑏𝑆�𝜔𝑆, 𝑡� � 𝑏𝑆�𝜔𝑆, 𝑇𝑖𝑛�𝑒𝑖𝜔𝑆�𝑇𝑖𝑛�𝑡� � 𝑖�
+𝛾
+𝜋 �
+𝜎↑𝑒�𝑡′�𝑒𝑖𝜔𝑆�𝑡′�𝑡�𝑑𝑡′
+𝑡
+𝑇𝑖𝑛
+.  
+(B8) 
+If we assume that the atom‐light interaction ends at time 𝑇𝑜𝑢𝑡 � 𝑡, we have 
+ 
+𝑏𝐽�𝜔𝐽, 𝑡� � 𝑏𝐽�𝜔𝐽, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝐽�𝑇𝑜𝑢𝑡�𝑡� � �
+𝜅𝐽
+𝜋 �
+𝑎�𝑡′�𝑒𝑖𝜔𝐽�𝑡′�𝑡�𝑑𝑡′
+𝑇𝑜𝑢𝑡
+𝑡
+,  
+(B9) 
+ 
+𝑏𝑆�𝜔𝑆, 𝑡� � 𝑏𝑆�𝜔𝑆, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝑆�𝑇𝑜𝑢𝑡�𝑡� � 𝑖�
+𝛾
+𝜋 �
+𝜎↑𝑒�𝑡′�𝑒𝑖𝜔𝑆�𝑡′�𝑡�𝑑𝑡′
+𝑇𝑜𝑢𝑡
+𝑡0
+.  
+(B10) 
+By integrating Eq. (B7) with frequency, it is not difficult to obtain that 
+�
+𝜅𝐽
+𝜋 �
+𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽
+∞
+－∞
+  
+� �
+��
+� �
+𝑏��𝜔�, 𝑇���𝑒����������𝑑𝜔�
+�
+－�
+� 2𝜅� �
+𝑎�𝑡��𝑑𝑡�
+�
+���
+�
+�� �
+𝑒���������𝑑𝜔�
+�
+－�
+  
+ 
+� �2𝜅�𝑎�,���𝑡� � 𝜅�𝑎�𝑡�.  
+(B11) 
+where we have used the relation [4] 
+ 
+� 𝑓�𝑡′�𝛿�𝑡 � 𝑡′�𝑑𝑡′
+𝑡
+𝑡0
+� �
+𝑓�𝑡′�𝛿�𝑡 � 𝑡′�𝑑𝑡′
+𝑡1
+𝑡
+� 1
+2 𝑓�𝑡�,    �𝑡0 � 𝑡 � 𝑡1�,  
+(B12) 
+and the assumptions (𝐽 � 𝑅, 𝑇) 
+ 
+𝑎𝐽,𝑖𝑛�𝑡� �
+1
+√2𝜋 �
+𝑏𝐽�𝜔𝐽, 𝑇𝑖𝑛�𝑒𝑖𝜔𝐽�𝑇𝑖𝑛�𝑡�𝑑𝜔𝐽
+∞
+－∞
+, 
+(B13) 
+ 
+𝑎𝐽,𝑜𝑢𝑡�𝑡� �
+1
+√2𝜋 �
+𝑏𝐽�𝜔𝐽, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝐽�𝑇𝑜𝑢𝑡�𝑡�𝑑𝜔𝐽
+∞
+－∞
+. 
+(B14) 
+ 
+𝑎𝑆,𝑖𝑛�𝑡� �
+1
+√2𝜋 �
+𝑏𝑆�𝜔𝑆, 𝑇𝑖𝑛�𝑒𝑖𝜔𝑆�𝑇𝑖𝑛�𝑡�𝑑𝜔𝑆
+∞
+－∞
+, 
+(B15) 
+ 
+𝑎𝑆,𝑜𝑢𝑡�𝑡� �
+1
+√2𝜋 �
+𝑏𝑆�𝜔𝑆, 𝑇𝑜𝑢𝑡�𝑒𝑖𝜔𝑆�𝑇𝑜𝑢𝑡�𝑡�𝑑𝜔𝑆
+∞
+－∞
+. 
+(B16) 
+Similarly, from Eqs. (B8)‐(B10), we have 
+ 
+�
+𝛾
+𝜋 �
+𝑏𝑆�𝜔𝑆, 𝑡�𝑑𝜔𝑆
+∞
+－∞
+� �2𝛾𝑎𝑆,𝑖𝑛�𝑡� � 𝑖𝛾𝜎↑𝑒�𝑡�,  
+(B17) 
+
+ 
+�
+𝜅𝐽
+𝜋 �
+𝑏𝐽�𝜔𝐽, 𝑡�𝑑𝜔𝐽
+∞
+－∞
+� �2𝜅𝐽𝑎𝐽,𝑜𝑢𝑡�𝑡� � 𝜅𝐽𝑎�𝑡�,  
+(B18) 
+ 
+�
+𝛾
+𝜋 �
+𝑏𝑆�𝜔𝑆, 𝑡�𝑑𝜔𝑆
+∞
+－∞
+� �2𝛾𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝑖𝛾𝜎↑𝑒�𝑡�.  
+(B19) 
+Then, by substituting Eqs. (B11)(B18) into (B2), we can obtain the dynamic equations 
+ 
+𝑑𝑎�𝑡�
+𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑖𝑛�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑖𝑛�𝑡� � 𝜅𝑇𝑎�𝑡�,   (B20) 
+ 
+𝑑𝑎�𝑡�
+𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑜𝑢𝑡�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑖𝑛�𝑡� � 𝜅𝑇𝑎�𝑡�,   (B21) 
+ 
+𝑑𝑎�𝑡�
+𝑑𝑡 � �𝑖𝜔𝑐𝑎�𝑡� � 𝑖𝑔𝜎↑𝑒�𝑡� � �2𝜅𝑅𝑎𝑅,𝑖𝑛�𝑡� � 𝜅𝑅𝑎�𝑡� � �2𝜅𝑇𝑎𝑇,𝑜𝑢𝑡�𝑡� � 𝜅𝑇𝑎�𝑡�.   (B22) 
+By substituting Eqs. (B17)(B19) into (B3), we have 
+ 
+𝑑
+𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖�2𝛾𝑎𝑆,𝑖𝑛�𝑡� � 𝛾𝜎↑𝑒�𝑡�,  
+(B23) 
+ 
+𝑑
+𝑑𝑡 𝜎↑𝑒�𝑡� � �𝑖�𝜔𝑒 � 𝜔↑�𝜎↑𝑒�𝑡� � 𝑖𝑔𝑎�𝑡� � 𝑖�2𝛾𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝛾𝜎↑𝑒�𝑡�.  
+(B24) 
+In  the  following,  we  assume  that  only  the  input  on  𝐶𝑀�  side  is  none‐zero,  i.e.,  𝑎𝑇,𝑖𝑛�𝑡� �
+𝑎𝑆,𝑖𝑛�𝑡� � 0. Then, by subtracting (B20) and (B21), we can get the relation between the input 
+𝑎𝑅,𝑖𝑛�𝑡� and output 𝑎𝑅,𝑜𝑢𝑡�𝑡�, 
+ 
+𝑎𝑅,𝑖𝑛�𝑡� � �2𝜅𝑅𝑎�𝑡� � 𝑎𝑅,𝑜𝑢𝑡�𝑡�.  
+(B25) 
+By subtracting (B20) and (B22), we have 
+ 
+𝑎𝑇,𝑜𝑢𝑡�𝑡� � �2𝜅𝑇𝑎�𝑡�. 
+(B26) 
+By subtracting (B23) and (B24), we have 
+ 
+𝑎𝑆,𝑜𝑢𝑡�𝑡� � �𝑖�2𝛾𝜎↑𝑒�𝑡�.  
+(B27) 
+In addition to the above relations, we next calculate the steady‐state solution of the dynamic 
+equations (B20)(B21)(B23) by assuming that the cavity‐atom system is driven by the input light 
+field with frequency 𝜔. We suppose that  
+𝑎𝑅,𝑖𝑛�𝑡� � 𝛼𝑖,1↑𝑒�𝑖𝜔𝑡, 
+𝑎�𝑡� � 𝛼𝑒����, 
+𝜎↑𝑒�𝑡� � 𝜎�𝑒�𝑖𝜔𝑡, 
+𝑎𝑅,𝑜𝑢𝑡�𝑡� � 𝛼𝑅,1↑𝑒�𝑖𝜔𝑡, 
+𝑎𝑇,𝑜𝑢𝑡�𝑡� � 𝛼𝑇,1↑𝑒�𝑖𝜔𝑡,  
+                                                                   𝑎𝑆,𝑜𝑢𝑡�𝑡� � 𝛼𝑆,1↑𝑒�𝑖𝜔𝑡.                                
+(B28) 
+Then, we can obtain that 
+ 
+��𝑖�𝜔𝑐 � 𝜔� � 𝜅𝑅 � 𝜅𝑇�𝛼 � 𝑖𝑔𝜎� � �2𝜅𝑅𝛼𝑖,1↑ � 0,  
+(B29) 
+ 
+��𝑖�𝜔𝑐 � 𝜔� � 𝜅𝑅 � 𝜅𝑇�𝛼 � 𝑖𝑔𝜎� � �2𝜅𝑅𝛼𝑅,1↑ � 0,  
+(B30) 
+
+ 
+��𝑖�𝜔𝑒 � 𝜔↑ � 𝜔� � 𝛾�𝜎� � 𝑖𝑔𝛼 � 0.  
+(B31) 
+It is not difficult to get that 
+ 
+𝛼𝑖,1↑ � �
+�𝑖�𝜔𝑐�𝜔��𝜅𝑅�𝜅𝑇��𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾��𝑔2
+�2𝜅𝑅�𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾�
+𝛼,  
+(B32) 
+ 
+𝛼𝑅,1↑ � �
+�𝑖�𝜔𝑐�𝜔��𝜅𝑅�𝜅𝑇��𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾��𝑔2
+�2𝜅𝑅�𝑖�𝜔𝑒�𝜔↑�𝜔��𝛾�
+𝛼.  
+(B33) 
+Eq. (B32) shows that when the input increases, the intensity of the cavity field also increase, 
+resulting in the condition (B4) not being satisfied. 
+With Eqs. (B32) and (B33), we can calculate Eq. (2) in the main text. Suppose that the cavity and 
+the external field are resonant, i.e., 𝜔 � 𝜔�, and 𝛥 � 𝜔� � 𝜔↑ � 𝜔�, we obtain  
+ 
+𝛼𝑅,1↑
+𝛼𝑖,1↑ � 1 �
+2𝜅𝑅�𝑖𝛥�𝛾�
+�𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2.  
+(B34) 
+By using Eqs. (B26) and (B27), we can also have Eqs. (3) and (4) in the main text, which are 
+ 
+𝛼𝑇,1↑
+𝛼𝑖,1↑ � �
+2√𝜅𝑅𝜅𝑇�𝑖𝛥�𝛾�
+�𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2,  
+(B35) 
+ 
+𝛼𝑆,1↑
+𝛼𝑖,1↑ �
+2�𝜅𝑅𝛾𝑔
+�𝜅𝑅�𝜅𝑇��𝑖𝛥�𝛾��𝑔2.  
+(B36) 
+ 
+ 
+ 
+[1] C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation 
+induced by a single‐electron spin in a quantum dot: Applications to entangling remote spins via a 
+single photon”, Phys. Rev. B 78, 085307 (2008). 
+[2] A. Reiserer, and G. Rempe, “Cavity‐ based quantum networks with single atoms and optical 
+photons”, Rev. Mod. Phys. 87, 1379 (2015). 
+[3] B. Hacker, S. Welte, S. Daiss, A. Shaukat, S. Ritter, L. Li, and G. Rempe, “Deterministic creation 
+of entangled atom–light Schrödinger‐cat states”, Nature Photonics 13, 110 (2019).  
+[4] D. F. Walls, and G. J. Milburn, Quantum Optics (Springer‐ Verlag, Berlin, 1994). 
+

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+Assessment of the reliability of Deconvolution 
+Procedures for RCF Spectroscopy of Laser-Driven 
+Ion Beams  
+S. McCalluma, b, G. Milluzzoc, a, M. Borghesia, A. Subielb, F. Romanod 
+a Centre for Plasma Physics, Queen’s University Belfast,  
+BT7 1NN, United Kingdom 
+b Medical Radiation Science, National Physical Laboratory,  
+Teddington, TW11 0LW, United Kingdom 
+c Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud,  
+Via S. Sofia 62, 95123 Catania, Italy 
+d Istituto Nazionale di Fisica Nucleare, Sezione di Catania,  
+Via S. Sofia 64, 95123 Catania, Italy 
+ 
+E-mail: smccallum05@qub.ac.uk 
+ABSTRACT: Laser-driven ion beams are defined by a number of unique features, including a large 
+spread in energy. A stack configuration of radiochromic film (RCF) can be utilized to characterize 
+such beams through measurements of their energy spectra. A spectroscopic procedure is reported 
+that allows the proton energy density within each active layer of a radiochromic film (RCF) stack 
+to be retrieved. This is based upon on a deconvolution algorithm developed through Geant4 Monte 
+Carlo simulations to correct the contributions of energy depositions within a given film layer. 
+Through Monte Carlo calculations, the spectrum retrieved from a simulated film stack can be 
+retrieved and compared with a known energy spectrum, providing an examination of the efficacy 
+of this tool. Application of the developed deconvolution procedure thus offers the potential to 
+correctly reconstruct the incident energy spectrum of a laser-driven proton and ion beam from a 
+stack of irradiated RCF.  
+KEYWORDS: Detector modelling and simulations I, dE/dx detectors, Plasma diagnostics - 
+charged-particle spectroscopy, Simulation methods and programs. 
+
+ 
+ 
+ 
+ 
+– 1 – 
+Contents 
+1. Introduction 
+1 
+2. Methodology 
+2 
+3. Monte Carlo Analysis 
+ 
+4. Conclusions 
+ 
+5. References  
+2 
+ 
+ 
+1. Introduction 
+Whilst laser-driven proton and light ion acceleration has attracted significant interest for over 20 
+years [1, 2], conducting accurate measurements of these beams has proven to be technically 
+challenging [3-5]. In particular, the ultra-high dose rates and wide spectral distributions make 
+conventional measurement techniques impracticable [6-8]. For applications, including clinical 
+and radiobiological ones requiring a precise energy selection, characterisation of such beams 
+through accurate measurement of their energy spectra is necessary. Spectroscopic methods reliant 
+on stacked configurations of radiochromic films (RCF) are well-established for measurements of 
+accelerated proton beams, with several approaches of radiochromic film imaging spectroscopy 
+(RIS) reported in the literature [9-14]. A stacked configuration of films placed perpendicularly to 
+the beam orientation can be used to perform an energy resolved measurement of an impinging ion 
+beam. Differential energy loss results in each particle depositing a fraction of its initial kinetic 
+energy on every film it passes before coming to arrest. For polyenergetic sources such as laser-
+driven beams, a superposition of kinetic energy contributions is amassed across the films, 
+requiring a calculation for correction of higher energies. This is achieved through a deconvolution 
+or unfolding of the energy transferred to each film in the stack, so that only the particles stopping 
+within a given film remain. The aim of the work reported here was to investigate and assess a 
+developed algorithm for spectroscopy of laser-driven proton and ion beams through Monte Carlo 
+simulations, studying the possible limitations. This procedure requires knowledge of the RCF 
+energy sensitivity values, and an algorithm to unfold the proton energy spectrum from the RCF 
+response, both of which have been evaluated using the Geant4 toolkit [15-17]. Further, the same 
+Monte Carlo methods were utilised to conduct analysis of the performance and limitations of the 
+developed technique in acquiring the energy spectrum. Once validated, the spectroscopic 
+procedure reported offers the potential to reliably extract the laser-driven proton spectra from a 
+stack of irradiated RCF. 
+ 
+2. Methodology 
+Energy resolved measurements of impinging proton and ion beams can be performed using 
+multiple RCF arranged into a stack configuration. The differing stopping positions for protons of 
+a given energy within an RCF stack, means each layer can be defined by a unique energy 
+
+ 
+ 
+– 2 – 
+sensitivity. This is chosen to correspond to the energy required to generate a Bragg peak at that 
+given depth, defining the energy of protons that will be referred to as peak region protons. Low 
+energy components stop in the first few layers of the stack, whilst higher energies penetrate 
+further downstream, giving a total energy composition of stopping protons, in addition to the 
+fractional contributions of those exceeding the energy sensitivity of a given film layer. Unfolding 
+the peak energy from the total energy deposited within any RCF can be achieved through the 
+development of a deconvolution procedure for proton spectroscopy. This relies on an algorithm 
+utilising weight factors to describe the fractional contributions of each energy component within 
+every film. This process is detailed in figure 1.  
+ 
+ 
+ 
+The developed algorithm performs a backwards weighted subtraction of contributions, starting 
+from the final layer, as a singular energy is contained on this film. Careful subtraction of weighted 
+components discriminates the energy of stopping protons within each film from passing energies. 
+This remaining peak or stopping energy is then converted into a measurement of the stopping 
+particle fluence through the corresponding stopping power of every given layer. 
+  
+𝑁!"#$#%& =
+'!"
+!#'
+$%&
+'()*+,(!
+' !"
+!#../'
+0(12,3
+	(𝐸𝑞. 1), 
+ 
+The numerator of Eq. 1 represents the remaining peak stopping energy within every active layer 
+after the deconvolution algorithm has been applied to the total deposited energy within each. The 
+denominator denotes the energy transfer as a function of the thickness of film material crossed, 
+found through Monte Carlo simulation. A processing script was written using the MATLAB 
+software [18], that compiles all of the required input parameters and procedures of this 
+spectroscopic method into a single program. This provides the possibility to directly input scanned 
+Figure 1. Visual representation of the calculation of weight factors. The water equivalent depths of the 
+active layers, in addition to the energy required to produce a Bragg peak at the depth of each, are both 
+well-known. Extrapolating the peak contributions allowed weighting factors to be calculated through 
+normalization of the deposited energy contribution to that of the respective peak value. For example, to 
+calculate the weighting factor provided by peak B to peak A, the ratio of the energy deposited by peak B 
+at the position of peak A, EdepB(x), to the maximum ionization of B itself, EdepB(peak), is found. This 
+process is performed for each energy component, at each active layer depth, and a matrix of weight factors 
+is then constructed. 
+
+0.0007
+0.0006
+Edepa(peak)
+0.0005
+Edep(peak)
+Dose [a.u.]
+0.0004
+Edepc(peak)
+Peak (A)
+0.0003
+Peak (B)
+Peak (C)
+0.0002
+EdepB(x)
+0.0001
+Edepc(x)
+0
+0
+0.5
+1
+1.5
+2
+2.5
+3
+Depthin water[mm] 
+ 
+– 3 – 
+RCF images, and through simple modification, data from simulation, for a direct reconstruction 
+of the proton energy spectrum. A typical reconstructed spectrum is highlighted in figure 2, with 
+data obtained at a laser-driven proton facility.  
+ 
+ 
+The resultant energy spectrum displays an exponentially decreasing behaviour typical of laser-
+driven beams produced through the target normal sheath acceleration process [1, 2]. The fact 
+that this is observed in the data in figure 2 gives some confidence that the procedure can 
+reproduce the expected spectral profile.  
+ 
+3. Monte Carlo Analysis 
+To further assess the effectiveness of the developed deconvolution procedure, a Monte Carlo 
+analysis was conducted through Geant4. A replicated RCF stack configuration of the model 
+GafChromic EBT3, with symmetrical structure of a 28 𝜇𝑚 active layer, sandwiched between two 
+polyester dead layers, was constructed as outlined within the manufacturer’s specifications [19]. 
+Detailed simulation of this film stack is a vital first step in the spectroscopic procedure 
+development, allowing the energy sensitivity and corresponding stopping power values to be 
+evaluated for each film layer, in addition to the weighting factors required in the deconvolution 
+algorithm.  
+ 
+The reliability of the developed deconvolution procedure as a tool for spectroscopy was assessed 
+through examination of the retrieved deconvolution spectrum, with one that is known. Through 
+Geant4 simulation, a proton source with tailored energy could be sent into the constructed RCF 
+stack, and the deposited energy converted to a measurement of the particle number (energy 
+fluence) at each active layer node using the deconvolution algorithm developed. This arrangement 
+was used for input proton sources with both exponential and flat energy spectra. The latter of 
+these source spectra proved more useful in highlighting potential discrepancies between the actual 
+and expected spectra. From analysis, it was noticed that particularly for laser-driven energy 
+spectra, with particle numbers extending orders of magnitude, the differences between the 
+Figure 2. Proton energy spectrum found from an irradiated stack of RCF of the model GafChromic 
+HDV2. This data was taken from a laser-plasma experiment at the LULI facility (Laboratoire pour 
+l'Utilisation des Lasers Intenses, École Polytechnique, France). The stack was placed immediately after 
+the target, from which protons were generated with the typical TNSA exponential behaviour.  
+
+14
+1010
+12
+10
+Number of Protons
+8
+9
+4
+2
+0
+0
+5
+10
+15
+20
+25
+30
+Energy [MeV] 
+ 
+– 4 – 
+retrieved spectral particle numbers can be quite large, whilst still maintaining an apparently good 
+degree of agreement. During cross-comparison, the potential to disguise discrepancies between 
+spectra was reduced with the use of a flat spectrum. Once the proton energy spectrum had been 
+recovered from the energy deconvolution data, it was cross-compared with the original energy 
+spectrum. A measurement of the spectrum that originates at the source can be obtained from the 
+simulation through examination of the particle flux at a thin region coinciding with the front face 
+of the film stack. This eliminates interaction of the impinging proton beam with the RCF material, 
+and potential errors induced through conversion of the measured deposited energy to particle flux. 
+Analysis of the retrieved spectrum through application of the deconvolution algorithm in Geant4 
+is shown in figure 3. 
+ 
+ 
+ 
+A reasonable agreement between the deconvolution and entrance spectra is observed from fig. 4, 
+outlining the accuracy of the developed procedure in obtaining the correct particle flux at each 
+measurement node. This systematic Monte Carlo investigation thus gives an insight into the 
+working order of the algorithm for deconvolution, providing an indication of its reliability in 
+correctly reconstructing the energy spectrum. Within previous works concerning RCF 
+spectroscopy, the final spectrum is often assumed to be correct, with no such systematic check 
+performed. Analysis has shown that this cannot be taken for granted, and so by carrying out this 
+procedure some confidence is gained concerning the reliability of this spectroscopic tool. 
+ 
+Conclusions 
+A spectroscopic procedure for the measurement of laser-driven proton energy spectra based on 
+the use of a stacked configuration of radiochromic films has been developed and reported here. A 
+deconvolution algorithm that operates through an iterative backwards weighted subtraction of 
+energy components from successive films has been developed to unfold the stopping proton 
+energy from the total energy deposited in each film layer. Initial tests demonstrated reconstruction 
+of a typical exponential-like spectrum with large energy spread for films irradiated using a laser-
+driven proton beam. Further analysis of the developed spectroscopic procedure was conducted 
+through Monte Carlo methods utilising the Geant4 particle simulation toolkit. Comparison of the 
+Figure 3. Cross-comparison of the proton energy spectrum obtained through a deconvolution of the total 
+energy deposited in each film layer, with the proton fluence spectrum originating at the source as measured 
+at the stack entrance. 
+
+250000
+200000
+150000
+100000
+Energydeconvolution
+Spectrum originatingatsource
+50000
+0
+0
+10 
+20
+30
+40
+50
+60
+Kineticenergy[MeV] 
+ 
+– 5 – 
+spectrum retrieved through deconvolution of the energy transferred to each film, to that 
+originating at the source for a flat energy spectrum showed a good agreement, indicating the 
+applicability of this tool in the spectral reconstruction of a laser-driven proton source. Although 
+the analysis reported is promising, a thorough examination of experimental data should be carried 
+out to validate the developed procedure. A reasonable result would outline the potential of this 
+tool in deriving a fast measurement of the energy spectrum from an irradiated stack of 
+radiochromic films. Nonetheless, this systematic investigation based on analysis of spectral 
+deconvolution through detailed Monte Carlo simulations represents one that has not been tried 
+before. Through cross-comparison within simulation, this has allowed an effective evaluation of 
+the performance of such a spectroscopic tool required for accurate measurement of the proton 
+energy spectrum generated through laser-driven beams.  
+ 
+References 
+[1] Macchi, A., et al, (2013), Rev. Mod. Phys. 85, 751  
+[2] Gibbon, P., (2005), Imperial College Press 
+[3] Badziak, J., et al, (2010), Appl. Phys. Lett. 96, 251502  
+[4] Bolton, P., et al, (2018), CRC Press 
+[5] Borghesi, M., (2014), NIMA, 740;6-9 
+[6] Bolton, P., et al, (2014), Physica Medica, 30, 3;255-270 
+[7] Schreiber, J., et al, (2016), Review of Scientific Instruments, 87, 7 
+[8] Margarone, D., et al, (2018), Quantum Beam Sci. 2(2), 8 
+[9] Breschi, E., et al., (2004), Laser Part. Beams, 22, 393.  
+[10] Schollmeier, M., et al., (2008), Phys. Rev. Lett. 101, 055004 
+[11] Cowan, T. E., et al., (2004), Phys. Rev. Lett. 92, 204801 
+[12] Hey, D. S., et al, (2008), Rev. Sci. Instrum. 79, 053501 
+[13] Nuernberg, F., et al., (2009), Rev. Sci. Instrum. 80, 033301 
+[14] Kirby, D., et al., (2011), Laser and Particle Beams 29(02) 
+[15] Agostinelli, S., et al. Nuclear Methods and Instruments in Physics Research, 506(3), (2003) 250 
+[16] Allison, J., et al. Nucl. Instrum. Meth. A 835 (2016) 186-225 
+[17] http://geant4.web.cern.ch 
+[18] MATLAB, R2019b, (www.mathworks.com)  
+[19] Ashland ISP Advanced Materials, NJ, USA, (www.gafchromic.com)  
+

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+page_content=' Milluzzoc, a, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Borghesia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Subielb, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Romanod a Centre for Plasma Physics, Queen’s University Belfast, BT7 1NN, United Kingdom b Medical Radiation Science, National Physical Laboratory, Teddington, TW11 0LW, United Kingdom c Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, Via S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Sofia 62, 95123 Catania, Italy d Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Sofia 64, 95123 Catania, Italy  E-mail: smccallum05@qub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='uk ABSTRACT: Laser-driven ion beams are defined by a number of unique features, including a large spread in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A stack configuration of radiochromic film (RCF) can be utilized to characterize such beams through measurements of their energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A spectroscopic procedure is reported that allows the proton energy density within each active layer of a radiochromic film (RCF) stack to be retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This is based upon on a deconvolution algorithm developed through Geant4 Monte Carlo simulations to correct the contributions of energy depositions within a given film layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Through Monte Carlo calculations, the spectrum retrieved from a simulated film stack can be retrieved and compared with a known energy spectrum, providing an examination of the efficacy of this tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Application of the developed deconvolution procedure thus offers the potential to correctly reconstruct the incident energy spectrum of a laser-driven proton and ion beam from a stack of irradiated RCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' KEYWORDS: Detector modelling and simulations I, dE/dx detectors, Plasma diagnostics - charged-particle spectroscopy, Simulation methods and programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  – 1 –  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Methodology  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Monte Carlo Analysis  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Conclusions  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' References  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Introduction Whilst laser-driven proton and light ion acceleration has attracted significant interest for over 20 years [1, 2], conducting accurate measurements of these beams has proven to be technically challenging [3-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' In particular, the ultra-high dose rates and wide spectral distributions make conventional measurement techniques impracticable [6-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' For applications, including clinical and radiobiological ones requiring a precise energy selection, characterisation of such beams through accurate measurement of their energy spectra is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Spectroscopic methods reliant on stacked configurations of radiochromic films (RCF) are well-established for measurements of accelerated proton beams, with several approaches of radiochromic film imaging spectroscopy (RIS) reported in the literature [9-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A stacked configuration of films placed perpendicularly to the beam orientation can be used to perform an energy resolved measurement of an impinging ion beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Differential energy loss results in each particle depositing a fraction of its initial kinetic energy on every film it passes before coming to arrest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' For polyenergetic sources such as laser- driven beams, a superposition of kinetic energy contributions is amassed across the films, requiring a calculation for correction of higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This is achieved through a deconvolution or unfolding of the energy transferred to each film in the stack, so that only the particles stopping within a given film remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  The aim of the work reported here was to investigate and assess a developed algorithm for spectroscopy of laser-driven proton and ion beams through Monte Carlo simulations, studying the possible limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This procedure requires knowledge of the RCF energy sensitivity values, and an algorithm to unfold the proton energy spectrum from the RCF response, both of which have been evaluated using the Geant4 toolkit [15-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Further, the same Monte Carlo methods were utilised to conduct analysis of the performance and limitations of the developed technique in acquiring the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Once validated, the spectroscopic procedure reported offers the potential to reliably extract the laser-driven proton spectra from a stack of irradiated RCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Methodology Energy resolved measurements of impinging proton and ion beams can be performed using multiple RCF arranged into a stack configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' The differing stopping positions for protons of a given energy within an RCF stack, means each layer can be defined by a unique energy  – 2 – sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This is chosen to correspond to the energy required to generate a Bragg peak at that given depth, defining the energy of protons that will be referred to as peak region protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Low energy components stop in the first few layers of the stack, whilst higher energies penetrate further downstream, giving a total energy composition of stopping protons, in addition to the fractional contributions of those exceeding the energy sensitivity of a given film layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Unfolding the peak energy from the total energy deposited within any RCF can be achieved through the development of a deconvolution procedure for proton spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This relies on an algorithm utilising weight factors to describe the fractional contributions of each energy component within every film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This process is detailed in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  The developed algorithm performs a backwards weighted subtraction of contributions, starting from the final layer, as a singular energy is contained on this film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Careful subtraction of weighted components discriminates the energy of stopping protons within each film from passing energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This remaining peak or stopping energy is then converted into a measurement of the stopping particle fluence through the corresponding stopping power of every given layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  𝑁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' "#$#%& =  \'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='"  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content="#'  $%&  '() +,(!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content="  ' !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='"  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content="./'  0(12,3  (𝐸𝑞." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' 1),  The numerator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' 1 represents the remaining peak stopping energy within every active layer after the deconvolution algorithm has been applied to the total deposited energy within each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' The denominator denotes the energy transfer as a function of the thickness of film material crossed, found through Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A processing script was written using the MATLAB software [18], that compiles all of the required input parameters and procedures of this spectroscopic method into a single program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This provides the possibility to directly input scanned Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Visual representation of the calculation of weight factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' The water equivalent depths of the active layers, in addition to the energy required to produce a Bragg peak at the depth of each, are both well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Extrapolating the peak contributions allowed weighting factors to be calculated through normalization of the deposited energy contribution to that of the respective peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' For example, to calculate the weighting factor provided by peak B to peak A, the ratio of the energy deposited by peak B at the position of peak A, EdepB(x), to the maximum ionization of B itself, EdepB(peak), is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This process is performed for each energy component, at each active layer depth, and a matrix of weight factors is then constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='0007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='0006  Edepa(peak)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='0005  Edep(peak)  Dose [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=']  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='0004  Edepc(peak)  Peak (A)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='0003  Peak (B)  Peak (C)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='0002  EdepB(x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='0001  Edepc(x)  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='5  3  Depthin water[mm]  – 3 – RCF images, and through simple modification, data from simulation, for a direct reconstruction of the proton energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A typical reconstructed spectrum is highlighted in figure 2, with data obtained at a laser-driven proton facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  The resultant energy spectrum displays an exponentially decreasing behaviour typical of laser- driven beams produced through the target normal sheath acceleration process [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' The fact that this is observed in the data in figure 2 gives some confidence that the procedure can reproduce the expected spectral profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Monte Carlo Analysis To further assess the effectiveness of the developed deconvolution procedure, a Monte Carlo analysis was conducted through Geant4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A replicated RCF stack configuration of the model GafChromic EBT3, with symmetrical structure of a 28 𝜇𝑚 active layer, sandwiched between two polyester dead layers, was constructed as outlined within the manufacturer’s specifications [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Detailed simulation of this film stack is a vital first step in the spectroscopic procedure development, allowing the energy sensitivity and corresponding stopping power values to be evaluated for each film layer, in addition to the weighting factors required in the deconvolution algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  The reliability of the developed deconvolution procedure as a tool for spectroscopy was assessed through examination of the retrieved deconvolution spectrum, with one that is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Through Geant4 simulation, a proton source with tailored energy could be sent into the constructed RCF stack, and the deposited energy converted to a measurement of the particle number (energy fluence) at each active layer node using the deconvolution algorithm developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This arrangement was used for input proton sources with both exponential and flat energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' The latter of these source spectra proved more useful in highlighting potential discrepancies between the actual and expected spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' From analysis, it was noticed that particularly for laser-driven energy spectra, with particle numbers extending orders of magnitude, the differences between the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Proton energy spectrum found from an irradiated stack of RCF of the model GafChromic HDV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=" This data was taken from a laser-plasma experiment at the LULI facility (Laboratoire pour l'Utilisation des Lasers Intenses, École Polytechnique, France)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' The stack was placed immediately after the target, from which protons were generated with the typical TNSA exponential behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  14  1010  12  10  Number of Protons  8  9  4  2  0  0  5  10  15  20  25  30  Energy [MeV]  – 4 – retrieved spectral particle numbers can be quite large, whilst still maintaining an apparently good degree of agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' During cross-comparison, the potential to disguise discrepancies between spectra was reduced with the use of a flat spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Once the proton energy spectrum had been recovered from the energy deconvolution data, it was cross-compared with the original energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A measurement of the spectrum that originates at the source can be obtained from the simulation through examination of the particle flux at a thin region coinciding with the front face of the film stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This eliminates interaction of the impinging proton beam with the RCF material, and potential errors induced through conversion of the measured deposited energy to particle flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Analysis of the retrieved spectrum through application of the deconvolution algorithm in Geant4 is shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  A reasonable agreement between the deconvolution and entrance spectra is observed from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' 4, outlining the accuracy of the developed procedure in obtaining the correct particle flux at each measurement node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' This systematic Monte Carlo investigation thus gives an insight into the working order of the algorithm for deconvolution, providing an indication of its reliability in correctly reconstructing the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Within previous works concerning RCF spectroscopy, the final spectrum is often assumed to be correct, with no such systematic check performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Analysis has shown that this cannot be taken for granted, and so by carrying out this procedure some confidence is gained concerning the reliability of this spectroscopic tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  Conclusions A spectroscopic procedure for the measurement of laser-driven proton energy spectra based on the use of a stacked configuration of radiochromic films has been developed and reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A deconvolution algorithm that operates through an iterative backwards weighted subtraction of energy components from successive films has been developed to unfold the stopping proton energy from the total energy deposited in each film layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Initial tests demonstrated reconstruction of a typical exponential-like spectrum with large energy spread for films irradiated using a laser- driven proton beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Further analysis of the developed spectroscopic procedure was conducted through Monte Carlo methods utilising the Geant4 particle simulation toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Comparison of the Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Cross-comparison of the proton energy spectrum obtained through a deconvolution of the total energy deposited in each film layer, with the proton fluence spectrum originating at the source as measured at the stack entrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  250000  200000  150000  100000  Energydeconvolution  Spectrum originatingatsource  50000  0  0  10  20  30  40  50  60  Kineticenergy[MeV]  – 5 – spectrum retrieved through deconvolution of the energy transferred to each film, to that originating at the source for a flat energy spectrum showed a good agreement, indicating the applicability of this tool in the spectral reconstruction of a laser-driven proton source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Although the analysis reported is promising, a thorough examination of experimental data should be carried out to validate the developed procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' A reasonable result would outline the potential of this tool in deriving a fast measurement of the energy spectrum from an irradiated stack of radiochromic films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Nonetheless, this systematic investigation based on analysis of spectral deconvolution through detailed Monte Carlo simulations represents one that has not been tried before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content=' Through cross-comparison within simulation, this has allowed an effective evaluation of the performance of such a spectroscopic tool required for accurate measurement of the proton energy spectrum generated through laser-driven beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
+page_content='  References [1] Macchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
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+page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}
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+page_content='com)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dFAT4oBgHgl3EQfnh1q/content/2301.08629v1.pdf'}

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+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+Abstract. For a nice-enough category C, we construct both the morphism category H(C) of
+C and the category mod-C of all finitely presented contravariant additive functors over C with
+values in Abelian groups. The main theme of this paper, is to translate some representation-
+theoretic attributes back and forth from one category to the other. This process is done by
+using an appropriate functor between these two categories, an approach which seems quite
+promising in particular when we show that many of almost split sequences are preserved by
+this functor. We apply our results to the case of wide subcategories of module categories to
+obtain certain auto-equivalences over them. Another part of the paper deals with Auslander
+algebras arising from algebras of finite representation type. In fact, we apply our results to
+study the Auslander-Reiten translates of simple modules over such algebras. In the last parts,
+we try to recognize particular components in the stable Auslander-Reiten quiver of Auslander
+algebras arising from self-injective algebras of finite representation type.
+1. Introduction
+As a popular belief, it is said that the introduction of the language of functor categories to
+the study of categories of modules over rings dates back to Auslander and his colleagues’ works.
+These works trace back mainly to the papers [A65, A71, A76, AR74, AR78].
+In particular
+Auslander’s Formulae [A65] that suggests to recover the category mod-Λ of finitely generated
+modules over an Artin algebra Λ as the quotient
+mod-Λ ≃ mod-(mod-Λ)
+{F : F(Λ) = 0}
+deserves attention; here and throughout, mod-(mod-Λ) denotes the category of additive con-
+travariant coherent functors on mod-Λ with values in Ab, the category of Abelian groups. While
+talking about the exchange between two categories consisting objects that are apparently of
+different types, one expects to encounter with functors transferring from one category to the
+other. Concerning the morphism categories and the functor categories, such a study has initi-
+ated probably in [A71]. Roughly, the general theme of the current paper is to figure out how some
+representation-theoretic attributes transfer between functor and morphism categories. However,
+to be more precise, we prefer to provide a layout of the paper section by section. Prior to this,
+we want to point out that the morphism category of Λ has on its own right been systematically
+studied from various aspects: deriving its Auslander-Reiten theory in the language of AR-theory
+of Λ [RS, XZZ, E, HE], establishing its links to Gorenstein homological algebra [Z, LZ, ZX], and
+looking at a particular subcategory of it, namely the monomorphism category, in order to study
+the so-called Auslander algebras [AR76, HM].
+2020 Mathematics Subject Classification. 18A25, 16G70, 16G10.
+Key words and phrases. Functor Category, Morphism Category, Auslander-Reiten Components.
+1
+arXiv:2301.00534v1  [math.RT]  2 Jan 2023
+
+2
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+First of all, to keep the results as general as possible, we try to deal with the morphism category
+H(C) of a nice-enough category C (definitions are recalled later on). Namely, if we assume that
+C is an idempotent-complete additive category that admits pseudokernels then, in Section 3, we
+endow H(C) with an exact structure defined by degree-wise split exact sequences in C, denoted
+Hcw(C). Even though such constructions have been considered in some particular cases, e.g.
+in [Ba] where the category of morphisms between projective modules over an Artin algebra
+have come to play, we do it in a most general possible circumstance as declared above. The
+motivation behind such considerations comes from two origins. Firstly, we look for a reasonable
+structure on H(C) with respect to which one may define almost split sequences. Note, secondly,
+that if one imposes tougher conditions on C, for instance taking C to be an extension-closed
+subcategory of mod-Λ, then H(C) inherits an exact structure as an extension-closed subcategory
+of the morphism category of Λ. So now a natural question arises: What are intrinsic similarities
+between these two exact structures on H(C)?
+To get more involved with the aforementioned question, we need to take a glance at the
+contents of Section 4. For, we recall form [A71] that there exists a functor Θ : H(C) −→ mod-C,
+where mod-C is the category of contravariant additive coherent functors on C. The objective in
+Section 4 is to study Θ form the point of view of Auslander-Reiten theory. We show that Θ
+induces an equivalence H(C)/
+�
+(M → 0), (M
+1→ M)
+�
+≃ mod-C where M runs through the objects
+of C. Using this, we show that Hcw(C) admits almost split sequences whenever C is assumed
+to be a dualizing variety. Furthermore, to conquer the question posed above, it is shown that
+if C is an extension-closed dualizing subvariety of mod-Λ then, in many cases, the almost split
+sequences in Hcw(C) and H(C) coincide. Not going off-topic, one more thing will be proved: Θ
+respects almost split sequences.
+In Section 5, we turn to apply some of the results to the case of wide subcategories. To
+illuminate the role and importance of wide subcategories of mod-Λ, we must point out that such
+subcategories arise naturally in the study of τ-tiling theory of Λ [AIR] and in connection with
+determination of certain torsion classes in mod-Λ [MS]. These also play significant role in the
+study of certain classes of universal localizations over Λ [MS, HMV1, HMV2]. Such classes of
+modules also appear in classification problems for the so-called τ-tilting finite algebras. Among
+other things, for a given functorially finite wide subcategory X of mod-Λ we construct, based on
+our previous results, an auto-equivalence σX : X → X which fulfills the exact sequence
+0 → (−, σX τX (X)) → D(P, −) → D(Q, −) → D(X, −)
+in mod-X for every indecomposable module X which is not projective in X; here τX denotes the
+Auslander-Reiten translation of X and P → Q → X → 0 is the minimal projective presentation
+of X with respect to X. In this regard, recall that for a non-projective Λ-module M with minimal
+projective presentation P → Q → M → 0, there exists an exact sequence 0 → τ(M) → ν(P) →
+ν(Q) → ν(M) → 0 where τ and ν stand respectively for the Auslander-Reiten translation and the
+Nakayama functor over mod-Λ. Hence the aforementioned exact sequence of functors resembles,
+and generalizes, the latter one. This is more clarified by showing that when X is the whole
+category mod-Λ, then σX is nothing but the identity functor. We believe that this observation
+is convincing-enough to say that the rich treasury behind functorially finite wide subcategories
+of mod-Λ might be discovered by applying some instruments from functor categories.
+In Section 6, we switch to algebras Λ of finite representation type. The main impetus for such
+a study comes from the fact that in this case, one may construct the Auslander algebra A of Λ
+which is, by definition, the endomorphism algebra of a representation-generator M of Λ. Then
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+3
+there is a nice interpretation of the category mod-A in terms of the functor category; namely,
+there is a categorical equivalence mod-A ≃ mod-(mod-Λ). In the meanwhile, it is known [A76]
+that simple functors over mod-Λ correspond bijectively to indecomposable Λ-modules. Hence
+this categorical equivalence provides a framework in which one tries to understand in more details
+the simple modules over A and its projectively stable version A. The results presented in this
+section come up by analyzing certain almost split sequences mainly provided in [HE] and also in
+[HZ]. The main results discover a relation between the (inverse) Auslander-Reiten translation of
+simple A-(resp. A-) modules and the cosyzygies (resp. syzygies) of simple A-(resp. A-) modules.
+The last section is devoted to study certain components in the (stable) Auslander-Reiten
+quiver ΓA of the Auslander algebra A whenever Λ is self-injective of finite representation type.
+Note that recognition of such components have already been the subject of some earlier researches
+[IPTZ].
+To this end, we firstly deal with τH-periodic objects by invoking some almost split
+sequences already obtained in [HE]. In this direction, it turns out that the auto-equivalence
+A = ντ 3 of the stable category mod-Λ, as defined in [HE], plays a significant role. In fact,
+we show that the existence of certain A -periodic Λ- modules makes ΓA into a finite oriented
+cycle, and in particular, makes A into an algebra of finite representation type. Another result
+asserts that for Λ self-injective of finite representation type, any component Ξ of the stable
+Auslander-Reiten quiver of A that contains a certain simple module is either infinite or is of the
+form Z∆/G for a Dynkin quiver ∆ and an automorphism group G of Z∆; this is based on a
+structural theorem due to Liu [L].
+2. preliminaries and notation
+In this section, we collect very briefly some necessary background material of the paper. When
+required, explicit references are provided.
+2.1. Functor Categories. Let k be a commutative Artinian ring and let C be a k-linear Krull-
+Schmidt category. A C-module is a contravariant additive functor from C to the category Ab
+of Abelian groups.
+We denote by Mod-C the category of all C-modules, and by mod-C the
+full subcategory of Mod-C consisting of finitely presented modules. Recall from [A65] that a
+C-module M is called finitely presented if there exists an exact sequence
+HomC(−, A) → HomC(−, B) → M → 0
+in Mod-C, for some objects A, B of C. Moreover, proj-C and inj-C denote the full subcategories of
+mod-C consisting of projective and injective objects in mod-C, respectively. The category mod-C
+is an abelian category if and only if C admits pseudokernels; see page 315 of [AR74]. We shall
+sometimes write (−, X) instead of the representable functor HomC(−, X).
+2.2. Dualizing k-varieties. Let r be the radical of k and E(k/r) be the injective envelope of
+the k-module k/r. A Hom-finite k-linear Krull-Schmidt category C is called a dualizing k-variety
+[AR74] if the k-dual functors D : Mod-C → Mod-(Cop) and D : Mod-(Cop) → Mod-C given by
+D(F)(C) = Homk(F(C), E(k/r)) for every object C of C and F ∈ Mod-(C) or Mod-(Cop) induce
+dualities
+D : mod-C → mod-(Cop) and D : mod-(Cop) → mod-C.
+In this case, it turns out that mod-C is an abelian subcategory of Mod-C that admits enough pro-
+jective and enough injective objects [AR74, Theorem 2.4]. As an example, proj-Λ, the category
+of finitely generated projective modules over an Artin k-algebra Λ, is a dualizing k-variety. We
+
+4
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+note from [AR74, Proposition 2.6] that if C is a dualizing k-variety then so is mod-C. Further-
+more, any functorially finite subcategory of a dualizing k-variety is itself a dualizing k-variety
+by [AS81, Theorem 2.3].
+2.3. Morphism Categories. Let C be a category.
+The morphism category H(C) of C is a
+category whose objects are morphisms f : X → Y in C, and whose morphisms are given by
+commutative diagrams. If we regard the morphism f : X → Y as an object in H(C), we will
+usually present it as (X
+f→ Y ). However, due to typographical considerations, we have to use
+also the vertical notation ( X
+Y )f. A morphism between the objects (X
+f→ Y ) and (X′ f ′
+→ Y ′) is
+presented as (σ1, σ2) : (X
+f→ Y ) → (X′ f ′
+→ Y ′) or, ( σ1
+σ2 ) : ( X
+Y )f →
+� X′
+Y ′
+�
+f ′, where σ1 : X → X′
+and σ2 : Y → Y ′ are morphisms in C with σ2f = f ′σ1.
+Adapting the notation, the morphism category raised from C = mod-Λ, the category of finitely
+generated right modules over an Artin k-algebra Λ, will be denoted simply by H; this will cause
+no ambiguity. The same rule also applies to the monomorphism category S of Λ whose objects
+are just monic Λ-maps.
+2.4. Auslander-Reiten-Serre Duality. Let (C, E) be an exact category in the sense of Quillen
+[Q, K] (see next section for an introduction).
+Recall that a morphism v: E → Y in C is
+called right almost split if it is not a retraction and each f : Z → Y which is not a retraction
+factors through v. Dually, a morphism u: X → E in C is called left almost split if it is not a
+section and each f : X → Z which is not a section factors through u. An admissible sequence
+δ: 0 → X
+u−→ E
+v−→ Y → 0 in E is an almost split sequence if u is left almost split and v is right
+almost split. Since δ determines X and Z in a unique way, we call X the Auslander-Reiten
+translation X = τC(Y ) of Y in C.
+A non-zero object X ∈ C is said to be endo-local if its
+endomorphism ring EndC(X) is local. Following [INY, Definition 3.1], we say that C has almost
+split sequences if endo-local non projective objects of C and endo-local non-injective objects of
+C are respectively the terminal and the initial terms of some almost split sequence in E.
+Assume now that C is further a k-linear category and let D be the k-dual functor. Put C and
+C denote respectively the projectively and the injectively stable categories of C. An Auslander-
+Reiten-Serre duality (ARS duality, in brief) is a pair (τC, η) consisting of an equivalence functor
+τC : C → C together with a bi-natural isomorphism
+ηX,Y : HomC(X, Y ) ≃ DExt1
+C(Y, τC(X))
+for any X, Y ∈ C.
+The following lemma, taken from [INY, Theorem 3.6] (see also [J]), provides a close connection
+between the existence of almost split sequences in C and the existence of an ARS-duality. Let us
+recall that under the above hypothesis, C is Ext-finite if the k-modules Ext1
+C(X, Y ) are finitely
+generated.
+Lemma 2.1. Let C be a k-linear Ext-finite Krull-Schmidt exact category. Then the following
+conditions are equivalent.
+(1) C has almost split sequences.
+(2) C has an Auslander-Reiten-Serre duality.
+(3) The stable category C is a dualizing k-variety.
+(4) The stable category C is a dualizing k-variety.
+Throughout the paper, Λ will stand for a fixed Artin k-algebra and modules are, by default,
+finitely generated right modules.
+The Auslander-Reiten translation, the Nakayama functor,
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+5
+the syzygy and the cosyzygy functor of Λ are respectively denoted by τ, ν, Ω, and Ω−1. If
+we deal with an algebra other than Λ or with a category, these functors will be accompanied
+with necessary subscripts. The symbols Ker, Coker, and Im, used freely in all contexts, stand
+respectively for the kernel, cokernel, and the image of morphisms.
+3. Exact structures on the morphism category
+An exact category (C, E) is formed by an additive category C, and a class E of composable
+pairs of morphisms in C (also called kernel-cokernel pairs) satisfying certain axioms that we
+refrain to exhibit here and refer the reader e.g. to [K]. The composable pair (i, p) in E is usually
+denoted by 0 → A′
+i→ A
+p→ A′′ → 0, where i : A′ → A and p : A → A′′ are respectively called
+an E-admissible monic and an E-admissible epic.
+Composable pairs, admissible monics and
+admissible epics are sometimes referred to respectively as conflations, inflations and deflations.
+The notion of an exact category was first introduced by Quillen in [Q] and then Keller [K] proved
+the redundancy of some axioms.
+Let C be an additive category. In this section, we shall put an exact structure on the morphism
+category H(C) of C [Ba]. For let Ecw be the class of all pairs of composable morphisms
+δ :
+� X1
+X2
+�
+f
+� φ1
+φ2
+�
+�� Z1
+Z2
+�
+h
+� ψ1
+ψ2
+�
+�� Y1
+Y2
+�
+g
+such that the induced composable morphisms Xi
+φi
+→ Zi
+ψi
+→ Yi split in C for i = 1, 2. It can be
+easily seen that any pair of composable morphisms in Ecw is isomorphic to a pair of composable
+morphisms of the form
+δ′ :
+� X1
+X2
+�
+f
+�[ 1
+0]
+[ 1
+0]
+�
+�� X1⊕Y1
+X2⊕Y2
+�
+h
+� [0 1]
+[0 1]
+�
+�� Y1
+Y2
+�
+g
+where h =
+�
+f q
+0 g
+�
+and q : Y1 → X2 is a possibly non-zero morphism in C. Regarding this easy
+observation, without loss of generality, we usually take all kernel-cokernel pairs in H(C) to be of
+this form; this is justified by the following lemma.
+Lemma 3.1. Any object in Ecw is a kernel-cokernel pair in H(C).
+Proof. Take the element δ′ of Ecw and assume that the composite of the morphisms (σ1, σ2) :
+(X1 ⊕ Y1
+h→ X2 ⊕ Y2) → (V
+s→ W) and ([ 1
+0 ], [ 1
+0 ]) vanishes. This means that the restriction of σi
+on Xi, for i = 1, 2, is the zero map. This enables us to define the morphisms σ1|Y1 and σ2|Y2 and
+it readily follows that (σ1, σ2) factors uniquely over ([0 1], [0 1]) via the morphism (σ1|Y1, σ2|Y2).
+The remaining axioms are verified similarly.
+□
+Recall that an additive category D is called idempotent-complete if every idempotent endo-
+morphism in D admits a kernel.
+Proposition 3.2. Assume C is idempotent-complete and admits pseudokernels. Then Ecw de-
+fines an exact structure on the additive category H(C).
+Proof. Since C is idempotent-complete, it is known that the Yoneda functor gives an equivalence
+C ≃ proj-C. This equivalence is naturally extended to an equivalence between corresponding
+morphism categories; i.e., H(C) ≃ H(proj-C). One observes that, under this equivalence, the
+
+6
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+kernel-cokernel pairs in Ecw provided by Lemma 3.1 correspond bijectively to the short exact se-
+quences in the abelian category H(mod-C) whose terms lie inside H(proj-C). But the subcategory
+H(proj-C) is closed under extensions and inherits an exact structure from H(mod-C).
+□
+Remark 3.3. To make our arguments work, we had to impose some restrictions on the additive
+category C to get a suitable exact structure out of H(C). However, it may be the case that
+the aforementioned set of requirements is not minimal in the sense that the above family of
+kernel-cokernel pairs may equip H(C) with an exact structure even if some of the hypothesis in
+Proposition 3.2 are dropped.
+From now on we assume that C is idempotent-complete and admits pseudokernels, and the
+symbol Hcw(C) stands for the exact category (H(C), Ecw), sometimes also called the cw-exact
+category. The following proposition is recorded for future use.
+Proposition 3.4. Suppose (X1
+f→ X2) is an object in Hcw(C).
+(1) f defines an indecomposable projective object in Hcw(C) if and only if it is isomorphic
+either to (X
+1→ X) or (0 → X) for some indecomposable object X in C.
+(2) f defines an indecomposable injective object in Hcw(C) if and only if it is isomorphic
+either to (X
+1→ X) or (X → 0) for some indecomposable object X in C.
+Furthermore, Hcw(C) has enough projectives and enough injectives.
+Proof. This should be compared to [Ba, Corollary 3.2].
+We just remark that the last claim
+follows from the short exact sequences
+0
+�� 0
+X1
+�
+0
+�
+0
+� f
+−1
+�
+�
+�� 0
+X2
+�
+0 ⊕
+� X1
+X1
+�
+1
+�
+1
+[ 1 f ]
+�
+�� X1
+X2
+�
+f
+�0
+and
+0
+�� X1
+X2
+�
+f
+� � f
+1
+�
+1
+�
+�� X2
+X2
+�
+1 ⊕
+� X1
+0
+�
+0
+� [ −1 f ]
+0
+�
+�� X2
+0
+�
+�0
+in Ecw.
+□
+Now assume C is an extension-closed subcategory of mod-Λ for an Artin algebra Λ. We may
+consider C as an exact category through the structure induced by the abelian category mod-Λ.
+Then also H(C), as an extension-closed subcategory of the abelian category H is endowed with
+the canonical exact structure inherited from H, still denoted by H(C). We also keep the cw-
+exact structure Hcw(C) defined by degree-wise split sequences. It will be indicated in the next
+section that if C is a k-dualizing variety, then Hcw(C) admits almost split sequences. Further,
+the canonical exact category H(C) admits almost split sequences provided C is a k-dualizing
+subvariety of mod-Λ.
+It also becomes clear how the canonical exact category H(C) inherits
+almost split sequence from Hcw(C) in the latter case. However, for technical reasons, we have to
+defer the proofs until next section.
+Remark 3.5. Suppose for a moment that C is further functorially finite in mod-Λ. In this case,
+another approach one may take to show that H(C) has almost split sequences is to explore when
+H(C) is functorially finite in H. This seems natural in view of the fact that, by [AS81, Theorem
+2.4], any functorially finite extension-closed subcategory of H admits almost split sequences.
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+7
+Restricting to the case where C = mod-Λ, in the last part of this section, we put a third exact
+structure on H that will turn out in Section 7 to be in connection with the stable Auslander-
+Reiten quiver of Auslander algebras; see Remark 7.8. An indecomposable object in H is said
+to be of type (a) (resp. (b), or (c)) provided it is isomorphic to (0 → M) (resp. (M
+1→ M),
+or (M → 0)) for some Λ-module M. Further, an indecomposable object is said to be of type
+(d) if it is isomorphic to (P
+f→ Q) where P, Q are projective Λ-modules. Let X be the smallest
+subcategory of H containing all objects of types (a), (b), (c) and (d). Let also EX be the class of
+all short exact sequences 0 → X → Y → Z → 0 in H such that the induced sequence
+0 → HomH(V, X) → HomH(V, Y) → HomH(V, Z) → 0
+is exact for every V ∈ X. We know from [AS93] and [Bu] that EX induces an exact structure on
+H denoted by HX = (H, EX ). One infers from [AS93, Theorem 1.12] that the exact category HX
+has enough projectives and enough injectives. Denote by P(HX ) (resp. I(HX )) the subcategory
+of projective (resp. injective) objects in HX . In view of [AS93, Corollary 1.6 and Proposition
+1.10], we have P(HX ) = X ∪ proj-H and I(HX ) = τH(X) ∪ inj-H, where proj-H and inj-H
+stand respectively for the subcategories of projective and injective objects in H and τH is the
+Auslander-Reiten translation of H. We exploit [AS93, Proposition 1.9] to examine the almost
+split sequences in HX ; it turns out that an almost split sequence 0 → X → Y → Z → 0 in H is
+an almost split sequence in HX if and only if neither X ∈ I(HX ) nor Z ∈ P(HX ).
+4. Interplay between morphism and functor categories
+Until further notice, we assume throughout the section that C is a dualizing k-variety. In
+this section, we will be involved with a functor going from morphism category to the functor
+category, originally defined and studied in [A71] and then reconsidered in [HM]. This functor
+is our main tool to exchange between these two categories. The construction is based on the
+Yoneda functor.
+Construction 4.1. Let (X1
+f→ X2) be an object of H(C). Define
+(X1
+f→ X2)
+Θ
+�→ Coker(C(−, X1)
+C(−,f)
+−→ C(−, X2)).
+If h =
+� h1
+h2
+�
+: X =
+� X1
+X2
+�
+f →
+�
+X′
+1
+X′
+2
+�
+f ′ = X′ is a morphism in H(Λ), then we let Θ(h) be the
+unique morphism σ that makes the following diagram commute.
+HomC(−, X1)
+HomC(−,f) �
+HomC(−,h1)
+�
+HomC(−, X2)
+�
+HomC(−,h2)
+�
+Θ(X)
+�
+σ
+�
+0
+HomC(−, X′
+1)
+HomC(−,f ′) � HomC(−, X′
+2)
+� Θ(X′)
+� 0.
+It is routine to verify that this rules introduce a well-defined functor Θ : H(C) → mod-C. The
+purpose of this section is to study this functor from the perspective of almost split sequences.
+It turns out that Θ behaves well over such sequences. Firstly, we need to recall some facts on
+objective functors; more details are provided in the Appendix of [RZ]. Let F : C −→ D be an
+additive functor between additive categories. F is called an objective functor if any morphism
+f in C with F(f) = 0 factors through an object K of C with F(K) = 0; such a K is then called
+a kernel object of F. We say that the kernel of a functor F is generated by a subcategory X of
+C if add-X, the additive closure of X in C, is the class of all kernel objects of F.
+
+8
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+Let F : C −→ D be a full, dense and objective functor and let the kernel of F be generated
+by X. Then F induces an equivalence F : C/X −→ D where the additive quotient category C/X
+of C with respect to X has the same objects as C and the morphisms are defined via the rule
+C/X(X, Y ) := C(X, Y )/{φ | φ factors through an object in add-X}
+for any pair of objects X, Y of C.
+Theorem 4.2. The functor Θ : H(C) −→ mod-C is full, dense and objective. Thus, there exists
+an equivalence Θ of categories that makes the following diagram commute.
+H(C)
+Θ �
+π
+�
+mod-C
+H(C)
+V
+Θ
+�
+Here, π is the natural quotient map and V is the full subcategory of H(C) generated by all finite
+direct sums of objects of type (b) or (c), that is to say, objects of the form (M
+1
+−→ M) and
+(M −→ 0), where M runs through the objects of C.
+Proof. Θ is dense; for take F ∈ mod-C with a projective presentation (−, X)
+(−,g)
+→
+(−, Y ) →
+F → 0. It is plain that Θ(X
+g→ Y ) = F. To see the fullness of Θ, take two objects (X
+g→ Y )
+and (X′
+g′
+�� Y ′) of H(C). As the representable functors (−, Y ) and (−, Y ′) are projective, it
+follows that any morphism σ : F = Θ(X
+g→ Y ) → Θ(X′
+g′
+→ Y ′) = F ′ in mod-C might be
+lifted to a map from the augmented projective presentation (−, X)
+(−,g)
+→
+(−, Y ) → F → 0
+to (−, X′)
+(−,g′)
+→
+(−, Y ′) → F ′ → 0.
+Then using Yoneda’s Lemma and the aforementioned
+construction, one obtains a morphism h : (X
+g→ Y ) → (X′ g′
+→ Y ′) in H(C) with σ = Θ(h).
+Now assume Θ(X
+g→ Y ) = 0, for some object (X
+g→ Y ). Then by the construction, we have
+the exact sequence 0 → (−, X)
+(−,g)
+−→ (−, Y ) → 0 in mod-C. One then observes that the identity
+map 1 : Y → Y factors over g via, say, h : Y → X. Therefore, X = Im(h) ⊕ Ker(g). This leads
+to the decomposition (X
+g→ Y ) = (Ker(g) → 0) ⊕ (Im(h)
+g|
+→ Y ) where g| is the restricted map
+which must be an isomorphism since gh = 1Y . This settles that the kernel of Θ is generated by
+V.
+Finally, suppose Θ(h) = 0, for h = (h, h′) : (X
+g→ Y ) → (X′
+g′
+→ Y ′) in H(C).
+Setting
+F = Θ(X
+g→ Y ) and F ′ = Θ(X′ g′
+→ Y ′), this induces a chain map between complexes of functors
+· · ·
+� (−, Z0)
+(−,α0)
+�
+� (−, X)
+(−,h)
+�
+(−,g) � (−, Y )
+(−,h′)
+�
+� F
+0
+�
+� 0
+· · ·
+� (−, Z′
+0)
+� (−, X′)
+(−,g′)� (−, Y ′)
+� F ′
+� 0
+raised by taking projective presentations of F and F ′.
+Evidently, this chain map is null-
+homotopic and, according to [G, Corollary 3.5], factors through a contractible complex.
+As
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+9
+any contractible complex of functors might be imagined to be a direct sum of complexes of the
+form
+· · · → 0 → (−, W)
+1→ (−, W) → 0 → · · ·
+for various objects W of C, this induces a commutative diagram
+· · ·
+� (−, Z0)
+(−,α0)
+�
+�
+� (−, X)
+(−,h)
+�
+�
+(−,g)
+� (−, Y )
+(−,h′)
+�
+�
+� 0
+· · ·
+� (−, Z0 ⊕ X′)
+�
+�
+(−, X′ ⊕ Y ′)
+�
+�
+(−, Y ′)
+�
+� 0
+· · ·
+� (−, Z′
+0)
+� (−, X′)
+(−,g′) � (−, Y ′)
+� 0.
+Therefore, there exists a factorization of h through the object (X′ → 0) ⊕ (Y ′
+1→ Y ′), which is
+a kernel object according to the above paragraph. This shows that Θ is an objective functor.
+Now the existence of the equivalence Θ comes up from observations prior to the theorem.
+□
+Let us record here that applying a dual construction to the opposite category Cop results in
+a contravariant functor
+Θ′ : H(C) → mod-Cop,
+(X
+f→ Y ) �→ Coker(C(Y, −)
+C(f,−)
+−→ C(X, −))
+which is seen to induce a duality Θ′ that makes the diagram
+H(C)
+Θ′ �
+π′
+�
+mod-Cop
+H(C)
+V′
+Θ′
+�
+commute. Here, V′ is the full subcategory of H(C) generated by all finite direct sums of objects
+of type (a) or (b).
+Consider the morphism category H(C), endowed with the exact structure given by Ecw. Ac-
+cording to Proposition 3.4, V (resp. V′) is nothing but the subcategory of injective (resp. pro-
+jective) objects of the exact category Hcw(C). Consequently, the factor categories H(C)/V′ and
+H(C)/V are equivalent respectively to the projectively and injectively stable categories Hcw(C)
+and Hcw(C). Hence the following proposition emerges to settle that Hcw(C) admits almost split
+sequences.
+Proposition 4.3. For a dualizing k-variety C, the following statements hold.
+(1) Hcw(C) admits almost split sequences.
+(2) Hcw(C) has an Auslander-Reiten-Serre duality.
+Proof. The above observations along with Theorem 4.2 provide the equivalences H(C)/V ≃
+mod-C ≃ Hcw(C). Note that by [AR74, Proposition 2.6], mod-C is a dualizing k-variety as well.
+Hence Lemma 2.1 completes the proof.
+□
+We are now in a position to prove the assertion in previous section concerning the existence
+of almost split sequences in H(C) where C is an extension-closed dualizing subvariety of mod-Λ
+for an Artin algebra Λ. The following lemma which is taken from [MO, Proposition 3.1] will be
+
+10
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+fruitful. While the special case C = mod-Λ has been dealt with in [MO], the same proof still
+works for general C as we consider here.
+Lemma 4.4. Assume δ : 0 → A
+f→ B
+g→ C → 0 is an almost split sequence in C. Then
+(1) The almost split sequence in H(C) ending at (0 → C) is of the form
+0
+�( A
+A )1
+� 1
+f
+�
+�( A
+B )f
+� 0
+g
+�
+�( 0
+C )0
+�0.
+(2) The almost split sequence in H(C) ending at (C
+1→ C) is of the form
+0
+�( A
+0 )0
+� f
+0
+�
+�( B
+C )g
+( g
+1)
+�( C
+C )1
+�0.
+Proposition 4.5. Let C be an extension-closed k-dualizing subvariety of mod-Λ.
+Then the
+canonical exact category H(C) admits almost split sequences.
+Proof. Let Z be an indecomposable non-projective object in H(C). Assume first that Z is of either
+types (0 → X) or (X
+1→ X), for an object X ∈ C. Then since C admits almost split sequences
+by [AS81, Theorem 1.1], from Lemma 4.4 we infer that Z is the end term of an almost split
+sequence in H(C). Otherwise, Z is not projective in the exact category Hcw(C) by Proposition
+3.4. Hence, by Proposition 4.3, there exists an almost split sequence ending at Z in the exact
+category Hcw(C). However, following the definitions, it is easy to verify that this is an almost
+split sequence in H(C) as well.
+□
+The following corollary is an immediate consequence of the arguments above.
+Corollary 4.6. Let C be an extension-closed k-dualizing subvariety of mod-Λ and let
+0
+�� X1
+X2
+�
+f
+� φ1
+φ2
+�
+�� Z1
+Z2
+�
+h
+� ψ1
+ψ2
+�
+�� Y1
+Y2
+�
+g
+�0
+be an almost split sequence in H(C). Then the sequences 0 → Xi
+φi
+→ Zi
+Ψi
+→ Yi → 0, i = 1, 2, split
+provided that either of the following situations occur.
+(1) The terminal term (Y1
+g→ Y2) is not of type (a) or (b).
+(2) The initial term (X1
+f→ X2) is not of type (b) or (c).
+We now turn to show that Θ respects almost split sequences; so we return to the setting that
+C is a dualizing k-variety. Let Y = (Y1
+g→ Y2) be an indecomposable non-projective object in
+H(C). Take
+δ :
+� X1
+X2
+�
+f
+� φ1
+φ2
+�
+�� Z1
+Z2
+�
+h
+� ψ1
+ψ2
+�
+�� Y1
+Y2
+�
+g
+to be the almost split sequence in Hcw(C) ending at Y. For simplicity, set Z = (Z1
+h→ Z2),
+X = (X1
+f→ X2), φ = (φ1, φ2) and ψ = (ψ1, ψ2). Note that δ induces degree-wise split sequences
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+11
+and, applying Θ, one gets the commutative diagram with exact rows
+0
+�
+0
+�
+0
+�
+0
+� K1
+�
+� K2
+i
+�
+η
+� K3
+λ
+�
+0
+� (−, X1)
+(−,f)
+�
+� (−, Z1)
+(−,h)
+�
+(−,ψ1)� (−, Y1)
+(−,g)
+�
+� 0
+0
+� (−, X2)
+�
+� (−, Z2)
+�
+(−,ψ2)� (−, Y2)
+�
+� 0
+Θ(X)
+Θ(φ)
+�
+�
+Θ(Z)
+Θ(ψ) �
+�
+Θ(Y)
+�
+�
+0
+0
+0
+0
+in mod-C whose bottom row is indeed the image Θ(δ) of δ under the functor Θ.
+Lemma 4.7. The map η in the above diagram is an epimorphism. As an upshot, Θ(δ) is a
+short exact sequence in mod-C.
+Proof. Let (−, P)
+σ→ K3 → 0 be an epimorphism in mod-C and let d : P → Y1 be a morphism
+in C which represents the composite λσ : (−, P) → K3 → (−, Y1). Since gd = 0, Yoneda’s
+lemma induces a morphism (d, 0) : (P → 0) → Y which is plainly not a retraction. Hence it
+must factor over the right almost split map (ψ1, ψ2) via, say, (a, 0) for some a : P → Z1 in
+C. Consequently, the map (−, a) in mod-C satisfies (−, h)(−, a) = 0. Adding that (−, P) is a
+projective functor, this gives a map γ : (−, P) → K2 in mod-C with (−, a) = iγ. Note that
+ληγ = (−, ψ1)iγ = (−, ψ1)(−, a) = λσ. But λ is a monomorphism; thus ηγ = σ whence the
+surjectivity of η. The second claim comes up immediately from the Snake Lemma.
+□
+The following theorem is another main result of the section.
+Theorem 4.8. Under the above notation, Θ(δ) is an almost split sequence in mod-C.
+Proof. The indecomposability of X and Y imply that Θ(X) and Θ(Y) are indecomposable. By
+previous lemma, Θ(δ) is an exact sequence that, moreover, does not split. Indeed, if it did, then
+(−, f) ⊕ (−, g) would be a minimal projective presentation for Θ(Z) which should comply with
+the one provided by the middle column of the above diagram. In view of the form of kernel
+elements of the functor Θ declared by Theorem 4.2, an application of Yoneda’s lemma gives
+that, for some objects A, B of C, there should exist an isomorphism
+(Z1
+h→ Z2) = (X1
+f→ X2) ⊕ (Y1
+g→ Y2) ⊕ (A
+1→ A) ⊕ (B → 0)
+of objects in H(C). As stated before, we may assume Zi ≃ Xi ⊕ Yi, i = 1, 2 since δ belongs to
+Ecw. However, C being a Krull-Schmidt category implies A = B = 0 which makes δ split. This
+contradiction shows that Θ(δ) does not split.
+
+12
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+Now, as [AR74, Theorem 2.4] guarantees that mod-C is abelian in this case, invoking [AR77,
+Theorem 2.14], it suffices to show that Θ(ψ) is right almost split. For let q : F → Θ(Y) be a
+non-retraction in mod-C and take a projective presentation (−, W1)
+(−,d)
+→ (−, W2) → F → 0 of
+F. Note that by definition, Θ(W1
+d→ W2) = F. The morphism q lifts to a morphism between the
+projective presentations (−, W1)
+(−,d)
+→ (−, W2) → F → 0 and (−, Y1)
+(−,g)
+→ (−, Y2) → Θ(Y ) → 0.
+The lifted morphism induces, again by the Yoneda lemma, a map
+( σ1
+σ2 ) :
+� W1
+W2
+�
+d →
+� Y1
+Y2
+�
+g
+in H(C) such that Θ(σ1, σ2) = q.
+Then (σ1, σ2) is not a retraction since otherwise q would
+be so.
+Now, δ being an almost split sequence in Hcw(C), (σ1, σ2) factors over ψ via some
+(η1, η2) : (W1
+d→ W2) → (Z1
+h→ Z2). Then applying Θ, we see that the morphism q factors over
+Θ(ψ) via Θ(η1, η2).
+□
+5. The case of wide subcategories
+Our objective in this section is to study the morphism categories raised by functorially finite
+wide subcategories of mod-Λ.
+Some results from previous section will come to play.
+After-
+wards, we shall switch to functor categories and obtain some results in this direction that extend
+others from the module category. So let firstly X be a functorially finite idempotent-complete
+subcategory of mod-Λ. By [AS81, Theorem 2.3], X itself is a dualizing variety.
+Following Proposition 4.3, for a dualizing k-variety C, there is an equivalence τH(C) : Hcw(C) →
+Hcw(C) that, based on what we said in previous section, might be considered as an equivalence
+from H(C)/V′ to H(C)/V. Pictorially, there exists a composition of equivalences
+H(C)/V
+τ −1
+H(C) � H(C)/V′
+Θ′
+�
+mod-C
+(Θ)−1
+�
+� mod-Cop
+D
+� mod-C
+denoted throughout by ∆C, or simply by ∆. Applied to the functorially finite subcategory X
+of mod-Λ, this yields an equivalence ∆X : mod-X → mod-X which is restricted to the category
+proj-X of projective functors. Since X is idempotent-complete, the Yoneda functor induces an
+equivalence proj-X ≃ X. Summing up, one obtains an equivalence σX : X → X which agrees
+with the restricted equivalence ∆X via the latter identification. We notice that, going through
+the definitions, one figures out that for an object X of X, there are A, B ∈ X and an exact
+sequence
+(B, −)
+(f,−)
+→ (A, −) → D(−, σX (X)) → 0
+in mod-X such that τ −1
+H(X)(0 → X) = (A
+f→ B).
+Definition 5.1. A minimal projective presentation of an object C ∈ X with respect to X is
+an exact sequence P1
+f→ P0
+h→ C with P1, P0 ∈ P(X), the class of projective objects of X,
+and is computed by taking minimal right P(X)-approximations consecutively. Minimal injective
+presentations with respect to X are defined dually via I(X), the class of injective objects of X.
+Recall that a subcategory M of mod-Λ is said to be closed under kernels (resp. cokernels,
+images) if for every morphism X
+f→ Y in M also Ker(f) (resp. Coker(f), Im(f)) belongs to
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+13
+M. Further, M is called a wide subcategory of mod-Λ if it is closed under extensions, kernels
+and cokernels. It is clear that a wide subcategory is closed under images and is automatically
+idempotent-complete.
+The following couple of propositions are quite useful.
+Proposition 5.2. Assume X is a functorially finite wide subcategory of mod-Λ and δ : 0 →
+A
+f→ B
+g→ C → 0 is an almost split sequence in X. Let also A
+d→ I0
+q→ I1 be a minimal injective
+presentation with respect to X, where b : Coker(d) → I1 is a minimal left I(X)-approximation,
+a : I0 → Coker(d) is the canonical quotient map and q = ba. Then the exact sequence
+0
+�� I0
+I1
+�
+q
+( u
+1 )
+�� W
+I1
+�
+br
+( v
+0 )
+�( C
+0 )0
+�0
+in H(X) raised by forming the push out diagram
+A
+d
+�
+f
+� B
+h
+�
+g
+� C
+I0
+a
+�
+u
+� W
+r
+�
+v
+� C
+Coker(d)
+Coker(d)
+in the exact category X, is almost split.
+Proof. Using that A
+d→ I0
+s→ I1 is a minimal injective presentation and A is indecomposable,
+we deduce that (I0
+s→ I1) is indecomposable. Hence it suffices to show that any non-retraction
+(φ, 0) : (M
+p→ N) → (C → 0) in X factors over (v, 0). If φ is a non-retraction, then, since δ is
+an almost split sequence, it factors in X over g via, say, w : M → B. Then it is easy that the
+morphism (hw, 0) : (M
+p→ N) → (W
+br
+→ I1) factors the morphism (φ, 0) over (v, 0). So now take
+φ to be a retraction. Without loss of generality, we may assume M = C and φ = 1C. Two cases
+might be distinguished:
+Case 1: p is a monomorphism. Since v is a retraction in X, there exists s : C → W such that
+vs = 1. As p : C → N is a monomorphism in X, there exists an extension of brs : C → I1 to a
+map z : N → I1; that is to say, zp = brs. It follows then that (s, z) : (C
+p→ N) → (W
+br
+→ I1)
+produces the desired factorization.
+Case 2: Assume Ker(p) ̸= 0. Note that since X is a wide subcategory, Ker(p) lies in X. The
+fact that (φ, 0) is a non-retraction implies that Ker(p) is a proper submodule of C and thus the
+canonical inclusion i : Ker(p) → C is a non-retraction in X. According to the hypothesis, we
+infer the existence of a map y : Ker(p) → B such that gy = i. Note further that, since v is
+retraction, one may write W = Im(s)⊕Ker(v) and, consequently, present h as h = [l1, l2]t, where
+l1 : B → Im(s) and l2 : B → Ker(v). Using the injectivity of Ker(v) in X yields an extension of
+l2y : Ker(p) → Ker(v) to C; that is, there exists y′ : C → Ker(v) such that y′i = l2y. Putting
+together, we get a diagram
+
+14
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+Ker(p)
+y
+�
+i
+� C
+[s y′]t
+�
+� Im(p)
+� 0
+B
+h
+� W
+r � Coker(h)
+� 0
+with commutative left part. This induces a map y′′ : Im(p) → Cok(h) completing the diagram.
+Again, as X is wide, the monomorphism Im(p)
+i′
+→ N lies inside X and, hence, the injectivity
+of I1 in X gives a map z′ : N → I1 with z′i′ = by′′. Finally, one verifies that the morphism
+([s y′]t, z′) : (C
+p→ N) → (W
+br
+→ I1) gives the required factorization.
+□
+As a dual statement, we record the following proposition.
+Proposition 5.3. Assume X is a functorially finite wide subcategory of mod-Λ and δ : 0 → A
+f→
+B
+g→ C → 0 is an almost split sequence in X. Let also P1
+ℓ→ P0
+h→ C be a minimal projective
+presentation with respect to X, where k : P1 → Ker(h) is a minimal right P(X)-approximation,
+i : Ker(h) → P0 is the canonical inclusion and ℓ = ik. Then the exact sequence
+0
+�( 0
+A )
+( 0
+u)
+�� P1
+Z
+�
+wk
+( 1
+v )
+�� P1
+P0
+�
+ℓ
+�0
+in H(X) raised by forming the pull back diagram
+Ker(h)
+w
+�
+Ker(h)
+i
+�
+A
+u
+� Z
+r
+�
+v
+� P0
+h
+�
+A
+f
+� B
+g
+� C
+in the exact category X, is almost split.
+Any functorially finite wide subcategory X of mod-Λ admits almost split sequences by [AS81,
+Theorem 2.4]. Hence, following Lemma 2.1, we let τX denote the Auslander-Reiten translation
+over X.
+Corollary 5.4. Let X be a functorially finite wide subcategory of mod-Λ and consider the auto-
+equivalence σX : X → X introduced earlier. Assume that X ∈ X is an indecomposable module
+not belonging to P(X), and that P
+f→ Q → X is a minimal projective presentation with respect
+to X. Then there is an exact sequence
+0 → (−, σX τX (X)) → D(P, −) → D(Q, −) → D(X, −)
+in mod-X
+Proof. Proposition 5.3 yields that the inverse Auslander-Reiten translation τ −1
+H(X)(0 → τX (X))
+of (0 → τX (X)) in H(X) coincides with (P
+f→ Q). Taking into account our previous observations
+on the functor σX gives the result.
+□
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+15
+Remark 5.5. For every non-projective indecomposable Λ-module M, we know that there exists
+an exact sequence
+0 → τ(M) → ν(P) → ν(Q) → ν(M) → 0
+where P → Q → M → 0 is the minimal projective presentation of M. In some sense, the
+sequence presented by Corollary 5.4 goes parallel to, and generalizes this observation. This is
+more justified as we show in the sequel that for the case X = mod-Λ, σX is just the identity
+functor. However, it would be interesting to explore σX further by considering other functorially
+finite wide subcategories X.
+Denote by σ := σX : mod-Λ → mod-Λ the auto-equivalence obtained above in the case
+where X = mod-Λ. We refer e.g. to [HE] for further explanation on how the Auslander-Reiten
+translation τH and its inverse τ −1
+H work in this case and suffice to recall that the standard duality
+functor DH might be computed in a local manner in terms of the standard duality D of Λ. That
+is to say, DH(X
+f→ Y ) = (D(Y )
+D(f)
+→ D(X)).
+Note that since σ is an equivalence, it clearly restricts to an equivalence σ′ : inj-Λ → inj-Λ on
+the subcategory of injective modules. In the sequel, it will be shown that σ′, and consequently
+σ, are nothing but the identity functor on the corresponding categories.
+Lemma 5.6. The restricted equivalence σ′ is isomorphic to the identity functor on inj-Λ.
+Proof. Let I be an injective Λ-module. There exists a minimal injective resolution in H
+0 → ( 0
+I )0 → ( I
+I )1 → ( I
+0 )0 → 0
+of the object (0 → I). Applying the duality DH leads to the projective presentation in Hop
+0 → ( 0
+DI )0 → ( DI
+DI )1 → ( DI
+0 )0 → 0
+of the object DH(0 → I). Then, we compute the transpose and deduce that τ −1
+H (0 → I) ≃
+(ν−1(I) → 0). As we pointed out earlier in this section, this results in an equivalence (ν−1(I), −) ≃
+D(−, σ(I)) in mod-(mod-Λ)op. Hence, evaluating on the regular module Λ, yields a natural iso-
+morphism σ(I) ≃ νν−1(I) ≃ I.
+□
+Theorem 5.7. The equivalence σ is isomorphic to the identity functor on mod-Λ.
+Proof. According to Lemma 5.6, the restricted equivalence σ′ is naturally isomorphic to the
+identity functor on the subcategory of injective modules. Using injective resolutions, it is then
+straightforward to see that the same holds for σ itself.
+□
+In the rest of this section, we will provide some applications of the aforementioned theorem.
+Corollary 5.8. Let F be a functor in mod-(mod-Λ) with a minimal projective presentation
+(−, X)
+(−,f)
+→ (−, Y ) → F → 0. Then there is an exact sequence
+(Y ′, −)
+(g,−)
+→ (X′, −) → DF → 0
+in mod-(mod-Λ)op where (X′
+g→ Y ′) is the inverse Auslander-Reiten translation of (X
+f→ Y ) in
+H.
+Proof. Again we specify our constructions to the dualizing variety C = mod-Λ. By virtue of
+Theorem 5.7, the functor ∆ := ∆mod-Λ acts identically on projective functors. Using projective
+presentations, it follows that ∆ is isomorphic to the identity functor on the whole mod-(mod-Λ).
+Definition of ∆ then implies that the duality functor D : mod-(mod-Λ) → mod-(mod-Λ)op is
+
+16
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+isomorphic to Θ′ ◦τ −1
+H ◦(Θ)−1. This proves the claim by following the definitions of the functors
+involved.
+□
+Corollary 5.9. Let M be an indecomposable Λ-module with a minimal projective presentation
+P
+f→ Q → M → 0. Then there is an exact sequence
+0 → (−, τ(M)) → D(P, −) → D(Q, −) → D(M, −) → 0
+in mod-(mod-Λ).
+Proof. If M is projective, then such a sequence exists trivially. Otherwise, applying Corollary
+5.4 for X = mod-Λ, there exists an exact sequence
+0 → (−, σX τX (M)) → D(P, −) → D(Q, −) → D(M, −) → 0
+in mod-(mod-Λ). Now Theorem 5.7 settles the statement.
+□
+Let us exploit Corollary 5.9 to observe a connection between the inverse Auslander-Reiten
+translation of an indecomposable non-projective Λ-module M with the second syzygies of injec-
+tive functors. For, replace M in Corollary 5.9 by τ −1M to get the exact sequence
+0 → (−, M) → D(P, −) → D(Q, −) → D(τ −1(M), −) → 0
+in mod-(mod-Λ)op in which P → Q → τ −1(M) → 0 is a minimal projective presentation.
+Applying the duality D : mod-(mod-Λ)op → mod-(mod-Λ) gives the exact sequence
+0 → (τ −1(M), −) → (Q, −) → (P, −) → D(−, M) → 0.
+This shows that the functor (τ −1(M), −) might be interpreted as a second syzygy of the injective
+functor D(−, M).
+6. simple modules over (stable) Auslander algebra
+Assume Λ is of finite representation type and let M be a basic representation generator of
+mod-Λ; that is, M is the direct sum of all pairwise non-isomorphic indecomposable finitely
+generated Λ-modules.
+The endomorphism algebra A(Λ) = EndΛ(M), simply denoted by A
+throughout the section, is called the Auslander algebra of Λ. Moreover, the stable Auslander
+algebra of Λ is by definition A = EndΛ(M)/P, where P is the ideal in EndΛ(M) consisting of
+those endomorphisms factoring through a projective module. In this case, we can identify mod-A
+with mod-(mod-Λ) via the equivalence induced by the evaluation functor eM : mod-(mod-Λ) →
+mod-A, F �→ F(M). It is also easy to see that eM induces an equivalence between mod-(mod-Λ)
+and mod-A.
+It is known [A76] that indecomposable modules in mod-Λ correspond bijectively to sim-
+ple functors in mod-(mod-Λ) by sending an indecomposable module M to the simple functor
+SM := (−, M)/rad(−, M). Further, for any indecomposable non-projective module M, there is
+a minimal projective resolution
+0 → (−, N)
+(−,f)
+→ (−, K)
+(−,g)
+→ (−, M) → SM → 0
+of SM such that 0 → N
+f→ K
+g→ M → 0 is an almost split sequence in mod-Λ ([A76, §2]).
+Combined to the above observations on the Auslander algebra A, one may identify simple A-
+modules (resp. simple A-modules) and indecomposable (resp. indecomposable non-projective)
+Λ-modules.
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+17
+Specializing [H, Construction 3.1] to the module category mod-Λ gives a functor Ψ : S →
+mod-(mod-Λ), S being the monomorphism category of Λ. This is defined by sending (X
+f→ Y )
+in S to the functor F ∈ mod-(mod-Λ) lying in the exact sequence
+0 → (−, X)
+(−,f)
+→ (−, Y ) → (−, Coker(f)) → F → 0
+in mod-(mod-Λ).
+As the following result says, Ψ behaves well with respect to almost split
+sequences.
+Lemma 6.1. ([H, Proposition 5.7]) Let 0 → U → V → W → 0 be an almost split sequence
+in S. Assume W is neither of types (a) or (b), nor of the form (Ω(X) → P), where X is a
+non-projective indecomposable Λ-module with projective cover P. Then
+0 → Ψ(U) → Ψ(V) → Ψ(W) → 0
+is an almost split sequence in mod-(mod-Λ).
+The following theorem is one of the main results in this section.
+Theorem 6.2. Assume Λ is of finite representation type and A is its stable Auslander algebra.
+Let S be a simple non-projective A-module. Then, exactly one of the followings hold:
+(1) the Auslander-Reiten translate τA(S) is projective.
+(2) there exists a simple A-module S′ such that τA(S) ≃ Ω−1
+A (S′). In this case, Ext2
+A(S, S′) ≃
+DHomA(S, S).
+Proof. According to aforementioned remarks, the simple non-projective module S corresponds
+to a simple functor (−, C)/rad(−, C) lying in the exact sequence
+0 → (−, A)
+(−,f)
+→ (−, B)
+(−,g)
+→ (−, C) → (−, C)/rad(−, C) → 0
+in mod-(mod-Λ) in such a way that λ : 0 → A
+f→ B
+g→ C → 0 is an almost split sequence
+in mod-Λ. Note that the middle term B may not be projective since otherwise there exists an
+isomorphism (−, C)/rad(−, C) ≃ (−, C) which is against non-projectivity of S.
+We distinguish two cases: Assume first that A is projective. So by Proposition 3.3 of [HE],
+there exists an almost split sequence
+0
+�� rad(A)
+A
+�
+i
+�( A
+A )1 ⊕
+� rad(A)
+B
+�
+fi
+�( A
+B )f
+�0
+in S(Λ). Hence, in view of Lemma 6.1, we get the almost split sequence
+0 → Ψ
+� rad(A)
+A
+�
+i → Ψ
+� rad(A)
+B
+�
+fi → Ψ( A
+B )f → 0
+in mod-(mod-Λ).
+Since A is projective, the definition of Ψ shows that Ψ(rad(A)
+i→ A) ≃
+(−, A/rad(A)). Likewise, as λ does not split, we have Ψ(A
+f→ B) ≃ (−, C)/rad(−, C). Hence
+τA((−, C)/rad(−, C)) ≃ (−, A/rad(A)) that proves the claim in this case.
+Assume next that A is not projective. Then there exists an almost split sequence
+ϵ : 0 → A′ → B′ → A → 0
+in mod-Λ. Applying [H, Lemma 6.3] on λ and ϵ, one infers the almost split sequences
+0
+�( A
+A )1
+�( A
+B )f
+�( 0
+C )0
+�0 and
+
+18
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+0
+�� A′
+I
+�
+e
+�� B′
+I⊕A
+�
+h
+�( A
+A )1
+�0
+in S where the second one is obtained from the push-out diagram
+A′
+e
+�
+� B′
+h
+�
+� A
+(†)
+I
+e
+�
+� I ⊕ A
+d
+�
+� A
+Ω−1
+Λ (A)
+Ω−1
+Λ (A)
+in which e : A′ → I is the injective envelope. From [HZ, Lemma 3.3], we can write (B′
+h→
+I ⊕ A) ≃ X ⊕ (J
+1→ J), where X is an indecomposable non-projective object and J is either zero
+or isomorphic to I. It follows then that τS(A
+f→ B) ≃ X. Accordingly, by taking into account
+that (−, C)/rad(−, C) ≃ Ψ(A
+f→ B) by the exact sequence mentioned at the beginning of the
+proof, another application of Lemma 6.1 shows that
+(6.1)
+F := τmod-(mod-Λ)((−, C)/rad(−, C)) ≃ Ψ(X).
+However, the definition of Ψ yields F = Ψ(B′
+h→ I ⊕ A). Therefore, abusing the notation, we
+may write τA(S) ≃ F.
+Regarding the definition of Ψ, the middle column of (†) gives the long exact sequence
+0
+� (−, B′)
+� (−, I ⊕ A)
+� (−, Ω−1
+Λ (A))
+�
+�
+Ext1
+Λ(−, B′)
+� Ext1
+Λ(−, I ⊕ A)
+F
+�
+in mod-(mod-Λ) that implies F = Ker(Ext1
+Λ(−, B′) → Ext1
+Λ(−, A)) because I is injective. On
+the other hand, since ϵ is an almost split sequence, our previous considerations show that there
+exists an exact sequence
+0
+� (−, A′)
+� (−, B′)
+� (−, A)
+�
+�
+Ext1
+Λ(−, A′)
+�
+�
+Ext1
+Λ(−, B′)
+� Ext1
+Λ(−, A)
+(−, A)/rad(−, A)
+�
+F
+�
+of functors. Invoking [AR74, Proposition 7.4], we see that Ext1
+Λ(−, A′) is an injective functor in
+mod-(mod-Λ) and so the induced short exact sequence 0 → (−, A)/rad(−, A) → Ext1
+Λ(−, A′) →
+F → 0 gives F = Ω−1
+A ((−, A)/rad(−, A)). Now it suffices to set S′ = (−, A)/rad(−, A). Notice
+that the last assertion in the theorem is an upshot of the Auslander-Reiten formula.
+□
+Based on previous theorem, in the following result we establish a bijection between certain
+simple modules over A and Λ. This provides an interesting application concerning the stable
+equivalences of Artin algebras.
+Corollary 6.3. Let Λ be of finite representation type and A be its stable Auslander algebra.
+There exists a bijection between
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+19
+(1) the set of isomorphism classes of non-projective simple modules S ∈ mod-A whose
+Auslander-Reiten translate τA(S) is projective; and
+(2) the set of isomorphism classes of indecomposable non-injective projective modules P ∈
+mod-Λ such that the middle term of the almost split sequence starting from P is not
+projective; and
+(3) the set of isomorphism classes of simple modules S ∈ mod-Λ whose projective cover P(S)
+is non-injective, and the middle term of the almost split sequence starting from P(S) is
+not projective.
+Proof. The bijection between (2) and (3) might be shown by restricting the well-known bijection
+between simple and indecomposable projective modules. The map from (2) to (1) is given by
+sending P to (−, τ −1(P))/rad(−, τ −1(P)), which is well-defined due to the argument given in
+Theorem 6.2. Let now S be a simple non-projective module in mod-A with τA(S) projective.
+We already know that there is an indecomposable non-projective Λ-module C such that S ≃
+(−, C)/rad(−, C). We claim that τ(C) is projective; otherwise, as in Theorem 6.2, there exists
+a simple A-module S′ such that τA(S) ≃ Ω−1
+A (S′). Hence the short exact sequence 0 → S′ →
+I → Ω−1
+A (S′) → 0, in which I′ is the injective envelop of S′, splits. This means that Ω−1
+A (S′) = 0
+and so τA(S) = 0 which is against non-projectivity of S. Thus τ(C) is projective and setting
+P := τ(C) completes the proof.
+□
+Recall that two Artin algebras Λ and Λ′ are said to be stably equivalent if there is an equiv-
+alence of categories mod-Λ ≃ mod-Λ′.
+Denote by n(Λ) the number of iso classes of simple
+Λ-modules satisfying the third condition of the above corollary. As a byproduct, we show that
+n(Λ) is an invariant of the stable equivalences.
+Proposition 6.4. Let Λ and Λ′ be of finite representation type and stably equivalent. Then
+n(Λ) = n(Λ′).
+Proof. Since Λ and Λ′ are stably equivalent, it follows that the corresponding stable Auslander
+algebras A and A′ are Morita equivalent. By Corollary 6.3, we see that simple modules in mod-Λ
+(resp. mod-Λ′) that satisfy condition (3) correspond bijectively to non-projective simple modules
+over the stable Auslander algebra A (resp. A′) with projective Auslander-Reiten translates. We
+are done since the modules of latter type are preserved under Morita equivalences.
+□
+The following lemma is taken from [HE, Proposition 3.2].
+Lemma 6.5. Let δ : 0 → A
+f→ B
+g→ C → 0 and δ′ : 0 → A′ f ′
+→ B′ g′
+→ A → 0 be almost split
+sequences in mod-Λ. Then
+0
+�� B′
+A
+�
+g′
+�
+�
+� g′
+1
+�
+� 1
+f
+�
+�
+�
+�( A
+A )1 ⊕
+� B′
+B
+�
+fg′
+� [ −1 g′ ]
+[ −f 1 ]
+�
+�( A
+B )f
+�0,
+is an almost split sequence in H. Further,
+� B′
+B
+�
+fg′ is an indecomposable object.
+The following theorem should be served as the second main result of this section.
+Theorem 6.6. Assume Λ is a self-injective algebra of finite representation type and let A be
+its Auslander algebra. Let also S be a simple A-module of projective dimension two. Then there
+exists a simple A-module S′ of projective dimension two such that ΩA(S′) ≃ τ −1
+A (S). In this
+case, Ext2
+A(S′, S) ≃ DHomA(S, S).
+
+20
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+Proof. We may identify the simple module S by the simple functor (−, A)/rad(−, A) lying in
+the exact sequence
+0 → (−, A′)
+(−,f ′)
+→
+(−, B′)
+(−,g′)
+→
+(−, A) → (−, A)/rad(−, A) → 0
+(†)
+in mod-(mod-Λ) in such a way that δ : 0 → A′ f ′
+→ B′ g′
+→ A → 0 is an almost split sequence in
+mod-Λ. Let also δ′ : 0 → A
+f→ B
+g→ C → 0 be an almost split sequence in mod-Λ. Then by
+Lemma 6.5 there exists an almost split sequence
+0
+�� B′
+A
+�
+g′
+�
+�
+� g′
+1
+�
+� 1
+f
+�
+�
+�
+�( A
+A )1 ⊕
+� B′
+B
+�
+fg′
+� [ −1 g′ ]
+[ −f 1 ]
+�
+�( A
+B )f
+�0
+in H. Thanks to Theorem 4.8, this induces the almost split sequence
+0 → Θ(B′ g′
+→ A) → Θ(B′ fg′
+→ B) → Θ(A
+f→ B) → 0
+(††)
+in mod-(mod-Λ). Note that (†) implies Θ(B′
+g′
+→ A) = (−, A)/rad(−, A) = S. Hence by (††),
+τ −1
+A (S) = Θ(A
+f→ B). Set now W = (A
+f→ B). Then, by definitions, there exists an exact
+sequence
+(−, A)
+� (−, B)
+�
+�
+(−, C)
+�
+�
+Ext1
+Λ(−, A)
+� Ext1
+Λ(−, B).
+Θ(W)
+�
+(−, C)/rad(−, C)
+�
+Set S′ be the simple functor (−, C)/rad(−, C). Then the short exact sequence 0 → Θ(W) →
+(−, C) → S′ → 0 proves the claim.
+□
+7. Auslander-Reiten components of Auslander algebras
+Throughout the section, we assume that Λ is a non-semisimple self-injective algebra of finite
+representation type and A denotes its Auslander algebra. In the whole section, we use the iden-
+tification mod-A ≃ mod-(mod-Λ) described earlier. Once more, in this section, the quadruple
+family of objects in H of types (a), (b), (c), and (d) become important. We aim to identify cer-
+tain components of the (stable) Auslander-Reiten quiver of A. To this end, we need firstly study
+particular τH-periodic objects in H and their periodicity.
+7.1. τH-periodic objects. As we observed in Theorem 4.8, the functor Θ : H → mod-A behaves
+well with respect to almost split sequences in the sense that if there exists an almost split sequence
+0 → X → Y → Z → 0 in H where Z is not of type (b) or (c), then 0 → Θ(X) → Θ(Y) → Θ(Z) → 0
+is also an almost split sequence in mod-A. Also we have seen in Theorem 4.2 that one is given an
+equivalence H/V ≃ mod-A where V is generated by the objects of type (b) or (c). Therefore, the
+Auslander-Reiten quiver ΓA of the Auslander algebra A might be computed via the Auslander-
+Reiten quiver ΓH of H by removing vertices corresponding to iso-classes of indecomposable
+objects of either types (b) or (c).
+The following construction is vital for the rest of this section. It is mainly based on an analysis
+of various almost split sequences already obtained in [HE]. For the sake of brevity, we prefer not
+to rewrite most of them here and suffice to give the precise reference number therein.
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+21
+Construction 7.1. Let C be an indecomposable non-projective Λ-module. There exist almost
+split sequences ϵ1 : 0 → τ(C)
+f→ B
+g→ C → 0 and ϵ2 : 0 → τ 2(C)
+f ′
+→ B′ g′
+→ τ(C) → 0 in mod-Λ.
+Applying Lemmas 6.5 and 4.4 we deduce the almost split sequences
+0
+�
+�
+B′
+τ(C)
+�
+g′
+�
+�
+� g′
+1
+�
+� 1
+f
+�
+�
+�
+�
+�
+τ(C)
+τ(C)
+�
+1 ⊕
+� B′
+B
+�
+fg′
+� [ −1 g′ ]
+[ −f 1 ]
+�
+�� τ(C)
+B
+�
+f
+�0,
+0
+�
+�
+τ(C)
+τ(C)
+�
+1
+� 1
+f
+�
+�� τ(C)
+B
+�
+f
+� 0
+g
+�
+�( 0
+C )0
+�0,
+and
+0
+�� τ(C)
+0
+�
+0
+� f
+0
+�
+�( B
+C )g
+( g
+1)
+�( C
+C )1
+�0
+in H. Evidently, the indecomposable object (B′ fg′
+→ B) is not projective; so let X := τH(B′ fg′
+→ B).
+Also, as (B′ g′
+→ τ(C)) is not projective, we let Y := τH(B′ g′
+→ τ(C)) and note that X and Y are
+not projective. In view of [HE, Propositions 2.2, 4.1], there exists an almost split sequence
+0
+�
+�
+ν(P )
+ν(Q)
+�
+ν(h)
+�Y ⊕ ( I
+0 )0
+�
+�
+τ 2(C)
+0
+�
+0
+�0
+in H where P
+h→ Q → τ 2(C) → 0 is the minimal projective presentation, and I is an injective
+module. On the other hand, by [HE, Propositions 2.4, 4.2], we have the almost split sequence
+0
+�
+�
+0
+τντ 2(C)
+�
+0
+�τH(Y) ⊕ ( 0
+P )0
+�
+�
+ν(P )
+ν(Q)
+�
+ν(h)
+�0
+in H where P is projective. Putting all together, one obtains the mesh
+τ(X)
+�
+X
+�
+�
+� B′
+B
+�
+fg′
+�
+�
+τ(Y)
+�
+�
+Y
+�
+�
+�
+�
+B′
+τ(C)
+�
+g′
+�
+�
+�
+� τ(C)
+B
+�
+f
+�
+�
+�
+0
+ντ 3(C)
+�
+0
+�
+�
+ν(P )
+ν(Q)
+�
+ν(h)
+�
+�
+�
+τ 2(C)
+0
+�
+0
+�
+�
+�
+τ(C)
+τ(C)
+�
+1
+�
+�
+( 0
+C )0
+�
+in the Auslander-Reiten quiver ΓA of A in which the vertices (I → 0) and (0 → P) have been
+ignored.
+Let us recall from [HE, Remark 5.7] that A = ντ 3 defines an auto-equivalence on the stable
+category mod-Λ. As is expected, the A -orbit of an indecomposable non-projective Λ-module M
+consists of the modules A m(M) where m ranges over the integer numbers.
+Proposition 7.2. Suppose every indecomposable non-projective Λ-module possesses a finite A -
+orbit. Then any indecomposable non-projective object X in H of either types (a), (b), (c) or (d)
+is of τH-periodicity a multiple of 4.
+
+22
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+Proof. According to our previous observations, all mentioned objects lie in the τH-orbit of some
+indecomposable object of type (a). Hence it suffices to prove the statement only for X = (0 → N)
+with N an indecomposable non-projective module. Justified by the hypothesis, choose a least
+integer n with A n(N) = N. Considering the particular mesh in ΓA as illustrated in Construction
+7.1, we get τ 4
+H(0 → N) ≃ (0 → A (N)) and thus τ 4n
+H (0 → N) ≃ (0 → A n(N)) = (0 → N).
+□
+Based on previous proposition, we are now able to prove the following theorem which will
+prove useful later on.
+Theorem 7.3. Assume every indecomposable non-projective Λ-module possesses a finite A -
+orbit. Then every simple A-module of projective dimension 2 is τA-periodic of periodicity divided
+by 4.
+Proof. Recall that such simple A-modules might be identified with simple functors SM =
+(−, M)/rad(−, M) where M is an indecomposable non-projective Λ-module lying in an almost
+split sequence 0 → τ(M)
+g→ N
+f→ M → in mod-Λ. By Proposition 7.2, (0 → τ −1(M)) is of
+τH-periodicity 4n for a suitable integer n. Since (0 → τ −1(M)) and (M
+1→ M) lie in the same
+τH-orbit, if it follows that (M
+1→ M) is also of the same periodicity 4n. Thus the irreducible
+morphism (N
+f→ M) → (M
+1→ M) in ΓH remains fixed after 4n applications of τH and ac-
+cordingly, (N
+f→ M) should be of τH-periodicity 4n. Consequently, according to Theorem 4.8,
+τ 4n
+A (SM) = τ 4n
+A Θ(N
+f→ M) = Θτ 4n
+H (N
+f→ M) = Θ(N
+f→ M) = SM.
+□
+7.2. Modules M with τ(M) = Ω(M).
+Definition 7.4. Let M be an indecomposable non-projective module.
+We say M has the
+property (∗) if 0 → Ω(M) → P(M) → M → 0 is an almost split sequence in mod-Λ where
+P(M) is the projective cover of M.
+Modules satisfying this property have already been classified in [ARS, Theorem V.3.3]: these
+are exactly non-injective simple Λ-modules M that are not a composition factor of rad(I)/soc(I)
+for every injective Λ-module I. This clearly yields that such modules are necessarily A-periodic.
+Note also that in the situation of the definition, τ(M) = Ω(M). The goal in this subsection is to
+see that existence of modules with this property may heavily affect the shape of the AR-quiver
+of A and in particular cases may even make it into an algebra of finite representation type. As
+a first pace to study modules with property (∗), the following lemma shows that this property
+carries over from a module to its (co)syzygies.
+Lemma 7.5. Let M be an indecomposable non-projective Λ-module. If M has the property (∗),
+then so do all its syzygies (resp. cosyzygies). In particular, the short exact sequences
+0 → Ωi+1(M) → P i → Ωi(M) → 0 for i ≥ 0, and
+0 → Ωi(M) → Ii → Ωi−1(M) → 0 for i ≤ 0
+in mod-Λ induced by the minimal projective (resp. injective) resolution of M are almost split.
+Proof. We prove the lemma for integers i ≥ 0 by using an inductive argument whose basis
+i = 0 is satisfied by the assumption; so we put i > 0.
+Consider the almost split sequence
+0 → τΩi(M) → B → Ωi(M) → 0 in mod-Λ.
+We claim that B is projective.
+Assume to
+the contrary that B has a non-projective indecomposable direct summand C. The induction
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+23
+hypothesis then implies that τ −1(C) is a non-projective direct summand of P i−1, which is
+absurd. Now use the fact that the morphisms involved in an almost split sequence are minimal
+to deduce that τΩi(M) = Ωi+1(M).
+□
+The following lemma shows a property of the modules M for which (∗) is satisfied; this will
+be used later on in this section.
+Lemma 7.6. Under the hypothesis of Lemma 7.5, one has ν(P i+1) ≃ P i for i ≥ 0 and
+ν−1(Ii−1) ≃ Ii for i ≤ 0.
+Proof. We prove the first assertion. The minimal projective presentation P 1 → P 0 → M → 0
+induces the short exact sequence 0 → τ(M) → ν(P 1) → ν(P 0) → ν(M) → 0 in mod-Λ.
+Note that, as ν is an auto-equivalence of mod-Λ, the map τ(M) → ν(P 1) is minimal and
+thus defines the injective envelope of τ(M) as ν(P 1) is injective. However, by definition, the
+monomorphism τ(M) → P 0 obtained by composing the isomorphism τ(M) ≃ Ω(M) and the
+inclusion Ω(M) → P 0 is also minimal with P 0 injective.
+Therefore ν(P 1) ≃ P 0 since the
+injective envelope is unique up to isomorphism.
+Now we deduce the result by applying an
+inductive argument in conjunction with Lemma 7.5.
+□
+The following theorem is the promised one.
+Theorem 7.7. Assume there exists an indecomposable non-projective Λ-module M with the
+property (∗). Then the Auslander-Reiten quiver ΓA of A is a finite oriented cycle. In particular,
+the Auslander algebra A is of finite representation type.
+Proof. The minimal projective presentation
+· · · → P n wn
+→ P n−1 → · · · P 1 w1
+→ P 0 → M → 0
+of M induces, according to Lemma 7.5, the almost split sequences
+ϵi : 0 → Ωi+1(M)
+vi
+→ P i ui
+→ Ωi(M) → 0
+in mod-Λ. Applying Lemma 6.5 on ϵ0 and ϵ1 gives the almost split sequence
+0
+�
+�
+P 1
+Ω(M)
+�
+u1
+�
+�
+Ω(M)
+Ω(M)
+�
+1 ⊕
+�
+P 1
+P 0
+�
+w1
+�
+�
+Ω(M)
+P 0
+�
+v0
+�0.
+Note that, by Lemma 4.4, τH(( 0
+M )) =
+�
+Ω(M)
+Ω(M)
+�
+1 and τH
+��
+Ω(M)
+Ω(M)
+�
+1
+�
+=
+�
+Ω2(M)
+0
+�
+. Moreover, in
+light of [HE, Proposition 2.4], we get τH(P 1 w1
+→ P 0) = (0 → Ω(M)) and so there exists an almost
+split sequence
+0
+��
+0
+Ω(M)
+�
+0
+�
+�
+P 1
+Ω(M)
+�
+u1
+⊕
+� 0
+P 0
+�
+0
+�
+�
+P 1
+P 0
+�
+w1
+�0.
+Furthermore, an application of [HE, Proposition 3.5] provides us with another almost split
+sequence
+0
+�
+�
+Ω2(M)
+P 1
+�
+v1
+��
+0
+Ω(M)
+�
+0 ⊕
+�
+P 1
+P 1
+�
+1 ⊕
+�
+Ω2(M)
+0
+�
+0
+�
+�
+P 1
+Ω(M)
+�
+u1
+�0.
+Also [HE, Proposition 2.2] combined to Lemma 7.6 results in the almost split sequence
+0
+�
+�
+ν(P 3)
+ν(P 2)
+�
+ν(w3)
+�
+�
+Ω2(M)
+P 1
+�
+v1 ⊕
+�
+ν(P 3)
+0
+�
+0
+�
+�
+Ω2(M)
+0
+�
+0
+�0.
+
+24
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+It is easy to see that, as Ω2(M) satisfies (∗), so does νΩ2(M) and consequently, τνΩ2(M) =
+ΩνΩ2(M) ≃ νΩ3(M) = A (M). Therefore, by [HE, Proposition 2.4], we have τH(ν(P 3)
+ν(w3)
+→
+ν(P 2)) = (0 → A (M)).
+Continuing in this manner, one obtains the following mesh in the
+Auslander-Reiten quiver of H.
+�
+0
+P 0
+�
+0
+�
+�
+Ω2(M)
+Ω2(M)
+�
+1
+�
+�
+0
+Ω(M)
+�
+0
+�
+�
+�
+�
+P 1
+P 0
+�
+w1
+�
+�
+�
+P 2
+Ω2(M)
+�
+u2
+�
+�
+�
+�
+ν(P 3)
+ν(P 3)
+�
+1
+�
+�
+Ω2(M)
+P 1
+�
+v1
+�
+�
+�
+�
+P 1
+P 1
+�
+1
+�
+�
+P 1
+Ω(M)
+�
+u1
+�
+�
+� Ω(M)
+P 0
+�
+v0
+�
+�
+�
+0
+A (M)
+�
+0
+�
+�
+ν(P 3)
+ν(P 2)
+�
+ν(w3)
+�
+�
+�
+�
+Ω2(M)
+0
+�
+0
+�
+�
+� Ω(M)
+Ω(M)
+�
+1
+�
+�
+� 0
+M
+�
+0
+�
+�
+ν(P 3)
+0
+�
+0
+�
+Starting then with the vertex (0 → A (M)) and iterating the above arguments, one may
+calculate the vertices lying on the left side of (0 → M) in ΓH. Also, by considering the minimal
+injective resolution of M, the vertices on the right part appear. Summarizing, it follows that the
+component in ΓH containing the vertex (0 → M) is obtained by putting together all parts of the
+above shape corresponding to modules in the A -orbit of M. By virtue of previous considerations,
+ΓA comes up from ΓH by removing vertices of types (b) and (c). Hence the AR-quiver ΓA is
+obtained by gluing together all pieces of the following shape.
+�
+0
+Ω(M)
+�
+0
+�
+�
+P 1
+P 0
+�
+w1
+�
+�
+�
+P 2
+Ω2(M)
+�
+u2
+�
+�
+Ω2(M)
+P 1
+�
+v1
+�
+�
+�
+P 1
+Ω(M)
+�
+u1
+�
+�
+� Ω(M)
+P 0
+�
+v0
+�
+�
+�
+0
+A (M)
+�
+0
+�
+�
+ν(P 3)
+ν(P 2)
+�
+ν(w3)
+�
+�
+� 0
+M
+�
+0
+Now, since M is A -periodic, we get a finite oriented cycle as a component in ΓA that by [ARS,
+Theorem VII.2.1] must be the whole of ΓA.
+□
+7.3. Components of the stable Auslander-Reiten quiver of A. We let Γs
+A, the stable
+Auslander-Reiten quiver of A, be the subquiver of ΓA obtained by removing projective vertices
+and their τA-orbits.
+It should be clarified that here, we distinguish with a usual custom in
+the corresponding literature where this terminology applies while removing vertices that are
+both projective and injective. Also we notice that, generally, this has nothing to do with the
+Auslander-Reiten quiver of the stable Auslander algebra A. For instance, despite A which is
+self-injective in this case, A can not be self-injective since it is of global dimension 2.
+Below, we use results from [AS93] to get a nice intuition of the stable Auslander-Reiten quiver
+Γs
+A of A in terms of the AR quiver of a triangulated category.
+Remark 7.8. Recall from Section 3 that the class X in H consisting of all objects of type
+(a), (b), (c), or (d) determines an exact structure HX on H which has enough projectives and
+enough injectives.
+We claim that P(HX ) = X ∪ proj-H and I(HX ) = τH(X) ∪ inj-H, the
+subcategories of projectives and injectives of HX , coincide.
+Indeed, as in Construction 7.1,
+
+FROM MORPHISM CATEGORIES TO FUNCTOR CATEGORIES
+25
+τH(X) ⊆ X. Since every indecomposable injective object in H is of type (b) or (c), we get
+I(HX ) ⊆ P(HX ). To settle the reverse inclusion, note that by [HE, Proposition 3.6], for an
+indecomposable projective Λ-module P there exists a projective Λ-module Q with τH(Q →
+0) = (0 → P). Besides that projective objects of type (P
+1→ P) lie in inj-H, this ensures that
+proj-H ⊆ I(HX ). Now take a non-injective object M of X. If M is of type (a), then M = (0 → M)
+for an indecomposable Λ-module M. If M is projective then as above, M = τH(Q → 0) for some
+Q; otherwise M is non-injective and Proposition 2.4 of [HE] shows that M = τH(P1 → P0)
+where P1 → P0 → τ −1(M) → 0 is the minimal projective presentation. If M = (M
+1→ M) is of
+type (b) with M non-injective, then Lemma 4.4 gives M ≃ τH(0 → τ −1(M)). Furthermore, if
+M = (M → 0) is of type (c), then again Lemma 4.4 shows that M ≃ τH(τ −1(M)
+1→ τ −1(M))
+as M is non-injective. Finally if M = (P
+f→ Q) is of type (d) then, setting N = Coker(ν(f)),
+we deduce from [HE, Proposition 2.2] that M = τH(N → 0). Summarizing, these imply that
+X ⊆ I(HX ) and the above claim follows; that is to say, HX is a Frobenius exact category and,
+consequently, the stable category HX is triangulated.
+On the other hand, according to [AS93, Proposition 1.9], we infer that an almost split sequence
+0 → X → Y → Z → 0 in H is an almost split sequence in HX if and only if neither X ∈ I(HX )
+nor Z ∈ P(HX ). Thus, in order to get the Auslander-Reiten quiver of the triangulated category
+HX , it is enough to remove the iso-classes of indecomposable objects in X and arrows attached
+to them from the Auslander-Reiten quiver ΓH of H. But, as stated before, what remains after
+deleting, is exactly the stable Auslander-Reiten quiver Γs
+A of A.
+The following theorem is the main result in this subsection. Note that if Λ admits a module
+M with property (∗) then, according to Theorem 7.7, Γs
+A is just a set of single vertices. That’s
+why one has to exclude this case from the hypothesis below.
+Theorem 7.9. Assume Λ is self-injective of finite representation type and Ξ is a component
+of Γs
+A containing a simple module SM for an indecomposable non-projective Λ-module M not
+fulfilling (∗). Then
+(i) If Ξ is finite, then Ξ = Z∆/G, where ∆ is a Dynkin quiver and G is an automorphism
+group of Z∆ containing a positive power of the translation. Moreover, Ξ is Γs
+A itself if
+we further assume that Λ is indecomposable.
+(ii) If Ξ is infinite, then it is a stable tube.
+Proof. We have seen before that the AR-quiver ΓA of the Auslander algebra A is obtained from
+ΓH by removing the vertices of types (b) and (c). Note that indecomposable objects of type (a)
+correspond to indecomposable projective A-modules and those of type (d) lie in the τA-orbit of
+indecomposable projective A-modules by Construction 7.1. So in fact, Ξ emerges by deleting
+vertices of either types (a), (b), (c) and (d) and the arrows attached to them from a component
+Ξ′ of ΓH containing the vertex (0 → M).
+Note that such a Ξ is connected as M does not
+satisfy the property (∗). Note also that all vertices in Γs
+A are stable in the sense that τ m
+A (−) is
+well-defined over them for arbitrary integers m. Therefore Theorem 7.3 implies that the vertex
+SM in Ξ is τA-periodic and both the assertions in (i) and (ii) follow from [L, Theorem 5.5]. For
+the second statement in (i), note that indecomposability of Λ implies that the lower triangular
+matrix algebra T2(Λ) = ( Λ 0
+Λ Λ ) is also indecomposable and recall that H is naturally equivalent to
+the category mod-T2(Λ). Now if Ξ is finite, then so is Ξ′. As such, Ξ′ itself is a finite component
+of ΓH which, according to [ARS, Theorem VII.2.1], should be the whole of ΓH. Hence Ξ = Γs
+A,
+as desired.
+□
+
+26
+HOSSEIN ESHRAGHI AND RASOOL HAFEZI
+We conclude this section by quoting an observation from [HE]. Assume M is an indecom-
+posable non-projective Λ-module. Denote by [M]A the A -orbit of M. Let also ΓH(M) be the
+unique component of ΓH containing the vertex (0 → M). Moreover, we set
+T = {ΓH(M) | M is indecomposable non-projective}
+E = {[M]A | M is indecomposable non-projective}
+Then there exists a well-defined map δ : E → T which is given by sending [M]A to ΓH(M).
+Further, if we let T∞ denote the subset of T consisting of all infinite components, and E∞ be the
+inverse image of T∞ under δ, then by [HE, Proposition 5.8], δ is surjective and the restricted map
+δ |: E∞ → T∞ is a bijection whenever Λ is indecomposable self injective of finite representation
+type.
+Inspired by this result, we let L be the set of all components of ΓA containing a simple
+module. Define λ : T −→ L by λ(ΓH(M)) = ΓA(SM) where ΓA(SM) is the component of ΓA
+that contains the simple vertex SM.
+Proposition 7.10. Suppose Λ is indecomposable self-injective of finite representation type.
+(i) The map λ is well-defined and surjective.
+(ii) λ restricts to a bijection λ |: T∞ → L∞, where L∞ is the subset of L consisting of all
+infinite components.
+(iii) The sets E∞ and L∞ are in bijection.
+Proof. (i). Take M0 and M1 to be indecomposable non-projective Λ-modules with ΓH(M0) =
+ΓH(M1). Then there exist almost split sequences 0 → A0
+f0
+→ B0
+g0
+→ M0 → 0 and 0 → A1
+f1
+→
+B1
+g′
+1
+→ M1 → 0 in mod-Λ. By Lemma 4.4, there exist almost split sequences
+0
+�� Ai
+0
+�
+0
+� fi
+1
+�
+�� Bi
+Mi
+�
+gi
+� 0
+gi
+�
+�� 0
+Mi
+�
+0
+�0
+in H for i = 0, 1. Accordingly, (B0
+g0
+→ M0) and (B1
+g1
+→ M1) lie inside ΓH(M0) and the connected-
+ness of ΓH(M0) gives the existence of a walk (B0
+g0
+→ M0) ←→ x1 ←→ · · · xn−1 ←→ (B1
+g1
+→ M1)
+in ΓH. If none of the xi is of the form (b) or (c), then using Theorem 4.8 we get a walk in ΓA be-
+tween Θ(B0
+g0
+→ M0) = SM0 and Θ(B1
+g1
+→ M1) = SM1. Consequently, λ(ΓH(M0)) = λ(ΓH(M1)).
+Otherwise we may, without loss of generality, assume that the xi are all non-projective and apply
+the arguments used in Construction 7.1 to obtain a walk in ΓH passing through (Bi
+gi
+→ Mi),
+i = 0, 1, none of the vertices over which are of the forms (b) or (c).
+To settle (ii), assume
+ΓA(SM) is an infinite component of ΓA for an indecomposable non-projective Λ-module M. By
+Theorem 7.9, ΓA(SM) is a stable tube and the τA-orbit of SM generates the mouth of ΓA(SM).
+The fact that the mouth of a stable tube is unique reveals that ΓA(SM) is uniquely determined
+by ΓH(M). The last statement is a combination of (ii) and [HE, Proposition 5.8].
+□
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+Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, PO
+Box 87317-51167, Kashan, Iran
+Email address: eshraghi@kashanu.ac.ir
+School of Mathematics and Statistics, Nanjing University of Information Science & Technology,
+Nanjing, Jiangsu 210044, P. R. China
+Email address: hafezi@nuist.edu.cn
+

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@@ -0,0 +1,1867 @@
+Information Entropy-based Camera Path Estimation for In-Situ
+Visualization
+Ken Iwata*
+Kobe University
+Naohisa Sakamoto†
+Kobe University
+Jorji Nonaka‡
+RIKEN R-CCS
+Chongke Bi§
+Tianjin University
+Information Entropy
+(for viewpoint selection)
+Depth and Lightness Information
+𝑑!𝑑"𝑑#⋯ 𝑑"$%
+⋯
+𝐷𝑒𝑝𝑡ℎ
+	𝑙!	𝑙"	𝑙#	⋯	𝑙"$%
+⋯
+𝐿𝑖𝑔ℎ𝑡𝑛𝑒𝑠𝑠
+Selected Viewpoint “A”
+Camera Path
+Selected Viewpoint “A”
+Selected Viewpoint “B”
+Intermediate Images
+Selected
+Image “A”
+Selected
+Image “B”
+Selected Image “A”
+︓Viewpoints
+Rendering Image
+(Information Source)
+Figure 1: Depth and lightness entropy-based viewpoint selection and camera path estimation for generating a smooth video, with as
+much information as possible, to assist the rapid understanding of the underlying simulation phenomena.
+ABSTRACT
+In-situ processing has widely been recognized as an effective ap-
+proach for the visualization and analysis of large-scale simulation
+outputs from modern HPC systems. One of the most common
+approaches for batch-based in-situ visualization is the image- or
+video-based approach. In this kind of approach, a large number of
+rendered images are generated from different viewpoints at each
+time step and has proven useful for detailed analysis of the main
+simulation results. However, during test runs and model calibration
+runs before the main simulation run, a quick overview might be
+sufficient and useful. In this work, we focused on selecting the
+viewpoints which provide as much information as possible by using
+information entropy to maximize the subsequent visual analysis task.
+However, by simply following the selected viewpoints at each of
+the visualization time steps will probably lead to a rapidly changing
+video, which can impact the understanding. Therefore, we have
+also worked on an efficient camera path estimation approach for
+connecting selected viewpoints, at regular intervals, to generate a
+smooth video. This resulting video is expected to assist in rapid
+understanding of the underlying simulation phenomena and can
+be helpful to narrow down the temporal region of interest to min-
+imize the turnaround time during detailed visual exploration via
+image- or video-based visual analysis of the main simulation run.
+We implemented and evaluated the proposed approach using the
+OpenFOAM CFD application, on an x86-based Server and an ARM
+A64FX-based supercomputer (Fugaku), and we obtained positive
+evaluations from domain scientists.
+Index Terms:
+Human-centered computing—Visualization—
+*e-mail: 228x202x@stu.kobe-u.ac.jp
+†e-mail: naohisa.sakamoto@people.kobe-u.ac.jp
+‡e-mail: jorji@riken.jp
+§e-mail: bichongke@tju.edu.cn
+Visualization systems and tools—Visualization toolkits
+1
+INTRODUCTION
+High-end high performance computing (HPC) systems have contin-
+uously become more and more capable with higher computational
+capacity with every new system replacement. This was the case for
+the replacement of the K computer to the supercomputer Fugaku at
+the RIKEN R-CCS. The increased number of CPUs and computa-
+tional cores have been applied for Capability Computing to tackle
+even larger numerical simulations with higher spatio-temporal reso-
+lutions. In addition, this also has been used for Capacity Computing
+to handle an even larger number of parameters and members during
+parametric sweep and ensemble simulations. On the other hand,
+this proportionately generates even larger simulation outputs, thus,
+making the visualization and analysis tasks even more challenging.
+As a result, the importance of in-situ visualization and analysis has
+continuously become even more evident.
+A variety of approaches have already been proposed and applied
+for the in-situ visualization and analysis as discussed in [6]. We can
+also verify that there are also a variety of existing applications and
+libraries for realizing in-situ visualization and analysis. However,
+since in-situ processing is executed simultaneously with the simu-
+lation, it becomes highly important to collaborate with the domain
+scientists . We have been working with domain scientists working
+with computational fluid dynamics (CFD) simulation of the sound
+generation mechanisms [27], and we already worked on an adaptive
+in situ time-step sampling approach [26]. In this work, we have
+used the same OpenFOAM CFD application and simulation model
+and obtained assistance from them for necessary technical feedback
+during the developments.
+Probably the most widely used image-based in-situ visualization
+approach is ParaView Cinema [1]. In that approach, a large set of
+pre-computed images are generated in-situ on the HPC system side
+for the interactive post-hoc visual exploration on a local machine
+such as desktop PC and laptop. There is also an image-based in-situ
+visualization approach that generates a set of images from omni-
+directional camera positions [9], and its extension for video-based
+arXiv:2301.11591v1  [cs.GR]  27 Jan 2023
+
+in-situ visualization [8]. These image- or video-based in-situ visual-
+ization approaches have proven useful for detailed analysis of the
+main simulation results. In this work, we have focused on rapid un-
+derstanding of the underlying simulation during test runs and model
+calibration runs before the main simulation run. For this purpose,
+we focused on selecting the most appropriate viewpoints, based on
+information entropy, at regular time intervals of the simulation in
+order to obtain as much information as possible trying to facilitate
+the rapid understanding of such kinds of simulations.
+2
+RELATED WORK
+There is an extensive work that culminated in the creation of a classi-
+fication and terminology for the in-situ visualization approaches [6].
+Here, we will only focus on related works for realizing tightly cou-
+pled in-situ visualization, and techniques for selecting time steps
+and viewpoints that can be used for minimizing the amount of im-
+ages for the image- or video-based in-situ visualization. VTK-based
+ParaView and VisIt are probably the most widely used visualization
+application for large data visualization. Both applications provide in-
+situ visualization APIs, ParaView Catalyst [2] and VisIt LibSim [11],
+for integrating to the simulation code. In a batch-based in-situ visu-
+alization, a large amount of images can be generated for assisting
+the post-hoc visualization [9]. To facilitate this post-hoc visual anal-
+ysis, Ahrens et al. [1] proposed an image-based approach for the
+in-situ visualization and analysis, and was implemented as Paraview
+Cinema. In this approach a large set of images are generated in-
+situ, and a custom visualization application is used, on the local
+machine, to perform interactive visual analysis by automatically
+switching between the generated set of images. Similar to this ap-
+proach, Kageyama et al. [8] proposed a video-based approach by
+generating an omnidirectional animated video, from the set of in-situ
+generated images, which are explorable from a custom visualization
+application. Although these approaches have proven efficient, most
+of the generated images may have small or even no contribution to
+the visual analysis, thus it may be unnecessarily increasing the time
+spent on the post-hoc visual analysis task.
+An approach to minimize the aforementioned amount of gen-
+erated images is the selection of the most valuable time steps for
+rendering the images. For this purpose, Ling et al. [12] proposed a
+method to estimate the probability density function of the simulation
+field, at each time step, by using the kernel density estimation. They
+also applied machine learning for extracting feature quantity from
+the obtained estimation, and detected potentially valuable time steps
+where an important phenomena may occur. However, this method
+can cause false detections depending on the high correlation among
+the physical quantities on the simulation field as mentioned by the
+authors. Yamaoka et al. [26] extend the aforementioned work, and
+proposed an adaptive time sampling method for in-situ visualization.
+In this method, kernel density function and Kullback–Leibler diver-
+gence is applied to estimate the amount of change on the simulation
+field. The sampling intervals are adaptively changed according to
+the estimated amount of change in the simulation.
+Another approach for reducing the amount of images is the se-
+lection of viewpoints for generating the images. For this purpose,
+Kamada et al. [10] considered the viewpoints capable of minimizing
+the number of degenerated face as being the optimal viewpoints.
+However, they did not extend their work for the case when there
+exist multiple viewpoints with the same number of degenerated
+faces. Barral et al. [3] solved this problem by adding the projected
+area as a weight to the number of degenerated faces. However,
+there still remains a problem on how to properly set these weights.
+Vazquez et al. [23] proposed a method to select the optimal view-
+point defined by the viewpoint entropy based on the information
+entropy. Since this method cannot handle the movement of view-
+points, the authors improved the viewpoint entropy and applied it
+to molecular objects [25] as well as to image-based modeling [24].
+Page et al. [18] proposed a method to analyze the object shape by
+calculating the entropy for the silhouette and surface curvature of
+the model. Polonsky et al. [19] discussed evaluation indices for the
+viewpoint selection, and concluded that none of them could make
+the best choice in any situation. However, they also said that by
+improving each of these indices, it will become possible to make a
+better choice by using a combination of them.
+Secord et al. [21] proposed some evaluation indices for the view-
+point selection, and showed that optimal viewpoints can be selected
+by combining these metrics. Takahashi et al. [22] proposed a method
+for estimating the optimal viewpoint for volume data by using in-
+formation entropy. Bordoloi et al. [5] proposed an information
+entropy-based evaluation metric for the viewpoints during volume
+rendering by using the transfer function, data distribution, and voxel
+visibility information. Zhang et al. [28] also proposed an evaluation
+metric for volume rendering based on the opacity, brightness, and
+structural features. Ji et al. [7] proposed a method to find the optimal
+time-varying views by using the viewpoint selection method to max-
+imize the amount of information for time-series volume data. They
+showed that it is possible to create an animation with the largest
+amounts of information. This was realized by searching for a move-
+ment route with the largest amounts of information using dynamic
+programming. Marsaglia et al. [14] proposed a viewpoint quality
+evaluation metric based on information entropy involving the visible
+field data, depth, and shading values belonging to each of the pixels
+in the image. In another work, they also utilized a trigger-based ap-
+proach in combination with information entropy to determine when
+to search for a new camera position as a simulation evolves [15].
+Our work was inspired in their viewpoint quality evaluation metric,
+which we extended with the lightness information for evaluating
+the viewpoint quality. We will detail the methodology behind our
+proposed method in the next section.
+3
+METHODOLOGY
+3.1
+Overview
+In this work, we focused on a viewpoint selection approach, based
+on information entropy, and on a camera path estimation approach,
+based on quaternion interpolation. The viewpoints selected at regular
+intervals are used as markers to estimate the smooth camera path.
+Following are the necessary requirements to meet this goal:
+R1. Images from the selected viewpoints should capture important
+phenomenon from the underlying simulation.
+R2. The resulting video generated from the rendered outputs should
+be smooth for post-hoc analysis.
+Below is the adopted approach to satisfy the aforementioned
+requirements, and they are divided into the following three parts:
+A. Viewpoint evaluation
+The viewpoint quality will be evaluated using information
+entropy and will be used to select the most appropriate image
+for each evaluated time step. Only images from the selected
+viewpoints will be output (R1).
+B. Camera path estimation
+The camera path connecting these selected viewpoints will be
+estimated, and the rendered images through this camera path
+will also be output as intermediate images (R2).
+C. Video generation
+At the end of the simulation, these output images will be se-
+quentially concatenated to produce a video (R2).
+Regarding part A, the simulation state usually does not often
+significantly change within a single simulation time step. Therefore,
+
+Simulation time step
+• Execute simulation
+Visualization time step
+• Store simulation data
+Entropy evaluation time step
+• Select optimal viewpoint
+• Calculate camera path 
+Times and Intervals
+• Simulation time
+𝑇! = 𝑖∆𝑇
+• Visualization time
+𝑇"
+! = 𝑖∆𝑇" = 𝑖𝑁"∆𝑇
+• Entropy evaluation time
+𝑇#
+! = 𝑖∆𝑇# = 𝑖𝑁"𝑁#∆𝑇
+data1
+time [ 𝑇" ]
+∆𝑇"
+time [ 𝑇# ]
+∆𝑇#
+time [	𝑇	]
+∆𝑇
+data2
+data3
+Rendering
+Selected Viewpoint “A”
+Selected Viewpoint “B”
+Intermediate Images
+Figure 2: Different time step intervals for the simulation, visualization and entropy evaluation.
+there is usually no need to visualize at every simulation time step,
+and the visualization can be performed at every set of simulation time
+steps. In the same manner, the viewpoint evaluation for viewpoint
+selection will be performed at every set of visualization time steps to
+satisfy R1. In this paper, as shown in Fig. 2, we use T to represent
+the simulation time step, TV to represent the visualization time step,
+and TE to represent the entropy evaluation time step, ∆T to represent
+the simulation time step interval, ∆TV to represent the visualization
+time step interval, and ∆TE to represent the entropy evaluation time
+step interval. The elapsed time for the ith simulation time step T i can
+be expressed as T i = i∆T. In the same manner, the visualization time
+step and entropy evaluation time steps can be expressed respectively
+as T i
+V = i∆TV and T i
+E = i∆TE. Considering that the simulation is
+performed NV times for every visualization, and the visualization is
+performed NE times for every optimal viewpoint selection, then we
+can express these intervals as ∆TV = NV ∆T and ∆TE = NENV ∆T.
+Regarding part B, a camera path connecting viewpoints selected at
+every ∆TE entropy evaluation time step will be estimated. Although
+visualization is not performed during the visualization time steps
+in between the entropy evaluation time steps, the simulation data
+for each ∆TV visualization time step is stacked. From the obtained
+camera path, the rendered images at the intermediate visualization
+time steps will be output as the intermediate images for generating a
+smooth video, and this satisfies R2. It is worth noting that it is also
+possible to generate the full set of images from the entire viewpoints
+for the detailed post-hoc analysis when necessary.
+Regarding part C, the set of output images generated at each ∆TV
+visualization time step will be joined sequentially to create a video
+file. For this purpose, we can use existing tools such as the well-
+known FFmpeg available to a variety of platforms. In the resulting
+video, the camera will automatically move and capture the important
+phenomenon, and this allows the R2 to be satisfied. Traditional
+approach requires the user to search for the best location, in the
+trial-and-error manner, to visually explore when searching for an
+important phenomenon during the simulation. However, by using
+the proposed method, this search for the best camera position may
+be alleviated and may facilitate narrowing down the spatio-temporal
+region of interest for the detailed visual analysis.
+3.2
+Viewpoint Selection
+In this section, we will detail the utilized viewpoint selection ap-
+proach. The evaluation of the viewpoints is based on information
+entropy. We used depth and lightness values from the rendered
+images for calculating the associated information entropy, that is,
+the depth entropy and lightness entropy.
+3.2.1
+Information Entropy
+Information entropy used in this work can be defined as the expected
+value for the amount of information obtained from a certain infor-
+mation source [4]. The information entropy H(X) from a source
+X given by the set of probabilities P(x1),P(x2),··· ,P(xn) corre-
+sponding respectively to the set of information x1,x2,··· ,xn can be
+represented as follows:
+H(X) = −
+n
+∑
+i=1
+P(xi)logP(xi)
+(1)
+Here, when P(xi) = 0, we will consider P(xi)logP(xi) = 0. Re-
+garding the selection of the logarithm’s base, the base influences
+the multiplication factor and, thus, is arbitrary. Base 2 is commonly
+used in information theory, and was used in this work. Taking into
+consideration the probability distribution of the information source,
+the larger the information bias, the smaller the value of information
+entropy, and vice versa.
+3.2.2
+Depth Entropy
+Depth entropy used in this work is based on the viewpoint quality
+evaluation metric proposed by Marsaglia et al. [14]. The information
+entropy is calculated by considering the image as the source of
+information, and by using the depth values belonging to each of the
+pixels in the image. The depth values can vary in the range of 0 ∼ 1,
+and the closer the distance to the object, the smaller the value. The
+background portion in the image where there is no object is set to
+infinity and will have their depth values corresponding to 1.
+The depth values from all pixels of the rendered image are binned
+into 256 groups d0,d1,··· ,d255 for creating a discrete probability
+distribution D, which will be used to calculate the information en-
+tropy. At this time, the background portion in the image is considered
+to have no information, and only the pixels with some information
+will be used in the calculation. By using the discrete probability
+distribution, the depth entropy Hd can be calculated as follows:
+Hd = −
+255
+∑
+i=0
+D(di)log2 D(di)
+(2)
+Here, D(di) corresponds to the probability for a given value,
+selected based on the probability distribution D, being di. When
+evaluating a viewpoint using depth entropy, the resulting value will
+be larger for images with large dispersion in the distribution of depth
+values. Therefore, the viewpoints of images showing objects with
+high undulations will be highly evaluated.
+
+3.2.3
+Lightness Entropy
+In this work, in addition to the depth entropy, we propose the use of
+lightness entropy to also take into consideration the color informa-
+tion in the image. Diverging color maps, proposed by Moreland [16],
+have become prevalent in scientific visualization as the substitute
+for the traditional but problematic rainbow color map. Although the
+change in color values, such as RGB values, in a color map may
+follow different behavior depending on the color map, diverging
+color maps usually show similar behavior in the lightness values in
+CIELAB color space as shown in the Fig. 3. Therefore, lightness
+entropy is expected to work robustly for the diverging color maps.
+The proposed lightness entropy can be defined as an information en-
+tropy using the lightness values from the target image as the source
+of information. The lightness value is calculated from RGB values
+and can vary in the range of 0 ∼ 100.
+PiYG
+RdBu
+PuOr
+Lightness L*
+Figure 3: Lightness values for different diverging color maps.
+The lightness values, as well as the depth values, obtained
+from all pixels of the rendered image, are binned into 256 groups
+l0,l1,··· ,l255 for creating a discrete probability distribution L, which
+will be used to calculate the information entropy. At this time, the
+background portion in the image is considered to have no infor-
+mation, and only the pixels with some information will be used in
+the calculation. By using the discrete probability distribution, the
+lightness entropy Hl can be calculated as follows:
+Hl = −
+255
+∑
+i=0
+L(li)log2 L(li)
+(3)
+Here, L(li) corresponds to the probability for a given value, se-
+lected based on the probability distribution L, being li. When a
+viewpoint is evaluated using the lightness entropy, the resulting
+value will be larger for images with large dispersion in the distribu-
+tion of lightness values. Therefore, the viewpoints of images with
+clear brightness and darkness will be highly evaluated.
+3.3
+Path Estimation between Selected Viewpoints
+In this section, we will detail the utilized camera path estimation
+between the viewpoints selected by using the depth and/or lightness
+entropy. In this work, we considered that the viewpoints are pre-
+arranged in a spherical surface as shown in Fig. 4, and the camera
+path from one viewpoint to another will move over this spherical
+surface. More specifically, the position and orientation of a given
+viewpoint will be represented as a quaternion, and the movement
+from one to another viewpoint will be obtained by using quaternion
+interpolation. In this work, we investigated two quaternion interpola-
+tion methods: spherical linear interpolation (SLERP) and spherical
+quadrangle interpolation (SQUAD). In the following subsections,
+we will explain about spherical linear interpolation and spherical
+quadrangle interpolation.
+3.3.1
+Spherical Linear Interpolation (SLERP)
+SLERP is an abbreviation for spherical linear interpolation, and is
+a quaternion interpolation method for connecting two points over a
+sphere in the straight line direction, or the shortest path, as shown
+in Fig. 5. SLERP-based interpolation from a quaternion qA to the
+quaternion qB can be calculated by using time t ∈ [0, 1] as follows:
+: Viewpoints
+Figure 4: Viewpoint distribution over spherical surface.
+slerp(qA, qB, t) = sin(1−t)φ
+sinφ
+qA + sintφ
+sinφ qB
+(4)
+Here, φ = arccos⟨qA, qB⟩. In addition, in the case of ⟨qA, qB⟩ < 0,
+the interpolation will be interpolated in the contrary direction over
+the sphere surface, that is by the longest path in the straight-line
+direction. To interpolate by the shortest path, then either qA or qB
+should be replaced with a quaternion with same rotation but in the
+opposite direction. For instance, by replacing qA with −qA.
+3.3.2
+Spherical Quadrangle Interpolation (SQUAD)
+SQUAD is an abbreviation for spherical quadrangle interpolation,
+and is a quaternion interpolation method to connect multiple points
+in a smoothness way so that the derivatives are continuous in the
+neighborhood of the points (Fig. 5). Considering a quaternion list
+{q1, q2, ... qn}, then the SQUAD-based interpolation from qi to
+qi+1 can be calculated by using the time t ∈ [0, 1] as follows:
+squad(qi, qi+1, ai, ai+1, t)
+= slerp(slerp(qi, qi+1, t), slerp(ai, ai+1, t), 2t(1−t))
+(5)
+ai = qi exp
+�
+−logq∗
+i qi−1 +logq∗
+i qi+1
+4
+�
+(6)
+Here, the exponential of the quaternion exp(q) and the logarithm
+of the quaternion logq for the quaternion q = a+bi+c j +dk are
+defined as follows:
+exp(q) = ea
+�
+cos∥bi+c j +dk∥+ bi+c j +dk
+∥bi+c j +dk∥ sin∥bi+c j +dk∥
+�
+(7)
+logq = log∥q∥+ bi+c j +dk
+∥bi+c j +dk∥ arctan ∥bi+c j +dk∥
+a
+(8)
+In addition, in the case of performing SQUAD-based interpolation
+from q1 to q2, and from qn−1 to qn, we consider q0 = q1 and qn+1 =
+qn.
+Unit Sphere in Quaternion Space
+: SLERP 
+: SQUAD
+: Rotation Quaternion
+Figure 5: Comparison of SLERP and SQUAD interpolation methods.
+
+7
+100
+80
+60
+40
+RdBu
+20
+PiYG
+PuOr3.3.3
+Implementation
+We utilized the Kyoto Visualization System (KVS) [20] for imple-
+menting the proposed viewpoint selection approach, based on depth
+and lightness entropy, as well as the sequential camera path between
+the selected viewpoints via SLERP- and SQUAD-based interpolation
+methods. KVS is a cross-platform, open-source C++ visualization
+library capable of running on a variety of hardware systems from
+traditional x86/GPU systems to GPU-less HPC systems including
+IBM Blue Gene L/P (PowerPC), K computer (SPARC VIIIfx), and
+Fugaku (ARM A64FX). KVS supports hybrid MPI/OpenMP paral-
+lelism and implements a sort-last parallel image composition method
+based on Binary-Swap [13], with an extension to support non-power-
+of-two number of nodes, which is named 234Compositor [17].
+The pseudocode of our implementation, using the SQUAD-based
+interpolation, is described in Algorithm 1. In this pseudocode, I[],
+V[], and Q[] respectively represent the queues for storing the out-
+put image, simulation data, and the quaternion for the selected
+viewpoint. In addition, is initial step(t) and is final step(t) are
+functions that respectively return the true information in the first
+and the final time step. The Vis(V, q) function renders the simu-
+lation data V from the viewpoint represented by the quaternion of
+q. Entropy Evaluation(V) is the function that calculates the en-
+tropy for the simulation data V at all pre-arranged viewpoints on
+the spherical surface, and returns the quaternion information of the
+viewpoint with highest entropy value. Its pseudocode is described in
+Algorithm 2. In this pseudocode, L represents the set of viewpoints
+and the read back(V, l) is a function that renders the simulation data
+V from the viewpoint l and returns its frame buffer. entropy( f) is
+a function that calculates the entropy for the frame buffer f and re-
+turns its value. quaternion() is a function that returns the quaternion
+information from a given viewpoint.
+4
+EXPERIMENTAL EVALUATIONS
+We used the OpenFOAM CFD code and model for the experimental
+evaluations. The simulation model used for the evaluations was
+obtained from our collaborators [27], and refers to a sound propaga-
+tion in the oral cavity by using irregular volume data composed of
+3,197,279 hexahedral elements. We integrated the in situ KVS mod-
+ule to the OpenFOAM code, and evaluated on two systems shown
+in Tables 1 and 2. The irregular volume data was decomposed into 8
+blocks for the x86 Server, and up to 1,024 blocks for the Fugaku.
+Table 1: x86/GPU-based Server System.
+Nodes
+1
+CPU
+Intel Xeon Gold 6238R 2.20GHz 28Core×2
+Cores
+28×2 = 56
+RAM
+384 GB DRAM
+GPU
+NVIDIA Quadro RTX8000
+Compiler
+GCC version 7.5.0
+MPI
+OpenMPI 2.1.1
+Table 2: ARM-based Supercomputer Fugaku.
+Nodes
+158,976
+CPU
+Fujitsu A64FX (Armv8.2-A SVE)
+Cores
+48 + 2 Assistant Cores
+RAM
+32GB HBM2
+Compiler
+GCC-based Fujitsu Compiler Ver. 4.8.0
+MPI
+OpenMPI with Fujitsu expensions for Tofu
+Algorithm 1 In-situ visualization (using SQUAD interpolation).
+1: function IN SITU VISUALIZATION(∆TV , ∆TE)
+2:
+I[], V[], Q[], t;
+3:
+while t ≤ tend do
+4:
+Vt = Sim();
+5:
+if is initial step(t) then
+6:
+Qt = Entropy Evaluation(Vt);
+7:
+Q.push(Qt); Q.push(Qt);
+8:
+else if t%∆TV == 0 then
+9:
+if t%∆TE == 0 then
+10:
+Qt = Entropy Evaluation(Vt);
+11:
+Q.push(Qt);
+12:
+if Q.size() == 4 then
+13:
+q1 = Q.front(); Q.pop();
+14:
+q2 = Q.front(); Q.pop();
+15:
+q3 = Q.front(); Q.pop();
+16:
+q4 = Q.front(); Q.pop();
+17:
+for i = 0, 1, ··· , ∆TE −1 do
+18:
+s = i/TE;
+19:
+qs = squad(q1, q2, q3, q4, s);
+20:
+Vs = V. front(); V.pop();
+21:
+Is = Vis(Vs, qs);
+22:
+I.push(Is);
+23:
+end for
+24:
+end if
+25:
+end if
+26:
+end if
+27:
+if is final step(t) then
+28:
+q1 = Q.front(); Q.pop();
+29:
+q2 = Q.front(); Q.pop();
+30:
+q3 = Q.front(); Q.pop();
+31:
+q4 = q3;
+32:
+for i = 0, 1, ··· , ∆TE −1 do
+33:
+s = i/TE;
+34:
+qs = squad(q1, q2, q3, q4, s);
+35:
+Vs = V. front(); V.pop();
+36:
+Is = Vis(Vs, qs);
+37:
+I.push(Is);
+38:
+end for
+39:
+while V.size() > 0 do
+40:
+Vs = V. front(); V.pop();
+41:
+Is = Vis(Vs, q3);
+42:
+I.push(Is);
+43:
+end while
+44:
+end if
+45:
+t ++;
+46:
+end while
+47:
+return I;
+48: end function
+4.1
+Some Results
+For the initial experimental evaluations, we selected the pressure
+variable and used multi-isosurface rendering with three distinct iso-
+values that are rendered as different colors. The total number of
+simulation time steps for the utilized CFD model was 15,000, and
+we used the parameters shown in Table 3 for the evaluations. We
+evaluated the use of our proposed lightness entropy (Lightness) in ad-
+dition to the depth entropy (Depth) proposed by Marsaglia et al. [14],
+and also the use of the average of depth and lightness entropy (Depth
+& Lightness). For the use of only lightness entropy, we experi-
+mented with three diverging color maps (RdBu, PiYG, PuOr). We
+selected three entropy evaluation intervals (10, 30, 50), which repre-
+sent the visualization time step interval for performing the entropy
+calculation. We also selected three sets of viewpoints with differ-
+
+Algorithm 2 Entropy Evaluation.
+1: function ENTROPY EVALUATION(V)
+2:
+E = 0.0;
+3:
+Q = 1+0i+0j +0k;
+4:
+for l ∈ L do
+5:
+f = read back(V, l);
+6:
+e = entropy( f);
+7:
+if e > E then
+8:
+E = e;
+9:
+Q = l.quaternion();
+10:
+end if
+11:
+end for
+12:
+return Q;
+13: end function
+ent numbers of viewpoints in the latitude and longitude directions
+(latitude × longitude). Both SLERP- and SQUAD-based quaternion
+interpolation methods were also evaluated for estimating the camera
+path between the selected viewpoints. The x86/GPU Server was
+used for the detailed evaluation using these different parameters, and
+the supercomputer Fugaku was used for the scalability analysis by
+using up to 1024 nodes, that is, 49,152 cores in hybrid MPI/OpenMP
+parallelism.
+Table 3: Parameters used for the experimental evaluations.
+Entropy source
+Depth; Lightness; Depth & Lightness
+Color maps
+RdBu; PiYG; PuOr
+# of viewpoints
+15×30; 25×50; 35×70
+Intervals (NE)
+10; 30; 50
+Interpolation
+SLERP; SQUAD
+Fig. 6 shows some entropy heatmaps evaluated by using all three
+entropy sources at different simulation time steps (2400, 6000, 9600,
+and 13200). The set of viewpoints evenly distributed on the spherical
+surface is mapped onto the 2D heatmap where the viewpoints on
+the same latitude are placed on the same horizontal axis, and in the
+same manner, the viewpoints on the same longitude are placed on
+the same vertical axis. The blue-colored regions show the portions
+where the evaluated entropy has low value, and on the other hand,
+the red-colored regions show the portions where the evaluated en-
+tropy has high value. Fig. 7 shows the multi-isosurface rendered
+results from the selected viewpoints obtained in Fig. 6; Fig. 9 shows
+the multi-isosurface rendered results from the selected viewpoints
+obtained in Fig. 8; Fig. 11 shows the multi-isosurface rendered
+results from the selected viewpoints obtained in Fig. 10. Table 4
+shows the average elapsed time of entropy calculation per image for
+different entropy sources when using an image size of 512 × 512
+on the x86/GPU-based Server System. Compared to depth entropy,
+we can observe that the computational costs when using lightness
+become much higher. In addition, we can verify that the number
+of viewpoints directly influences the computational cost. Code op-
+timizations and the use of parallel processing for trying to reduce
+this computational cost are planned as future works.Table 5 shows
+a comparison of output images’ average entropy when varying the
+number of viewpoints. Here, the utilized entropy source is Depth &
+Lightness, the entropy evaluation interval is 30, and the interpolation
+method is SQUAD. We can observe that the difference in the average
+entropy when varying the number of viewpoints is small.
+Fig. 12 shows a comparison of the accumulative distance from
+the estimated camera path position to the viewpoint with the high-
+est entropy, at each visualization time step, for different entropy
+evaluation intervals; Fig. 13 shows a comparison of the accumula-
+tive distance for different interpolation methods.Table 6 shows a
+comparison of output images’ average entropy for different entropy
+Table 4: Average elapsed time of entropy calculation for different
+entropy sources (x86 System).
+Entropy Sources
+Average elapsed time [s]
+Depth
+2.24e-4
+Lightness
+1.30e-3
+Depth & Lightness
+1.53e-3
+Table 5: Average entropy when varying the number of viewpoints.
+# of viewpoints
+Average entropy
+15×30
+3.09
+25×50
+3.10
+35×50
+3.09
+evaluation intervals. Here, the utilized entropy source is Depth &
+Lightness, the number of viewpoints is 25×50, and the interpolation
+method is SQUAD. In the case of NE = 1, rendered images of the
+viewpoint with the highest entropy at each visualization time step
+are output. From this figure and table, we can observe that as the
+entropy evaluation interval increases, the accumulative distance also
+increases, and the amount of average entropy decreases. Table 7
+shows the average elapsed time for path calculation between two
+selected viewpoints for different interpolation methods. We can
+observe that the computational cost is proportional to the number
+of intervals, and the cost of SQUAD is much higher than that of
+SLERP. However, it is worth noting that the influence on the total
+computational cost compared to the entropy calculation cost is small
+and almost neglectable. Table 8 shows a comparison of output im-
+ages’ average entropy for different interpolation methods. Here, the
+entropy source is Depth & Lightness, the number of viewpoints is
+25×50, and the entropy evaluation interval is 30. We can observe
+that when selecting SQUAD, the accumulative distance becomes
+smaller and achieves a slight increase in average entropy.
+Table 6: Average entropy for different entropy evaluation intervals.
+Intervals (NE)
+Average entropy
+1
+3.17
+10
+3.12
+30
+3.10
+50
+3.08
+Table 7: Average elapsed time for path calculation between two se-
+lected viewpoints using different interpolation methods (x86 System).
+Interpolation
+Intervals (NE)
+method
+10
+30
+50
+SLERP
+2.80e-6
+5.16e-6
+6.93e-6
+SQUAD
+9.80e-6
+3.04e-5
+3.54e-5
+Table 8: Average entropy for different interpolation methods.
+Interpolation method
+Average entropy
+SLERP
+3.09
+SQUAD
+3.10
+4.2
+Discussions
+Regarding the influence of different entropy sources (Fig. 6), we
+observed that the lightness entropy has a higher influence than the
+
+Simulation step:
+2400
+6000
+9600
+13200
+Depth:
+Depth & Lightness:
+Lightness:
+Figure 6: Entropy heatmaps for different entropy sources.
+Depth:
+Simulation step:
+2400
+6000
+9600
+13200
+Lightness:
+Depth & Lightness:
+Figure 7: Rendered images from the selected viewpoints.
+depth entropy, and has an even higher influence when using both
+depth and lightness entropy. Therefore, we opted to add both depth
+and lightness entropies after normalization. In addition, due to a
+large number of viewpoints with high entropy, the sequentially se-
+lected viewpoints can be separated far apart from each other thus
+resulting in an intense camera movement over the entire visualization
+time steps. We also observed that when using the lightness entropy,
+the entropy calculation took a little more time than using the depth
+entropy. This was because of the necessary conversion from RGB
+values to lightness values. It is worth noting that the selection of the
+viewpoint evaluation metric will depend on the targeted simulation,
+visualization method, and users’ analysis goals. Therefore, to satisfy
+R1, it becomes important to implement a variety of viewpoint evalu-
+ation metrics to handle different use case combinations. In addition,
+depending on the use case, it may be helpful that different evaluation
+metrics are interchangeable at run time in an adaptive manner.
+Regarding the influence of the diverging color maps for the light-
+ness entropy, we initially perceived almost no difference between
+the heatmaps. However, there was a slight difference among them,
+and at certain time steps, we observed that the selected viewpoints
+were also different. Among the color maps, heatmaps for the PuOr
+was especially different from the others. This may be because the
+change in the lightness of the PuOr was also different from the other
+diverging color maps.
+Regarding the influence of the entropy evaluation intervals, as this
+interval becomes smaller there will be fewer complementary images
+between the selected viewpoints. As a result, changes in viewpoint
+may become intense in a short period of time, this will lead to a non-
+smooth video which affects the users’ post-hoc visual analysis tasks.
+It is worth noting that when the simulation state is not expected
+to change rapidly, there will be no necessity to frequently evaluate
+the viewpoints. However, when utilizing larger entropy evaluation
+intervals, a larger amount of memory will be required for temporarily
+storing the simulation data. That is, there is a trade-off between the
+entropy evaluation intervals and the memory consumption, and as a
+result, depending on the simulation time step range and simulation
+data size, large entropy evaluation intervals, such as the utilized
+NE = 30 and NE = 50, may be sufficient to satisfy the R2.
+Regarding the influence of the number of viewpoints on the spher-
+ical surface, we verify that there was no significant difference for
+
+1.8 
+2.6
+37.5
+2.4
+Colatitude
+75.0
+2.2
+- 2.0
+112.5
+1.8
+150.0
+1.6
+1.4
+0.0
+72.0
+144.0
+216.0
+288.0
+Longitude1.8
+6.8
+37.5
+6.6
+latitude
+75.0
+6.4
+Col
+112.5
+6.2
+150.0
+6.0
+5.8
+0.0
+70.6
+141.2
+211.8
+282.4
+352.9
+Longitude1.8 
+6.9
+37.5
+6.8
+Colatitude
+6.7
+75.0
+6.6
+112.5
+6.5
+150.0
+6.4
+6.3
+0.0
+70.6
+141.2
+211.8
+282.4
+352.9
+Longitude1.8
+6.75
+37.5
+6.50
+6.25
+latitude
+75.0
+6.00
+Col
+112.5
+5.75
+5.50
+150.0
+5.25
+0.0
+70.6
+141.2
+211.8
+282.4
+352.9
+Longitude1.8
+3.00
+2.75
+37.5
+2.50
+Colatitude
+75.0
+2.25
+2.00
+112.5
+- 1.75
+150.0
+1.50
+1.25
+0.0
+72.0
+144.0
+216.0
+288.0
+Longitude1.8 
+2.8
+37.5
+2.6
+Colatitude
+75.0
+2.4
+- 2.2
+112.5
+2.0
+150.0
+1.8
+0.0 
+72.0
+144.0
+216.0
+288.0
+Longitude1.8
+2.8
+37.5
+2.6
+Colatitude
+75.0
+ 2.4
+112.5
+2.2
+2.0
+150.0
+1.8
+0.0
+72.0
+144.0
+216.0
+288.0
+Longitude1.8
+4.6
+37.5
+4.4
+latitude
+4.2
+75.0
+4.0
+Col
+112.5
+3.8
+150.0
+ 3.6
+3.4
+0.0
+70.6
+141.2
+211.8
+282.4
+352.9
+Longitude1.8
+4.8
+37.5
+4.6
+latitude
+4.4
+75.0
+4.2
+Col
+112.5
+4.0
+150.0
+3.8
+3.6
+0.0
+70.6
+141.2
+211.8
+282.4
+352.9
+Longitude1.8
+4.8
+37.5
+4.7
+Colatitude
+4.6
+75.0
+4.5
+112.5
+4.4
+150.0
+4.3
+4.2
+0.0
+70.6
+141.2
+211.8
+282.4
+352.9
+Longitude1.8
+4.6
+37.5
+latitude
+- 4.4
+75.0
+4.2
+Col
+112.5
+4.0
+150.0
+3.8
+0.0
+70.6
+141.2
+211.8
+282.4
+352.9
+Longitude1.8
+6.8
+6.6
+37.5
+6.4
+Colatitude
+75.0
+6.2
+- 6.0
+112.5
+5.8
+150.0
+5.6
+5.4
+0.0
+70.6
+141.2
+211.8
+282.4
+352.9
+LongitudeCSimulation step:
+2400
+6000
+9600
+13200
+25x50:
+15x30:
+35x70:
+Figure 8: Lightness Entropy heatmaps for different diverging color maps.
+varying number of viewpoints. However, it is worth noting that the
+computational time required to select the viewpoints will increase
+proportionately with the increase in the number of viewpoints.
+Regarding the influence of the quaternion interpolation method
+for estimating the camera path between selected viewpoints, we
+observed that the camera path using SQUAD-based interpolation
+passes closer to the viewpoint with highest entropy at the interme-
+diate time steps. We also observed that jerky camera movements
+tend to occur when using the SLERP-based interpolation. On the
+other hand, smoother camera movement was observed when using
+the SQUAD-based interpolation, and as a result, we can consider
+that it will cause less discomfort to the user when seeing the ani-
+mated rendering results since the camera movement will be more
+natural. Therefore, we can consider that SQUAD-based quaternion
+interpolation satisfies the R2.
+Moreover, we carried out some evaluations with the domain sci-
+entists who assisted in the development of previous work on in-situ
+adaptive timestep selection [26]. We obtained technical feedback
+from the generated visualization results in the form of animated
+videos. According to them, the video generated by using the pro-
+posed method seems to present more information than the video
+generated by using fixed viewpoint camera settings, which has tradi-
+tionally been used in their simulation analysis. However, they also
+pointed out that the proposed video gives the impression of exces-
+sive movement and sometimes tracking phenomena that do not need
+much attention. As some suggestions, they mentioned that it would
+be better to slightly reduce high viewpoint variations or suppress
+unnecessary movement, and to improve evaluation methods for the
+viewpoint selection. As an additional suggestion, they would prefer
+to have the ability to zoom in on the target object to enable closer
+observation. These suggestions will be taken into consideration for
+further developments planned as future works.
+In our current implementation, the set of volume data in the en-
+tropy evaluation interval needs to be stored in the memory before
+the processing, and this memory cost can become an impediment
+for memory-hungry simulations. However, we consider that this
+approach can be useful during test runs and model calibration runs,
+before the main simulation run, when smaller models are usually
+sufficient. In addition, the in-transit approach for flushing the simu-
+lation data from the memory to another node or even system can be
+considered helpful for minimizing this problem and is planned for
+future work. Another planned future work is the application of the
+adaptive timestep sampling [26] where larger time intervals will be
+assigned to timestep regions with small variations between the simu-
+lation results. This larger entropy evaluation time step by skipping
+some simulation results may be helpful for accelerating the visu-
+alization processing as well as reducing the excessive movements
+pointed out by the domain scientists.
+5
+CONCLUSIONS
+In this work, we proposed an information entropy-based camera path
+estimation method for in-situ visualization. Considering that most
+of the images generated by traditional batch-based tightly coupled
+in-situ visualization may have small or even no contribution for the
+post-hoc visual analysis, we focused on generating a smooth video
+that tries to provide as much information as possible to facilitate the
+rapid understanding of the simulation or to narrow down the spatio-
+temporal region of interest for posterior detailed analysis such as by
+using traditional image-based visualization. The proposed method
+focuses on selecting the most appropriate viewpoints, based on in-
+formation entropy, at regular intervals. Intermediate images are
+generated from the estimated camera path connecting these selected
+viewpoints, and the produced smooth video that is produced is ex-
+pected to be helpful for understanding the underlying simulation
+phenomena. From the experimental evaluations and feedback from
+domain scientists, we can confirm that the video generated by the
+proposed approach provides more information compared to those
+generated by using fixed viewpoint camera settings. However, there
+is still need for improvements, and we can cite the following targets
+for future works: implementation of better evaluation methods for
+the viewpoint selection; implementation of zoom in and out func-
+tionalities; integration with the adaptive timestep sampling (irregular
+time intervals); improvement of computational performance such as
+by applying parallel processing; and estimation of the focal point
+for the camera.
+ACKNOWLEDGMENTS
+The authors are grateful to Tsukasa Yoshinaga (Toyohashi Univer-
+sity of Technology) and Kazunori Nozaki (Osaka University) for
+the simulation model and technical feedback. This work was par-
+tially supported by JSPS KAKENHI (Grant Numbers: 20H04194,
+21H04903, 22H03603), and the National Key R&D Program of
+China under Grant No. 2021YFE0108400. This work used compu-
+tational resources of supercomputer Fugaku provided by the RIKEN
+Center for Computational Science.
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+211.8
+282.4
+352.9
+Longitude1.8
+4.8
+37.5
+4.7
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+4.6
+75.0
+4.5
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+4.4
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+4.3
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+0.0
+70.6
+141.2
+211.8
+282.4
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+Longitude1.8
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+3.6
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+0.0
+72.0
+144.0
+216.0
+288.0
+Longitude1.8 -
+4.8
+38.6
+4.6
+Colatitude
+4.4
+77.1
+4.2
+115.7
+4.0
+154.3
+3.8
+3.6
+0.0
+72.0
+144.0
+216.0
+288.0
+Longitude1.8 -
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+Colatitude
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+77.1
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+154.3
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+4.6
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+111.2
+3.8
+ 3.6
+148.2
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+- 4.4
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+arXiv:2301.01474v1  [eess.SY]  4 Jan 2023
+1
+UAV-aided Metaverse over Wireless
+Communications: A Reinforcement Learning
+Approach
+Peiyuan Si1, Wenhan Yu1, Jun Zhao1, Kwok-Yan Lam1, Qing Yang2
+1School of Computer Science & Engineering
+Nanyang Technological University, Singapore
+2University of North Texas, United States
+{peiyuan001, wenhan002}@e.ntu.edu.sg, {junzhao, kwokyan.lam}@ntu.edu.sg, Qing.yang@unt.edu
+Abstract—Metaverse is expected to create a virtual world
+closely connected with reality to provide users with immersive
+experience with the support of 5G high data rate communication
+technique. A huge amount of data in physical world needs to be
+synchronized to the virtual world to provide immersive experi-
+ence for users, and there will be higher requirements on coverage
+to include more users into Metaverse. However, 5G signal suffers
+severe attenuation, which makes it more expensive to maintain
+the same coverage. Unmanned aerial vehicle (UAV) is a promising
+candidate technique for future implementation of Metaverse
+as a low-cost and high-mobility platform for communication
+devices. In this paper, we propose a proximal policy optimization
+(PPO) based double-agent cooperative reinforcement learning
+method for channel allocation and trajectory control of UAV
+to collect and synchronize data from the physical world to the
+virtual world, and expand the coverage of Metaverse services
+economically. Simulation results show that our proposed method
+is able to achieve better performance compared to the benchmark
+approaches.
+Index
+Terms—Metaverse,
+UAV,
+cooperative
+reinforcement
+learning, PPO
+I. INTRODUCTION
+The proposal of Metaverse has been promoted by the im-
+plementation of 5G communication technology and maturing
+AR/VR devices in recent years [1]–[4]. Metaverse aims to
+create a virtual world for all kinds of activities, including
+education, trading and gaming, and is considered the next
+generation of the Internet [?], [5], [7], [8]. With the support of
+AR/VR applications, online users are provided with immersive
+services that are similar to in-person activities, and the trading
+of virtual items brings job opportunities.
+To support the Metaverse applications, data synchronization
+and wide wireless network coverage are two practical prob-
+lems to be solved as the Metaverse services usually involve
+wearable wireless devices. For the first problem, 5G commu-
+nication technology is able to provide high-speed and low-
+latency data transmission, but it is not necessary to update all
+the collected data immediately, e.g., environment information
+to build the background of Metaverse and offline trading
+records [9]–[11]. For the second problem, 5G network suffers
+higher costs for the same coverage area due to severe signal
+attenuation. Thus, it is not economically efficient to deploy
+base stations in suburban with low population density, and in
+wild areas it is not even applicable to traditional base stations
+[12].
+Unmanned aerial vehicle (UAV) is a cheaper substitution
+solution to set up network coverage for Metaverse data syn-
+chronization in the suburban area due to its ability to carry
+communication devices. The UAV technique has been fully
+studied and commercialized, and there are numerous works
+on UAV-based communication scenarios for traditional appli-
+cations, e.g., research on communication resource allocation,
+UAV trajectory control and the internet of vehicles [13]–
+[15]. The UAV-based optimization problems which take the
+trajectory of UAV into consideration usually segment the flight
+time of UAV into discrete time slots for the convenience of
+computation. The resource allocation variables need to be
+optimized in each time slot to obtain the global or local
+optimal. Although these methods ensure the convergence of
+the solution, the increasing number of time slots results to
+the increment of algorithm complexity. Besides, the integer
+characteristic of channel allocation variables results to mixed
+integer programming problems, which can be hard to solve if
+the variables are inseparable.
+Related Work. In some cases, reinforcement learning (RL)
+is more suitable for UAV-based optimization problems than
+convex methods because it gives a feasible solution with
+relatively good performance even if the global optimal is
+extremely hard to find, and it can handle time-sequential
+problems without increasing the number of variables. Cui et
+al. [16] proposed multi-agent reinforcement learning resource
+allocation algorithm for multi-UAV networks, and showed fast
+convergence with the basic Q-learning algorithm. Luong et
+al. [17] utilized the deep Q-learning algorithm to learn the
+network state for the decision of the movement of UAV, and
+improved the network performance by up to 70%. Rodriguez-
+Ramos et al. [18] implemented a versatile Gazebo-based rein-
+forcement learning framework for UAV landing on a moving
+platform, which is a novel experiment of DDPG on UAV
+controlling research.
+For communication optimization problems with discrete
+channels and continuous resource allocation, both discrete and
+continuous action spaces need to be considered. To solve
+
+2
+discrete-continuous hybrid action space reinforcement learning
+problems, multi-agent architecture is commonly adopted. Fu
+et al. [19] proposed two multi-agent reinforcement learning
+architectures for hybrid action spaces based on deep Q-
+learning (DQN), where agents work in a parallel manner to
+generate joint actions. Jiang et al. [20] designed a hybrid action
+algorithm for massive access control, which optimized the
+discrete action selection for back-off and distributed queuing
+problems and generate continuous action for access class
+barring.
+The agents of most existing hybrid action space reinforce-
+ment learning algorithms work in a parallel manner, which
+does not build the inter-agent relationship. In this paper,
+we propose a hybrid reinforcement learning architecture to
+optimize the discrete channel allocation variable and the
+continuous trajectory controlling variable. Two agents work
+in a sequential manner motivated by the alternative optimiza-
+tion algorithms, i.e., the output of an agent is the input of
+another agent. Compared to the existing works, our paper
+considers the inter-agent relationship for better convergence
+performance. The advantage of our scenario over traditional
+convex optimization is that the number of variables does not
+increase when the number of time slots increases, which is
+more friendly to time-sequential problems.
+Contribution. The contributions of this paper are as fol-
+lows:
+• A PPO-based double-agent cooperative hybrid action
+reinforcement learning architecture (PPO-PPO) for UAV-
+enabled Metaverse data synchronization is proposed.
+• Proximal policy optimization (PPO) algorithm is imple-
+mented in both discrete action agents and continuous ac-
+tion agents, and two agents work in a sequential manner.
+• The simulation shows the comparison between the pro-
+posed algorithm and two baselines (DQN and duelling
+DQN), which verifies the advantage of our proposed
+PPO-PPO algorithm.
+The rest of this paper is organized as follows. Section
+II introduces the proposed system model. The double-agent
+policy generation model and its implementation are presented
+in Section III and Section IV, respectively. Section V shows
+the simulation results and the corresponding explanation. The
+conclusion of this paper is discussed in Section VI.
+II. SYSTEM MODEL
+As shown in Fig. 1, we consider a UAV-based uplink data
+collection system for Metaverse service. In a given L × L
+area which is beyond the coverage of 5G base station, N
+Metaverse data collectors (MDCs) are deployed to collect
+delay-insensitive local data, such as offline digital currency
+trading and weather information, which are generated by
+Metaverse users or the sensors [21], [22]. The location of
+MDC n is denoted by (xn, yn, 0). MDCs are assumed to have
+enough energy but limited transmission power.
+To synchronize the local data with the Metaverse server,
+one mobile base station (MBS) carried by UAV is deployed
+to collect the local data saved at MDCs through M channels.
+Each MDC can occupy only one channel, but multiple MDCs
+are able to share one channel. The set of MDCs in channel m is
+denoted by Nm, and the number of MDCs in the set is denoted
+as Nm. We assume that the UAV flies at a fixed height H, and
+the location of UAV is denoted by (xuav[t], yuav[t], H). Once
+the data is received by the MBS, MDCs clear the historical
+data and get ready for the future data collection. In this paper,
+we assume that the local data size of each receiver is U.
+A. Channel Settings
+According to the experimental characterization of the
+vehicle-to-infrastructure radio channels in suburban environ-
+ments implemented by M. Yusuf et al, the small-scale fading
+of the strongest path is found to be Rician distributed [23].
+The channel gain between UAV and MDC n in channel m
+and time slot t is given by [24]
+hn,m[t] =
+�
+βn[t]gn,m[t],
+(1)
+where βn[t] denotes the large-scale average channel gain
+at time slot t, and gn,m[t] denotes the small-scale fading
+coefficient, which is modelled as Rician fading. βn[t] and
+gn,m[t] are given by
+βn[t] = β0d−α
+n [t],
+(2)
+and
+gn,m[t] =
+�
+K
+K + 1g +
+�
+1
+K + 1 ˜g,
+(3)
+where β0 denotes the channel gain at the reference distance
+d0 = 1m, α denotes the path loss exponent, which varies from
+2 to 6 (in this paper we assume that α = 2). g denotes the
+deterministic LoS channel component with |g| = 1, which
+denotes the randomly scattered component. The Rician factor
+is denoted by K. dn[t] denotes the distance from UAV to MDC
+n in time slot t, which is given by
+dn[t] =
+�
+(xn − xuav[t])2 + (yn − yuav[t])2 + H2.
+(4)
+The channel-to-noise-ratio (CNR) is given by
+Γn,m[t] = hn,m[t]
+Bσ2
+(5)
+where σ2 denotes the power of additive white Gaussian noise
+(AWGN) at the receiver. The signal to interference plus noise
+ratio (SINR) of MDC n in channel m in time slot t is given
+by
+γn,m[t] =
+pn,m[t]Γn,m[t]
+1 +
+|Nm|−1
+�
+i=1
+pi,m[t]Γi,m[t]
+,
+(6)
+where pn,m denotes the transmission power of MDCs. Thus,
+the transmission rate of MDC n in channel m and time slot t
+is given by
+Rn,m[t] = Blog2(1 + γ).
+(7)
+
+3
+Fig. 1: System model.
+Channel Allocation
+Trajectory Control
+Environment
+UAV
+MDR
+MDR
+Discrete PPO
+Continuous PPO
+Combined Action
+Reward
+uav
+uav
+ˆ
+{ [ ],
+[
+1],
+[
+1]}
+I t
+x
+t
+y
+t
+�
+�
+Critic
+ch
+ta
+Actor
+BP
+Critic
+ch
+ta
+Actor
+BP
+Critic
+ch
+ta
+Actor
+BP
+Critic
+ch
+ta
+Actor
+BP
+ch
+ta
+Fig. 2: Double-agent policy generation model.
+III. DOUBLE-AGENT POLICY GENERATION MODEL
+In this section, we introduce the double-agent policy gener-
+ation model based on PPO (PPO-PPO) for channel allocation
+and UAV trajectory control, which is shown in Fig. 2.
+The objective is to minimize the total required time for UAV
+to finish collecting the data saved at MDCs with the constraint
+of maximum UAV speed by optimizing channel allocation
+indicator matrix I[t], and UAV trajectory {xuav[t], yuav[t]}.
+Each agent only focuses on a specific type of variable, and
+the values of other variables are loaded from the results
+of another agent in the previous step. In each step, the
+discrete proximal policy optimization (PPO) agent generates
+the channel allocation according to its policy, and forwards the
+result to the continuous PPO agent for trajectory generation.
+The combined action is generated by concatenating the output
+of two RL agents which interact with the environment to get
+reward for both RL agents.
+A. Discrete Agent for Channel Allocation
+In this subsection, we will introduce the action space, state
+space and reward settings of the discrete agent for channel
+allocation.
+1) Action of the Discrete Agent: Intuitively, the channel
+allocation indicator I[t] can be defined as an one-hot matrix,
+i.e., In,m[t] ∈ {0, 1} denotes if channel m is selected by MDC
+n. An example with the number of users N = 4 and number
+of channels M = 3 is given by
+I[t] =
+
+
+I1,1[t]
+I1,2[t]
+I1,3[t]
+I2,1[t]
+I2,2[t]
+I2,3[t]
+I3,1[t]
+I3,2[t]
+I3,3[t]
+I4,1[t]
+I4,2[t]
+I4,3[t]
+
+ ,
+(8)
+whose dimension is N × M. The one-hot definition of I[t] is
+intuitive but increases the dimension of action space. To reduce
+the dimension, we re-define the channel allocation indicator
+matrix as ˆI[t], whose elements are ˆIn[t] ∈ {0, 1, .., M}. Under
+this definition, ˆIn[t] = m indicates that MDC n is assigned
+with channel m, and ˆIn[t] = 0 indicates that it is not assigned
+with any channel.
+ˆ[ ]
+I t
+1ˆ [ ]
+I t
+2ˆ [ ]
+I t
+3ˆ [ ]
+I t
+4ˆ [ ]
+I t
+5ˆ [ ]
+I t
+6ˆ [ ]
+I t
+Action
+�
+�
+�
+0
+M
+1
+�
+�
+�
+�
+�
+1
+M
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2
+M
+1
+�
+�
+�
+3
+M
+1
+�
+�
+�
+4
+M
+1
+�
+�
+�
+5
+M
+1
+�
+Fig. 3: Action encoding.
+As shown in Fig. 3, the action of the agent is encoded
+according to the channel allocation indicator matrix. The
+encoded action is given by
+ach
+t =
+N
+�
+n=1
+ˆIn[t](M + 1)n−1
+(9)
+2) State of the Discrete Agent: The decisions of RL agents
+are generated based on the current state. In this paper, the
+state of the discrete agent includes the channel gain and the
+remaining data at MDCs in the current step. The state of the
+discrete agent is concatenated by two parts, which is given by
+Sch
+t = {Ures[t], h[t]},
+(10)
+where Ures denotes the matrix of remaining data in MDCs,
+and h[t] denotes the matrix of channel gain at tth step.
+3) Reward of the Discrete Agent: The optimization objec-
+tive in this paper is the required time for UAV to finish the
+data collection mission, i.e., to minimize the number of steps
+in each episode. Intuitively, the more steps the agent takes, the
+
+181(0)UAV
+MDCT
+BS4
+less reward it should receive. Thus, we set a time-based penalty
+rtime
+t
+with negative value in each step to build the connection
+between reward and our objective. If the agent fails to finish
+the mission in given time limit Tmax, it will receive a failure
+penalty rfail
+t .
+The time-based penalty rtime
+t
+is further modified according
+to the data size collected by UAV in the current step to give
+higher reward to the actions which result to larger transmission
+rate. The reward of the discrete agent is given by
+rch
+t =
+
+
+
+
+
+rtime
+t
+U
+N
+�
+n=1
+M
+�
+m=1
+tslotRn,m[t], if t ≤ Tmax
+rfail
+t , if t > Tmax
+(11)
+B. Continuous Agent for Trajectory Optimization
+The trajectory of UAV is optimized by a continuous RL
+agent, whose action, state and reward are defined as follows.
+Mission Area
+slot
+max
+2t
+V
+slot
+max
+2t
+V
+Action 
+Space (t+1)
+Traj
+ta
+Action 
+Space (t)
+Fig. 4: Action space of the continuous agent.
+1) Action of the continuous agent: As shown in Fig. 4, the
+action of the continuous agent atraj
+t
+determines the location of
+UAV in the next step. atraj
+t
+is defined as
+atraj
+t
+= {ax
+t , ay
+t }, ax
+t , ay
+t ∈ [−tslotVmax, tslotVmax] ,
+(12)
+where ax
+t , ay
+t denote the movement of UAV on the x-axis and
+y-axis respectively.
+2) State of the Continuous Agent: The state of the con-
+tinuous agent is similar to the discrete agent, which includes
+the current channel gain h[t] and remaining data at MDCs
+Ures[t]. In addition, the current horizontal location of UAV
+(xuav[t], yuav[t]) is also included in the state Straj
+t , which is
+given by
+Straj
+t
+= {Ures[t], h[t], (xuav[t], yuav[t])}
+(13)
+3) Reward of the Continuous Agent: The reward of the
+continuous agent is modified based on rch
+t . We give additional
+penalty to the agent if the location of UAV exceeds reasonable
+region to regularize the trajectory decision. The reward of the
+continuous agent is given by
+rtraj
+t
+=
+�
+rch
+t , if xuav[t] ∈ [xmin, xmax], yuav[t] ∈ [ymin, ymax]
+rch
+t + rpenalty
+t
+, otherwise
+(14)
+IV. IMPLEMENTATION OF PROXIMAL POLICY
+OPTIMIZATION (PPO)
+PPO is a state-of-art on policy reinforcement learning al-
+gorithm which supports both discrete and continuous actions
+spaces. In this section, we introduce the preliminary and
+implementation of PPO algorithm for discrete agent (channel
+allocation) and continuous agent (UAV trajectory optimiza-
+tion).
+A. Implementation of Continuous and Discrete PPO
+1) Critic Network: The critic network is responsible to
+give scores to the actor according to the current state. The
+architectures of both discrete and continuous critic networks
+are the same, which consists of multiple fully connected layers.
+Loss function of continuous and discrete critic networks
+are given by
+Jtraj(φ) =
+�
+V traj
+φ (straj
+t ) −
+�
+rtraj
+t
++ γV traj
+φ′ (straj
+t+1)
+��2
+,
+(15)
+Jch(φ) =
+�
+V ch
+φ (sch
+t ) −
+�
+rch
+t + γV ch
+φ′ (sch
+t+1)
+��2,
+(16)
+where Ltraj
+t (φ) and Lch
+t (φ) denote the loss function for the
+critic network of continuous and discrete agent respectively.
+V traj
+φ′ (straj
+t+1) and V ch
+φ′ (sch
+t+1) are the state value estimations
+generated by the old critic networks φ
+′
+traj and φ
+′
+ch respectively,
+which are saved in during the interaction with environment.
+V traj
+φ (straj
+t ) and V ch
+φ (sch
+t ) are the state value estimations gener-
+ated by the current critic networks φtraj and φch , which are
+updated in each training iteration.
+2) Actor Network: As shown in Fig. 5, the architecture of
+discrete and continuous actor network are different due to the
+difference in action space.
+The continuous actor network for trajectory control is a
+network for value approximation, which outputs a µ head and
+a σ head which denotes the mean and variance of Gaussian dis-
+tributions respectively. Each head includes two variables, i.e.,
+{µx, µy} and {σx, σy}, which denotes the x-axis and y-axis
+respectively. The action {ax[t], ay[t]} is generated by sampling
+from the obtained distribution N(µx, σ2
+x) and N(µy, σ2
+y).
+The discrete actor network for channel allocation is a
+network for classification, which outputs the probabilities
+Pr(a) of each action. The agent sample its action from the
+obtained action probabilities with ε-greedy, i.e., the output
+action is generated by sampling from Pr(a) with probability
+1−ǫ, and selected randomly with probability ǫ for exploration.
+The output action of the discrete actor network is encoded,
+which will be decoded into one-hot indicators before being
+utilized for further calculation.
+Loss functions of actor networks in our implementation
+adopt the trick of clipping to simplify the calculation, which
+is proposed by J. Schulman et al [25].
+The PPO-PPO algorithm is summarized in Algorithm 1.
+
+(0)5
+Fully Conneted Layers
+Softmax Layer
+State
+Head
+� Head
+�
+Head
+� Head
+�
+Trajectory
+2
+( ,
+)
+N � �
+Sample From
+2
+( ,
+)
+N � �
+Sample From
+Fully Conneted Layers
+Softmax Layer
+State
+Channel Allocation
+Pr( )
+a
+Action Probability
+�
+Sample with -Greedy
+Fig. 5: Actor Network Architecture.
+Algorithm 1 PPO-PPO
+Initiate: Remaining data at MDCs, UAV location, network
+parameters of discrete and continuous agent
+1: for iteration t = 1, 2, .. do
+2:
+Discrete agent execute action according to the current
+state and policy πch
+θ′
+�
+ach
+t |sch
+t
+�
+to obtain the channel
+allocation indicator matrix ˆI[t]
+3:
+With given ˆI[t], the continuous agent for trajectory
+control execute action according to the current state and
+policy πtraj
+θ′
+�
+atraj
+t |straj
+t
+�
+ˆI[t]
+4:
+Agent interact with environment to get reward rch
+t and
+rtraj
+t
+for discrete agent and continuous agent respectively
+5:
+Update state straj
+t
+← straj
+t+1, sch
+t ← sch
+t+1
+6:
+Save
+trajectory
+�
+sch
+t , ach
+t , rch
+t , sch
+t+1, V ch
+φ′ (sch
+t )
+�
+and
+�
+straj
+t , atraj
+t , rtraj
+t , straj
+t , V traj
+φ′ (straj
+t )
+�
+7:
+for every i iterations do
+8:
+Shuffle data order and make batch with size bs.
+9:
+for j=0, 1, ..., T
+bs − 1 do
+10:
+Calculate loss functions of critic and actor net-
+works and update network parameters by gradient
+ascent
+11:
+end for
+12:
+end for
+13: end for
+V. SIMULATION RESULTS
+The performance of our proposed double-agent reinforce-
+ment learning approach for Metaverse data collecting is tested
+and compared with two benchmark scenarios (DQN-PPO and
+duelling DQN-PPO), whose discrete agents are replaced with
+DQN or duelling DQN algorithm respectively. The simulation
+settings are given in Table I.
+Fig. 6 presents the required time to complete data collecting
+mission of our proposed algorithm and two benchmark algo-
+rithms with given data size U = 50Mb. At the beginning of
+the training process (0-1000 episodes), all three algorithms
+TABLE I: Constant Parameter Setting
+Parameter and Physical Meaning
+Value
+Number of channels(M)
+3
+Default number of users (N)
+5
+Bandwidth (B)
+5MHz
+Transmission power of MDCs
+5W
+Frequency (f)
+28GHz (5G spectrum)
+Power of Gaussian noise (σ2)
+5 × 10−8W
+Maximum speed of UAV
+10m/s
+Mission area size (L)
+200m
+are unstable because the reasonable policy has not been
+established, and the agents are exploring the environment fre-
+quently. From 1000 episodes to 2000 episodes, our proposed
+PPO-PPO algorithm shows the tendency of convergence while
+the benchmark DQN-PPO algorithm is still very unstable.
+The duelling DQN-PPO algorithm also starts to finish the
+mission within a shorter time period, but is less stable than
+the PPO-PPO algorithm. DQN-PPO algorithm shows poor
+convergence performance in this task, but both PPO-PPO and
+duelling DQN-PPO algorithms are able to converge within
+5000 episodes with similar performance due to their common
+implementation of the advantage function.
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Episodes
+30
+35
+40
+45
+50
+55
+60
+Time (s)
+DQN-PPO
+Duelling DQN-PPO
+PPO-PPO
+Fig. 6: Comparison of required time to finish mission with
+data size U = 50Mb.
+Fig. 7 presents the mission completing time experiment with
+a similar parameter setting as in Fig. 6, but the data size is
+increased to U = 100Mb. All three algorithms need more
+time to finish the data collecting mission due to larger data
+size, and the PPO-PPO algorithm shows similar convergence
+performance as in Fig. 6. However, the dueling DQN-PPO
+algorithm becomes unstable in the training process, i.e., some
+sudden increase in the required time. The superior stability
+of PPO over dueling DQN can be attributed to its policy
+update constraint by equipping it with a KL-divergence penalty
+between the old policy (the policy for sampling data) and the
+updated policy (the policy used for training and evaluating).
+Fig. 8 and Fig. 9 are the corresponding rewards in the train-
+ing processes of Fig. 6 and Fig. 7 respectively. We consider
+the reward given to the agent as guidance but not the exact
+objective function in the implementation of reinforcement
+
+6
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Episodes
+60
+70
+80
+90
+100
+110
+120
+130
+140
+150
+160
+Time (s)
+DQN-PPO
+Duelling DQN-PPO
+PPO-PPO
+Fig. 7: Comparison of required time to finish mission with
+data size U = 100Mb.
+learning algorithm. The tendencies of the reward and the
+required time are highly similar although they are generated
+from different formulas, which indicates that our reward design
+successfully leads the agent to learn a better policy.
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Episodes
+-3000
+-2500
+-2000
+-1500
+-1000
+-500
+0
+Reward
+DQN-PPO
+Duelling DQN-PPO
+PPO-PPO
+Fig. 8: Comparison of reward with data size U = 50Mb.
+The mission completing time comparison for the case with
+eight users is shown in Fig. 10. The duelling DQN-PPO
+algorithm shows similar average performance as the PPO-PPO
+algorithm but less stability, i.e., the required time sometimes
+jumps to extremely large values. Taking the stability into
+consideration, the PPO-PPO algorithm is better than duelling
+DQN-PPO algorithm in general. The DQN-PPO algorithm is
+obviously not able to converge in this experiment, so we do
+not consider it a candidate for our double-agent reinforcement
+learning algorithm.
+VI. CONCLUSION
+In this paper, we propose a double-agent reinforcement ar-
+chitecture for data collecting and synchronization in Metavese,
+and adopt PPO algorithm for both discrete and continuous
+agents. Two agents with different action space and state space
+work in a cascade manner for channel allocation and UAV
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Episodes
+-3000
+-2500
+-2000
+-1500
+-1000
+-500
+Reward
+DQN-PPO
+Duelling DQN-PPO
+PPO-PPO
+Fig. 9: Comparison of reward with data size U = 100Mb.
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Episodes
+50
+100
+150
+200
+Time (s)
+DQN-PPO
+Duelling DQN-PPO
+PPO-PPO
+Fig. 10: Comparison of reward with data size U = 50Mb and
+8 users.
+trajectory control to form a combined action in each iteration.
+Our experiments indicate the advantage of the PPO-PPO in
+both the required time for the mission and the stability. In
+future work, we will consider transmission power allocation
+and test the performance of other state-of-art reinforcement
+learning algorithms in our proposed architecture.
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+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='01474v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='SY]  4 Jan 2023  1  UAV-aided Metaverse over Wireless  Communications: A Reinforcement Learning  Approach  Peiyuan Si1, Wenhan Yu1, Jun Zhao1, Kwok-Yan Lam1, Qing Yang2  1School of Computer Science & Engineering  Nanyang Technological University, Singapore  2University of North Texas, United States  {peiyuan001, wenhan002}@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='sg, {junzhao, kwokyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='lam}@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='sg, Qing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='yang@unt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='edu  Abstract—Metaverse is expected to create a virtual world  closely connected with reality to provide users with immersive  experience with the support of 5G high data rate communication  technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' A huge amount of data in physical world needs to be  synchronized to the virtual world to provide immersive experi-  ence for users, and there will be higher requirements on coverage  to include more users into Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' However, 5G signal suffers  severe attenuation, which makes it more expensive to maintain  the same coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Unmanned aerial vehicle (UAV) is a promising  candidate technique for future implementation of Metaverse  as a low-cost and high-mobility platform for communication  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In this paper, we propose a proximal policy optimization  (PPO) based double-agent cooperative reinforcement learning  method for channel allocation and trajectory control of UAV  to collect and synchronize data from the physical world to the  virtual world, and expand the coverage of Metaverse services  economically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Simulation results show that our proposed method  is able to achieve better performance compared to the benchmark  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Index  Terms—Metaverse,  UAV,  cooperative  reinforcement  learning, PPO  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' INTRODUCTION  The proposal of Metaverse has been promoted by the im-  plementation of 5G communication technology and maturing  AR/VR devices in recent years [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Metaverse aims to  create a virtual world for all kinds of activities, including  education, trading and gaming, and is considered the next  generation of the Internet [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' ], [5], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' With the support of  AR/VR applications, online users are provided with immersive  services that are similar to in-person activities, and the trading  of virtual items brings job opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  To support the Metaverse applications, data synchronization  and wide wireless network coverage are two practical prob-  lems to be solved as the Metaverse services usually involve  wearable wireless devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' For the first problem, 5G commu-  nication technology is able to provide high-speed and low-  latency data transmission, but it is not necessary to update all  the collected data immediately, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', environment information  to build the background of Metaverse and offline trading  records [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' For the second problem, 5G network suffers  higher costs for the same coverage area due to severe signal  attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Thus, it is not economically efficient to deploy  base stations in suburban with low population density, and in  wild areas it is not even applicable to traditional base stations  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Unmanned aerial vehicle (UAV) is a cheaper substitution  solution to set up network coverage for Metaverse data syn-  chronization in the suburban area due to its ability to carry  communication devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The UAV technique has been fully  studied and commercialized, and there are numerous works  on UAV-based communication scenarios for traditional appli-  cations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', research on communication resource allocation,  UAV trajectory control and the internet of vehicles [13]–  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The UAV-based optimization problems which take the  trajectory of UAV into consideration usually segment the flight  time of UAV into discrete time slots for the convenience of  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The resource allocation variables need to be  optimized in each time slot to obtain the global or local  optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Although these methods ensure the convergence of  the solution, the increasing number of time slots results to  the increment of algorithm complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Besides, the integer  characteristic of channel allocation variables results to mixed  integer programming problems, which can be hard to solve if  the variables are inseparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In some cases, reinforcement learning (RL)  is more suitable for UAV-based optimization problems than  convex methods because it gives a feasible solution with  relatively good performance even if the global optimal is  extremely hard to find, and it can handle time-sequential  problems without increasing the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Cui et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' [16] proposed multi-agent reinforcement learning resource  allocation algorithm for multi-UAV networks, and showed fast  convergence with the basic Q-learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Luong et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' [17] utilized the deep Q-learning algorithm to learn the  network state for the decision of the movement of UAV, and  improved the network performance by up to 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Rodriguez-  Ramos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' [18] implemented a versatile Gazebo-based rein-  forcement learning framework for UAV landing on a moving  platform, which is a novel experiment of DDPG on UAV  controlling research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  For communication optimization problems with discrete  channels and continuous resource allocation, both discrete and  continuous action spaces need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' To solve  2  discrete-continuous hybrid action space reinforcement learning  problems, multi-agent architecture is commonly adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Fu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' [19] proposed two multi-agent reinforcement learning  architectures for hybrid action spaces based on deep Q-  learning (DQN), where agents work in a parallel manner to  generate joint actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' [20] designed a hybrid action  algorithm for massive access control, which optimized the  discrete action selection for back-off and distributed queuing  problems and generate continuous action for access class  barring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The agents of most existing hybrid action space reinforce-  ment learning algorithms work in a parallel manner, which  does not build the inter-agent relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In this paper,  we propose a hybrid reinforcement learning architecture to  optimize the discrete channel allocation variable and the  continuous trajectory controlling variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Two agents work  in a sequential manner motivated by the alternative optimiza-  tion algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', the output of an agent is the input of  another agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Compared to the existing works, our paper  considers the inter-agent relationship for better convergence  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The advantage of our scenario over traditional  convex optimization is that the number of variables does not  increase when the number of time slots increases, which is  more friendly to time-sequential problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The contributions of this paper are as fol-  lows:  A PPO-based double-agent cooperative hybrid action  reinforcement learning architecture (PPO-PPO) for UAV-  enabled Metaverse data synchronization is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Proximal policy optimization (PPO) algorithm is imple-  mented in both discrete action agents and continuous ac-  tion agents, and two agents work in a sequential manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The simulation shows the comparison between the pro-  posed algorithm and two baselines (DQN and duelling  DQN), which verifies the advantage of our proposed  PPO-PPO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Section  II introduces the proposed system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The double-agent  policy generation model and its implementation are presented  in Section III and Section IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Section V shows  the simulation results and the corresponding explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The  conclusion of this paper is discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' SYSTEM MODEL  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 1, we consider a UAV-based uplink data  collection system for Metaverse service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In a given L × L  area which is beyond the coverage of 5G base station, N  Metaverse data collectors (MDCs) are deployed to collect  delay-insensitive local data, such as offline digital currency  trading and weather information, which are generated by  Metaverse users or the sensors [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The location of  MDC n is denoted by (xn, yn, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' MDCs are assumed to have  enough energy but limited transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  To synchronize the local data with the Metaverse server,  one mobile base station (MBS) carried by UAV is deployed  to collect the local data saved at MDCs through M channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Each MDC can occupy only one channel, but multiple MDCs  are able to share one channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The set of MDCs in channel m is  denoted by Nm, and the number of MDCs in the set is denoted  as Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' We assume that the UAV flies at a fixed height H, and  the location of UAV is denoted by (xuav[t], yuav[t], H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Once  the data is received by the MBS, MDCs clear the historical  data and get ready for the future data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In this paper,  we assume that the local data size of each receiver is U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Channel Settings  According to the experimental characterization of the  vehicle-to-infrastructure radio channels in suburban environ-  ments implemented by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Yusuf et al, the small-scale fading  of the strongest path is found to be Rician distributed [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The channel gain between UAV and MDC n in channel m  and time slot t is given by [24]  hn,m[t] =  �  βn[t]gn,m[t],  (1)  where βn[t] denotes the large-scale average channel gain  at time slot t, and gn,m[t] denotes the small-scale fading  coefficient, which is modelled as Rician fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' βn[t] and  gn,m[t] are given by  βn[t] = β0d−α  n [t],  (2)  and  gn,m[t] =  �  K  K + 1g +  �  1  K + 1 ˜g,  (3)  where β0 denotes the channel gain at the reference distance  d0 = 1m, α denotes the path loss exponent, which varies from  2 to 6 (in this paper we assume that α = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' g denotes the  deterministic LoS channel component with |g| = 1, which  denotes the randomly scattered component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The Rician factor  is denoted by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' dn[t] denotes the distance from UAV to MDC  n in time slot t, which is given by  dn[t] =  �  (xn − xuav[t])2 + (yn − yuav[t])2 + H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  (4)  The channel-to-noise-ratio (CNR) is given by  Γn,m[t] = hn,m[t]  Bσ2  (5)  where σ2 denotes the power of additive white Gaussian noise  (AWGN) at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The signal to interference plus noise  ratio (SINR) of MDC n in channel m in time slot t is given  by  γn,m[t] =  pn,m[t]Γn,m[t]  1 +  |Nm|−1  �  i=1  pi,m[t]Γi,m[t]  ,  (6)  where pn,m denotes the transmission power of MDCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Thus,  the transmission rate of MDC n in channel m and time slot t  is given by  Rn,m[t] = Blog2(1 + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  (7)  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 1: System model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Channel Allocation  Trajectory Control  Environment  UAV  MDR  MDR  Discrete PPO  Continuous PPO  Combined Action  Reward  uav  uav  ˆ  { [ ],  [  1],  [  1]}  I t  x  t  y  t  �  �  Critic  ch  ta  Actor  BP  Critic  ch  ta  Actor  BP  Critic  ch  ta  Actor  BP  Critic  ch  ta  Actor  BP  ch  ta  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 2: Double-agent policy generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' DOUBLE-AGENT POLICY GENERATION MODEL  In this section, we introduce the double-agent policy gener-  ation model based on PPO (PPO-PPO) for channel allocation  and UAV trajectory control, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The objective is to minimize the total required time for UAV  to finish collecting the data saved at MDCs with the constraint  of maximum UAV speed by optimizing channel allocation  indicator matrix I[t], and UAV trajectory {xuav[t], yuav[t]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Each agent only focuses on a specific type of variable, and  the values of other variables are loaded from the results  of another agent in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In each step, the  discrete proximal policy optimization (PPO) agent generates  the channel allocation according to its policy, and forwards the  result to the continuous PPO agent for trajectory generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The combined action is generated by concatenating the output  of two RL agents which interact with the environment to get  reward for both RL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Discrete Agent for Channel Allocation  In this subsection, we will introduce the action space, state  space and reward settings of the discrete agent for channel  allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  1) Action of the Discrete Agent: Intuitively, the channel  allocation indicator I[t] can be defined as an one-hot matrix,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', In,m[t] ∈ {0, 1} denotes if channel m is selected by MDC  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' An example with the number of users N = 4 and number  of channels M = 3 is given by  I[t] =  \uf8ee  \uf8ef\uf8ef\uf8f0  I1,1[t]  I1,2[t]  I1,3[t]  I2,1[t]  I2,2[t]  I2,3[t]  I3,1[t]  I3,2[t]  I3,3[t]  I4,1[t]  I4,2[t]  I4,3[t]  \uf8f9  \uf8fa\uf8fa\uf8fb ,  (8)  whose dimension is N × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The one-hot definition of I[t] is  intuitive but increases the dimension of action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' To reduce  the dimension, we re-define the channel allocation indicator  matrix as ˆI[t], whose elements are ˆIn[t] ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='., M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Under  this definition, ˆIn[t] = m indicates that MDC n is assigned  with channel m, and ˆIn[t] = 0 indicates that it is not assigned  with any channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  ˆ[ ]  I t  1ˆ [ ]  I t  2ˆ [ ]  I t  3ˆ [ ]  I t  4ˆ [ ]  I t  5ˆ [ ]  I t  6ˆ [ ]  I t  Action  �  �  �  0  M  1  �  �  �  �  �  1  M  1  �  �  �  �  �  �  �  �  �  �  �  2  M  1  �  �  �  3  M  1  �  �  �  4  M  1  �  �  �  5  M  1  �  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 3: Action encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 3, the action of the agent is encoded  according to the channel allocation indicator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The  encoded action is given by  ach  t =  N  �  n=1  ˆIn[t](M + 1)n−1  (9)  2) State of the Discrete Agent: The decisions of RL agents  are generated based on the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In this paper, the  state of the discrete agent includes the channel gain and the  remaining data at MDCs in the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The state of the  discrete agent is concatenated by two parts, which is given by  Sch  t = {Ures[t], h[t]},  (10)  where Ures denotes the matrix of remaining data in MDCs,  and h[t] denotes the matrix of channel gain at tth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  3) Reward of the Discrete Agent: The optimization objec-  tive in this paper is the required time for UAV to finish the  data collection mission, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', to minimize the number of steps  in each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Intuitively, the more steps the agent takes, the  181(0)UAV  MDCT  BS4  less reward it should receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Thus, we set a time-based penalty  rtime  t  with negative value in each step to build the connection  between reward and our objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' If the agent fails to finish  the mission in given time limit Tmax, it will receive a failure  penalty rfail  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The time-based penalty rtime  t  is further modified according  to the data size collected by UAV in the current step to give  higher reward to the actions which result to larger transmission  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The reward of the discrete agent is given by  rch  t =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  rtime  t  U  N  �  n=1  M  �  m=1  tslotRn,m[t], if t ≤ Tmax  rfail  t , if t > Tmax  (11)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Continuous Agent for Trajectory Optimization  The trajectory of UAV is optimized by a continuous RL  agent, whose action, state and reward are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Mission Area  slot  max  2t  V  slot  max  2t  V  Action  Space (t+1)  Traj  ta  Action  Space (t)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 4: Action space of the continuous agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  1) Action of the continuous agent: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 4, the  action of the continuous agent atraj  t  determines the location of  UAV in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' atraj  t  is defined as  atraj  t  = {ax  t , ay  t }, ax  t , ay  t ∈ [−tslotVmax, tslotVmax] ,  (12)  where ax  t , ay  t denote the movement of UAV on the x-axis and  y-axis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  2) State of the Continuous Agent: The state of the con-  tinuous agent is similar to the discrete agent, which includes  the current channel gain h[t] and remaining data at MDCs  Ures[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In addition, the current horizontal location of UAV  (xuav[t], yuav[t]) is also included in the state Straj  t , which is  given by  Straj  t  = {Ures[t], h[t], (xuav[t], yuav[t])}  (13)  3) Reward of the Continuous Agent: The reward of the  continuous agent is modified based on rch  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' We give additional  penalty to the agent if the location of UAV exceeds reasonable  region to regularize the trajectory decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The reward of the  continuous agent is given by  rtraj  t  =  �  rch  t , if xuav[t] ∈ [xmin, xmax], yuav[t] ∈ [ymin, ymax]  rch  t + rpenalty  t  , otherwise  (14)  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' IMPLEMENTATION OF PROXIMAL POLICY  OPTIMIZATION (PPO)  PPO is a state-of-art on policy reinforcement learning al-  gorithm which supports both discrete and continuous actions  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In this section, we introduce the preliminary and  implementation of PPO algorithm for discrete agent (channel  allocation) and continuous agent (UAV trajectory optimiza-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Implementation of Continuous and Discrete PPO  1) Critic Network: The critic network is responsible to  give scores to the actor according to the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The  architectures of both discrete and continuous critic networks  are the same, which consists of multiple fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Loss function of continuous and discrete critic networks  are given by  Jtraj(φ) =  �  V traj  φ (straj  t ) −  �  rtraj  t  + γV traj  φ′ (straj  t+1)  ��2  ,  (15)  Jch(φ) =  �  V ch  φ (sch  t ) −  �  rch  t + γV ch  φ′ (sch  t+1)  ��2,  (16)  where Ltraj  t (φ) and Lch  t (φ) denote the loss function for the  critic network of continuous and discrete agent respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  V traj  φ′ (straj  t+1) and V ch  φ′ (sch  t+1) are the state value estimations  generated by the old critic networks φ  ′  traj and φ  ′  ch respectively,  which are saved in during the interaction with environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  V traj  φ (straj  t ) and V ch  φ (sch  t ) are the state value estimations gener-  ated by the current critic networks φtraj and φch , which are  updated in each training iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  2) Actor Network: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 5, the architecture of  discrete and continuous actor network are different due to the  difference in action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The continuous actor network for trajectory control is a  network for value approximation, which outputs a µ head and  a σ head which denotes the mean and variance of Gaussian dis-  tributions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Each head includes two variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=',  {µx, µy} and {σx, σy}, which denotes the x-axis and y-axis  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The action {ax[t], ay[t]} is generated by sampling  from the obtained distribution N(µx, σ2  x) and N(µy, σ2  y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The discrete actor network for channel allocation is a  network for classification, which outputs the probabilities  Pr(a) of each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The agent sample its action from the  obtained action probabilities with ε-greedy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', the output  action is generated by sampling from Pr(a) with probability  1−ǫ, and selected randomly with probability ǫ for exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The output action of the discrete actor network is encoded,  which will be decoded into one-hot indicators before being  utilized for further calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Loss functions of actor networks in our implementation  adopt the trick of clipping to simplify the calculation, which  is proposed by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Schulman et al [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The PPO-PPO algorithm is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  (0)5  Fully Conneted Layers  Softmax Layer  State  Head  � Head  �  Head  � Head  �  Trajectory  2  ( ,  )  N � �  Sample From  2  ( ,  )  N � �  Sample From  Fully Conneted Layers  Softmax Layer  State  Channel Allocation  Pr( )  a  Action Probability  �  Sample with -Greedy  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 5: Actor Network Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Algorithm 1 PPO-PPO  Initiate: Remaining data at MDCs, UAV location, network  parameters of discrete and continuous agent  1: for iteration t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' do  2:  Discrete agent execute action according to the current  state and policy πch  θ′  �  ach  t |sch  t  �  to obtain the channel  allocation indicator matrix ˆI[t]  3:  With given ˆI[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' the continuous agent for trajectory  control execute action according to the current state and  policy πtraj  θ′  �  atraj  t |straj  t  �  ˆI[t]  4:  Agent interact with environment to get reward rch  t and  rtraj  t  for discrete agent and continuous agent respectively  5:  Update state straj  t  ← straj  t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' sch  t ← sch  t+1  6:  Save  trajectory  �  sch  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' ach  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' rch  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' sch  t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' V ch  φ′ (sch  t )  �  and  �  straj  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' atraj  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' rtraj  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' straj  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' V traj  φ′ (straj  t )  �  7:  for every i iterations do  8:  Shuffle data order and make batch with size bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  9:  for j=0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', T  bs − 1 do  10:  Calculate loss functions of critic and actor net-  works and update network parameters by gradient  ascent  11:  end for  12:  end for  13: end for  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' SIMULATION RESULTS  The performance of our proposed double-agent reinforce-  ment learning approach for Metaverse data collecting is tested  and compared with two benchmark scenarios (DQN-PPO and  duelling DQN-PPO), whose discrete agents are replaced with  DQN or duelling DQN algorithm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The simulation  settings are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 6 presents the required time to complete data collecting  mission of our proposed algorithm and two benchmark algo-  rithms with given data size U = 50Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' At the beginning of  the training process (0-1000 episodes),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' all three algorithms  TABLE I: Constant Parameter Setting  Parameter and Physical Meaning  Value  Number of channels(M)  3  Default number of users (N)  5  Bandwidth (B)  5MHz  Transmission power of MDCs  5W  Frequency (f)  28GHz (5G spectrum)  Power of Gaussian noise (σ2)  5 × 10−8W  Maximum speed of UAV  10m/s  Mission area size (L)  200m  are unstable because the reasonable policy has not been  established,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' and the agents are exploring the environment fre-  quently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' From 1000 episodes to 2000 episodes, our proposed  PPO-PPO algorithm shows the tendency of convergence while  the benchmark DQN-PPO algorithm is still very unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The duelling DQN-PPO algorithm also starts to finish the  mission within a shorter time period, but is less stable than  the PPO-PPO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' DQN-PPO algorithm shows poor  convergence performance in this task, but both PPO-PPO and  duelling DQN-PPO algorithms are able to converge within  5000 episodes with similar performance due to their common  implementation of the advantage function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  3000  3500  4000  4500  5000  Episodes  30  35  40  45  50  55  60  Time (s)  DQN-PPO  Duelling DQN-PPO  PPO-PPO  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 6: Comparison of required time to finish mission with  data size U = 50Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 7 presents the mission completing time experiment with  a similar parameter setting as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 6, but the data size is  increased to U = 100Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' All three algorithms need more  time to finish the data collecting mission due to larger data  size, and the PPO-PPO algorithm shows similar convergence  performance as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' However, the dueling DQN-PPO  algorithm becomes unstable in the training process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', some  sudden increase in the required time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The superior stability  of PPO over dueling DQN can be attributed to its policy  update constraint by equipping it with a KL-divergence penalty  between the old policy (the policy for sampling data) and the  updated policy (the policy used for training and evaluating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 9 are the corresponding rewards in the train-  ing processes of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' We consider  the reward given to the agent as guidance but not the exact  objective function in the implementation of reinforcement  6  0  500  1000  1500  2000  2500  3000  3500  4000  4500  5000  Episodes  60  70  80  90  100  110  120  130  140  150  160  Time (s)  DQN-PPO  Duelling DQN-PPO  PPO-PPO  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 7: Comparison of required time to finish mission with  data size U = 100Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The tendencies of the reward and the  required time are highly similar although they are generated  from different formulas, which indicates that our reward design  successfully leads the agent to learn a better policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  3000  3500  4000  4500  5000  Episodes  3000  2500  2000  1500  1000  500  0  Reward  DQN-PPO  Duelling DQN-PPO  PPO-PPO  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 8: Comparison of reward with data size U = 50Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  The mission completing time comparison for the case with  eight users is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The duelling DQN-PPO  algorithm shows similar average performance as the PPO-PPO  algorithm but less stability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=', the required time sometimes  jumps to extremely large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Taking the stability into  consideration, the PPO-PPO algorithm is better than duelling  DQN-PPO algorithm in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' The DQN-PPO algorithm is  obviously not able to converge in this experiment, so we do  not consider it a candidate for our double-agent reinforcement  learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' CONCLUSION  In this paper, we propose a double-agent reinforcement ar-  chitecture for data collecting and synchronization in Metavese,  and adopt PPO algorithm for both discrete and continuous  agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' Two agents with different action space and state space  work in a cascade manner for channel allocation and UAV  0  500  1000  1500  2000  2500  3000  3500  4000  4500  5000  Episodes  3000  2500  2000  1500  1000  500  Reward  DQN-PPO  Duelling DQN-PPO  PPO-PPO  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 9: Comparison of reward with data size U = 100Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  3000  3500  4000  4500  5000  Episodes  50  100  150  200  Time (s)  DQN-PPO  Duelling DQN-PPO  PPO-PPO  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' 10: Comparison of reward with data size U = 50Mb and  8 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  trajectory control to form a combined action in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content='  Our experiments indicate the advantage of the PPO-PPO in  both the required time for the mission and the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
+page_content=' In  future work, we will consider transmission power allocation  and test the performance of other state-of-art reinforcement  learning algorithms in our proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfgvyq/content/2301.01474v1.pdf'}
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+arXiv:2301.01492v1  [cs.IT]  4 Jan 2023
+A Pulse-Shape Binary Multiplex Modulation
+Pavel Loskot, Senior Member, IEEE
+Abstract
+The root raised-cosine pulse commonly used in linear digital modulations yields exactly two
+intersymbol interference components from the preceding and the subsequent data symbols, provided
+that the roll-off factor is 100% and the modulation packing factor is set to 50%. This can be exploited
+to symmetrically multiplex two data streams of transmitted symbols. Hence, the proposed scheme is
+referred to as pulse-shape binary multiplex modulation. The demodulation of the two multiplexed
+data streams at the receiver can be aided by making the streams mutually orthogonal. It can be
+achieved by superposition modulation with symbol-by-symbol interference cancellation, proper design
+of transmission sequences interleaving pilot and data symbols in order to also enable channel estimation,
+and using orthogonal spreading sequences. The presented numerical results indicate that the proposed
+modulation scheme can outperform Nyquist signaling in terms of transmission reliability or the time
+required for transmitting the whole sequence of data symbols. For instance, differentially encoded
+modulation symbols can be transmitted twice as fast by the proposed modulation scheme with a 3
+dB penalty in signal-to-noise ratio over additive white Gaussian noise channels.
+Index Terms
+Intersymbol-interference; linear modulation; Nyquist signaling; partial response signaling; root raise
+cosine pulse; sequence multiplexing.
+I. INTRODUCTION
+The spectrum scarcity necessities the use of spectrally efficient modulations. The Nyquist
+signaling is a well established and robust technique for constructing linear digital modulations
+which are employed in a vast majority of today’s communication systems. These modulation
+schemes are often combined with channel encoding to improve the transmission reliability and
+even approach the channel capacity. An alternative strategy is to assume modulations having
+a controlled level of intersymbol interference (ISI), which can increase the rate of information
+transmission as well as act as a form of information encoding for improving the transmission
+The author is with ZJU-UIUC Institute, Haining, China (e-mail: pavelloskot@intl.zju.edu.cn).
+This work was supported by a research grant from Zhejiang University.
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+1
+reliability [1], albeit at the cost of increased detection complexity at the receiver. Such so-called
+faster-than-Nyquist (FTN) schemes are linear modulations that can be used over band-limited
+channels [2].
+The renewed interest in FTN signaling schemes goes back to the early 2000’s [3]. However,
+a closely related idea of partial response linear modulations with controlled ISI appeared much
+earlier [4]. The achievable spectral efficiency of coded and uncoded FTN schemes is evaluated
+in [2], [5], and [6]. The observation that up to 25% increase in the transmission rate is possible
+without deteriorating the error performance is known as the Mazo limit [3], [2], [7], [6]. The
+energy and complexity costs of FTN signaling are reviewed in [2].
+The FTN schemes can be implemented both in time and in frequency domains [2], [8]. An
+orthogonal FTN scheme based on OFDM was designed in [9]. Alternatively, Nyquist signaling
+with dual root raised-cosine (RRC) pulses akin to duobinary modulation has been investigated
+in [10]. This scheme was further refined for the RRC pulses with zero roll-off in [11]. The link
+between duobinary modulation and FTN signaling has been pointed out in [1].
+An important issue is how to efficiently perform the detection of transmitted symbols at the
+receiver. Unlike the ISI due to multipath propagation, the ISI created by FTN signaling also
+correlates samples of additive noise. The optimum detection necessitates the use of whitening
+matched filter (WMF) prior to symbol decisions. The ISI at the detector input can be equivalently
+represented as an auxiliary channel [5], [6]. The output signal of such channel has a trellis-like
+structure, which can be optimally equalized by the Viterbi, BCJR and other such algorithms
+with varying complexity [2], [7], [6]. These decoding methods can approach the performance
+of zero-ISI (Nyquist) modulations over additive white Gaussian noise (AWGN) channels [2].
+The symbol-by-symbol detector for FTN signals was devised in [6] and [12]. The detection of
+FTN signals with oversampling and one-bit quantization was developed in [5]. A low complexity
+linear equalization for FTN signaling was designed in [13]. The joint channel estimation and
+decoding of FTN signals was studied in [14] and in [15].
+Nearly all investigations of FTN signaling schemes in the literature assume the RRC modulation
+pulse. The RRC pulse is parameterized by a time period, Tp, and a roll-off factor, α. Linear
+modulations combine the RRC pulses weighted by data symbols, which are then transmitted
+once every symbol period, Ts. The packing factor defines the relationship between Tp and Ts,
+i.e., τ = 1 − Ts/Tp. The design and analysis of FTN signaling in the literature usually assumes
+arbitrary values of 0 ≤ α ≤ 1 and 0 ≤ τ < 1. The search for good values of α and τ over an
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+2
+entire α − τ plane to allow symbol-by-symbol decisions was carried out in [12]. However and
+importantly, the case of 100% bandwidth roll-off is rarely explicitly considered in the literature
+[6]. The authors in [5] noticed that, for α = 1 and an arbitrary value of τ, the ISI is approximately
+limited to the two previous and the two subsequent symbol samples.
+In this paper, we show that the RRC pulse with 100% roll-off and 50% packing has a well-
+defined ISI, which is exactly and symmetrically constrained to one previous and one subsequent
+symbol. Such a unique property of the RRC pulse appears to remain unnoticed in the literature.
+Interestingly, reference [1] states that ISI with only two components can be obtained with 100%
+roll-off and 50% packing assuming prolate spheroidal wave pulses, but not RRC pulses. Although
+such a modulation scheme can be assumed to be a special case of FTN signaling, it is argued
+that RRC pulses having 100% roll-off and 50% packing offers symmetric multiplexing of the
+two transmitted data streams. For this reason, such a partial response signaling is referred to
+in this paper as a pulse-shape binary multiplexing (PSBM) modulation. The main task then
+is how to separate the two multiplexed data streams at the receiver with acceptable reliability
+and complexity. As with other partial response signalings, the modulation constellation and
+the dependency between transmitted symbols must be carefully selected in order to trade-
+off the performance and the decoding complexity. We design several transmission sequences
+interleaving pilot and data symbols, discuss superposition modulation with symbol-by-symbol
+sequential interference cancellation (SIC), and also consider orthogonal spreading sequences to
+aid separation of the data streams at the receiver. In addition, the performance of multiplexed
+differentially encoded phase-shift keying (PSK) modulation symbols is evaluated numerically.
+The numerical results identify several cases when the proposed PSBM modulation outperforms
+the Nyquist signaling in terms of either transmission reliability or the time required to transmit
+a given number of data symbols.
+The rest of this paper is organized as follows. Linear modulation schemes that are related to
+the proposed pulse-shape multiplexing signaling are outlined in Section II. System model and
+the received signal structure are described in Section III. The proposed pulse-shape multiplexing
+modulation is defined in Section IV including the design of transmitted symbol sequences.
+Numerical results are presented in Section V. Section VI concludes the paper.
+We adopt the following notations: E[·] is expectation, ⊛ is convolution, | · | is absolute value,
+(·)∗ is complex conjugate, Re{·} and Im{·}, respectively, denote the real and imaginary part
+of a complex number, Card{·} is cardinality of a set, (·)T is matrix transpose, (·)−1 is matrix
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+3
+inverse, and ∥·∥ is the Euclidean norm of a matrix or vector.
+II. RELATED LINEAR MODULATION SCHEMES
+A linearly modulation signal is constructed as,
+x(t) =
+�
+k
+sk p(t − kTs)
+(1)
+where sk are M-ary modulation symbols transmitted every symbol period, Ts, and p(t) denotes
+a deterministic pulse-shape, which is also known at the receiver. The stationary sequence of
+transmitted symbols, sk, has zero-mean, and the variance, E[|sk|2] = Es. The symbols are usually
+obtained as output of a finite-state modulator, i.e.,
+sk = s(qk, ck)
+(2)
+where the states, qk, represent modulation memory, and the data symbols, ck, each carry, log2 M,
+bits of input information. In this paper, p(t) is assumed to be the unit-energy RRC pulse, [4]
+p(t) = rrcα(t/Ts)
+√Ts
+(3)
+where
+rrcα(t) =
+1
+1 − 16α2t2
+�sin ((1 − α)πt)
+πt
++ 4α cos ((1 + α)πt)
+π
+�
+.
+(4)
+The roll-off factor, 0 ≤ α ≤ 1, however, it is possible to also consider pulse shapes having a
+roll-off greater than 100%.
+Since the sequence of symbols, sk, is stationary, the auto-correlation, Rs(i − j) = E
+�
+sis∗
+j
+�
+.
+The corresponding power-spectrum density (PSD) of signal (1) is computed as, [4]
+Sx(f) = 1
+Ts
+|P(f)|2 �
+k
+Rs(k) ej2πfkTs
+(5)
+where P(f) denotes the Fourier transform of p(t).
+Correlative coding assumes the discrete modulator (2) to be a finite impulse response (FIR)
+filter, i.e.,
+sk =
+K−1
+�
+i=0
+vick−i.
+(6)
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+4
+The filter weights, vi, are normalized, so that, �
+i |vi|2 = 1. More importantly, with a change in
+indices, modulated signal (1) with symbols (6) can be rewritten as,
+x(t) =
+�
+k
+K−1
+�
+i=0
+vi ck−ip(t − kTs)
+=
+�
+k
+ck
+K−1
+�
+i=0
+vi p(t − (k + i)Ts) =
+�
+k
+ck ˜p(t − kTs)
+(7)
+where the compound pulse, ˜p(t) = �K−1
+i=0 vi p(t − iTs).
+Duobinary modulation is a special case of correlative coding, such that the FIR filter has
+only two non-zero weights, vo = v1 = 1/
+√
+2, the modulation symbols are binary, i.e., ck ∈
+{−√Es, +√Es}, and the RRC pulse has the smallest possible roll-off, α = 0. Modified duobinary
+modulation assumes instead the weights, v0 = 1/
+√
+2, v1 = 0, and v2 = −1/
+√
+2.
+The following modulations assume the RRC pulse-shape with an arbitrary roll-off value.
+Differential PSK constructs the transmitted symbols as,
+sk = ck sk−1
+(8)
+where the data symbols, ck ∈ {√Es ej2π(i−1)/M}, i = 1, 2, . . . , M. Generalized shift-keying
+extends the modulation alphabet of amplitude or phase shift-keying modulations with a zero
+symbol [16]. Offset-quadrature (M = 4) PSK delays the imaginary part of the modulated signal
+by half a symbol period, i.e.,
+x(t) =
+�
+k
+Re{ck} p(t − kTs) + j Im{ck} p(t − kTs − Ts/2).
+(9)
+Finally, FTN signaling is a linear modulation described by eq. (1). More importantly, the RRC
+pulse-shape in (3) can now be scaled by, Tp = Ts/(1 − τ), instead of Ts, where 0 ≤ τ < 1 is
+so-called the packing factor, i.e.,
+x(t) =
+�
+k
+skp(t − k(1 − τ)Tp) =
+�
+k
+skp(t − kTs)
+(10)
+so that Tp is a design parameter of the pulse, p(t), whereas, Ts = (1 −τ)Tp, denotes the symbol
+period. Thus, τ = 0 packing corresponds to a conventional Nyquist signaling, whereas τ = 1
+packing would completely overlap the transmitted symbols. More importantly, the PSD of (10)
+is still given by eq. (5), and it is otherwise completely independent of the packing factor, τ.
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+5
+III. RECEIVED SIGNAL
+The standard wireless channel model with L propagation paths is an FIR filter with the impulse
+response,
+˜h(t) =
+L
+�
+l=1
+hl(t)δ(t − τl).
+(11)
+The signal delays, τl, are assumed to be constant. The path attenuations, hl(t), are zero-mean
+circularly symmetric Gaussian processes. These processes are stationary, and generally mutually
+correlated. They have a defined auto-correlation, Rh(∆t), which determines the coherence bandwidth.
+For narrow-band signals, the number of paths, L, is small. For L = 1, the channel model (11)
+becomes frequency non-selective. In low-mobility scenarios, the channel attenuations, hl(t), are
+often assumed to be constant over blocks of transmitted symbols, and independent between the
+successive blocks, which is often referred to as a block fading model.
+The received signal corresponding to multi-path propagation model (11) is written as,
+y(t) = ˜h(t) ⊛ x(t) + w(t)
+=
+L
+�
+l=1
+hl(t)x(t − τl) + w(t)
+(12)
+where w(t) is a zero-mean stationary circularly symmetric AWGN with the variance, σ2
+w =
+E[|w(t)|2].
+The received signal is filtered through a filter matched to the transmitted pulse, p(t), and
+synchronously sampled at a rate, 1/Ts. In particular, assuming RRC pulses, the matched filter,
+p∗(−t) = p(t), and provided that the channel attenuations are constant over blocks of transmitted
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+6
+symbols, the received samples are modeled as,
+rn =y(t) ⊛ p∗(−t)
+���
+t=nTs+τ0
+=
+L
+�
+l=1
+hl x(t − τl) ⊛ p(t)
+���
+t=nTs+τ0 + w(t) ⊛ p(t)
+���
+t=nTs+τ0
+=
+L
+�
+l=1
+hl
+�
+k
+sk
+� ∞
+−∞
+p(ζ + (n − k)Ts − τl)p(ζ − τ0) dζ
++
+� ∞
+−∞
+w(ζ + nTs)p(ζ − τ0) dζ
+=
+�
+k
+sk
+L
+�
+l=1
+hl pn−k,l + wn =
+�
+k
+sk˜pn−k + wn
+=sk˜p0 +
+�
+k
+n̸=k
+sk˜pn−k
+�
+��
+�
+ISI
++wn.
+(13)
+The timing offset, τ0, at the receiver can be optimized to minimize the ISI term (in some sense)
+in (13) defined as,
+˜pn−k =
+L
+�
+l=1
+hl
+� ∞
+−∞
+p(ζ + (n − k)Ts − τl)p(ζ − τ0) dζ, n ̸= k.
+(14)
+Thus, the ISI arises when the orthogonality between the transmitter and the receiver pulses
+is violated, for example, due to multi-path propagation, time-synchronization errors between
+transmitter and receiver, and also due to symbol-period compression in FTN signaling schemes
+[4].
+An interesting question is how much ISI is produced for different combinations of parameters
+α and τ in FTN signaling schemes using RRC pulses. Hence, define the function,
+ISI(µ) = Card{|˜pk| > µ, k ̸= 0}
+(15)
+to be the number of ISI components that are greater than a given threshold, µ. Note that,
+ISI(µ) ∈ {0, 2, 4, . . .}, due to even symmetry of the RRC pulses. Assuming different thresholds,
+µ, the roll-off, 0 ≤ α ≤ 2, and the RRC pulses truncated to (−4Ts, +4Ts), the values ISI(µ) = 0
+(red points) and ISI(µ) = 2 (blue points) in the α − τ plane are shown in Fig. 1. The empty
+(white) spaces in Fig. 1 indicate the values, ISI(µ) > 2. It can be observed that by decreasing
+the threshold, µ, several cases of interest for designing FTN signaling schemes start to emerge.
+In particular, exactly two ISI components can be obtained for these parameters: α = 1.0 and
+τ = 0.5, α = 1.07 and τ ∈ (0.70, 0.71), and α ∈ (1.65, 1.85) and τ ∈ (0.47, 0.50).
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+7
+0
+� ��
+1
+�
+�
+2
+=0.01
+�
+�� �
+�
+
+
+
+=0.005
+
+ 
+
+
+
+
+=0.003
+0
+0.5
+1
+0
+1
+2
+=0.002
+1−τ
+1−τ
+α
+α
+Fig. 1. The ISI(µ) = 0 components (red points) and ISI(µ) = 2 components (blue points) for four different thresholds, µ.
+IV. PULSE-SHAPE BINARY MULTIPLEX MODULATION
+As indicated in Fig. 1, the RRC pulse with 100% roll-off and 50% packing has well-defined
+and finite ISI components. In particular, the RRC pulse (4) for α = 1 becomes,
+rrc1(t) = 4 cos(2πt)
+π(1 − 16t2).
+(16)
+This pulse has the following ISI components in an AWGN channel without multi-path. Such a
+fundamental property appears to remain unnoticed in the literature.
+Lemma 1: Let n be a non-negative integer. The ISI integral involving the RRC pulse, rrc1(t),
+with 100% roll-off has the exact solution,
+� ∞
+−∞
+rrc1(t) × rrc1(t − n/4) dt
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+8/(3π)
+n = 1
+8
+π(n−2)n(n+2)
+n − odd, n > 1
+1
+n = 0
+1/2
+n = 2
+0
+n − even, n > 2.
+(17)
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+8
+n=0
+n=2
+n=4
+1
+2
+3
+4
+5
+0.998
+0.999
+1.000
+1.001
+1.002
+d
+truncated integral values
+Fig. 2. Numerically computed integral (17) truncated to interval, (−d, +d), as a function of d (solid lines). The exact values
+for an infinite interval are shifted to be all equal to unity in order to enable comparison. The dashed lines are mirrored solid
+lines about the unit value.
+Lemma 1 can be proved by solving the integral for the first few values of n (for example, using
+Mathematica software), and then using induction.
+However, the result (17) is exact only when the integration is performed over an infinite
+interval. In practice, the pulse shapes must be truncated to a finite interval. The numerically
+computed values of integral (17) when the interval of integration is truncated to (−d, +d) are
+shown in Fig. 2. It can be observed that the RRC pulse shape, rrc1(t), should not be truncated
+to the intervals shorter than, (−4, +4), in order to achieve the RRC property given in Lemma 1
+with at least 99.9% accuracy.
+Definition 2: The modulated signal of pulse-shape binary multiplex modulation is written as,
+x(t) =
+�
+k
+sk
+rrc1
+�
+t−kTs
+2Ts
+�
+√2Ts
+.
+(18)
+The synchronously sampled matched filter output of modulated signal (18) received in AWGN,
+w(t), is,
+rn =(x(t) + w(t)) ⊛
+rrc1
+�
+t
+2Ts
+�
+√2Ts
+���
+t=nTs
+=
+�
+k
+sk
+� ∞
+−∞
+rrc1(ζ + (n − k)/2) rrc1(ζ) dζ + wn
+=
+�1
+2sn−1 + sn + 1
+2sn+1
+�
++ wn.
+(19)
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+9
+a1
+a1+a2
+2
+a3
+a2+a3
+2
+a4
+a3+a4
+2
+a2
+Stream #1:
+Stream #2:
+b1
+b1+b2
+2
+b3
+b2+b3
+2
+b3+b4
+2
+b2
+T1
+T2
+T3
+T4
+T5
+T6
+T7
+Fig. 3. A visualization of pulse-shape multiplex modulated signal.
+The noise samples, wn, in (19) are zero-mean, have the variance, E[|wn|2] = E[|w(t)|2] = σ2
+w,
+and their stationary auto-correlation is,
+Rw(n − m) = E[wnw∗
+m] =
+
+
+
+
+
+
+
+
+
+σ2
+w
+n = m
+σ2
+w/2
+|n − m| = 1
+0
+|n − m| > 1.
+(20)
+Such noise samples can be equivalently modeled by a simple FIR filter,
+wn = un + un−1
+√
+2
+(21)
+where un are the samples of a zero-mean, circularly symmetric Gaussian process having the
+variance, E[|un|2] = σ2
+w. In addition, it is straightforward to show that the variance of the sum
+of N noise samples having the correlations (20) is,
+var
+� N
+�
+n=1
+wn
+�
+= (2N − 1)σ2
+w
+(22)
+which is greater than the variance, Nσ2
+w, of the sum of N uncorrelated samples.
+The modulated signal (18) in Definition 2 can be visualized as shown in Fig. 3. In particular,
+the transmitted data symbols can be viewed as consisting of two multiplexed streams of data
+symbols, ak, and, bk, which are each transmitted with a period 2Ts, but mutually shifted by Ts.
+The corresponding received symbol samples after the matched filtering are,
+rn =
+
+
+
+an + bn−1+bn
+2
++ wn
+n − odd
+bn + an+an+1
+2
++ wn
+n − even.
+(23)
+Using (5), the PSD of modulated signal (18) is computed as,
+Sx(f) = 2|RRC1(2Tsf)|2 �
+k
+E[s0s∗
+k] ej2πfkTs
+(24)
+where the Fourier transform of the pulse, rrc1(t), is,
+RRC1(f) =
+
+
+
+cos(πf/2)
+|f| ≤ 1
+0
+otherwise.
+(25)
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+10
+A. Transmitted Sequence Design
+The optimum detection of transmitted symbols in the presence of ISI must consider complete
+sequences of received samples. However, in the absence of multi-path, the received samples
+have structure (19), and the transmitted sequences can be designed, so that the complexity of
+detection at the receiver can be reduced.
+The key strategy for reducing the detection complexity is to exploit orthogonality among
+sub-sequences of transmitted symbols. Offset-quadrature PSK modulation (9) alternates one-
+dimensional modulation symbols along the in-phase and quadrature components, which allows
+the optimum symbol-by-symbol decisions.
+Multiplexing two data streams as described by (23) can exploit the design principles of
+superposition modulation and multiuser detection. In such a case, symbol-by-symbol decisions
+can be performed by SIC. Specifically, provided that symbols, bn, can be reliably detected, even
+if the symbols, (an + an+1)/2, are not yet known, then the symbol, an, can be reliably detected
+after canceling the ISI term, (bn−1 + bn)/2.
+In the sequel, three other sequence design strategies are discussed in more detail. The first
+strategy combines pilot and data symbols to aid the data detection and channel estimation. The
+second strategy employs orthogonal spreading codes in order to separate the two multiplexed
+data sequences. The third strategy adopts the differential encoding of transmitted symbols.
+B. Sequences with Interleaved Pilot Symbols
+In general, pilot symbols for channel estimation can be interleaved with data symbols or
+superimposed onto data symbols [17]. Here, the more common former approach is adopted.
+Thus, consider a transmitted sequence consisting of alternating groups of Ld data symbols and
+Lp ≪ Ld pilot symbols, which are separated by a single zero-symbol as shown in Fig. 4.
+For instance, the following sub-sequences with reduced or no ISI can be considered with pilot
+symbol, p, and arbitrary data symbols, d1, and, d2: (0, p, 0), (d1, p, −d1), (d1, p, −d1, −p, d1),
+and (d1, p, −d1, −p, d2, p, −d2). These sub-sequences enable ISI-free data detection and channel
+estimation, as can be deduced from eq. (23) and Fig. 3. Recall also that the noise samples, wn,
+and, wn±2, are uncorrelated, i.e., independent.
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+11
+0 pilots
+0
+data
+0 pilots
+0
+data
+0
+Lp
+separator
+Ld
+Fig. 4. The transmitted sequence with interleaved sub-sequences of data and pilot symbols and a single zero-symbol separator.
+In order to illustrate the ISI-free channel estimation, consider the sequence, (d1, p, p, −d1).
+The received samples corresponding to the two pilot symbols in the middle are,
+rn =h3
+2p + h1
+2d1 + wn
+rn+1 =h3
+2p − h1
+2d1 + wn+1
+(26)
+where h denotes the complex-valued channel attenuation (i.e., frequency non-selective slow
+fading). The samples, rn, and, rn+1, can be simply combined as,
+rn + rn+1 = 3hp + wn + wn+1
+(27)
+where the total variance of the additive noise samples is equal to 3σ2
+w due to correlations (20).
+More generally, the transmitted sequence,
+(−p, d1, p, d2, −p, d3, p, d4, −p, d5, . . . , dN, ±p)
+(28)
+where the last pilot symbol is p, if N is odd, and −p, if N is even, allows the ISI-free symbol-
+by-symbol decisions of all data symbols. Moreover, assuming again a slow fading channel, the
+received samples corresponding to the pilot symbols can be summed up to obtain,
+N
+�
+n=1
+(−1)n r2n−1 = N h p +
+√
+Nw
+where the noise sample, w, has the variance, σ2
+w, so the signal-to-noise ratio (SNR) for estimating
+the channel coefficient, h, has been improved N-times. Note also that once the channel has been
+estimated, the pilot symbols can be subtracted from the received samples in order to aid decisions
+of the remaining data symbols.
+Finally, consider the case of a symbol repetition diversity. The transmitted sequence, (d, 0, d, 0, . . ., 0, d),
+of a data symbol, d, repeated (N ≥ 2)-times corresponds to the canonical Nyquist signaling.
+The pulse-shape multiplex modulation instead transmits the sequence, (d, d, . . . , d), of N1-times
+repeated data symbol, d. For the same sequence length, N1 = 2N − 1. Assuming slowly
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+12
+fading channel, the detector combines the received samples for the two modulation schemes,
+respectively, as,
+r =N h d +
+√
+Nw
+r =(2 · 3/2 + 2(N1 − 2)) h d/
+√
+2 +
+�
+2N1 − 1w
+(29)
+where the scaling by
+√
+2 was introduced for the second modulation in order to account for the
+larger number of symbols in its transmitted sequence. The resulting SNR of these two schemes
+is proportional to, γ ∝ N, and, γ ∝ 2N −3/2, respectively. Consequently, for symbol repetition
+diversity, the SNR gain of the pulse-shape binary multiplexing is asymptotically 3 dB larger than
+for the Nyquist signaling.
+C. Sequences with Orthogonal Spreading
+Another strategy for transmitting interleaved, but orthogonal symbols in modulated signal (18)
+is to use orthogonal spreading codes. In particular, assume transmitted symbols,
+an = d1c(1)
+n ,
+bn = d2c(2)
+n
+(30)
+where d1 and d2 are two data symbols, and, c(1)
+n and c(2)
+n , n = 1, 2, . . . , N, are generally complex-
+valued, orthogonal spreading sequences, so that,
+N
+�
+n=1
+c(i)
+n c∗(j)
+n
+=
+
+
+
+N,
+i = j
+0,
+i ̸= j.
+(31)
+Then, the sequences of received samples (23) are linearly combined as,
+N
+�
+n=1
+r2n−1c∗(1)
+n
+=d1 + d2
+N
+�
+n=1
+c(2)
+n + c(2)
+n+1
+2
+c∗(1)
+n
++
+N
+�
+n=1
+w2n−1c∗(1)
+n
+=d1 + ˜w1
+N
+�
+n=1
+r2nc∗(2)
+n
+=d2 + d1
+N
+�
+n=1
+c(1)
+n + c(1)
+n+1
+2
+c∗(2)
+n
++
+N
+�
+n=1
+w2nc∗(2)
+n
+=d2 + ˜w2
+(32)
+provided that the spreading sequences, c(1)
+n , and, c(2)
+n , are exactly orthogonal. In such a case, the
+SNR improvement for transmitting two data symbols with orthogonal spreading sequences using
+the pulse-shape binary multiplex modulation (18) is proportional to,
+γ ∝
+N2
+2N − 1.
+(33)
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+13
+0
+50
+100
+150
+200
+250
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+r=10%
+r=5%
+N
+Probability
+Fig. 5. The probability (35) vs. the spreading sequence length, N, assuming κ = 5% and κ = 10%, respectively.
+For instance, if the spreading symbols, cn, are generated independently at random and with an
+equal probability from the set, {−1, +1}, the probability that two such sequences are orthogonal
+is,
+Pr
+� N
+�
+n=1
+c(1)
+n c∗(2)
+n
+= 0
+�
+=
+� N
+N/2
+� �1
+2
+�N/2 �1
+2
+�N−N/2
+=
+� N
+N/2
+�
+2−N.
+(34)
+Since the probability (34) of exact orthogonality asymptotically goes to zero with large N,
+consider instead the probability,
+Pr
+�
+−⌈κ N/2⌋ ≤
+N
+�
+n=1
+c(1)
+n c∗(2)
+n
+≤ ⌈κ N/2⌋
+�
+=
+⌈κ N/2⌋
+�
+n=−⌈κ N/2⌋
+�N
+n
+� �1
+2
+�N
+(35)
+for some small κ ≥ 0. The probabilities (35) as a function of N for two different values of
+factor, κ, are shown in Fig. 5. These probabilities are indicative of how many random spreading
+sequences need to be generated in order to select the required number of such sequences having
+an acceptable level of mutual orthogonality.
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+14
+c0
+c0
+2 (1 + c1)
+c0
+2 (1 + c1c2 + 2c1)
+c0
+2 (1 + c0c1 + 2c0)
+c0
+2 (1 + c1c2)
+c0c1
+c0c1
+2 (1 + c2c3 + 2c2)
+c0c1
+2 (1 + c2c3)
+c0c1c2
+c0c1c2
+2
+(1 + c3c4 + 2c3)
+c0c1c2c3
+c0c1c2
+2
+(1 + c3c4)
+c0
+c0c1c2c3c4
+T1
+T0
+T2
+T3
+T5
+T4
+Stream #1:
+Stream #2:
+Tx symbols:
+Fig. 6. Differentially encoded M-ary PSK symbols transmitted via pulse-shape binary multiplex modulation.
+D. Sequences with Differential Encoding
+Differential PSK is a popular modulation scheme for fast fading channels, since it alleviates
+the need for recovering the absolute phase reference. Fig. 6 shows differentially encoded M-ary
+PSK symbols (8) transmitted via pulse-shape binary multiplex modulation. In particular, the n-th
+transmitted symbol is,
+sn =
+�n−2
+�
+k=0
+ck
+�
+1 + cn−1cn + 2cn−1
+2
+=1
+2
+�n−2
+�
+k=0
+ck
+�
++ 1
+2
+�n−1
+�
+k=0
+ck
+�
+cn +
+�n−1
+�
+k=0
+ck
+�
+.
+(36)
+Consequently, the differential decoding can be performed as,
+cn =
+�
+2sn −
+�n−2
+�
+k=0
+ck
+�
+− 2
+�n−1
+�
+k=0
+ck
+�� �n−1
+�
+k=0
+ck
+�∗
+=2sn
+�n−1
+�
+k=0
+c∗
+k
+�
+− c∗
+n−1 − 2.
+(37)
+The performance of this modulation scheme is evaluated in the next section.
+V. NUMERICAL EXAMPLES
+It is convenient to use a vector notation to generate samples of pulse-shape binary multiplex
+modulation (18) received over a frequency non-selective fading channel. The vector, r, of N
+received samples corresponding to the vector, s, of N transmitted symbols can be obtained as,
+r = s A diag(h) + w AT
+0
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+15
+where h is a vector of fading channel coefficients, w are samples of AWGN, and the (N × N)
+ISI matrix,
+A =
+
+
+1
+1/2
+1/2
+1
+1/2
+...
+...
+1/2
+1
+
+
+= A0 AT
+0 .
+The optimum detection requires that the additive noise is first whitened as, [4]
+r A−T
+0
+= s A diag(h) A−T
+0
++ w.
+Then the maximum likelihood (ML) detection of sequence s is,
+ˆs = arg min
+s
+���r A−T
+0
+− s A diag
+�
+ˆh
+�
+A−T
+0
+���
+2
+(38)
+where ˆh is the estimate of h representing channel state information (CSI).
+An uncoded binary phase shift keying (BPSK) modulation and Rayleigh-distributed fading
+amplitudes, h, are assumed for simplicity. The transmitted sequence interleaves pilot symbols and
+data symbols as shown in Fig. 4. The pilot symbols are used to estimate the channel coefficients,
+h, by linear minimum mean-square error (LMMSE) algorithm. The spectral efficiency of pulse-
+shape binary multiplexing is, 2, which is always larger than the spectral efficiency of the Nyquist
+signaling being equal to, 2/(1 + α).
+The BER curves, Pe, for short data sequences of Ld = 4 and Ld = 8 binary symbols,
+respectively, separated by a single zero-symbol are shown in Fig. 7 and Fig. 8. The SNR is
+defined as, γb = 1/(2σ2
+w). Both cases of perfect and estimated CSI are considered. The Nyquist
+signaling (no ISI) with symbol-by-symbol decisions is assumed as a reference. The ML data
+detector (38) is used for pulse-shape multiplexing signaling.
+It can be observed that the performance penalty due to channel estimation is much larger for
+pulse-shape multiplexing than for the Nyquist signaling, which is to be expected. The WMF
+improves the performance by several dB’s for both signaling schemes. More importantly, the
+performance of pulse-shape multiplexing improves with the data block length by exploiting the
+time diversity over a fading channel, so it can significantly outperform the Nyquist signaling
+at medium to large SNR values. It is likely that by employing more sophisticated channel
+estimation and equalization techniques, the performance of pulse-shape multiplexing can be
+further improved. In order to demonstrate the effect of time diversity, Fig. 9 shows that, over
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+16
+0
+5
+10
+15
+20
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+estim.
+CSI
+perfect
+CSI
+No-ISI
+ISI
+estim. CSI
+perfect CSI
+ML
+WMF-ML
+γb [dB]
+Pe
+LLLddd === 444
+Fig. 7. The BER of BPSK vs. SNR over Rayleigh fading channel for sequences of 4 binary symbols.
+an AWGN channel, the performance of pulse-shape multiplexing is worse than that of Nyquist
+signaling, even though some performance loss can be recovered by WMF.
+Lastly, the BER performance of Nyquist modulation and pulse-shape multiplex modulation
+transmitting differentially encoded quadrature PSK (QPSK) symbols over an AWGN channel is
+compared in Fig. 10. It can be observed that even though the pulse-shape multiplexing suffers
+asymptotically a 3 dB penalty in SNR, it reduces the time required for transmitting the whole
+symbol sequence to one half.
+VI. CONCLUSION
+The paper introduced a pulse-shape binary multiplex modulation. Such a modulation scheme is
+akin to partial-response signaling, correlative coding, offset-QPSK modulation and FTN signaling.
+It combines two data streams under controlled ISI created by the RRC pulses having 100%
+roll-off, and transmitted at twice the Nyquist rate. The ISI analysis showed that this is unique
+property among all the roll-off factors being at most 100% and the packing factors greater
+than 5%. However, the successive samples of additive noises at the output of matched filter at
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+17
+0
+5
+10
+15
+20
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+estim.
+CSI
+perfect
+CSI
+No-ISI
+ISI
+estim. CSI
+perfect CSI
+ML
+WMF-ML
+γb [dB]
+Pe
+LLLddd === 888
+Fig. 8. The BER of BPSK vs. SNR over Rayleigh fading channel for sequences of 8 binary symbols.
+the receiver become correlated, which incurs a SNR performance penalty. This penalty could
+be reduced or even removed by using more complex sequence-based detection schemes as
+shown elsewhere in the literature. The BER performance as well as decoding complexity of
+the proposed pulse-shape binary multiplexing modulation scheme is critically affected by the
+choice of transmitted sequences. One can consider superposition modulation with SIC decoding,
+interleave data symbols with pilot and zero-symbols to aid channel estimation and data decoding,
+and also employ orthogonal spreading sequences to separate the multiplexed data streams. The
+numerical results indicate that pulse-shape binary multiplexing can exploit time-diversity in
+fading channels to outperform the Nyquist signaling. In addition, it has been shown numerically
+that a sequence of differentially encoded PSK symbols can be transmitted twice as fast by the
+proposed modulation scheme compared to canonical Nyquist signaling, although with a 3 dB
+SNR penalty over AWGN channels.
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+18
+4
+5
+6
+7
+8
+9
+10
+11
+10 -5
+N=2
+ML
+WHF-ML
+BPSK
+4
+5
+6
+7
+8
+9
+10
+11
+10-5
+N=4
+γb [dB]
+Pe
+Pe
+Fig. 9. The BER of BPSK vs. SNR over AWGN channel for sequences of 2 and 4 binary symbols, respectively.
+6
+8
+10
+12
+14
+16
+18
+10-6
+10-5
+10-4
+10-3
+10-2
+DQPSK
+DQPSK-PSBM
+γb [dB]
+Pe
+Fig. 10.
+The BER comparison of differentially encoded QPSK with Nyquist and pulse-shape binary multiplexing (PSBM)
+modulation transmitted over an AWGN channel.
+
+PULSE-SHAPE BINARY MULTIPLEX MODULATION
+19
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+

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+arXiv:2301.01035v1  [math.FA]  3 Jan 2023
+BOUNDARY REPRESENTATIONS OF INTERMEDIATE FORMS
+BETWEEN A REGULAR DIRICHLET FORM AND ITS ACTIVE
+MAIN PART
+MATTHIAS KELLER, DANIEL LENZ, MARCEL SCHMIDT, MICHAEL SCHWARZ,
+AND MELCHIOR WIRTH
+Abstract. We characterize all semigroups sandwiched between the semigroup of
+a Dirichlet form and the semigroup of its active main part. In case the Dirichlet
+form is regular, we give a more explicit description of the quadratic forms of the
+sandwiched semigroups in terms of pairs consisting of an open set and a measure
+on an abstract boundary.
+Introduction
+One prime example of different self-adjoint realizations of the same differential
+expression are the Dirichlet and Neumann Laplacian on a bounded domain, i.e. two
+operators that only differ by the choice of boundary conditions.
+More generally
+one may ask which self-adjoint realizations of a differential expression arise from
+choosing boundary conditions.
+For the Laplacian, one possible answer was given by Arendt and Warma in [AW03]:
+If Ω is a domain with Lipschitz boundary, a self-adjoint positive operator L on L2(Ω)
+is a Laplacian with Robin-type boundary conditions if and only if the associated
+semigroup (e−tL) is sandwiched between the Dirichlet and Neumann heat semigroup
+in the sense that
+et∆(D)f ≤ e−tLf ≤ et∆(N)f
+for all f ≥ 0 and t > 0. Here the Laplacians with Robin-type boundary conditions
+can best be described in terms of associated quadratic forms: The Dirichlet form Q
+associated with L satisfies D(Q) = {f ∈ H1(Ω) | f = 0 quasi everywhere on Ω \ O}
+and
+Q(f) =
+�
+Ω |∇f|2dx +
+�
+∂Ω | ˜f|2dµ
+for some open O ⊆ ∂Ω and a measure µ on ∂Ω not charging sets of capacity zero.
+Here ˜f denotes a quasi-continuous modification of f.
+Note that in the original work of Arendt and Warma there was an additional
+condition that L be local, but this was later shown to be superfluous by Akhlil
+[Akh18].
+This result has been generalized in several directions. Chill and Warma [CW12]
+gave a similar characterization of (nonlinear) semigroups sandwiched between the
+1
+
+2
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+semigroup generated by the p-Laplacian with Dirichlet boundary conditions and the
+p-Laplacian with Neumann boundary conditions. Later this characterization was
+extended to semigroups associated with local nonlinear Dirichlet forms by Claus
+[Cla21]. In [ACD21] Arora, Chill and Djida do not study sandwiched semigroups
+but give a characterization of all semigroups dominating a semigroup induced by a
+regular form.
+A related problem was studied by Posilicano in [Pos14]. For a bounded domain
+with smooth boundary he characterizes all self-adjoint realizations of the Laplacian
+that generate Markovian semigroups via certain Dirichlet forms on the boundary of
+the domain. Applying his findings to realizations with sandwiched semigroups one
+obtains the result of Arendt and Warma under higher regularity assumptions on
+the boundary of the domain. For discrete Laplacians associated with infinite graphs
+similar characterizations of Markovian realizations were obtained by the first four
+authors in [KLSS19]. In this case, the employed boundary is the Royden boundary
+of the graph, which is defined using Gelfand theory.
+In this article we treat the question of sandwiched semigroups in the abstract
+context of Dirichlet forms. We start with a regular Dirichlet form without killing
+whose generator we take as an abstract analogue of the Dirichlet Laplacian.
+In
+this setting there is a natural analogue of the Neumann Laplacian, namely the
+generator of the active main part of our given regular Dirichlet form, which was
+introduced in [Sch17, Sch20a]. Our framework includes not only the Laplacian on
+domains treated by Arendt and Warma, but also various Laplace-like operators
+like fractional Laplacians, Laplacians on manifolds and metric measure spaces or
+Laplacians on weighted graphs and quantum graphs.
+We first give an abstract characterization of the generators of semigroups that are
+sandwiched between the semigroup associated with a regular Dirichlet form and the
+semigroup associated with its active main part in terms of order properties.
+To connect these sandwiched semigroups to boundary conditions, the first problem
+is to find a good notion of boundary in this setting. As all the quadratic forms
+involved are defined on the L2-space of some abstract topological measure space,
+there is no immediate geometric notion of boundary available. As in [KLSS19] and
+[ACD21] we introduce a notion of boundary that is defined using Gelfand theory
+and depends on the given regular Dirichlet form.
+With this notion of boundary, we can prove an abstract version of the main result
+of Arendt and Warma (Theorem 4.6):
+Theorem. Let Q be a regular Dirichlet form on L2(X, m) without killing and Q(M)
+its active main part. For a Dirichlet form Q′ on L2(X, m), the following assertions
+are equivalent:
+(i) There exists an open subset O of X ∪ ∂X and a measure µ on O ∩ ∂X that
+does not charge polar sets such that Q′ is the closure of the quadratic form Qc
+O,µ
+
+INTERMEDIATE DIRICHLET FORMS
+3
+given by D(Qc
+O,µ) = D(Q) ∩ Cc(O) and
+Qc
+O,µ(f) = Q(f) +
+�
+O∩∂X f 2 dµ.
+(ii) The semigroup associated with Q′ is sandwiched between the semigroup associ-
+ated with Q and the semigroup associated with Q(M), and D(Q′) ∩ Cc(X ∪ ∂X)
+is a form core for Q′.
+In other words, the Dirichlet forms sandwiched between Q and Q(M) (in the sense
+of domination of semigroups) are parametrized by measures on open subsets of an
+abstract boundary.
+In spirit our main result for regular Dirichlet forms is similar to the one of
+[ACD21], which treats an even more general setting without assuming the Markov
+property. The main differences are that for the first abstract part we need not as-
+sume any regularity of the forms and when we assume regularity, our results are
+more explicit.
+The article is organized as follows: In Section 1 we introduce the notation used
+throughout this article and recall some basic facts about Dirichlet forms and domina-
+tion of semigroups. In Section 2 we review the active main part of a regular Dirichlet
+form and give an abstract characterization of the Dirichlet forms sandwiched between
+the given regular Dirichlet form and its active main part (Theorem 2.8). In Section
+3 we study some properties of the forms Qc
+O,µ in the main theorem stated above,
+in particular their closability. In Section 4 we introduce our notion of boundary
+and show how sandwiched Dirichlet forms can be represented by measures on open
+subsets of the boundary (Theorem 4.6). Finally, in the appendix we collect some
+facts about bilinear forms on spaces of compactly supported continuous functions.
+Parts of this paper are based on the PhD thesis of the fourth-named author
+[Sch20b].
+Acknowledgments. The first three authors acknowledge financial support of the
+DFG within the priority programme Geometry at Infinity. M.W. acknowledges fi-
+nancial support by the German Academic Scholarship Foundation, by the Austrian
+Science Fund (FWF) through grant number F65 and the Esprit Programme [ESP
+156], and by the European Research Council (ERC) under the European Union’s
+Horizon 2020 research and innovation programme (grant agreement No 716117).
+For the purpose of Open Access, the authors have applied a CC BY public copy-
+right licence to any Author Accepted Manuscript (AAM) version arising from this
+submission.
+1. Dirichlet forms and domination of associated semigroups
+In this section we introduce notation and review some basic definitions and results
+about Dirichlet forms and domination of the associated semigroups. Unless stated
+otherwise, all functions are real-valued. Throughout (X, A, m) is a σ-finite measure
+
+4
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+space and Q denotes a nonnegative quadratic form with domain D(Q) ⊆ L2(X, m).
+We tacitly identify Q and the bilinear form it induces by polarization. In particular,
+we have the convention Q(f) = Q(f, f) for f ∈ D(Q). The form norm ∥·∥Q is the
+norm on D(Q) defined by
+∥f∥2
+Q = Q(f) + ∥f∥2,
+where ∥·∥ is the norm on L2(X, m). If Q′ is another quadratic form we write Q ⊑ Q′
+if D(Q) ⊆ D(Q′) and Q(f) ≥ Q′(f) for all f ∈ D(Q). The induced order relation
+⊑ on all quadratic forms is called the natural order.
+We say that a quadratic form is positive if Q(f, g) ≥ 0 for all nonnegative f, g ∈
+D(Q). It is called local if fg = 0 implies Q(f, g) = 0 for all f, g ∈ D(Q). Moreover,
+Q is called monotone if |f| ≤ |g| implies Q(f) ≤ Q(g) whenever f, g ∈ D(Q). In this
+case, Q(f) only depends on the absolute value of f and not on its sign. We discuss
+these properties for forms whose domains are continuous functions in Appendix A.
+1.1. (Regular) Dirichlet forms. A densely defined closed quadratic form Q on
+L2(X, m) is called Dirichlet form if f ∈ D(Q) implies f+∧1 ∈ D(Q) and Q(f+∧1) ≤
+Q(f). The second Beurling-Deny criterion [RS78, Theorem XIII.51] asserts that Q
+is a Dirichlet form if and only if the semigroup (e−tL) generated by the positive
+self-adjoint operator L associated with Q is Markovian, i.e., 0 ≤ f ≤ 1 implies
+0 ≤ e−tLf ≤ 1 for all t ≥ 0.
+If Q is a Dirichlet form, then D(Q) ∩ L∞(X, m) is an algebra with respect to
+pointwise multiplication and
+Q(fg)1/2 ≤ ∥g∥∞Q(f)1/2 + ∥f∥∞Q(g)1/2
+for all f, g ∈ D(Q) ∩ L∞(X, m), see [FOT11, Theorem 1.4.2].
+A Dirichlet form Q is called regular if the following are satisfied:
+• X is a locally compact separable metric space and m is a Radon measure of
+full support.
+• D(Q)∩Cc(X) is uniformly dense in Cc(X) and in D(Q) with respect to ∥·∥Q.
+In this case, the Q-capacity (or simply capacity if Q is fixed) of an open set O ⊆ X
+is defined by
+cap(O) = inf{∥f∥2
+Q | f ∈ D(Q) with f ≥ 1 m-a.e. on O}.
+Here we use the convention cap(O) = ∞ if there does not exist f ∈ D(Q) with
+f ≥ 1 on O. For an arbitrary set A ⊆ X, the capacity is defined by
+cap(A) = inf{cap(O) | O open with A ⊆ O}.
+The capacity is inner regular, i.e., for any Borel set A ⊆ X it satisfies
+cap(A) = sup{cap(K) | K compact with K ⊆ A},
+see [FOT11, Theorem 2.1.1]. Moreover, by [FOT11, Lemma 2.2.7], the capacity for
+compact K ⊆ X can alternatively be described as
+cap(K) = inf{∥f∥2
+Q | f ∈ D(Q) ∩ Cc(X) with f ≥ 1 on K}.
+
+INTERMEDIATE DIRICHLET FORMS
+5
+A subset A of X is called polar if Cap(A) = 0 holds. A property is said to hold
+quasi everywhere, abbreviated q.e., if it holds on the complement of a polar set.
+A measurable function f : X → [−∞, ∞] is said to be quasi continuous if for every
+ε > 0 there is an open set O with Cap(O) < ε such that f|X\O is finite-valued and
+continuous. If Q is a regular Dirichlet form, then every f in D(Q) has a unique (up
+to equality quasi everywhere) quasi continuous representative ˜f, cf. [CF12, Theorem
+2.3.4].
+1.2. Domination of Dirichlet forms and semigroups. If U, V are sublattices
+of L2(X, m), we say that U is an order ideal in V if f ∈ U, g ∈ V and |g| ≤ |f|
+implies g ∈ U.
+If U, V are subalgebras of L∞(X, m) we say U is an algebraic ideal in V if f ∈ U
+and g ∈ V implies fg ∈ U.
+We will frequently use the following characterization. The equivalence of (i) and
+(ii) is Ouhabaz’ domination criterion [Ouh96, Theorem 3.7], whereas the equivalence
+with (iii) is taken from [Sch20a, Lemma 2.2].
+Proposition 1.1 (Characterization of Domination). Let Q, Q′ be Dirichlet forms
+with associated self-adjoint operators L, L′. The following assertions are equivalent.
+(i) For all nonnegative f ∈ L2(X, m) and all t ≥ 0 we have
+e−tLf ≤ e−tL′f.
+(ii) D(Q) ⊆ D(Q′), D(Q) is an order ideal in D(Q′) and
+Q(f, g) ≥ Q′(f, g)
+for all non-negative f, g ∈ D(Q).
+(iii) D(Q) ⊆ D(Q′), D(Q) ∩ L∞(X, m) is an algebraic ideal in D(Q′) ∩ L∞(X, m)
+and
+Q(f, g) ≥ Q′(f, g)
+for all non-negative f, g ∈ D(Q).
+If Q and Q′ satisfy one of the conditions of this proposition, we say that Q′
+dominates Q and write Q ⪯ Q′. Similarly, in this situation we write (e−tL) ⪯ (e−tL′)
+and say that the semigroup (e−tL′) dominates the semigroup (e−tL).
+Domination also induces an order relation on the set of all Dirichlet forms on
+L2(X, m). Note that in general Q ⪯ Q′ does not imply Q ⊑ Q′ nor the other way
+round.
+2. The maximal dominating form and an abstract characterization
+of sandwiched semigroups
+For every Dirichlet form Q there is a maximal Dirichlet form Q(M) (with respect
+to the natural order) that dominates the given Dirichlet form Q. In this section we
+describe the construction of this maximal form and give an abstract characterization
+
+6
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+of all Dirichlet forms Q′ that satisfy Q ⪯ Q′ ⪯ Q(M) provided that Q(M) is an
+extension of Q.
+We denote by (Tt) the semigroup generated by Q and by (T (M)
+t
+) the semigroup
+generated by Q(M). According to the discussion in Subsection 1.2, any self-adjoint
+C0-semigroup (St) with
+(Tt) ⪯ (St) ⪯ (T (M)
+t
+)
+corresponds to a Dirichlet form Q′ with Q ⪯ Q′ ⪯ Q(M). Hence, our result can be
+seen as an abstract characterization of all semigroups sandwiched between (Tt) and
+(T (M)
+t
+).
+2.1. The active main part and the killing part. We will next recall the def-
+inition of the active main part and the killing part of a Dirichlet form. For two
+concrete examples see Examples 2.5, 2.6 below.
+Let Q be a Dirichlet form on L2(X, m). For ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1 we define
+the domain of the quadratic form ˜Qϕ on L2(X, m) by
+D( ˜Qϕ) = {f ∈ L2(X, m) ∩ L∞(X, m) | fϕ, f 2ϕ ∈ D(Q)},
+on which it acts by
+˜Qϕ(f) = Q(ϕf) − Q(ϕf 2, ϕ).
+Since D(Q) ∩ L∞(X, m) is an algebra, we have D(Q) ∩ L∞(X, m) ⊆ D( ˜Qϕ). The
+form ˜Qϕ is closable on L2(X, m). Indeed, [Sch20a, Theorem 3.1] shows that ˜Qϕ is
+lower semicontinuous on its domain with respect to local convergence in measure
+and hence it is lower semicontinuous on its domain with respect to L2-convergence.
+We denote its closure by Qϕ. The next proposition summarizes further important
+properties of Qϕ.
+Proposition 2.1. Let ϕ, ψ ∈ D(Q) with 0 ≤ ϕ ≤ ψ ≤ 1.
+(a) Qϕ is a Dirichlet form and its domain satisfies
+D(Qϕ) ∩ L∞(X, m) = D( ˜Qϕ) = {f ∈ L2(X, m) ∩ L∞(X, m) | fϕ ∈ D(Q)}.
+(b) D(Q) ⊆ D(Qϕ) and
+Qϕ(f) ≤ Q(f),
+f ∈ D(Q).
+(c) D(Qψ) ⊆ D(Qϕ) and
+Qϕ(f) ≤ Qψ(f),
+f ∈ D(Qψ).
+Proof. This follows from [Sch20a, Theorem 3.18]. The proofs given there treat an
+extension of Qϕ to all measurable m-a.e. defined functions that is lower semicontin-
+uous with respect to local convergence in measure. Restricting this form with larger
+domain to L2(X, m) yields all the claims.
+□
+
+INTERMEDIATE DIRICHLET FORMS
+7
+Remark 2.2. Part (a) of this proposition is important because it yields a formula
+for Qϕ for bounded functions in its domain. Namely, for f, g ∈ D(Qϕ) ∩ L∞(X, m)
+we have f, g ∈ D( ˜Qϕ) and hence
+Qϕ(f, g) = ˜Qϕ(f, g) = Q(ϕf, ϕg) − Q(ϕfg, ϕ).
+For the last equality, we used the definition of ˜Qϕ and polarization.
+Definition 2.3 (Active main part). The active main part Q(M) of Q is defined as
+follows: Its domain D(Q(M)) consists of all f ∈ L2(X, m) that satisfy f ∈ D(Qϕ)
+for all ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1 such that
+{ϕ ∈ D(Q) | 0 ≤ ϕ ≤ 1} → [0, ∞),
+ϕ �→ Qϕ(f)
+is bounded. On it Q(M) acts by
+Q(M)(f) = sup{Qϕ(f) | ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1}.
+Since ϕ �→ Qϕ(f) is monotone increasing, the form Q(M) is indeed a Dirichlet
+form, see [Sch20a, Theorem 3.6]. It turns out that Q(M) is the maximal Dirichlet
+form with respect to the natural order that dominates Q, i.e., Q ⪯ Q(M) and for all
+Dirichlet forms Q′ with Q ⪯ Q′ we have Q′ ⊑ Q(M), see [Sch20a, Theorem 3.19].
+However, Q(M) need not be an extension of Q and hence we introduce the following
+definition.
+Definition 2.4 (Killing part). The difference
+Q(k) = Q − Q(M)
+with domain D(Q(k)) = D(Q) is called the killing part of Q.
+The killing part is a local and positive quadratic form. Both properties are a
+consequence of Q(k) being monotone, see [Sch20a, Lemma 3.11] for monotonicity
+and [Sch20a, Lemma B.1] for how monotonicity implies the other properties. In
+particular, the value of Q(k)(f) only depends on |f| and not on the sign of f.
+We illustrate these objects with an example. It shows that the active main part is
+an abstract way of constructing operators with Neumann boundary conditions from
+the quadratic forms leading to Dirichlet boundary conditions.
+Example 2.5 (Dirichlet and Neumann Laplacian on domains). Let Ω ⊆ Rn be
+open (or more generally let Ω be a Riemannian manifold) and let V ∈ L1
+loc(Ω) be
+nonnegative. We consider the Dirichlet form E(N)
+V
+with domain D(E(N)
+V
+) = {f ∈
+H1(Ω) | V 1/2f ∈ L2(Ω)}, on which it acts by
+E(N)
+V
+(f) =
+�
+Ω |∇f|2dx +
+�
+Ω |f|2V dx.
+The associated operator is the self-adjoint realization of the Schrödinger operator
+H = −∆ + V with (abstract) Neumann boundary conditions, which we denote by
+H(N). Moreover, we let E(D)
+V
+be the restriction of E(N)
+V
+to D(E(D)
+V
+) = {f ∈ H1
+0(Ω) |
+
+8
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+V 1/2f ∈ L2(Ω)}. This is a regular Dirichlet form and the associated operator is the
+self-adjoint realization of the Schrödinger operator H = −∆ + V with (abstract)
+Dirichlet boundary conditions, which we denote by H(D).
+The active main part of E(D)
+V
+is given by E(N)
+0
+. Hence, the self-adjoint operator
+associated to the active main part is −∆(N). For f ∈ D(E(D)
+V
+), the killing part of
+E(D)
+V
+is given by
+(E(D)
+V
+)(k)(f) =
+�
+Ω |f|2V dx.
+In particular, if V = 0, this discussion shows that a Dirichlet form Q satisfies E(D)
+0
+⪯
+Q ⪯ (E(D)
+0
+)(M) if and only if the associated semigroup (St) satisfies (et∆(D)) ⪯ (St) ⪯
+(et∆(N)). Hence, forms sandwiched between E(D)
+0
+and (E(D)
+0
+)(M) = E(N)
+0
+correspond
+to semigroups sandwiched between the Dirichlet and the Neumann semigroup of the
+Laplacian. This is precisely the situation studied in [AW03].
+Proof. Here we only sketch the main ideas of the proof. For the details we refer to
+[Sch20a, Example 3.9]. We only consider bounded functions, the general case can
+be treated through approximations.
+Let f ∈ H1(Ω)∩L∞(Ω) and let ϕ ∈ C∞
+c (Ω) with 0 ≤ ϕ ≤ 1. A direct computation
+using the product rule for ∇ shows f ∈ D((E(D)
+V
+)ϕ) and
+(E(D)
+V
+)ϕ(f) =
+�
+Ω ϕ2|∇f|2dx.
+Letting ϕ ր 1 and taking into account that C∞
+c (Ω) is dense in D(E(D)
+V
+) yields
+f ∈ D((E(D)
+V
+)(M)) and the formula for the action of (E(D)
+V
+)(M).
+Similarly, if f ∈ D((E(D)
+V
+)(M))∩L∞(Ω), by the definition of (E(D)
+V
+)ϕ and the active
+main part, we have ϕf ∈ D(E(D)
+V
+) = H1
+0(Ω) ∩ L2(Ω, V · dx) for every ϕ ∈ C∞
+c (Ω).
+This yields ∇f ∈ ⃗L2
+loc(Ω). With this at hand, an application of the product rule for
+∇ as above shows (E(D)
+V
+)ϕ(f) =
+�
+Ω ϕ2|∇f|2dx. Since (E(D)
+V
+)ϕ(f) ≤ (E(D)
+V
+)(M)(f) and
+ϕ is arbitrary, we conclude ∇f ∈ ⃗L2(Ω) so that f ∈ H1(Ω).
+The statement on the killing part is an immediate consequence.
+□
+Example 2.6 (Fractional Laplacians). As above we let Ω ⊆ Rn be open. For a
+background on fractional Sobolev spaces we refer to [DNPV12]. For 0 < s < 1, we
+denote by Qs,(N) the Dirichlet form with domain D(Qs,(N)) = W s(Ω) on which it
+acts by
+Qs,(N)(f) = 1
+2
+�
+Ω×Ω
+|f(x) − f(y)|2
+|x − y|n+2s
+dx dy.
+The restriction of this form to W s
+0 (Ω) is denoted by Qs,(D), it is a regular Dirichlet
+form. Note that at least if Ω is bounded and has C∞-boundary, the spaces W s
+0 (Ω)
+and W s(Ω) coincide for 0 < s ≤
+1
+2 by [LM72, Theorem 11.1], which makes the
+problem of finding the Dirichlet forms sandwiched between Qs,(D) and Qs,(N) trivial.
+
+INTERMEDIATE DIRICHLET FORMS
+9
+It is well-known that the associated self-adjoint operators H(N)
+s
+and H(D)
+s
+are
+restrictions of the restricted fractional Laplacian Hs given by
+Hsf(x) = P.V.
+�
+Ω
+f(x) − f(y)
+|x − y|n+2s dy = lim
+ε→0+
+�
+Ω\Bε(x)
+f(x) − f(y)
+|x − y|n+2s dy.
+Hence, they can be viewed as realizations of Hs with abstract Neumann and Dirich-
+let boundary conditions.
+Note that we ignore a constant so that our fractional
+Laplacian is only a constant multiple of the ’usual’ restricted fractional Laplacian,
+cf. [DNPV12, Section 3]. Similar as in the previous example the active main part
+of Qs,(D) is Qs,(N).
+Proof. Here we only show the statement on the active main part of Qs,(D), the rest
+is well-known. Since Qs,(N) and (Qs,(D))(M) are Dirichlet forms, it suffices to prove
+D(Qs,(N)) ∩ L∞(Ω) = D((Qs,(D))(M)) ∩ L∞(Ω) and that Qs,(N) and (Qs,(D))(M) agree
+on these sets (use that bounded functions are dense in the domains of Dirichlet
+forms, see [FOT11, Theorem 1.4.2]).
+We first proof that Qs,(N) is a restriction of (Qs,(D))(M) (on L∞(Ω)). Let f ∈
+W s(Ω) ∩ L∞(Ω) and let ϕ ∈ W s
+0 (Ω) with 0 ≤ ϕ ≤ 1. Then fϕ ∈ W s
+0 (Ω). We infer
+Qs,(D)(ϕf) − Qs,(D)(ϕf 2, ϕ) = 1
+2
+�
+Ω×Ω
+(ϕ(x)f(x) − ϕ(y)f(y))2
+|x − y|n+2s
+dx dy
+− 1
+2
+�
+Ω×Ω
+(ϕ(x)f(x)2 − ϕ(y)f(y)2)(ϕ(x) − ϕ(y))
+|x − y|n+2s
+dx dy
+= 1
+2
+�
+Ω×Ω ϕ(x)ϕ(y)|f(x) − f(y)|2
+|x − y|n+2s
+dx dy.
+Taking the supremum over such ϕ yields f ∈ D((Qs,(D))(M)) and (Qs,(D))(M)(f) =
+Qs,(N)(f).
+It remains to prove D((Qs,(D))(M)) ∩ L∞(Ω) ⊆ W s(Ω). Let f ∈ D((Qs,(D))(M)) ∩
+L∞(Ω). For ϕ ∈ W s
+0(Ω) with 0 ≤ ϕ ≤ 1, we have by definition of the main part
+ϕf, ϕf 2 ∈ W s
+0 (Ω) and
+(Qs,(D))(M)(f) ≥ Qs,(D)(ϕf) − Qs,(D)(ϕf 2, ϕ)
+= 1
+2
+�
+Ω×Ω ϕ(x)ϕ(y)|f(x) − f(y)|2
+|x − y|n+2s
+dx dy.
+For the last equality we used the same computation as above. Since ϕ was arbitrary,
+this shows f ∈ W s(Ω).
+□
+Remark 2.7. These examples show that it is a good intuition to think of a regular
+Dirichlet form Q with Q(k) = 0 as being a form with ‘Dirichlet type’ boundary con-
+ditions and Q(M) being the ’same’ form with ‘Neumann type’ boundary conditions.
+
+10
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+2.2. An abstract characterization of sandwiched semigroups and forms.
+The main abstract result of this paper is the following characterization of Dirichlet
+forms sandwiched between a Dirichlet form without killing and its active main part.
+Theorem 2.8. Let Q, Q′ be Dirichlet forms on L2(X, m) with Q(k) = 0. The fol-
+lowing assertions are equivalent.
+(i) Q ⪯ Q′ ⪯ Q(M).
+(ii) (a) D(Q′) ⊆ D(Q(M)) and D(Q′) is an order ideal in D(Q(M)).
+(b) Q′ − Q(M) is a positive and local form on D(Q′).
+(c) Q′ is an extension of Q.
+Proof. (i) =⇒ (ii): (a) This is a consequence of Proposition 1.1.
+(b) The positivity of Q′ − Q(M) follows directly from Q′ ⪯ Q(M), cf. Proposi-
+tion 1.1. In order to see that Q′ − Q(M) is local, we let f, g ∈ D(Q′) with fg = 0.
+Without loss of generality we may assume f, g ≥ 0, for otherwise we can decom-
+pose f, g into positive and negative parts and use f±g± = 0. Since Q′ and Q(M)
+are Dirichlet forms, we can further assume that f, g are bounded. As we already
+established positivity, it remains to prove Q′(f, g) − Q(M)(f, g) ≤ 0.
+Let ϕ ∈ D(Q) with 0 ≤ ϕ ≤ 1. According to Proposition 1.1 we have fϕ, gϕ ∈
+D(Q), so that by Proposition 2.1 f, g ∈ D(Qϕ). Using fg = 0 and Q ⪯ Q′ we obtain
+Q′(f, g) − Qϕ(f, g) = Q′(f, g) − Q(ϕf, ϕg) + Q(ϕfg, ϕ)
+= Q′(f, g) − Q′(ϕf, ϕg)
+= Q′((1 − ϕ)f, g) + Q′(ϕf, (1 − ϕ)g).
+The functions η = (1 − ϕ)f and ζ = g are nonnegative and satisfy ηζ = 0. The
+Dirichlet form property of Q′ implies
+Q′(η + ζ) = Q′(|η + ζ|) = Q′(|η − ζ|) ≤ Q′(η − ζ),
+from which we deduce Q′(η, ζ) ≤ 0 by bilinearity. The same argument applies to
+η = ϕf and ζ = (1 − ϕ)g so that we obtain
+Q′(f, g) − Qϕ(f, g) = Q′((1 − ϕ)f, g) + Q′(ϕf, (1 − ϕ)g) ≤ 0.
+By the definition of Q(M) we can choose ϕ such that Qϕ(f, g) is arbitrarily close to
+Q(M)(f, g) and hence obtain locality.
+(c) The domination Q ⪯ Q′ ⪯ Q(M) and Q(k) = 0 yield for all nonnegative
+f, g ∈ D(Q) the inequality
+Q(f, g) = Q(M)(f, g) ≤ Q′(f, g) ≤ Q(f, g).
+By splitting functions into positive and negative parts this shows Q = Q′ on D(Q).
+(ii) =⇒ (i): Q′ ⪯ Q(M) follows directly from (a) and (b) and the characterization
+of domination Proposition 1.1.
+
+INTERMEDIATE DIRICHLET FORMS
+11
+Q ⪯ Q′: Since D(Q′) is contained in D(Q(M)) and D(Q) is an order ideal in
+D(Q(M)) we obtain that D(Q) is also an order ideal in D(Q′).
+Since Q′ is an
+extension of Q, this already implies domination.
+□
+We can rephrase this theorem slightly. Let Q be a Dirichlet form with Q(k) = 0.
+We say that a pair (F, q) consisting of a vector lattice F ⊆ D(Q(M)) that is an order
+ideal in D(Q(M)) and a quadratic form q with D(q) = F is an abstract admissible
+pair for Q, if it satisfies the following properties:
+• D(Q) ⊆ D(q) and q(f) = 0 for f ∈ D(Q),
+• q is local and positive,
+• the form QF,q = Q(M)|F + q is closed.
+Corollary 2.9. Let Q be Dirichlet forms with Q(k) = 0. The following assertions
+are equivalent.
+(i) Q′ is a Dirichlet form with Q ⪯ Q′ ⪯ Q(M).
+(ii) There exists an abstract admissible pair (F, q) such that Q′ = QF,q.
+Proof. (i) =⇒ (ii): This is a reformulation of the previous theorem.
+(ii) =⇒ (i): Using the previous theorem it suffices to show that QF,q is a Dirichlet
+form. Since closedness and density of D(QF,q) = F are part of the definition of
+abstract admissible pairs, it suffices to prove the Markov property. By assumption
+F is an order ideal in D(Q(M)) and for f ∈ F we have f+ ∧ 1 ∈ D(Q(M)) and
+|f+ ∧ 1| ≤ |f|. This shows f+ ∧ 1 ∈ F whenever f ∈ F. Moreover, as already
+discussed after introducing the killing part, q being local and positive yields that q is
+monotone, see [Sch20a, Lemma B.1]. These observations and Q(M) being Markovian
+imply
+QF,q(f+ ∧ 1) = Q(M)(f+ ∧ 1) + q(f+ ∧ 1) ≤ Q(M)(f) + q(f) = QF,q(f).
+□
+Remark 2.10. This corollary shows that in order to determine all sandwiched forms
+between Q and Q(M) we need to characterize all abstract admissible pairs. This is
+possible when Q(M) is a regular Dirichlet form on a metric space K containing X as
+a dense open subset. In the next section we will prove that in this case:
+(a) Positive and local forms correspond to measures if their domain contains suf-
+ficiently many continuous functions, see Appendix A. If these forms satisfy
+q(f) = 0 for f ∈ D(Q), the corresponding measure is supported on the boundary
+K \ X.
+(b) Closed order ideals in D(Q(M)) correspond to functions vanishing outside an
+open set (under some additional density assumption for continuous functions).
+This then allows us to identify abstract admissible pairs with pairs of open subsets
+of the boundary and certain measures on them.
+
+12
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+3. Domination for parts of regular Dirichlet forms
+Let Q be a regular Dirichlet form on L2(X, m). Let O ⊆ X be an open set and let
+µ be a Radon measure on the Borel σ-algebra of O. We define the quadratic form
+Qc
+O,µ by letting D(Qc
+O,µ) = D(Q) ∩ Cc(O) and
+Qc
+O,µ(f) = Q(f) +
+�
+O f 2dµ.
+Here Cc(O) is tacitly identified with {ϕ ∈ Cc(X) | supp ϕ ⊆ O}.
+Proposition 3.1. The following assertions are equivalent.
+(i) µ charges no sets of Q-capacity zero.
+(ii) The quadratic form Qc
+O,µ is closable.
+In this case, the closure QO,µ of Qc
+O,µ is given by
+D(QO,µ) = {f ∈ D(Q) | ˜f = 0 q.e. on X \ O and
+�
+O
+˜f 2dµ < ∞},
+QO,µ(f) = Q(f) +
+�
+O
+˜f 2dµ.
+Proof. (i) =⇒ (ii): This follows as in [Sto92, Theorem 1.2].
+(ii) =⇒ (i): By the inner regularity of the capacity and the inner regularity of
+the Radon measure µ it suffices to show for compact sets K ⊆ O that cap(K) = 0
+implies µ(K) = 0.
+Let now K ⊆ O be compact with cap(K) = 0. Since Q is regular, there exists a
+sequence (ϕn) in D(Q) ∩ Cc(X) such that ∥ϕn∥Q → 0, 0 ≤ ϕn ≤ 1 and ϕn ≥ 1 on
+K. Let G be open and relatively compact with K ⊆ G ⊆ O. Using regularity of Q
+again yields the existence of a function ψ ∈ D(Q) ∩ Cc(X) with 0 ≤ ψ ≤ 1, ψ = 1
+on K and supp ψ ⊆ G.
+We now consider fn := ψ · ϕn.
+Since Q is a Dirichlet form, it satisfies fn ∈
+D(Q) ∩ Cc(O) = D(Qc
+O,µ) and
+Q(fn)1/2 ≤ Q(ψ)1/2 + Q(ϕn)1/2.
+The inequality 0 ≤ fn ≤ 1 and supp fn ⊆ G imply ∥fn∥ → 0 as n → ∞ and
+�
+O |fn|2dµ ≤ µ(G),
+n ≥ 1.
+In particular, these estimates show that (fn) is bounded with respect to the form
+norm ∥·∥QO,µ. Let QO,µ be the closure of Qc
+O,µ, which exists by (ii). The Banach–Saks
+theorem implies that for some subsequence (fnk) the sequence of Césaro means
+gN := 1
+N
+N
+�
+k=1
+fnk
+converges to some g ∈ D(QO,µ) with respect to ∥·∥QO,µ. The form norm of QO,µ
+is larger than ∥·∥ and hence we obtain gn → g with respect to ∥·∥.
+But since
+
+INTERMEDIATE DIRICHLET FORMS
+13
+∥fn∥ → 0, we conclude g = 0. By the choice of (fn) we also have gN ∈ D(Q)∩Cc(O),
+0 ≤ gN ≤ 1 and gN ≥ 1 on K. Putting everything together we obtain
+µ(K) ≤
+�
+O |gN|2dµ ≤ QO,µ(gN) → 0 as N → ∞.
+This yields the desired µ(K) = 0.
+The proof of the formula for QO,µ follows as in [SV96, Proposition 1.1].
+□
+Definition 3.2. A pair (O, µ) satisfying one of the conditions of the previous the-
+orem is called an admissible pair for the form Q. In this case, we write QO,µ for the
+closure of the form Qc
+O,µ above.
+Proposition 3.3. Let (Oi, µi), i = 1, 2, be admissible pairs for Q. The following
+assertions are equivalent:
+(i) QO1,µ1 ⪯ QO2,µ2.
+(ii) cap(O1 \ O2) = 0 and µ2(A) ≤ µ1(A) for every Borel set A ⊆ O1 ∩ O2.
+Proof. (ii) =⇒ (i): This follows immediately from (ii) and the formula for QOi,µi
+given in Proposition 3.1.
+(i) =⇒ (ii): Since D(QO1,µ1) is a lattice and an order ideal in D(QO2,µ2), we have
+D(QO1,µ1) ⊆ D(QO2,µ2). Hence, every f ∈ D(QO1,µ1) satisfies ˜f = 0 q.e. on X \ O2.
+Now, suppose cap(O1 \ O2) > 0.
+By [FOT11, Theorem 2.1.1] there exists a
+compact set K ⊆ O1 \ O2 with cap(K) > 0 and by [FOT11, Theorem 2.1.5] there
+exists f ∈ D(Q) with 0 ≤ f ≤ 1 and ˜f = 1 q.e. on K. By the regularity of Q there
+exists ϕ ∈ D(Q) ∩ Cc(O1) with ϕ ≥ 1 on K. We obtain ϕf ∈ D(QO1,µ1) as ϕ ˜f = 0
+q.e. on X \ O1 and
+�
+O1
+|ϕ ˜f|2dµ1 ≤
+�
+O1
+|ϕ|2dµ1 < ∞.
+Furthermore, ϕ ˜f ≥ 1 q.e. on K ⊆ X \ O2. This and cap(K) > 0 are a contradiction
+to the fact that functions in D(QO1,µ1) vanish q.e. on X \ O2.
+It remains to prove the inequality for the measures. Domination implies
+Q(ϕ) +
+�
+O2
+|ϕ|2dµ2 ≤ Q(ϕ) +
+�
+O1
+|ϕ|2dµ1
+for all nonnegative ϕ ∈ D(Q) ∩ Cc(O1). For any compact set K ⊆ O1 ∩ O2 and any
+open neighborhood G of K with G ⊆ O1 ∩ O2 there exists ψ ∈ Cc(X) ∩ D(Q) with
+supp ψ ⊆ G, 0 ≤ ψ ≤ 1 and ψ ≥ 1 on K. Plugging this into the last inequality
+yields
+µ2(K) ≤
+�
+O2
+|ψ|2dµ2 ≤
+�
+O1
+|ψ|2dµ1 ≤ µ1(G).
+Thus we obtain µ2(K) ≤ µ1(K) from the outer regularity of the Radon measure
+µ1. By inner regularity of Radon measures this implies the statement for all Borel
+sets.
+□
+
+14
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+4. A boundary for regular Dirichlet forms and a characterization
+of sandwiched semigroups
+4.1. A boundary for regular Dirichlet forms. Let Q be a regular Dirichlet form
+on L2(X, m). In this subsection we introduce a locally compact separable metric
+space K that contains X as an open subset and extend m to a Radon measure ˆm
+on K such that Q(M) can be considered to be a regular form on L2(K, ˆm).
+The spaces Cc(X) and L2(X, m) are separable because X is locally compact sep-
+arable metric space and m is a Radon measure. The map
+L2(X, m) → D(Q(M)),
+f �→ (L(M) + 1)−1f
+is continuous with respect to the form norm ∥·∥Q(M) (here L(M) denotes the positive
+self-adjoint operator associated with Q(M)). It has dense image D(L(M)) in D(Q(M))
+with respect to ∥·∥Q(M), showing that (D(Q(M)), ∥·∥Q(M)) is also separable. Moreover,
+[Sch20a, Theorem 4.3] asserts that for a regular Dirichlet form Q the space D(Q(M))∩
+Cb(X) is dense in D(Q(M)) with respect to ∥·∥Q(M) (this is an abstract version of the
+Meyers-Serrin theorem).
+Combining these observations yields the existence of a subalgebra C of D(Q(M))∩
+Cb(X) with the following three properties:
+• C is countably generated.
+• C is ∥·∥Q(M)-dense in D(Q(M)).
+• C ∩ Cc(X) is uniformly dense in Cc(X).
+Let A be the uniform closure of C. Its complexification AC = {f + ig | f, g ∈ A}
+is a commutative C∗-algebra that satisfies C0(X; C) ⊆ AC ⊆ Cb(X; C). By Gelfand
+theory there exists a unique (up to homeomorphism) locally compact, separable
+Hausdorff space K with the following properties:
+• X is a dense and open subset of K.
+• Every f ∈ AC can be extended to a function ˆf ∈ C0(K; C) and
+C0(K; C) = { ˆf | f ∈ AC}.
+As C is countably generated, the space K is metrizable. Hence, K is Polish, that is,
+separable and completely metrizable, since every locally compact, separable, second
+countable space is Polish. Since X is dense in K, the continuous extension of a
+function from A to K is unique and we will therefore not distinguish between ele-
+ments of A and their extension. For real-valued functions this interpretation leads
+to A = C0(K)(= C0(K; R)).
+The measure m on X can be extended to a Borel measure ˆm on K by setting
+ˆm(A) = m(A ∩ X),
+A ∈ B(K).
+The measure ˆm is again a Radon measure of full support. By this definition the
+space L2(K, ˆm) can be naturally identified with L2(X, m) via the unitary map
+R: L2(K, ˆm) → L2(X, m),
+f �→ f|X.
+
+INTERMEDIATE DIRICHLET FORMS
+15
+Our discussion shows R−1(A∩L2(X, m)) = C0(K)∩L2(K, ˆm). Since R also preserves
+the order relation, any Dirichlet form on L2(X, m) can be viewed as a Dirichlet form
+on L2(K, ˆm) under this transformation. In particular, the form Q(M) is a regular
+Dirichlet form on L2(K, ˆm), see [Sch20a, Theorem 4.4].
+The following remark sketches the uniqueness of the space K. We leave details
+(especially the involved definitions, which can be found in [FOT11, Appendix A.4])
+to the reader.
+Remark 4.1 (Uniqueness of K). The space K depends on the choice of the algebra C.
+However, given two algebras C, C′ with the required properties and the corresponding
+spaces K, K′, there exists a unitary order isomorphism
+U : L2(K, ˆm) → L2(K′, ˆm′)
+such that 0 ≤ fn ≤ 1, fn ր 1 implies Ufn ր 1 and U intertwines Q(M) and Q(M)
+(when considererd as a form on the corresponding space). This implies that both
+forms are equivalent in the sense of [FOT11, Appendix A.4]. Since they are also
+regular, [FOT11, Theorem A.4.2] yields that K and K′ are quasi-homeomorphic (and
+establishes further properties of a corresponding quasi-homeomorphism).
+In view of the previous remark we make the following definition.
+Definition 4.2. The set ∂X = K \ X is called the boundary of X relative to the
+form Q.
+Example 4.3 (Dirichlet and Neumann Laplacian – continued). We use the situation
+and notation of Example 2.5 and assume that the potential vanishes, i.e., V = 0.
+As discussed in Example 2.5 we have (E(D)
+0
+)(M) = E(N)
+0
+so that D((E(D)
+0
+)(M)) =
+H1(Ω) and the standard Sobolev norm on H1(Ω) coincides with the form norm of
+(E(D)
+0
+)(M).
+If Ω ⊆ Rn has continuous boundary (for a precise definition see e.g.
+[EE18, Definition 4.1]), the space {f|Ω | f ∈ C∞
+c (Rn)} is dense in H1(Ω) with
+respect to the standard Sobolev norm. Hence, in this case we can choose the algebra
+C to be a subset of {f|Ω | f ∈ C∞
+c (Rn)} ⊆ Cc(Ω).
+Since by Stone-Weierstraß
+{f|Ω | f ∈ C∞
+c (Rn)} is dense in C0(Ω), we can further assume that C is dense
+in C0(Ω), so that the algebra A, the uniform closure of C, equals C0(Ω).
+Our
+construction then yields K = Ω (up to homeomorphism) and that the boundary of
+Ω relative to E(D)
+0
+coincides with the metric boundary ∂Ω = Ω \ Ω in Rn.
+Example 4.4 (Fractional Laplacian – continued). We use the situation and notation
+of Example 2.6. As discussed above we have D((Qs,(D))(M)) = W s(Ω). Moreover, if
+Ω has Lipschitz boundary, then {f|Ω | f ∈ C∞
+c (Rn)} is dense in W s(Ω) with respect
+to the form norm of Qs,(N), which coincides with the ususal norm on W s(Ω), see
+[DNPV12, Corollary 5.5]. With this at hand the same argument as in the previous
+example yields that we can choose K = Ω such that the boundary of Ω relative to
+Qs,(D) coincides with the metric boundary ∂Ω = Ω \ Ω in Rn.
+
+16
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+4.2. A characterization of sandwiched semigroups for regular Dirichlet
+forms. In this subsection we prove the main result of this paper.
+Let Q be a
+regular Dirichlet form on L2(X, m). We apply the theory developed in Section 3 to
+the form Q(M) when considered as a regular Dirichlet form on L2(K, ˆm). We start
+with a simple observation that follows from the previous discussion.
+Proposition 4.5. Let Q be a regular Dirichlet form. Then (K, 0) and (X, 0) are
+admissible pairs for the regular Dirichlet form Q(M) on L2(K, ˆm) and we have Q(M) =
+(Q(M))K, 0 and Q = (Q(M))X, 0. In particular,
+D(Q) = {f ∈ D(Q(M)) | ˜f = 0 q.e. on ∂X}.
+The following is the main result of the paper.
+Theorem 4.6. Let Q be a regular Dirichlet form with Q(k) = 0. For a Dirichlet
+form Q′ the following assertions are equivalent:
+(i) There exists an admissible pair (O, µ) for Q(M) with X ⊆ O and µ(X) = 0
+such that Q′ = (Q(M))O,µ.
+(ii) (a) Q ⪯ Q′ ⪯ Q(M)
+(b) D(Q′) ∩ Cc(K) is dense in D(Q′) with respect to ∥·∥Q′.
+Proof. (i) =⇒ (ii): (a) follows from Proposition 3.3 and the identities discussed in
+Propostion 4.5. The density of D((Q(M))O,µ) ∩ Cc(O) in D((Q(M))O,µ) with respect
+to ∥·∥(Q(M))O,µ is part of the definition of the form (Q(M))O,µ.
+(ii) =⇒ (i): Let D be the uniform closure of the algebra D(Q′) ∩ Cc(K). Since
+D(Q′) ∩ Cc(K) is an algebraic ideal in D(Q(M)) ∩ Cc(K) (here we use domination
+and Proposition 3.3), D is a uniformly closed ideal in
+D(Q(M)) ∩ Cc(K)
+∥·∥∞ = C0(K)
+(here we use the regularity of Q(M)). Moreover, by Theorem 2.8 we have D(Q) ⊆
+D(Q′) so that D(Q)∩Cc(X) ⊆ D(Q′)∩Cc(K). Since Q is regular on L2(X, m), this
+yields C0(X) ⊆ D. By the characterization of closed ideals in C0(K) there exists an
+open set X ⊆ O ⊆ K such that
+D = {f ∈ C0(K) | f = 0 on K \ O}.
+Altogether this discussion shows that D(Q′) ∩ Cc(O) is ∥·∥Q′ dense in D(Q′) and
+uniformly dense in Cc(O).
+Next, we show D(Q′) ∩ Cc(O) = D(Q(M)) ∩ Cc(O). Let ϕ ∈ D(Q(M)) ∩ Cc(O)
+and let K = supp ϕ ⊆ O.
+Since Q′ is a Dirichlet form and D(Q′) ∩ Cc(O) is
+uniformly dense in Cc(O), there exists ψ ∈ D(Q′) ∩ Cc(O) with ψ = 1 on K.
+We obtain ϕ = ψϕ ∈ D(Q′) ∩ Cc(O) since D(Q′) ∩ Cc(O) is an algebraic ideal in
+D(Q(M)) ∩ Cc(O) (here we use domination and Proposition 3.3).
+According to Theorem 2.8 the domination (a) implies that the form q = Q′−Q(M)
+with domain D(q) = D(Q(M)) ∩ Cc(O) is positive, local and satisfies q(f) = 0 for
+
+INTERMEDIATE DIRICHLET FORMS
+17
+all f ∈ D(Q) ∩ Cc(X). By Corollary A.4 there exists a Radon measure µ on O such
+that
+q(f) =
+�
+O |f|2dµ,
+f ∈ D(Q(M)) ∩ Cc(O).
+Since D(Q) ∩ Cc(X) is uniformly dense in Cc(X), the property q(f) = 0 for f ∈
+D(Q) ∩ Cc(X) implies µ(X) = 0.
+For f ∈ D(Q(M)) ∩ Cc(O), we have by definition of q that
+Q′(f) = Q(M)(f) +
+�
+O |f|2dµ = (Q(M))c
+O,µ(f).
+Since Q′ is closed and D(Q(M)) ∩ Cc(O) is ∥·∥Q′-dense in D(Q′), this implies that
+(O, µ) is an admissible pair for Q(M) and Q′ = (Q(M))O,µ.
+□
+We can reformulate this theorem as follows.
+Corollary 4.7. Let Q be a regular Dirichlet form with Q(k) = 0. For a Dirichlet
+form Q′, the following assertions are equivalent.
+(i) There exists an open subset ∂µX ⊆ ∂X and a Radon measure µ on ∂µX that
+does not charge sets of Q(M)-capacity zero such that
+D(Q′) = {f ∈ D(Q(M)) | ˜f = 0 q.e. on ∂X \ ∂µX and
+�
+∂µX | ˜f|2dµ < ∞}
+and
+Q′(f) = Q(M)(f) +
+�
+∂µX | ˜f|2dµ.
+(ii) (a) Q ⪯ Q′ ⪯ Q(M)
+(b) D(Q′) ∩ Cc(K) is dense in D(Q′) with respect to ∥·∥Q′.
+As an application of this result and our examples we obtain one of the main results
+of [AW03] under slightly less restrictive assumptions.
+Example 4.8. Again we use the situation of Schrödinger operators on Ω of Exam-
+ple 2.5 with V = 0. Assume further that Ω ⊆ Rn has continuous boundary and
+let Q be a Dirichlet form on L2(Ω) with associated Markovian semigroup (St). Let
+∂Ω = Ω \ Ω be the metric boundary of Ω. The disscusion in Example 2.5 and Ex-
+ample 4.3 combined with the previous corollary yield that the following assertions
+are equivalent.
+(i) There exists an open subset ∂µΩ ⊆ ∂Ω and a Radon measure µ on ∂µΩ that
+does not charge sets of E(N)
+0
+-capacity zero such that
+D(Q) = {f ∈ H1(Ω) | ˜f = 0 q.e. on ∂Ω \ ∂µΩ and
+�
+∂µΩ | ˜f|2dµ < ∞}
+and
+Q(f) =
+�
+Ω |∇f|2dx +
+�
+∂µΩ | ˜f|2dµ.
+(ii) (a) (et∆(D)) ⪯ (St) ⪯ (et∆(N))
+
+18
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+(b) D(Q) ∩ Cc(Ω) is dense in D(Q) with respect to ∥·∥Q.
+This is precisely the statement of [AW03, Theorem 4.1] under the slightly less restric-
+tive assumption of Ω having continuous boundary instead of Lipschitz boundary.
+Example 4.9. We use the situation of fractional Laplacians of Example 2.6. Assume
+further that Ω ⊆ Rn has Lipschitz boundary and let Q be a Dirichlet form on L2(Ω)
+with associated Markovian semigroup (St). Let ∂Ω = Ω \ Ω be the metric boundary
+of Ω. The disscusion in Example 2.6 and Example 4.4 combined with the previous
+corollary yield that the following assertions are equivalent.
+(i) There exists an open subset ∂µΩ ⊆ ∂Ω and a Radon measure µ on ∂µΩ that
+does not charge sets of Qs,(N)-capacity zero such that
+D(Q) = {f ∈ W s(Ω) | ˜f = 0 q.e. on ∂Ω \ ∂µΩ and
+�
+∂µΩ | ˜f|2dµ < ∞}
+and
+Q(f) = 1
+2
+�
+Ω×Ω
+|f(x) − f(y)|2
+|x − y|n+2s
+dx dy +
+�
+∂µΩ | ˜f|2dµ.
+(ii) (a) (e−tH(D)
+s
+) ⪯ (St) ⪯ (e−tH(N)
+s
+)
+(b) D(Q) ∩ Cc(Ω) is dense in D(Q) with respect to ∥·∥Q.
+The implication (i) =⇒ (ii) was also proved for Dirichlet forms associated with a re-
+lated, but different fractional Laplacian by Claus and Warma [CW20, Theorem 4.2].
+As mentioned in the introducion we wanted to provide a version of the results of
+[AW03] for general Dirichlet forms. In the abstract framework we were as general
+as possible but held back in generality for regular Dirichlet forms. In the following
+remarks we collect what else can be deduced from our general framework (at the
+cost of brevity and technical simplicity).
+Remark 4.10. (a) Another result of [AW03] is the descritption of the operators
+corresponding to semigroups (et∆(D)) ⪯ (St) ⪯ (et∆(N)) as Laplacians with Robin
+type boundary conditions. Something similar is possible here after equipping
+the abstract boundary ∂X with so-called harmonic measures. This allows for
+the definition of densities of normal derivatives and leads to abstract Robin
+boundary conditions. In the Euclidean setting with Ω having Lipschitz boundary
+the harmonic measures are mutually absolutely continuous with respect to the
+surface measure on ∂Ω and the abstract normal derivatives are given by the
+usual normal derivative.
+(b) In Theorem 4.6 we used that D(Q′) ∩ Cc(K) is dense in D(Q′) because we
+constructed the set O as the complement of the zero set of the closed ideal
+D(Q′) ∩ Cc(K)
+∥·∥∞
+in C0(K). One can drop the density assumption and replace this argument by
+the characterization of closed ideals in regular Dirichlet spaces given in [Sto93].
+
+INTERMEDIATE DIRICHLET FORMS
+19
+In this case, Theorem 4.6 remains true without assertion (ii)(b) but with O open
+replaced by O quasi-open.
+(c) We always assumed that the killing part vanishes. If Q(k) ̸= 0, then there are
+two possible choices of reference for the maximal form:
+(1) One can characterize all Dirichlet forms Q′ with Q ⪯ Q′ ⪯ Q(M) via abstract
+admissible pairs. Since in this case Q(M) is not an extension of Q, the form q
+of the abstract admissible pair corresponding to Q′ does not vanish on D(Q)
+but is bounded above by Q(k). In the regular setting this implies that the
+measure µ from the admissible pair corresponding to Q′ is not necessarily
+supported only on ∂X. It satisfies µ ≤ k on X, where k is the measure
+corresponding to the local and positive form Q(k) (cf. Appendix A).
+(2) Instead of comparing Q′ with Q(M) one can characterize Q ⪯ Q′ ⪯ Qref,
+where Qref is the active reflected Dirichlet form of Q. It arises by adding a
+suitable extension of Q(k) to Q(M), cf. [Sch20a, Section 3.3]. In this case, our
+main theorems still hold true but the proofs become substantially longer.
+Appendix A. Bilinear forms on Cc(X)
+Let X be a locally compact metric space. In this section we provide a character-
+ization of positive and local forms defined on Cc(X). First we show that densely
+defined positive forms on Cc(X) can be extended to the whole of Cc(X) if their
+domain is a lattice. In a second step we prove a representation theorem. Certainly
+both results are well-known to experts. Since we could not find a proper reference,
+we include the proofs for the convenience of the reader.
+In the following lemma we write C(K) for the subspace {f ∈ Cc(X) | supp f ⊆
+K}.
+Lemma A.1. Let q be a densely defined (with respect to the uniform norm) quadratic
+form on D(q) ⊆ Cc(X). Suppose q is positive and D(q) is a lattice.
+(a) For any compact K ⊆ X the restriction of q to D(q) ∩ C(K) is continuous.
+(b) q can be uniquely extended to a positive quadratic form on Cc(X).
+Proof. We first show that for any compact set K ⊆ X the restriction of q to D(q) ∩
+C(K) is continuous with respect to the supremum norm.
+Let f, g ∈ D(q) ∩ C(K) be nonnegative. Let θK ∈ D(q) be such that θK ≥ 0 and
+θK ≥ 1 on K. Such a functions exists because D(q) is a dense lattice in Cc(X).
+Without loss of generality we assume
+q(∥f∥∞θK, g) − q(∥g∥∞θK, f) ≤ 0,
+for otherwise we could interchange f and g. Then, using the positivity of q, we get
+0 ≤ q(∥f∥∞θK − f, ∥g∥∞θK + g)
+= −q(f, g) + ∥f∥∞∥g∥∞q(θK, θK) + q(∥f∥∞θK, g) − q(∥g∥∞θK, f)
+≤ −q(f, g) + ∥f∥∞∥g∥∞q(θK, θK).
+
+20
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+This implies
+0 ≤ q(f, g) ≤ ∥f∥∞∥g∥q(θK, θK).
+For arbitrary f, g ∈ D(q) ∩ C(K) we have f +, f −, g+, g− ∈ D(q) ∩ C(K) because
+D(q) is a lattice. We obtain
+|q(f, g)| ≤ q(f +, g+) + q(f +, g−) + q(f −, g+) + q(f −, g−)
+≤ 4∥f∥∞∥g∥∞q(θK, θK).
+Using this continuity in order to prove that q can be uniquely extended to a posi-
+tive quadratic form on Cc(X) it suffices to show the following: For every nonnegative
+ϕ ∈ Cc(X) there exists a compact K ⊆ X with supp ϕ ⊆ K such that ϕ can be
+approximated by nonnegative functions in D(q) ∩ C(K).
+To this end, we choose a nonnegative θ ∈ D(q) with θ ≥ ∥ϕ∥∞ on supp ϕ. Such a
+function exists because D(q) is a dense lattice. Let K = supp θ. Since q is densely
+defined, there exists ( ˜ϕn) in D(q) with ˜ϕn → ϕ uniformly, as n → ∞. Since D(q) is
+a lattice, the sequence
+ϕn = ( ˜ϕn)+ ∧ θ
+belongs to D(q).
+It is nonnegative and supp ϕn ⊆ supp θ = K for all n ≥ 1.
+Moreover, using that 0 ≤ ϕ ≤ θ, we obtain ϕn → ϕ uniformly, as n → ∞.
+□
+The following theorem provides a characterization of monotone quadratic forms
+on Cc(X).
+Theorem A.2. Let q: Cc(X) → [0, ∞) be a quadratic form. The following asser-
+tions are equivalent:
+(i) q is positive and local.
+(ii) For all f, g ∈ Cc(X) the inequality fg ≥ 0 implies q(f, g) ≥ 0.
+(iii) For all f, f ′, g, g′ ∈ Cc(X) the inequality fg ≥ f ′g′ implies q(f, g) ≥ q(f ′, g′).
+(iv) q is monotone.
+(v) There exists a Radon measure µ on X such that
+q(u) =
+�
+X f 2dµ,
+f ∈ Cc(X).
+In this case, the measure µ is unique.
+Proof. Clearly, (ii) implies (i), (iii) implies (ii) and (v) implies all other assertions.
+(i) =⇒ (iv): Let f, g ∈ Cc(X) with |g| ≤ |f|. The positivity of q yields
+q(|f|) = q(|g|, |f|) + q(|f| − |g|, |f|) ≥ q(|g|, |f|) = q(|g|) + q(|g|, |f| − |g|) ≥ q(|g|).
+It is left to show q(f) = q(|f|) for every f ∈ Cc(X). Since f +, f − ∈ Cc(X) and
+f +f − = 0, the locality of q implies q(f +, f −) = 0 and hence
+q(f) = q(f +) − 2q(f +, f −) + q(f −) = q(f +) + 2q(f +, f −) + q(f −) = q(|f|).
+
+INTERMEDIATE DIRICHLET FORMS
+21
+(iv) =⇒ (ii): Let f, g ∈ Cc(X) with fg ≥ 0. Then |f + g| ≥ |f − g| so that by
+monotonicty
+q(f) + q(g) − 2q(f, g) = q(f − g) ≤ q(f + g) ≤ q(f) + q(g) + 2q(f, g).
+This shows (ii).
+We already proved the equivalence of (i),(ii) and (iv) and that these assertions
+are implied by (iii). Next we prove that they imply (iii).
+Let f, f ′, g, g′ ∈ Cc(X) with fg ≥ f ′g′. If f = f ′, the inequality q(f, g) ≥ q(f ′, g′)
+directly follows from (ii). With the help of an approximation we reduce the case
+f ̸= f ′ to this one.
+We start with the following observation: Locality of q implies that for ϕ, χ ∈
+Cc(X) the value q(ϕ, χ) is independent of χ as long as χ = 1 on supp ϕ. In this
+case, we write I(ϕ) := q(ϕ, χ).
+Let ε > 0. By compacteness of the supports we can choose finitely many rela-
+tively compact open sets Gj, j = 1, . . . , N, that cover the union of the supports of
+f, f ′, g, g′, and choose ξj ∈ Gj, such that
+sup
+x∈Gj
+|f(x) − f(ξj)| < ε and sup
+x∈Gj
+|f ′(x) − f ′(ξj)| < ε.
+We let χj ∈ Cc(X), j = 1, . . ., N, be a subordinate partition of unity, i.e. 0 ≤ χj ≤ 1,
+suppχj ⊆ Gj and
+N
+�
+j=1
+χj = 1 on
+N
+�
+j=1
+Gj.
+Such a partition of unity exists because metric spaces are normal. We define
+˜f =
+N
+�
+j=1
+f(ξj)χj and ˜f ′ =
+N
+�
+j=1
+f ′(ξj)χj.
+Then, using
+�
+j χj = 1 on the supports of f, g, we obtain
+|q( ˜f, g) − q(f, g)| ≤
+N
+�
+j=1
+|q(χj(f − f(ξj)), g)| ≤ εq(
+N
+�
+j=1
+χj, |g|) = εI(|g|).
+For the second inequality we used |q(ϕ, ψ)| ≤ q(|ϕ|, |ψ|), which directly follows from
+the positivity of q, and the fact that |χj(f − f(ξj))| ≤ εχj.
+Similarly, we have
+
+22
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+|q( ˜f ′, g′) − q(f ′, g′)| ≤ εI(|g′|). Moreover,
+q( ˜f, g) − q( ˜f ′, g′) =
+N
+�
+j=1
+(q(f(ξj)χj, g) − q(f ′(ξj)χj, g′))
+=
+N
+�
+j=1
+q(χj, f(ξj)g − f ′(ξj)g′)
+≥
+N
+�
+j=1
+q(χj, fg − f ′g′) − εq(
+N
+�
+j=1
+χj, |g| + |g|′)
+≥ −εI(|g| + |g′|).
+For the first inequality we used (ii) and the estimate
+χj(f(ξj)g − f ′(ξj)g′) ≥ χj(fg − f ′g′ − ε(|g| + |g′|)).
+The last inequality follows from χj(fg − f ′g′) ≥ 0 and
+�
+j χj = 1 on the support of
+|g| + |g′|. Since ε > 0 was arbitrary, these estimates show (iii).
+(iii) =⇒ (v): As above we define I : Cc(X) → R by letting
+I(ϕ) = q(χ, ϕ)
+for some χ ∈ Cc(X) with χ = 1 on the support of ϕ. It follows from (iii) that this
+is well-defined and positive. Moreover, I is linear. By the Riesz-Markov-Kakutani
+representation theorem there exists a unique Radon measure µ such that
+I(ϕ) =
+�
+X ϕdµ
+for all ϕ ∈ Cc(X). Let now f, g ∈ Cc(X) an let χ ∈ Cc(X) such that χ = 1 on the
+supports of f and g. Since fg = χ(fg), property (iii) yields
+q(f, g) = q(χ, fg) = I(fg) =
+�
+X fgdµ.
+Thus, µ is the desired measure.
+□
+Remark A.3. The statement of the theorem is not only valid for quadratic forms
+on continuous functions. The equivalence of (ii) and (iv) was observed in [Sch20a,
+Appendix B] for quadratic forms on sublattices of L0(Y, m), where Y is an arbitrary
+set and m is a measure on Y . Indeed, the above proof yields the equivalence of (i),(ii)
+and (iv) in this situation. The equivalence with (iii) requires the existence of suitable
+partitions of unity in the domain of q and the equivalence with (v) requires that the
+domain of q is an algebra and a representation theorem for positive functionals.
+Corollary A.4. Let q be a densely defined positive and local quadratic form on
+Cc(X) such that D(q) is a lattice. Then there exists a unique Radon measure µ on
+X such that
+q(f) =
+�
+X f 2dµ,
+f ∈ D(q).
+
+INTERMEDIATE DIRICHLET FORMS
+23
+Proof. As noted in the previous remark the form q is also monotone. By Lemma A.1
+it can be uniquely extended to a positive quadratic form on Cc(X) and by the
+continuity of restrictions to compact sets this extension is also monotone. Hence,
+the statement follows from the previous theorem.
+□
+References
+[ACD21] Sahiba Arora, Ralph Chill, and Jean-Daniel Djida. Domination of semigroups generated
+by regular forms. 2021.
+[Akh18] Khalid Akhlil. Locality and domination of semigroups. Results Math., 73(2):Art. 59, 11,
+2018.
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+Wolfgang Arendt and Mahamadi Warma. Dirichlet and Neumann boundary conditions:
+What is in between? J. Evol. Equ., 3(1):119–135, 2003. Dedicated to Philippe Bénilan.
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+Zhen-Qing Chen and Masatoshi Fukushima. Symmetric Markov processes, time change,
+and boundary theory, volume 35 of London Mathematical Society Monographs Series.
+Princeton University Press, Princeton, NJ, 2012.
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+Burkhard Claus. Non-linear Dirichlet forms. PhD thesis, TU Dresden, Dresden, 2021.
+[CW12]
+Ralph Chill and Mahamadi Warma. Dirichlet and Neumann boundary conditions for the
+p-Laplace operator: what is in between? Proceedings of the Royal Society of Ediburgh:
+Section A Mathematics, 142(5):975–1002, 2012.
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+Burkhard Claus and Mahamadi Warma. Realization of the fractional Laplacian with
+nonlocal exterior conditions via form methods. J. Evol. Equ., 20(4):1597–1631, 2020.
+[DNPV12] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci. Hitchhiker’s guide to
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+D. E. Edmunds and W. D. Evans. Spectral theory and differential operators. Oxford
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+metric Markov processes, volume 19 of De Gruyter Studies in Mathematics. Walter de
+Gruyter & Co., Berlin, extended edition, 2011.
+[KLSS19] Matthias Keller, Daniel Lenz, Marcel Schmidt, and Michael Schwarz. Boundary repre-
+sentation of Dirichlet forms on discrete spaces. J. Math. Pures Appl. (9), 126:109–143,
+2019.
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+J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications.
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+Potential Analysis, 5(6):611–625, 1996.
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+Michael Reed and Barry Simon. Methods of modern mathematical physics. IV. Analysis of
+operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London,
+1978.
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+Marcel Schmidt. Energy forms. PhD thesis, Mar 2017.
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+[Sch20b] Michael Schwarz. Nodal Domains and Boundary Representation for DirichletForms. PhD
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+Peter Stollmann. Smooth perturbations of regular Dirichlet forms. Proc. Amer. Math.
+Soc., 116(3):747–752, 1992.
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+Peter Stollmann. Closed ideals in Dirichlet spaces. Potential Anal., 2(3):263–268, 1993.
+
+24
+M. KELLER, D. LENZ, M. SCHMIDT, M. SCHWARZ, AND M. WIRTH
+[SV96]
+Peter Stollmann and Jürgen Voigt. Perturbation of Dirichlet forms by measures. Potential
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+M.Keller, Institut für Mathematik, Universität Potsdam, Campus Golm, Haus 9,
+Karl-Liebknecht-Str. 24-25, 14476 Potsdam OT Golm, Germany
+Email address: matthias.keller@uni-potsdam.de
+D.Lenz, Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07737 Jena,
+Germany
+Email address: daniel.lenz@uni-jena.de
+M. Schmidt, Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109
+Leipzig, Germany
+Email address: marcel.schmidt@math.uni-leipzig.de
+M. Schwarz, dotSource GmbH, Goethestr. 1, 07743 Jena, Germany
+Email address: m.schwarz@dotSource.de
+M. Wirth, Institute of Science and Technology Austria (ISTA), Am Campus 1,
+3400 Klosterneuburg, Austria
+Email address: melchior.wirth@ist.ac.at
+

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+arXiv:2301.01735v1  [math.MG]  4 Jan 2023
+NORMED SPACES USING INTRINSICALLY LIPSCHITZ SECTIONS
+AND EXTENSION THEOREM FOR THE INTRINSICALLY HÖLDER
+SECTIONS
+DANIELA DI DONATO
+Abstract. The purpose of this article is twofold: first of all, we want to define two norms
+using the space of intrinsically Lipschitz sections. On the other hand, we want to generalize
+an Extension Theorem proved by the author in the context of the intrinsically Hölder sections
+with target a topological space Y. Here our target will be Y × Rs with s ≥ 1 instead of Y.
+Contents
+1.
+Introduction
+1
+2.
+Intrinsically Hölder sections
+3
+2.1.
+Intrinsically Hölder sections: when Y is bounded
+3
+2.2.
+Equivalent definition for intrinsic Hölder sections
+4
+3.
+Normed space for the intrinsically Lipschitz sections
+5
+3.1.
+Normed space: Version 1
+5
+3.2.
+Normed space: Version 2
+6
+4.
+Level sets and extensions
+7
+References
+9
+1. Introduction
+In this paper, we focus our attention on a new point of view for the intrinsically Lipschitz
+graphs in the Franchi-Serapioni-SerraCassano sense [FSSC01, FSSC03b, FSSC03a] (see also
+[SC16, FS16]) in metric spaces.
+They introduced and analized this notion in order to establish a good notion of rectifi-
+ability in a particular metric spaces called subRiemannian Carnot groups [ABB19, BLU07,
+CDPT07].
+In the usual way, Federer [Fed69] says that a subset of Rn is countably d-rectifiable if it is
+possible to cover it with a countable union of suitable graphs. More precisely, he considers
+graphs of Lipschitz maps f : Rd → Rn−d. However, Ambrosio and Kirchheim [AK00] (see
+also [Mag04]) proved that this definition does not work in Carnot groups and so many
+mathematicians give other notions of rectifiability. The reader can see [AM22a, AM22b,
+Bat21, DS91, DS93, CP06, Pau04, NY18]. As we said, another possible solution is given
+by Franchi-Serapioni-SerraCassano with the so-called "Intrinsic Lipschitz maps" in Carnot
+Date: 5th January 2023.
+1
+
+groups. The idea is similar to Euclidean case: firstly, they introduce suitable cones called
+intrinsic cones which are not equivalent with the Euclidean ones; and, then, they say that a
+map ϕ is intrinsic Lipschitz if it is possible to have for any point p belongs to the graph of ϕ
+an empty intersection between a suitable intrinsic cone with vertex p and the graph of this
+map.
+Recently, Le Donne and the author generalize this concept in metric spaces (see [DDLD22]).
+A basic difference is the following: in Franchi, Serapioni and Serra Cassano approach, they
+consider intrinsically Lipschitz maps. On the other hand, in our approach we consider the
+graphs and this a bit change is so important because:
+• The setting are more general: the class of the metric spaces is larger than the class
+of Carnot groups.
+• The proofs are elegantly short and simple.
+• We use basic mathematical tools in the proofs.
+In a natural way, in [DD22c] the author introduce the notion of intrinsically Hölder sections
+which extend the Lipschitz ones. Here, our setting is the following. We have a metric space
+X, a topological space Y , and a quotient map π : X → Y , meaning continuous, open, and
+surjective. The standard example for us is when X is a metric Lie group G (meaning that the
+Lie group G is equipped with a left-invariant distance that induces the manifold topology),
+for example a subRiemannian Carnot group, and Y is the space of left cosets G/H, where
+H < G is a closed subgroup and π : G → G/H is the projection modulo H, g �→ gH.
+Definition 1.1 (Intrinsic Hölder section). Let (X, d) be a metric space and let Y be a
+topological space. We say that a map ϕ : Y → X is a section of a quotient map π : X → Y
+if
+π ◦ ϕ = idY .
+Moreover, we say that ϕ is an intrinsically (L, α)-Hölder section with constant L > 0 and
+α ∈ (0, 1) if in addition
+(1)
+d(ϕ(y1), ϕ(y2)) ≤ Ld(ϕ(y1), π−1(y2))α + d(ϕ(y1), π−1(y2)),
+for all y1, y2 ∈ Y.
+Equivalently, we are requesting that
+d(x1, x2) ≤ Ld(x1, π−1(π(x2)))α + d(x1, π−1(π(x2))),
+for all x1, x2 ∈ ϕ(Y ).
+When α = 1, a section ϕ is intrinsic Lipschitz in the sense of [DDLD22]; and, if in addition,
+π is a Lipschitz quotient or submetry [BJL+99, VN88], the results trivialize, since in this
+case being intrinsically Lipschitz is equivalent to biLipschitz embedding, see Proposition
+2.4 in [DDLD22]. In a natural way, following the seminal papers [AGS14a, LV09, Sav22]
+(see also [AGS15, AGS14b, FSS22, Stu06, Vil09]), the author introduced and studied the
+link between the intrinsic sections/intrinsic Lipschitz sections and the intrinsic Hopf-Lax
+semigroups [DD22b, DD22e].
+The purpose of this article is twofold: first of all, we want to define two norms using
+the notion of Lipschitz sections. Second, we want to generalize an Extension Theorem with
+target Y which is a topological space; in this paper, our target will be Y × Rs instead of Y.
+More precisely, in Section 3, the main results are Theorem 3.1 and 3.4. Here, we define
+two possible normed spaces using the following ingredients:
+2
+
+• we know that there is a large class of intrinsically Hölder sections and so Lipschitz
+sections that is a vector space over R or C (see Theorem 2.7);
+• we can define two different norms noting the following simple fact: in the usual
+case, we have that d(x, y) = d(y, x) for any point x, y ∈ X; on the other hand, in
+our intrinsic context, in general, we have that d(f(x), π−1(y)) ̸= d(f(y), π−1(x)), for
+every x, y ∈ X.
+• we obtain the homogeneity of our norms defined in (5) and in (9) thanks to linearity
+of π and, in particular, to Lemma 3.2.
+Finally, in Section 4 the main result is Theorem 4.1 which generalizes Extension Theorem
+for the intrinsically Hölder sections proved in [DD22c, Theorem 1.3]. The main difference is
+that, in this project, the target space is a topological space Y × Rs with s ≥ 1 instead of Y.
+As in Vittone’s case, we build each component fi for i = 1, . . . , s separately and then join
+them without any additional assumptions. However, the final step when the target space is
+only Y does not provide a Lipschitz map f = (f1, . . . , fs).
+2. Intrinsically Hölder sections
+2.1. Intrinsically Hölder sections: when Y is bounded. Definition 1.1 it is very natural
+if we think that what we are studying graphs of appropriate maps. However, in the following
+proposition, we introduce an equivalent condition of (1) when Y is a bounded set.
+Proposition 2.1. Let π : X → Y be a quotient map between a metric space X and a
+topological and bounded space Y and let α ∈ (0, 1). The following are equivalent:
+(1) there is L > 0 such that
+d(ϕ(y1), ϕ(y2)) ≤ Ld(ϕ(y1), π−1(y2))α + d(ϕ(y1), π−1(y2)),
+for all y1, y2 ∈ Y.
+(2) there is K ≥ 1 such that
+(2)
+d(ϕ(y1), ϕ(y2)) ≤ Kd(ϕ(y1), π−1(y2))α,
+for all y1, y2 ∈ Y.
+We further rephrase the definition as saying that ϕ(Y ), which we call the graph of ϕ,
+avoids some particular sets (which depend on α, L and ϕ itself):
+Proposition 2.2. Let π : X → Y be a quotient map between a metric space and a topological
+space, ϕ : Y → X be a section of π, α ∈ (0, 1) and L > 0. Then ϕ is intrinsically (L, α)-
+Hölder if and only if
+ϕ(Y ) ∩ Rx,L = ∅,
+for all x ∈ ϕ(Y ),
+where
+Rx,L :=
+�
+x′ ∈ X | Ld(x′, π−1(π(x)))α + d(x′, π−1(π(x))) < d(x′, x)
+�
+.
+Proposition 2.2 is a triviality, still its purpose is to stress the analogy with the intrinsically
+Lipschitz sections theory introduced in [DDLD22] when α = 1. In particular, the sets Rx,L
+are the intrinsic cones in the sense of Franchi, Serapioni and Serra Cassano when X is a
+subRiemannian Carnot group and α = 1. The reader can see [DD22d] for a good notion of
+intrinsic cones in metric groups.
+3
+
+2.2. Equivalent definition for intrinsic Hölder sections.
+Definition 2.3 (Intrinsically Hölder with respect to a section). Given sections ϕ, ψ : Y → X
+of π. We say that ϕ is intrinsically (L, α)-Hölder with respect to ψ at point ˆx, with L >
+0, α ∈ (0, 1) and ˆx ∈ X, if
+(1) ˆx ∈ ψ(Y ) ∩ ϕ(Y );
+(2) ϕ(Y ) ∩ Cψ
+ˆx,L = ∅,
+where
+Cψ
+ˆx,L := {x ∈ X : d(x, ψ(π(x))) > Ld(ˆx, ψ(π(x)))α + d(ˆx, ψ(π(x)))}.
+Remark 2.4. Definition 2.3 can be rephrased as follows. A section ϕ is intrinsically (L, α)-
+Hölder with respect to ψ at point ˆx if and only if there is ˆy ∈ Y such that ˆx = ϕ(ˆy) = ψ(ˆy)
+and
+(3)
+d(x, ψ(π(x))) ≤ Ld(ˆx, ψ(π(x)))α + d(ˆx, ψ(π(x))),
+∀x ∈ ϕ(Y ),
+which equivalently means
+(4)
+d(ϕ(y), ψ(y)) ≤ Ld(ψ(ˆy), ψ(y))α + d(ψ(ˆy), ψ(y)),
+∀y ∈ Y.
+Notice that Definition 2.3 does not induce an equivalence relation because of lack of
+symmetry in the right-hand side of (4). However, following Cheeger theory [Che99] (see also
+[Kei04, KM16]), in [DD22c, Theorem 4.2] we give an equivalent property of Hölder section.
+Being intrinsically Lipschitz section is equivalent to the last definition as proved in [DD22c,
+Proposition 1.5]
+Proposition 2.5. Let X be a metric space, Y a topological and bounded space, π : X → Y
+a quotient map, L ≥ 1 and α, β, γ ∈ (0, 1). Assume that every point x ∈ X is contained in
+the image of an intrinsic (L, α)-Hölder section ψx for π. Then for every section ϕ : Y → X
+of π the following are equivalent:
+(1) for all x ∈ ϕ(Y ) the section ϕ is intrinsically (L1, β)-Hölder with respect to ψx at x;
+(2) the section ϕ is intrinsically (L2, γ)-Hölder.
+We conclude this section recall an important concept which we will be used later.
+Definition 2.6 (Intrinsic Hölder set with respect to ψ). Let α ∈ (0, 1] and ψ : Y → X a
+section of π. We define the set of all intrinsically Hölder sections of π with respect to ψ at
+point ˆx as
+Hψ,ˆx,α := {ϕ : Y → X a section of π : ϕ is intrinsically (˜L, α)-Hölder w.r.t. ψ at point ˆx
+for some ˜L > 0}.
+In particular, in [DD22c] we have the following statement regarding the set Hψ,ˆx,α.
+Theorem 2.7 (Theorem 3.5 [DD22c]). Let π : X → Y is a linear and quotient map from a
+normed space X to a topological space Y. Assume also that ψ : Y → X is a section of π and
+{λˆx : λ ∈ R+} ⊂ X with ˆx ∈ ψ(Y ).
+Then, for any α ∈ (0, 1], the set �
+λ∈R+ Hλψ,λˆx,α ∪ {0} is a vector space over R or C.
+Notice that it is no possible to obtain the statement for Hψ,ˆx,α since the simply observation
+that if ψ(ˆy) = ˆx then ψ(ˆy) + ψ(ˆy) ̸= ˆx.
+4
+
+3. Normed space for the intrinsically Lipschitz sections
+3.1. Normed space:
+Version 1. In this section, we consider the case of intrinsically
+Lipschitz sections, i.e., α = 1.
+Let π : Rκ → Y be a quotient map with Y ⊂ Rκ.
+Assume also that K ⊂ Y is a
+compact set and ψ|K : K → R is an intrinsically L-Lipschitz section of π with L ≥ 1 and
+ˆx = ψ(¯y) ∈ ψ(Y ). We will use the following notation
+ILSλψ|K,λˆx := Hλψ|K,λˆx,1.
+Here, we show that
+(L, ∥.∥) :=
+� �
+λ∈R+
+ILSλψ|K,λˆx ∪ {0}, ∥.∥
+�
+is a normed space for a suitable norm ∥.∥= ∥.∥ILSλψ,λˆx: L → R+ defined as for any ϕ ∈ L,
+(5)
+∥ϕ∥ILSλψ,λˆx:= ∥ϕ∥∞+[ϕ]λψ,λˆx,
+where ∥ϕ∥∞:= supy∈K|ϕ(y)| and
+[ϕ]λψ,λˆx := sup
+y∈K
+d(λϕ(y), (1/λπ)−1(π(ˆx))).
+Then, we are able to give the main result of this section.
+Theorem 3.1. Let π : Rκ → Y be a linear and quotient map with Y ⊂ Rκ. Assume also
+that ψ : K → Rκ is an intrinsically L-Lipschitz section of π with K ⊂ Y compact, L ≥ 1
+and ˆx ∈ X. Then, the set L endowed with ∥·∥ILSψ,ˆx is a normed space.
+We need the following lemma.
+Lemma 3.2 (Lemma 4.7 [DD22a]). Let X be a normed space, Y be a topological space and
+π : X → Y be a linear and quotient map. Then
+(6)
+|λ|d(x1, π−1(y2)) = d(λx1, (1/λπ)−1(y2)),
+∀x1 ∈ Rκ, y2 ∈ Y, λ ∈ R − {0}.
+Remark 3.3. An easy corollary of Lemma 3.2 when Y ⊂ R and X = Rκ is that
+lim
+h→0+
+d(hϕ(t + h), (1/hπ)−1(t)))
+h
+= 0,
+lim
+h→0+
+d(hϕ(t), (1/hπ)−1(t + h)))
+h
+= 0,
+for t ∈ Y. Indeed, for h > 0
+d(hϕ(t + h), (1/hπ)−1(t))
+h
+= d(ϕ(t + h), π−1(t)) ≤ d(ϕ(t + h), ϕ(t)),
+and so take to the limit for h → 0, we obtain the first limit thank to the continuity of ϕ. In
+a similar way, it is possible to see the second limit.
+At this point, we give the proof of Theorem 3.1.
+5
+
+Proof of Theorem 3.1. The fact ∥ϕ∥≡ 0 if and only if ϕ ≡ 0 follows because ∥.∥∞ is a norm.
+On the other hand, since π is linear map and thanks to Lemma 3.2, we have
+sup
+y∈K
+d(δϕ(y), (1/δπ)−1(π(ˆx))) = sup
+y∈K
+|δ|d(ϕ(y), π−1(π(ˆx)))
+for any δ ∈ R − {0} and so
+(7)
+∥δϕ∥∞+[δϕ]ψ,ˆx = |δ|(∥ϕ∥∞+[ϕ]ψ,ˆx),
+for any ϕ ∈ L.
+Finally, the triangle inequality follows using again the linearity of π. Indeed, if ϕ, η ∈
+L − {0} and, in particular, ϕ, η ∈ ILS(λ1+λ2)ψ|K,(λ1+λ2)ˆx then for xϕ, xη ∈ Rκ such that
+[ϕ](λ1+λ2)ψ,(λ1+λ2)ˆx = sup
+y∈K
+d((λ1 + λ2)ϕ(y), (1/(λ1 + λ2)π)−1(π(ˆx))) = d((λ1 + λ2)ϕ(y), xϕ)
+[η](λ1+λ2)ψ,(λ1+λ2)ˆx = sup
+y∈K
+d((λ1 + λ2)η(y), (1/(λ1 + λ2)π)−1(π(ˆx))) = d((λ1 + λ2)η(y), xη)
+one gets
+[ϕ + η](λ1+λ2)ψ,(λ1+λ2)ˆx = sup
+y∈K
+d((λ1 + λ2)(ϕ(y) + η(y)), (2/(λ1 + λ2)π)−1(π(ˆx)))
+≤ ∥(λ1 + λ2)ϕ(y) + (λ1 + λ2)η(y) − (xϕ + xη)∥
+≤ ∥(λ1 + λ2)ϕ(y) − xϕ∥+∥(λ1 + λ2)η(y) − xη∥,
+[ϕ](λ1+λ2)ψ,(λ1+λ2)ˆx + [η](λ1+λ2)ψ,(λ1+λ2)ˆx,
+where in the first equality, by linearity of π, we used the fact
+π((λ1 + λ2)(ϕ(y) + η(y))) = π((λ1 + λ2)ϕ(y)) + π((λ1 + λ2)η(y))
+= (λ1 + λ2)(π(ϕ(y)) + (π(η(y)))
+= 2(λ1 + λ2)y.
+Hence,
+[ϕ + η](λ1+λ2)ψ,(λ1+λ2)ˆx ≤ [ϕ](λ1+λ2)ψ,(λ1+λ2)ˆx + [η](λ1+λ2)ψ,(λ1+λ2)ˆx,
+and this complete the proof of the statement.
+□
+3.2. Normed space: Version 2. In the usual case, we have that d(x, y) = d(y, x) for any
+point x, y ∈ X; on the other hand, in our intrinsic context, in general, we have that
+d(f(x), π−1(y)) ̸= d(f(y), π−1(x)),
+for every x, y ∈ X. In particular, it holds
+(8)
+d(f(y), π−1(x)) − d(f(z), π−1(x)) ≤ d(f(y), f(z)),
+∀x, y, z ∈ Y
+d(f(x), π−1(y)) − d(f(x), π−1(z)) ≰ d(f(y), f(z)),
+for some x, y, z ∈ Y.
+In fact, for any fixed x, y, z ∈ Y, if we choose a ∈ π−1(x) such that
+d(f(z), π−1(x)) = d(f(z), a),
+6
+
+we deduce that
+d(f(y), π−1(x)) − d(f(z), π−1(x)) ≤ d(f(y), a) − d(f(z), π−1(x))
+≤ d(f(y), f(z)) + d(f(z), a) − d(f(z), π−1(x))
+= d(f(y), f(z)),
+i.e., the first inequality of (8) holds. On the other hand, for the second inequality in (8),
+we give the following example. let X ⊂ R2 the set given by the three lines with vertex
+(0, 8), (8, 8); (1, 4), (8, 6) and (0, 3), (8, 7) and the subset Y of R2 defined as the line with
+vertex (0, 0) and (8, 0). On X we consider the usual distance on R2. Then, if we consider a
+continuous section f : Y → X of the projection on the first component π : X → Y with
+f(x) = f((1, 0)) = (1, 4), f(y) = f((7, 0)) = (8, 7) and f(z) = f((6, 0)) = (8, 6), it is easy to
+see that
+d(f(x), π−1(y)) − d(f(x), π−1(z)) =
+�
+5
+4,
+d(f(y), f(z)) = 1,
+and so
+d(f(x), π−1(y)) − d(f(x), π−1(z)) ≰ d(f(y), f(z)).
+Then, it is not trivial to consider the norm |||.||| defined as
+(9)
+|||ϕ|||ILSψ,ˆx:= ∥ϕ∥∞+{ϕ}λψ,λˆx,
+where ∥ϕ∥∞:= supy∈K|ϕ(y)| and
+{ϕ}λψ,λˆx := sup
+y∈K
+d(λˆx, (1/λπ)−1(y)).
+and to prove as above the following statement.
+Theorem 3.4. Let π : Rκ → Y be a linear and quotient map with Y a metric space. Assume
+also that ψ : K → Rκ is an intrinsically L-Lipschitz section of π with K ⊂ Y compact, L ≥ 1
+and ˆx ∈ X. Then, the set L endowed with |||·|||ILSψ,ˆx as in (9) is a normed space.
+Proof. The proof follows in a similar way as in Theorem 3.1.
+□
+4. Level sets and extensions
+In this section we prove the following theorem.
+Theorem 4.1 (Extensions as level sets). Let π : X → Y × Rs be a quotient map between a
+metric space X and a topological space Y × Rs.
+Assume that X is geodesic and that there exist k ≥ 1, ρ : X × X → R k-biLipschitz
+equivalent to the distance of X, and τ = (τ1, . . . , τs) : X → Rs k-Lipschitz and k-biLipschitz
+on fibers such that for all τ0 ∈ Rs
+(1) the set τ −1
+1 (τ0) is an intrinsically k-Lipschitz graph of a section ϕ1,τ0 : Y × R ×
+{0}s−1 → X; the set τ −1
+2 (τ0) is an intrinsically k-Lipschitz graph of a section ϕ2,τ0 :
+Y × {0} × R × {0}s−2 → X; . . . , the set τ −1
+s (τ0) is an intrinsically k-Lipschitz graph
+of a section ϕs,τ0 : Y × {0}s−1 × R → X;
+7
+
+(2) for all x0 ∈ τ −1
+i
+(τ0) the map X → R, x �→ δi,τ0(x) := ρ(x0, ϕi,τ0(π(x))) is k-Lipschitz
+on the set {|τi|≤ δi,τ0}.
+Let Y ′ × Rs ⊂ Y × Rs a set and L ≥ 1. Then for every intrinsically L-Lipschitz section
+ϕ : Y ′ × Rs → π−1(Y ′ × Rs) of π|π−1(Y ′×Rs): π−1(Y ′ × Rs) → Y ′ × Rs, there exists a map
+f : X → Rs that is K-Lipschitz and K-biLipschitz on fibers, with K = 2k(Lk + 2), such that
+(10)
+ϕ(Y ′ × Rs) ⊆ f −1(0).
+In particular, each ‘partially defined’ intrinsically Lipschitz graph ϕ(Y ′ × Rs) is a subset of
+a ‘globally defined’ intrinsically Lipschitz graph f −1(0).
+We need to mention that there have been several earlier partial results on extensions of
+Lipschitz graphs, as for example in [FSSC06], [Vit, Theorem 1.5], [Mon14, Proposition 4.8],
+in the Heisenberg group with codimension larger than one; [Vit12, Proposition 3.4], for the
+case of codimension one in the Heisenberg groups; [FS16, Theorem 4.1], for the case of
+codimension one in Carnot groups. Finally, for the general metric spaces the reader can see
+[AP20, LN05, Oht09].
+Proof of Theorem 4.1 (4.1.i). It is proved in [DDLD22].
+□
+Proof of Theorem 4.1 (4.1.ii). Step 1. Fix i = 1, . . . , s and, for simplicity, we write τ −1, fx0
+instead of τ −1
+i
+and fx0,i. Fix x0 ∈ τ −1(τ0). We consider the map fx0 : X → R defined as
+(11)
+fx0(x) =
+
+
+
+2(τ(x) − τ(x0) − αδτ0(x))
+if |τ(x) − τ(x0)|≤ 2αδτ0(x)
+τ(x) − τ(x0)
+if τ(x) − τ(x0) > 2αδτ0(x)
+3(τ(x) − τ(x0))
+if τ(x) − τ(x0) < −2αδτ0(x)
+where α := kL + 1. We prove that fx0 satisfies the following properties:
+(i): fx0 is K-Lipschitz;
+(ii): fx0(x0) = 0;
+(iii): fx0 is biLipschitz on fibers.
+where K = max{3k, 2k + 2αk} = 2k + 2αk because α > 1. The property (i) derives using
+that τ, δτ0 are both Lipschitz and X is a geodesic space. On the other hand, (ii) is true
+noting that δτ0(x0) = ρ(x0, ϕτ0(π(x0))) = 0 because x0 ∈ ϕτ0(Y ).
+Finally, for any x, x′ ∈ π−1(y) we have that ρ(x0, ϕτ0(π(x))) = ρ(x0, ϕτ0(π(x′))) and so f
+is biLipschitz on fibers because τ is so too. Hence (iii) holds.
+At this point, we consider the map f : X → R given by
+f(x) :=
+sup
+x0∈ϕ(Y )
+fx0(x),
+∀x ∈ X,
+and we want to prove that it is the map we are looking for. The Lipschitz properties are
+true recall that the function δx0 is constant on the fibers. Consequently, the only non trivial
+fact to show is (10). Fix ¯x0 ∈ ϕ(Y ′). By (ii) we have that f¯x0(¯x0) = 0 and so it is sufficient
+to prove that fx0(¯x0) ≤ 0 for x0 ∈ ϕ(Y ′). Let x0 ∈ ϕ(Y ′). Then using in addition that τ is
+k-Lipschitz, and that ϕ is intrinsically L-Lipschitz, we have
+|τ(¯x0) − τ(x0)|≤ kd(¯x0, x0) ≤ Lkd(x0, π−1(π(¯x0))) ≤ Lkd(x0, ϕτ0(π(¯x0))) < αδτ0(¯x0),
+8
+
+and so
+fx0(¯x0) = 2(τ(¯x0) − τ(x0) − αδτ0(¯x0)) < 0,
+i.e., (10) holds.
+Step 2. We consider f = (f1, . . . , fs) where each fi is given by τ −1
+i
+. Roughly speaking,
+here the problem is that when we put together (f1, . . . , fs) using τ −1
+1 , . . . , τ −1
+s
+then f can not
+Lipschitz. But now f is Lipschitz thanks to the construction of Y × Rs and in particular of
+ϕ1,τ0 : Y × R × {0}s−1 → X; . . . ; ϕs,τ0 : Y × {0}s−1 × R → X.
+□
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+Daniela Di Donato: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Via
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+Email address: d.didonato@staff.univpm.it
+11
+